On the Determination of the Reaction Rate Constant and Selectivity in Gas and Liquid-Phase Organic Reactions: Temperature and Solvent Eﬀects

Home , Reaction rate constant

Aikaterini Diamanti

A thesis submitted for the Doctor of Philosophy degree of Imperial College London and the Diploma of Imperial College.

Centre for Process Systems Engineering Department of Chemical Engineering Imperial College London London SW7 2AZ United Kingdom

9th December 2016 Declaration of Originality

I, Aikaterini Diamanti, certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline.

The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or distribution, researchers must make clear to others the licence terms of this work.

2 Abstract

Chemical reactions occur abundantly in nature and the rates at which they proceed are critically influenced by factors such as the temperature and the solvent medium in which they take place. These two factors do not only affect the rate of the reaction but may also have a significant impact on other metrics, such as the selectivity and the catalytic activity, as well as on the overall performance of a process application. Faced with such a broad scope of considerations, the identification of an optimum reaction environment remains a challenge. This thesis provides an in-depth investigation of gas and liquid-phase reactions in the context of predicting selected reaction metrics under different thermodynamic and media conditions. In particular, the effect of temperature on the rate constant of a gas-phase reaction and the effects of solvent media on the rate constant and selectivity of a liquid-phase reaction are considered. A gas-phase hydrogen abstraction reaction between ethane and the hydroxyl radical is studied in a broad range of temperatures. A thorough computational investigation is performed of the temperature dependence of the reaction rate constant, assessing several ab initio and density functional theory methods with various basis sets. A novel hybrid strategy is proposed for the development of correlative kinetic models that incorporate information from quantum-mechanical calculations and experiments into classical Arrhenius expressions. The hybrid models derived bring new insight into the value and contribution of the data obtained via quantum-mechanical calculations and via measurements. The benefits of such models in the context of accuracy, statistical significance, applicability and practical importance on the study of reactions with scarce experimental data, are highlighted. A regioselective Williamson reaction between sodium β-naphthoxide and benzyl bromide is selected for the investigation of solvent effects on the reaction rate constant and product selectivity. The solvent medium has a key impact on the selectivity of the reaction for alkylation at two possible sites (an oxygen or a carbon atom) resulting in O- and C-alkylated products. For this reaction, a systematic study is performed combining detailed kinetic experiments and density functional theory calculations to determine the reaction rate constants in a set of solvents. The challenges in conducting reliable experiments using NMR spectroscopy are highlighted, and the performance of the various computational methods is scrutinized. Good agreement is obtained between computational predictions and experimental data for the reaction rate constants as well as for the ranking of solvents in terms of the product selectivity ratios for a number of the theoretical methods considered. These promising results pave the way for future computer-aided molecular design tools for the identification of solvents for improved reaction performance. Acknowledgements

The completion of my PhD would not have been possible without the support of many people. First of all, I am deeply grateful to my supervisors Claire Adjiman and Amparo Galindo for all their help during these years. I thank them for their invaluable advice, invariable good disposition, inspiring enthusiasm, and kind motivational comments. This work was supported financially by Syngenta and I would like to gratefully acknowledge the attention received from their team. Especially, I would like to thank Patrick M. Piccione and Anita M. Rea for our constructive meetings, and for giving me the opportunity to complete part of my research work in Syngenta’s International Research Centre at Jealott’s Hill. I feel privileged to be a member of the MSE group and I would like to thank everyone in the group for the happy moments we had together and especially Christianna, Sadia, Eirini, Isaac, Suela, Eliana. I would like to thank my friends Eva, Maria and Sara for being happy with me the good days and cheering me up the difficult days. Also, I would like to thank my friends Maria and Spyros for reminding me how it is having friends close to you and for bringing a little flavour of Greece in my daily life in London. Nothing would have been possible without the love, continuous support and encouragement of my parents and my brother, who, despite being far away, have always been by my side. Finally, the truth is I would not have the strength to complete this PhD without the love and support of my other half Alon. Thank you for believing in me wholeheartedly, inspiring me with your example and making every day of my life more beautiful!

4 Στoυς γoν´ις µoυ, Bασ´ιλιo και Γαλατια´ ,

και τoν αδλφo´ µoυ, Aθανασιo´ . Contents

Abstract 3

List of Figures 9

List of Tables 18

1 Introduction 32 1.1 Temperature and solvent eﬀects on chemical reactions...... 32 1.2 Predicting temperature and solvent eﬀects on reaction rates...... 34 1.3 Objectives of the thesis...... 35 1.4 Outline of the thesis...... 36

2 Determination of reaction kinetics: temperature and solvent effects 37 2.1 Basic kinetic concepts...... 37 2.2 Conventional transition state theory...... 39 2.3 Determination of reaction kinetics...... 41 2.3.1 Experimental techniques...... 41 2.3.2 Computational methods...... 43 2.4 Modelling solvent effects...... 45 2.4.1 Solvation models...... 46 2.4.2 Predictive capabilities of continuum solvation models...... 51 2.5 Modelling temperature effects...... 53 2.5.1 First order temperature dependence...... 54 2.5.2 Second order temperature dependence...... 55 2.5.3 Other approaches...... 57 2.6 Summary...... 58

6 3 Predicting temperature eﬀects on gas-phase reaction kinetics 60 3.1 Introduction...... 60 3.2 Computational methodology...... 63 3.2.1 Quantum-mechanical calculations...... 63 3.2.2 Rate-constant calculations...... 64 3.3 Computational investigation of the reaction kinetics...... 66 3.3.1 Activation energy barrier...... 66 3.3.2 Reaction rate constant at 298 K...... 70 3.3.3 Geometry and frequency calculations...... 72 3.3.4 Rate constants at diﬀerent temperatures...... 73 3.4 Hybrid correlative models...... 79 3.4.1 Methodology...... 80 3.4.2 Arrhenius-type hybrid models...... 83 3.4.3 Results...... 86 3.5 Summary...... 92

4 Kinetic investigation of a Williamson reaction in acetonitrile 94 4.1 Theoretical background for kinetic predictions in the liquid phase...... 95 4.1.1 Liquid-phase reaction rate constant by CTST...... 95 4.1.2 The SMD solvation model...... 97 4.1.3 Liquid-phase reaction rate constant and selectivity ratio by CTST and the SMD solvation model...... 100 4.2 The Williamson reaction of sodium β-naphthoxide and benzyl bromide...... 101 4.3 Experimental methodology...... 107 4.3.1 Monitoring technique...... 107 4.3.2 Proton-exchange experiment...... 108 4.3.3 Methodology for kinetic experiments...... 111 4.4 Experimental results...... 115 4.4.1 Kinetic experiments at 298, 313 and 323 K...... 115 4.4.2 Thermodynamic analysis based on experimental data...... 125 4.5 Computational methodology...... 129 4.6 Computational results...... 131 4.6.1 Structure search for stable conformations of sodium β-naphthoxide.... 131

7 4.6.2 Transition-state structures for the O- and C-alkylation...... 133 4.6.3 Selectivity and reaction kinetics at 298 K...... 139 4.6.4 Thermodynamic analysis based on QM-calculated data...... 147 4.7 Summary...... 152

5 Kinetic investigation of a Williamson reaction in methanol 155 5.1 Experimental methodology...... 155 5.1.1 Monitoring technique...... 155

5.1.2 Reaction of benzyl bromide with methanol-d4 ...... 156 5.1.3 Reaction for the formation of the double C-alkylated product...... 157 5.1.4 Methodology for kinetic experiments...... 158 5.2 Experimental results...... 163 5.2.1 Kinetic experiments at 298, 313 and 328 K...... 163 5.2.2 Thermodynamic analysis based on experimental data...... 171 5.3 Computational methodology...... 173 5.4 Computational results...... 174 5.4.1 Structure search for stable conformations of sodium β-naphthoxide.... 174 5.4.2 Selectivity and reaction kinetics at 298 K...... 175 5.4.3 Thermodynamic analysis based on QM-calculated data...... 182 5.5 Performance of levels of theory for acetonitrile and methanol...... 186 5.6 Summary...... 189

6 Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 192 6.1 Computational methodology...... 193 6.2 Computational results...... 195 6.2.1 Transition-state structures for the O- and C-alkylation...... 195 6.2.2 Selectivity and reaction kinetics at 298 K...... 197 6.3 Selection of available experimental data...... 212 6.4 Solvent ranking...... 216 6.5 Summary...... 219

7 Conclusions and future work 222 7.1 Summary...... 222

8 7.2 Main contributions...... 226 7.3 Future work...... 227

Bibliography 229

Appendix A Electronic structure calculations for the gas-phase reaction 257

Appendix B Electronic-structure calculations for the Williamson reaction 262 B.1 Optimized geometries...... 262 B.2 Optimized energies...... 284 B.3 Root-Mean-Square Errors...... 285

Appendix C Overall scheme of the Williamson reaction in methanol-d4 286

Appendix D Methanol-d4: 2-reaction model results 287

9 List of Figures

2.1 Illustration of the activated complex and the activation energy of a reaction... 38 2.2 Solvent representation by diﬀerent solvation model types: from discrete to continuum models. Empty dots represent atoms or molecules calculated by quantum mechanics, full dots by molecular mechanics. Blue represents a dielectric continuum. (Adapted from Jalan et al.1)...... 47

3.1 Rate constants of Reaction 1, calculated using Equation (3.3) at 298 K for various electronic structure levels of theory. Columns of the same colour correspond to the same method. For each method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The QM-calculated rate-constant values at 298 K are compared with the experimental value obtained by evaluating the recommended expression given by Atkinson2 at the same temperature. The black dashed line indicates the experimental reaction rate value so it may easily be compared to computed values...... 71 3.2 Optimized transition-state structure at the M05-2X/cc-pV5Z level of theory. The optimized values for the bond lengths, bond angles, and dihedral angles are reported in Table 3.4...... 74 3.3 Rate constants of Reaction 1 as a function of inverse temperature in the range IG 200-1250 K. The red squares () correspond to the k values calculated in this work (CTST/W), the green triangles (4) to the values computed by Hashimoto and Iwata3 using CTST-ZCT, and the blue circles ( ) to the values calculated by Melissas and Truhlar4 using CVT/SCT. The black curve ( ) represents the rate expression recommended by Atkinson.2 The black diamonds ( ) indicate the experimental values5–13 at the speciﬁc temperatures for which QM3 calculations are performed...... 75

10 3.4 Arrhenius plots of the various models developed in this work compared with experimental reaction rate data reported by Atkinson2 for Reaction 1 over a temperature range from 200 to 1250 K, as shown in Table 3.5, illustrated here by black diamonds ( ). (a) GA model ( ) and Arrhenius model (··· ). (b) GAQM 3 model ( ) and HM A model (··· ). (c) GA4T model ( ) and HM A4T model (··· ). 90

4.1 The reaction of sodium β-naphthoxide and benzyl bromide to form benzyl β- naphtyl ether (O-alkylated product) and 1-benzyl-2-naphthol (C-alkylated product). (Adapted from reference [14].)...... 102 4.2 The β-naphthoxide ion possesses two potential reactive sides due to delocalization of the negative charge. (Adapted from reference [14].)...... 102

4.3 Labelled aromatic carbon C2 for (a) sodium β-naphthoxide and (b) β-naphthol,

and (c) aromatic carbon C12 for the C-alkylated product, for monitoring using 13C NMR spectroscopy...... 109 4.4 The equilibrium between the protonated and deprotonated forms of sodium β- naphthoxide and the C-alkylated product...... 111

4.5 Experimental concentration data as a function of time in acetonitrile-d3 at 298 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS...... 117

4.6 Experimental concentration data as a function of time in acetonitrile-d3 at 313 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS...... 118

4.7 Experimental concentration data as a function of time in acetonitrile-d3 at 323 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS...... 119

11 4.8 Experimental concentration data as a function of time in acetonitrile-d3 at 298 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.42)-(4.51) for the proton-exchange model in gPROMS...... 124 4.9 Arrhenius plot of the logarithm of experimental reaction rate constants as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between

sodium β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K...... 127 4.10 Eyring plot of the logarithm of experimental reaction rate constants over temperature as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the

reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K...... 128 4.11 Electronic energy (Eel,IG) as a function of the sodium-oxygen distance (r) in sodium β-naphthoxide in vacuum performing a relaxed potential energy surface scan at the B3LYP/6-31+G(d) level of theory...... 132 4.12 Electronic energy (Eel) as a function of the sodium-oxygen distance (r) in sodium β-naphthoxide in acetonitrile performing a relaxed potential energy surface scan at the B3LYP/6-31+G(d) level of theory. The results from the calculations in vacuum are shown for comparison...... 133 4.13 A transition-state structure (TS 1) for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum...... 134 4.14 An alternative transition-state structure (TS 2), energetically more favoured, for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum...... 136 4.15 A transition-state structure for the C-alkylation pathway optimized at the B3LYP/6- 31+G(d) level of theory in vacuum...... 138

12 4.16 Gibbs free-energy diﬀerence between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at 298 K obtained for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 2-reaction model (cf. Table 4.8)...... 141 4.17 Selectivity ratio for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at 298 K calculated using Equation (4.56) for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 2-reaction model (cf. Table 4.4)...... 144 4.18 Rate constants for the O- and C-alkylation pathways of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at 298 K calculated using the CTST theory (Equation (4.20)) for various levels of theory and the SMD solvation model implemented in Gaussian 09. The columns of the same colour correspond to the same QM method. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The dashed lines indicate the experimental values at 298 K from the 2-reaction model (cf. Table 4.4)...... 145 4.19 Arrhenius plot of the logarithm of the QM-calculated reaction rate constants at the M05-2X/cc-pVTZ level of theory as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K. The experimental rate constants, illustrated by black symbols, are shown for comparison...... 149

13 4.20 Eyring plot of the logarithm of the QM-calculated reaction rate constants at the M05-2X/cc-pVTZ level of theory over temperature as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K. The experimental rate constants, illustrated by black symbols, are shown for comparison...... 150

5.1 The reaction of benzyl bromide and methanol-d4, resulting in the formation of benzyl methyl ether...... 157 5.2 The reaction of benzyl bromide with the C-alkylated product (deprotonated), resulting in the formation of 1,1-dibenzyl-2-(1H)-naphthalenone (double C-alkylated product)...... 158

5.3 Experimental concentration data as a function of time in methanol-d4 at 298 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (5.9)-(5.21) for the 4-reaction proton-exchange model in gPROMS...... 164

5.4 Experimental concentration data as a function of time in methanol-d4 at 313 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (5.9)-(5.21) for the 4-reaction proton-exchange model in gPROMS...... 165

5.5 Experimental concentration data as a function of time in methanol-d4 at 328 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (5.9)-(5.21) for the 4-reaction proton-exchange model in gPROMS...... 166

14 5.6 Arrhenius plot of experimental reaction rate constants as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium

β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K...... 171 5.7 Eyring plot of experimental reaction rate constants as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium

β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K...... 172 5.8 Electronic energy (Eel) as a function of the distance (r) of the sodium-oxygen atoms in sodium β-naphthoxide in methanol performing a relaxed potential energy surface scan at the B3LYP/6-31+G(d) level of theory. The results from the calculations in vacuum and acetonitrile are shown for comparison...... 175 5.9 Gibbs free-energy diﬀerence between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in methanol at 298 K obtained for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 4-reaction proton-exchange model (cf. Table 5.4)...... 179 5.10 Selectivity ratio for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at 298 K calculated using Equation (4.56) for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 4-reaction proton-exchange model (cf. Table 5.4)...... 180

15 5.11 Rate constants for the O- and C-alkylation pathways of the reaction between sodium β-naphthoxide and benzyl bromide in methanol at 298 K calculated using CTST theory (Equation (4.20)) for various levels of theory and the SMD solvation model implemented in Gaussian 09. Columns of the same colour correspond to the same QM method. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The dashed lines indicate the experimental values at 298 K from the 4-reaction proton-exchange model (cf. Table 5.4)...... 181 5.12 Arrhenius plot of the logarithm of QM-calculated reaction rate constants at the B3LYP/6-311G(2d,d,p) level of theory as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K. The experimental rate constants, illustrated by black symbols, are shown for comparison...... 184 5.13 Eyring plot of the logarithm of QM-calculated reaction rate constants at the B3LYP/6-311G(2d,d,p) level of theory over temperature as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K. The experimental rate constants, illustrated by black symbols, are shown for comparison...... 185

6.1 QM-calculated Gibbs free-energy difference between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in different solvents at 298 K obtained for various levels of theory, indicated by the different symbols and colours, and the SMD solvation model implemented in Gaussian 09...... 200 6.2 Gibbs free-energy difference between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in different solvents at 298 K obtained for selected levels of theory, indicated by the different symbols and colours, and the SMD solvation model implemented in Gaussian 09...... 201

16 6.3 Selectivity ratio of the reaction between sodium β-naphthoxide and benzyl bromide in different solvents at 298 K calculated using Equation (4.56) for various levels of theory, indicated by the different symbols and colours, and the SMD solvation model implemented in Gaussian 09...... 202 6.4 Rate constants for the O- and the C-alkylation pathways of the reaction between sodium β-naphthoxide and benzyl bromide in different solvents at 298 K calculated using Equation (4.20) for various levels of theory, indicated by the different symbols and colours, and the SMD solvation model implemented in Gaussian 09. 203

B.1 A transition-state structure (TS 1) for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The optimized values for the bond lengths, bond angles and dihedral angles are reported in Table B.1...... 262 B.2 A transition-state structure (TS 2) for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The optimized values for the bond lengths, bond angles and dihedral angles are reported in Table B.2...... 264 B.3 A transition-state structure for the C-alkylation pathway optimized at the B3LYP/6- 31+G(d) level of theory in vacuum. The optimized values for the bond lengths, bond angles and dihedral angles are reported in Table B.3...... 266 B.4 Root-mean-square error (RMSE) values for the structural changes between the transition-state structures obtained in diﬀerent solvents and in vacuum for the O- and C-alkylation pathways optimized at the B3LYP/6-31+G(d) level of theory obtained using the iterative closest point (ICP) algorithm. The solvents are presented in descending order of dielectric constant ε. For the O-alkylation pathway, only the transition-state structure TS 2 is considered...... 285

C.1 Overall reaction scheme of the Williamson reaction between sodium β-naphthoxide

and benzyl bromide in methanol-d4...... 286

D.1 Experimental concentration data as a function of time in methanol-d4 at 298 K measured using in situ 1H NMR. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS...... 288

17 D.2 Experimental concentration data as a function of time in methanol-d4 at 313 K measured using in situ 1H NMR. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS...... 289 D.3 Experimental concentration data as a function of time in methanol-d4 at 328 K measured using in situ 1H NMR. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS...... 290

18 List of Tables

3.1 Electronic activation energy barriers ∆‡Eel,IG, corrected for the zero-point vibrational energy, calculated for various electronic structure QM methods. Rate constants kIG are calculated using CTST (Equation (3.3)) at 298 K and compared with available experimental values kExpt. at 298 K from the literature. The uncertainty for the kExpt. values at 298 K is given within the parenthesis..... 68

3.2 Optimized bond lengths (r) and bond angles (∠) between various atoms, cal-

culated at the M05-2X/cc-pV5Z level of theory, for reactants C2H6 and ·OH in comparison with experimental values from the literature. QM=Quantum mechanics, SP=Spectroscopic techniques, ED=Diﬀraction method...... 72 3.3 Vibrational frequencies ν for the equilibrium optimized structures for the reac-

tants C2H6 and ·OH at the M05-2X/cc-pV5Z level of theory and from experimental studies...... 73 3.4 Optimized structure of the transition state at the M05-2X/cc-pV5Z level of theory. The bond lengths between various atoms are represented by r, the bond angles

are as speciﬁed in Figure 3.2, τ1 is the H(2)-C(1)-C(2)-H(1) dihedral angle, τ2 is

the H(1)-C(2)-C(1)-H(3) dihedral angle, φ is the H(4)-C(2)-C(1)-H(1) dihedral angle,

χ1 is the H(5)-C(2)-C(1)-H(4) dihedral angle, χ2 is the H(4)-C(1)-C(2)-H(6) dihedral

angle, ψ is the O(1)-H(4)-C(2)-C(1) dihedral angle and ω is the H(7)-O(1)-H(4)-C(2) dihedral angle...... 74

19 3.5 Rate constants for the kinetics of Reaction 1 obtained from diﬀerent computational and experimental studies. CTST/W refers to the calculations of this work obtained at the M05-2X/cc-pV5Z level of theory using the CTST theory (Equation (3.3)) with the Wigner tunnelling factor (Equation (3.4)). CTST/SCT refers to the calculations of Hashimoto and Iwata3 obtained using the CTST theory with zero-curvature tunnelling (ZCT) correction method for the geometries and frequencies optimized at the MP2/aug-cc-pVDZ level of theory and the barrier heights obtained at the CCSD(T)/aug-cc-pVDZ level of theory. CVT/SCT refers to the calculations of Melissas and Truhlar4 obtained using the canonical- variational theory (CVT) with small-curvature tunnelling (SCT) correction method for optimized structures at the PMP2//MP2/adj2-cc-pVTZ level of theory. Atkin- son’s recommended rate-constant expression2 (evaluated at the various temperatures) and individual experimental values5–13,15–22 (with the uncertainty given within the parenthesis) from the literature are also reported...... 77 3.6 Mean absolute percentage error (MAPE) (Equation (3.5)) and mean absolute

error of the logarithm (MAEln) (Equation (3.6)) for the QM-calculated rate- constant values with CTST/W for Reaction 1 compared with the correlated values given by Atkinson’s expression2 over temperature ranges 200-1250 K, 200-299 K, 300-499 K and 500-1250 K...... 79 3.7 Data sets used to estimate the parameters of the various Arrhenius-type models

developed in this study for the kinetics of Reaction 1. Parameters A1 and Eã refer ? to Equation (3.10). Parameters A2, m and Ea refer to Equation (3.13). Descriptor “Expt.” refers to experimental reaction rate data. Descriptor “QM” refers to QM-calculated reaction rate data obtained with CTST/W or activation energy values at the M05-2X/cc-pV5Z level of theory. Within the parenthesis following the descriptors , the range or the specific point(s) of temperature at which data are fitted is given. Regarding the experimental data, for the entire temperature range the equation of Atkinson2 is used while for the specific temperature points the experimental values of Talukdar et al.7 are used. HM=Hybrid Model..... 85

20 3.8 Comparison of parameters and their 95% conﬁdence intervals (CI)a for the various

Arrhenius-type models developed for the kinetics of Reaction 1. Parameter A1 3 −1 −1 3 −1 −1 −m is given in dm mol s , parameter A2 is given in dm mol s K and ˜ ? −1 parameters Ea and Ea are given in kJ mol . The MAEln (Equation (3.5)) is also reported, comparing with the experimental reaction rate data for Reaction 1, as shown in Table 3.5 for the range 200-1250 K. HM=Hybrid Model...... 88

4.1 Percentage product yield of the O- and the C-alkylated product (based on the isolated amounts) and the calculated percentage product O:C ratio of the reaction between sodium β-naphthoxide and benzyl bromide from available experimental studies in various experimental conditions. RT=room temperature...... 106 Expt. Expt. 4.2 The estimated reaction rate constants kO and kC and the percentage rate Expt. Expt. constant ratio kO :kC of the reaction between sodium β-naphthoxide and benzyl bromide in nine solvents at 298 K as reported by Ganase.23 The solvents are listed in descending order of dielectric constant ε. The experiments were performed using in situ 1H NMR...... 107 4.3 Initial concentrations at time zero of the reactants, sodium β-naphthoxide ([1])

and benzyl bromide ([2]), for the reaction studied in acetonitrile-d3 at three different temperatures. The reported concentration values correspond to the weighted amounts, indicated by the superscript “w”, and the estimated values, indicated by the superscript “e”, with their 95% confidence intervals (CI), obtained by implementing the 2-reaction model in gPROMS...... 121 Expt. Expt. 4.4 Estimated reaction rate constants kO and kC , with their 95% confidence intervals (CI), obtained by fitting the 2-reaction model to experimental concentration data for the components of the reaction between sodium β-naphthoxide

and benzyl bromide in acetonitrile-d3 at three diﬀerent temperatures. The per- Expt. Expt. centage rate constant ratio of kO :kC is also reported...... 121

4.5 Initial concentrations at time zero of the reactants, sodium β-naphthoxide ([1]0)

and benzyl bromide ([2]0), for the reaction studied in acetonitrile-d3 at 298 K. The reported concentration values correspond to the weighted amounts, indicated by the superscript “w”, and the estimated values, with their 95% conﬁdence intervals (CI), obtained by implementing the proton-exchange model in gPROMS, indicated by the superscript “e”...... 123

21 Expt. Expt. 4.6 Estimated reaction rate constants kO and kC , and the equilibrium constant Expt. Keq , with their 95% conﬁdence intervals (CI), obtained by ﬁtting the proton- exchange model to experimental concentration data for the components of the reaction between sodium β-naphthoxide and benzyl bromide and the equilibrium between the protonated and deprotonated forms of sodium β-naphthoxide and the Expt. Expt. C-alkylated product at 298 K. The percentage rate constant ratio of kO :kC is also reported...... 123 4.7 Estimated parameters of the Arrhenius equation (Equation (3.10)) for the re-

action between sodium β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K...... 128 4.8 Estimated thermodynamic parameters of the Eyring equation (Equations (4.54) and (4.4)) for the reaction between sodium β-naphthoxide and benzyl bromide in

acetonitrile-d3 at temperatures 298, 313 and 323 K...... 128 ◦,QM 4.9 QM-calculated values for the Gibbs free-energy diﬀerence (∆GC-O ), the selectiv- QM QM ity ratio (kO /kC ) and the rate constants of the O- and C-alkylation pathways (kQM) of the reaction in acetonitrile at 298 K at all the levels of theory tested. The corresponding experimental values obtained in this work are also reported for comparison...... 142 4.10 Absolute percentage error (APE) for the QM-calculated values for the Gibbs ◦,QM QM QM free-energy diﬀerence (∆GC-O ), the selectivity ratio (kO /kC ) and the rate constants of the two alkylation pathways (kQM) of the reaction in acetonitrile at 298 K at the M05-2X/6-31G(d) and M05-2X/cc-pVTZ levels of theory...... 147 4.11 QM-calculated rate constants for the O- and the C-alkylation pathways (kQM) of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at the M05-2X/cc-pVTZ level of theory using the CTST theory and the SMD model (Equation (4.20)) at temperatures 298, 313 and 323 K. The percentage Expt. Expt. rate constant ratio of kO :kC is also reported...... 148 4.12 Absolute percentage error (APE) and mean absolute percentage error (MAPE) values for the QM-calculated rate constants of the two pathways of the reaction (kQM) at the M05-2X/cc-pVTZ level of theory compared with the experimental values in acetonitrile at temperatures 298, 313 and 323 K...... 148

22 4.13 Estimated parameters obtained by ﬁtting the Arrhenius equation (Equation (3.10)) to QM-calculated rate constants at the M05-2X/cc-pVTZ level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K...... 151 4.14 Estimated thermodynamic parameters obtained by ﬁtting the Eyring equation (Equation (4.2)) to QM-calculated rate constants at the M05-2X/cc-pVTZ level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K...... 151

1 13 5.1 H NMR and C NMR chemical shifts for the aliphatic −CH2 group in benzyl bromide and in benzyl methyl ether resulted from a 2D HSQC experiment.... 157 1 5.2 Heteronuclear chemical shift correlation between the aliphatic −CH2 group ( H 13 NMR) and the −CD3 group ( C NMR) in benzyl methyl ether obtained from a 2D HMBC experiment...... 157

5.3 Initial concentrations at time zero of the reactants, sodium β-naphthoxide ([1]0)

and benzyl bromide ([2]0), for the reaction studied in methanol-d4 at 298 K. The reported concentration values correspond to the weighted amounts, indicated by the superscript “w”, and the estimated values, indicated by the superscript “e”, with their 95% conﬁdence intervals (CI), obtained by implementing the 4-reaction proton-exchange model in gPROMS...... 170 Expt. Expt. Expt. Expt. 5.4 Estimated reaction rate constants kO , kC , kDC , the term kBME ·[8], and the

equilibrium constant Keq, with their 95% conﬁdence intervals (CI), obtained by ﬁtting the 4-reaction proton-exchange model to experimental concentration data for the components of the reaction between sodium β-naphthoxide and benzyl bromide, the reaction for the formation of the double C-alkylated product, the

reaction between benzyl bromide and methanol-d4 and the equilibrium between the protonated and deprotonated forms of sodium β-naphthoxide and the C- Expt. Expt. alkylated product at 298 K. The percentage rate constant ratio of kO :kC is also reported...... 170 5.5 Estimated parameters of the Arrhenius equation (Equation (3.10)) for the re-

action between sodium β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K...... 173

23 5.6 Estimated thermodynamic parameters of the Eyring equation (Equations (4.54) and (4.4)) for the reaction between sodium β-naphthoxide and benzyl bromide in

methanol-d4 at temperatures 298, 313 and 328 K...... 173 ◦,QM 5.7 QM-calculated values for the Gibbs free-energy difference (∆GC-O ), the selectiv- QM QM ity ratio (kO /kC ) and the rate constants of the O- and C-alkylation pathways (kQM) of the reaction in methanol at 298 K at all the levels of theory tested. The corresponding experimental values obtained in this work are also reported for comparison...... 177 5.8 Absolute percentage error (APE) for the QM-calculated values for the Gibbs ◦,QM QM QM free-energy difference (∆GC-O ), the selectivity ratio (kO /kC ) and the rate constants of the two alkylation pathways (kQM) of the reaction in methanol at 298 K at the M05-2X/6-31G(d) and B3LYP/6-311G(2d,d,p) levels of theory... 182 5.9 QM-calculated rate constants for the O- and the C-alkylation pathways (kQM) of the reaction between sodium β-naphthoxide and benzyl bromide in methanol at the B3LYP/6-311G(2d,d,p) level of theory using CTST and the SMD model at QM QM temperatures 298, 313 and 328 K. The percentage rate constant ratio of kO :kC is also reported...... 183 5.10 Absolute percentage error (APE) and mean absolute percentage error (MAPE) values for the QM-calculated rate constants (kQM) of the two pathways of the reaction at the B3LYP/6-311G(2d,d,p) level of theory compared with the experimental values in methanol at temperatures 298, 313 and 328 K...... 183 5.11 Estimated parameters obtained by fitting the Arrhenius equation (Equation (3.10)) to QM-calculated rate constants at the B3LYP/6-311G(2d,d,p) level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K...... 185 5.12 Estimated thermodynamic parameters obtained by fitting the Eyring equation (Equation (4.2)) to QM-calculated rate constants at the M05-2X/cc-pVTZ level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K...... 186 5.13 Mean absolute percentage error (MAPE) values reported for the two levels of theory identified as the most accurate for QM predictions for selectivity (M05-2X/6- 31G(d)) and kinetics (B3LYP/6-31+G(d)) for solvents acetonitrile and methanol at 298 K...... 189

24 6.1 Macroscopic descriptors for the solvent 1,2-dimethoxyethane (1,2-DME) determined either from experiments (Expt.) or group-contribution (GC) techniques. The various symbols are defined in the text...... 194 6.2 Mean signed error (MSE) and mean unsigned error (MUE) in SMD calculations for solvation free energies and transfer free energies for neutral and ionic solutes per solvent, reported by Marenich et al.24 The SMD calculations have been performed at the M05-2X/6-31G(d) level of theory...... 195 6.3 Root-mean-square error (RMSE) values for the structural changes between the transition-state structures obtained in different solvents and in vacuum for the O- and C-alkylation pathways optimized at the B3LYP/6-31+G(d) level of theory obtained using the iterative closest point (ICP) algorithm. The solvents are listed in descending order of dielectric constant ε. For the O-alkylation pathway, only the transition-state structure TS 2 is considered...... 196 ◦,QM 6.4 QM-calculated values for the Gibbs free-energy difference (∆GC−O ), the selectiv- QM QM QM ity ratio (kO /kC ) and the rate constants of the two alkylation pathways (k ) of the reaction in ethanol at 298 K at all the levels of theory tested...... 206 ◦,QM 6.5 QM-calculated values for the Gibbs free-energy difference (∆GC−O ), the selectiv- QM QM QM ity ratio (kO /kC ) and the rate constants of the two alkylation pathways (k ) of the reaction in acetone at 298 K at all the levels of theory tested...... 207 ◦,QM 6.6 QM-calculated values for the Gibbs free-energy difference (∆GC−O ), the selectiv- QM QM QM ity ratio (kO /kC ) and the rate constants of the two alkylation pathways (k ) of the reaction in THF at 298 K at all the levels of theory tested...... 208 ◦,QM 6.7 QM-calculated values for the Gibbs free-energy difference (∆GC−O ), the selectiv- QM QM QM ity ratio (kO /kC ) and the rate constants of the two alkylation pathways (k ) of the reaction in 1,2-DME at 298 K at all the levels of theory tested...... 209 ◦,QM 6.8 QM-calculated values for the Gibbs free-energy difference (∆GC−O ), the selectiv- QM QM QM ity ratio (kO /kC ) and the rate constants of the two alkylation pathways (k ) of the reaction in ethyl acetate at 298 K at all the levels of theory tested..... 210 ◦,QM 6.9 QM-calculated values for the Gibbs free-energy difference (∆GC−O ), the selectiv- QM QM QM ity ratio (kO /kC ) and the rate constants of the two alkylation pathways (k ) of the reaction in 1,4-dioxane at 298 K at all the levels of theory tested...... 211

25 6.10 Graphically extracted range of variation for the product ratio of the O- over the C-alkylated product (O:C) – indicated as P1/P2 in the original study – based on the experimentally measured raw concentration data in six solvents tested in the study of Ganase.23 Only the solvents selected from that study for comparison with the QM-calculated values of this work are listed. The percentage product ratio O:C is also reported...... 213 6.11 Available experimental data of the Williamson reaction between sodium β-naphthoxide and benzyl bromide at 298 K for the set of solvents investigated in this work. The data are classiﬁed based on the experimental technique used. When work-up techniques were used, the percentage product ratio values (O:C) – based on the isolated product amounts – are reported. When 1H NMR spectroscopy was used, Expt. Expt. the percentage rate constant ratio values (kO :kC ) are reported...... 215 6.12 Solvent ranking according to the experimental percentage rate constant ratio of Expt. Expt. Expt. kO :kC in descending order of kO . The experimental values from this work and the study of Ganase23 are used...... 216 6.13 Ranking of high- and low-dielectric constant (ε) solvents according to the ex- Expt. Expt. perimental percentage rate constant ratio of kO :kC in descending order of Expt. 23 kO . The experimental values from this work and the study of Ganase are used...... 217 6.14 Solvent ranking according to the QM-calculated percentage rate constant ratio QM QM QM of kO :kC at the M05-2X/6-31G(d) level of theory in descending order of kO . The solvents are also ranked according to the corresponding experimental values for comparison. The experiments for solvents acetonitrile, acetone, methanol, ethanol, THF and 1,4-dioxane have been performed using deuterated solvents... 217 6.15 Ranking of high- and low-dielectric constant (ε) solvents according to the QM- QM QM calculated percentage rate constant ratio of kO :kC at the M05-2X/6-31G(d) QM level of theory in descending order of kO . The solvents are also ranked according to the corresponding experimental values for comparison. The experiments for solvents acetonitrile, acetone, methanol, ethanol, THF and 1,4-dioxane have been performed using deuterated solvents...... 218

A.1 QM-calculated ideal-gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of ethane

(C2H6) at all the levels of theory tested...... 258

26 A.2 QM-calculated ideal-gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of the hydroxyl radical (·OH) at all the levels of theory tested...... 259 A.3 QM-calculated imaginary frequency of the transition-state structure (ν‡), ideal- gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of the transition-state structure

of the reaction between ethane (C2H6) and the hydroxyl radical (·OH) at all the levels of theory tested...... 260 A.4 QM-calculated ideal-gas molecular partition functions (q0,IG) at diﬀerent tempera-

tures for the components of the reaction between ethane (C2H6) and the hydroxyl radical (·OH) at the M05-2X/cc-pV5Z level of theory. TS=Transition state.... 261

B.1 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 263 B.2 Internal coordinates (Z-matrix) of the transition-state structure TS 2 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 265 B.3 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).267

27 B.4 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetonitrile. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 268 B.5 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetonitrile. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).269 B.6 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in methanol. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 270 B.7 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in methanol. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).271 B.8 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethanol. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 272 B.9 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethanol. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).273

28 B.10 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetone. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 274 B.11 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetone. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).275 B.12 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in THF. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 276 B.13 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in THF. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).. 277 B.14 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,2- DME. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 278 B.15 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,2-DME. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).279

29 B.16 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethyl acetate. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 280 B.17 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethyl acetate. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).281 B.18 Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,4- dioxane. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle)...... 282 B.19 Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,4-dioxane. The internal coordinates for each atom are defined by the bond lengths, bond angles and dihedral angles with respect to previously defined atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).283 B.20 QM-calculated imaginary frequency of the transition-state structure (ν‡), ideal- gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of the transition-state structures for both the alkylation pathways optimized at the B3LYP/6-31+G(d) level of theory in vacuum. For the O-alkylation pathway, only the transition-state structure TS 2 is considered...... 284 B.21 QM-calculated imaginary frequency of the transition-state structure (ν‡), liquid- phase electronic energy (Eel,L), zero-point vibrational energy (ZPVE) and the non-electrostatic term of the solvation free energy (GCDS,L) at 298 K of the transition-state structure TS 2 for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in different solvents...... 284

30 Introduction 31

B.22 QM-calculated imaginary frequency of the transition-state structure (ν‡), liquid- phase electronic energy (Eel,L), zero-point vibrational energy (ZPVE) and the non-electrostatic term of the solvation free energy (GCDS,L) at 298 K of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6- 31+G(d) level of theory in diﬀerent solvents...... 284

D.1 Initial concentrations at time zero of the reactants, sodium β-naphthoxide ([1]) and benzyl bromide ([2]), for the reaction studied in methanol-d4 at temperatures 298, 313 and 328 K. The reported concentration values correspond to the weighted amounts, indicated by the superscript “w”, and the estimated values, indicated by the superscript “e”, with their 95% confidence intervals (CI), obtained by implementing the 2-reaction model in gPROMS...... 291 Expt. Expt. D.2 Estimated reaction rate constants kO and kC , with their 95% confidence intervals (CI), obtained by fitting the 2-reaction model to experimental concentration data for the components of the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4 at three different temperatures. The percent- Expt. Expt. age rate constant ratio of kO :kC is also reported...... 291 Chapter 1

Introduction

1.1 Temperature and solvent eﬀects on chemical reactions

Chemical reactions are the processes in which molecular species or groups of species change their chemical nature by transforming into other molecules. They occur lavishly in gas, liquid and solid phase and are present in every aspect of life. Our very own human existence is the wondrous result of intriguing, rather complex combinations of perpetually occurring chemical reactions.25 The speed at which a chemical reaction proceeds is called the reaction rate, and it is expressed as the product of the rate constant and the concentrations of the reacting substances (rate = rate constant × concentration).26 The knowledge of chemical reactions is intertwined with the knowledge of the factors that influence the rate of reactions and the laws that govern these rates. Acquiring such knowledge is utterly important in many engineering applications in which chemical reactions are involved as key constituents.27 Several factors impact the rates of chemical reactions; two of these factors that have attracted a great deal of scientific interest are the temperature and the solvent medium.27 The effect of temperature on chemical reaction rates may be illustrated using a simple example. At room temperature, the gaseous mixture of hydrogen and oxygen in stoichiometric ratio reacts at an extremely slow rate, undetectable even after hundreds of years. Based on reliable kinetic studies,26 the estimated half-life† of the reactants is more than 1025 years. However, in the presence of a spark, where the temperature may be more than 1000 K, the mixture explodes with a half-life of less than 10−6 seconds, suggesting an increase in the rate of more than 1038 times.26 This example, eloquent in its simplicity, describes the profound impact of temperature on the rate of a reaction.

†Half-life is the time required for half of a reactant to be consumed. 26

32 Introduction 33

Although the effect of temperature on reaction rates is easily perceived, (usually the higher the temperature, the faster the rate†) a quantitative law that links temperature alterations with rate variations has been the subject of significant research since the middle of the 19th century. A number of empirical approaches have been proposed for the relation of temperature with the rate constant (the most famous empirical relation is the Arrhenius equation29). These empirical approaches have paved the way to more elaborate and theoretically founded predictive kinetic models that today constitute valuable tools in the hands of scientists and engineers. Solvents are the components that are in excess in a solution30 and have been the subject of research for thousands of years. Already from the 15th century, alchemists used to believe that compounds do not react unless in fluid or if dissolved (which reads in Latin “Corpora non agunt nisi fluida seu soluta”)‡ and they were long searching for a universal solvent that dissolves all substances: the so-called “Alkahest”.30 Although the quest for a “universal” solvent was somewhat elusive, nowadays the search of an optimal solvent that meets a number of performance criteria for a given reaction is much more feasible and relevant to a broad range of industrial applications (e.g., crystallisation, CO2 capture, liquid-liquid extraction and chemical reactions).32 In terms of reaction rates, the effects of solvents were first noted in 1862 by Berthelot and Péande Saint-Gilles,33 who studied the esterification of acetic acid with ethanol. Nearly 30 years later, Menschutkin,34 who performed extensive experiments for the reaction of trialky- lamines with haloalkanes, concluded that “solvents can greatly influence the course of chemical reactions”.§ Since then, the study of solvents has significantly progressed, mostly experimentally, enabling researchers to have a good understanding about how solvents affect not only the rate but also the selectivity, the stability and the overall performance of a reaction.30 An example of solvent effects on the rate constant of a reaction may be found in the study of Fukuzumi and Kochi,35 who showed that the rate of iodinolysis of tetramethyltin is 105 times faster in acetonitrile than in tetrachloromethane. A more elaborate example may be found in the study of Adjiman et al.,36 in which the effect of solvent in the Grubbs II-catalysed ring- closing metathesis reaction was investigated on multiple aspects, such as the rate constant of the reaction, the solubility of the catalyst and the deactivation of the catalyst. For example, in acetone the reaction rate was relatively fast, but the quick deactivation of the catalyst was

†However, some types of reactions do not always follow this trend, for example enzymatic reactions. 28 ‡The Greek philosopher Aristotle (384-322 B.C.) was the first to proclaim this in his work De generatione et corruptione. 31 The original text is given in Greek as “T α υγρα´ µικτα´ µαλιστα´ των σωµατων´ ”. §This sentence is written in a letter that Menschutkin wrote to Prof. Louis Henry in 1890. 30 Introduction 34 found to impede total conversion of the reaction. In cyclohexane, the reaction rate was fast without observable catalyst deactivation, but the low catalyst solubility resulted in large reactor volumes. In dichloromethane, the reaction rate was slow but, since no catalyst deactivation occurred, complete conversion was achieved. Acetic acid was found to have similar performance to dichloromethane but less environmental impact. At the end, the final solvent selection is a challenging task involving trade-offs between performance, economic and environmental criteria.

1.2 Predicting temperature and solvent eﬀects on reaction rates

In a recent study of the American Chemical Society (ACS) Green Chemistry Institute (GCI), the need of an optimum reaction environment for the respective phase of a reaction has been emphasized as an essential element for securing low-cost and environmentally friendly reactive processes.37 The role of the solvent has been highlighted as crucial in various steps of many manufacturing processes, while temperature is also mentioned as an important control variable which may lead to improved yields and selectivity, and to reduced process variance.37 The first critical step towards designing an optimum reaction environment is the accurate prediction of the impact of reaction environment parameters, such as the temperature and the solvent medium, and such a task is by no means trivial. Temperature and solvent effects on reaction rates are currently accounted mostly based on experience and heuristic processes. There is a growing interest in developing predictive models that will enable fast and accurate predictions and reduce unnecessary and expensive experimentation.38 Ideally, such predictive models will guide targeted experiments, and potentially, in the future, precede experiments in the exploration of unknown or dangerous or counterintuitive options. A newly emerged concept is that of computer-aided molecular design39,40 (CAMD). CAMD techniques are employed for the design of optimal solvents for many processes (including chemical reactions) based on suitable performance criteria (e.g., solubility, selectivity, kinetics, environmental health or safety properties).38,41–45 Predictive models have been recently incorpo- rated into CAMD techniques for solvent design in reaction kinetics with promising results. For example, in the studies of Folićet al.42 and Struebing et al.45 a linear empirical model is used to relate reaction rate constants with solvent properties, based either on experiments (in the case of Folićet al.42) or quantum-mechanical methods (in the case of Struebing et al.45). Thanks to the steadily increasing computing power, even highly demanding computational methods, such as quantum mechanics, may now be employed in the quest for efficient predictive models. Introduction 35

The applicability range of the CAMD techniques leading to successful predictions is confined to the applicability range and accuracy of the predictive models used to compute the targeted properties.39 In this context, temperature is recognized as a crucial parameter, especially for the implementation of CAMD techniques for reaction kinetics, where accurate predictions for thermodynamic quantities may be required at varying temperature ranges. In a nutshell, the prediction of the effects of reaction environment conditions, such as the temperature and the solvent medium, on the rate (or any other metric of interest) of a reaction is, by no means, a trivial task; however, given the multiple benefits that this task is associated with in numerous engineering contexts and the variety of emerging powerful computational techniques, it is indeed worth researching.

1.3 Objectives of the thesis

In this thesis we aim to develop a systematic methodology for the prediction of reaction environment effects, such as temperature and solvent effects, on the kinetics and selectivity of a given reaction. The novelty of this methodology lies in the creative and beneficial combination of experimental and computational methods with the following objectives:

• To obtain a profound understanding of chemical phenomena, such as reaction rates and selectivity, relevant to a given reaction;

• To predict these chemical phenomena in an accurate qualitative and quantitative manner.

Key aspects considered in the development of this methodology are the following: the use of an accurate kinetic model to obtain reliable predictions; the suitability of various computational tools to balance computational cost and accuracy; and ﬁnally the availability of reliable experimental data to validate and complement the predictions. The proposed methodology covers the following two cases:

• The prediction of temperature eﬀects on the rate constant of a gas-phase reaction, followed by the development of correlative kinetic models using a number of experimental and predicted data.

• The prediction of solvent eﬀects on the rate constant and selectivity of a liquid-phase reaction, for which detailed kinetic experiments involving in situ 1H NMR acquisition are performed for comparative purposes.

In both studies, kinetic calculations are performed at the quantum-mechanical level of accuracy. Introduction 36

1.4 Outline of the thesis

In Chapter 2, we introduce some fundamental theoretical concepts on the field of reaction kinetics as well as the main experimental and computational tools available in this field. In addition, we provide an overview of the state of the art in modelling temperature and solvent effects on the thermodynamics of reaction kinetics in the liquid phase. In Chapter 3, we focus on the prediction of temperature effects on the kinetics of a hydrogen abstraction reaction in the gas phase employing quantum-mechanical methods. We also propose a methodology for the development of hybrid correlative kinetic models based on experimental and predicted data; this methodology is then applied to the gas-phase hydrogen abstraction reaction. In Chapter 4, we present an expression for the rate constant of a reaction in the liquid phase using conventional transition state theory and a continuum solvation model; this expression is used later in this chapter for liquid-phase reaction kinetic and selectivity predictions. With the aim of studying the impact of temperature and solvent on liquid-phase kinetics and selectivity, we select a Williamson reaction with two possible pathways. We perform experiments and quantum-mechanical calculations for the determination of the kinetics and selectivity of the reaction in the solvent acetonitrile at various temperatures. In Chapter 5, we extend the investigation of the Williamson reaction with methanol as the solvent medium. We perform more experiments and quantum-mechanical calculations for this system at various temperatures. Considering the computational methods that agree best with experiments, as identified in this and the previous chapter, we propose one computational method for selectivity predictions and one for kinetic predictions that perform best on average for both solvents (acetonitrile and methanol). In Chapter 6, we undertake a systematic computational investigation of the Williamson reaction in a diverse set of eight solvents of varying polarity. For these eight solvents, we perform quantum-mechanical calculations and obtain the selectivity as well as the kinetics of the reaction in each solvent. The best quantum-mechanical method for selectivity, as identified in the previous chapter, is used to rank the solvents according to selectivity criteria and the results are compared with the experimental ranking. Finally, in Chapter 7 we summarize the main conclusions of this thesis and we discuss possible directions for future research. Chapter 2

Determination of reaction kinetics: temperature and solvent eﬀects

2.1 Basic kinetic concepts

Chemical kinetics, or reaction kinetics, is the study of aspects related to chemical reactions and in particular, to the investigation of the experimental conditions that affect the rate of reactions in chemical processes.26 A plethora of experimental methods – some of them very elaborate – as well as many theoretical ones have been developed for studying and following chemical reactions.46,47 The current state of experimental and theoretical treatment of reaction kinetics allows us to have a reasonably good understanding of the main factors that influence the reaction rates. Naturally, unresolved issues exist and they direct the paths for future investigation in the field. Temperature was recognized early on as one of the most influential factors on the rate of a reaction. Deliberations on the temperature dependence of reaction rates lasted more than 60 years, from around 1850 until 1910, and caused a lot of controversy and confusion at the time. The controversial nature of the topic may be condensed in Mellor’s monograph on equilibrium and kinetics,48 published in 1904, which quotes Ostwald as saying that the temperature dependence of reaction rates is “one of the darkest chapters in chemical mechanics”. A major contribution to the topic was made in 1889 by S. Arrhenius29 with the development of his equation for the relation of a reaction rate constant k1 with temperature T :

! E˜ k = A exp − a , (2.1) 1 1 RT

37 Chapter 2. Determination of reaction kinetics: temperature and solvent effects 38 where A1 is referred to as the pre-exponential factor, Eã is the activation energy (or energy barrier) and R is the ideal gas-constant. (A derivation of the Arrhenius equation is presented in a later chapter and the terms A1 and Eã are more formally determined with respect to their temperature dependence.) Arrhenius established his theory based on van’t Hoff’s pioneering work on chemical dynamics49 in 1884. Inspired by van’t Hoff’s research, Arrhenius presented theoretical arguments regarding the physical evolution of a reaction. He introduced the concept of activation energy Eã, according to which the reactants must overcome an energy barrier before they form the final products, passing through a maximum at the top of the barrier, which today we call a transition state (TS). The molecules at the energy maximum are nowadays known as activated complexes. These concepts are graphically illustrated in Figure 2.1. Arrhenius suggested that an equilibrium is established between reactants and activated complexes and that the rate of a reaction is proportional to the concentration of the activated complexes.29 His explanation, rough as it might seem from today’s point of view, captured the fundamental concepts of reaction kinetics and became the stepping stone for the development of many future kinetic theories.26 One of these theories is the well-known transition state theory,50,51 which will be discussed in a Section 2.2.

Activated Complex

Activation Energy

Energy Reactants

Products Reaction coordinate

Figure 2.1: Illustration of the activated complex and the activation energy of a reaction.

The solvent is another factor that may have a profound effect on the rate of a reaction. De- pending on the nature of the solvent, the reactants and the activated complex of a reaction may be solvated in different extents. As a result, they will be stabilised at different potential energy (and free energy) levels, varying subsequently the value of the activation energy (cf. Figure 2.1). Chapter 2. Determination of reaction kinetics: temperature and solvent effects 39

Ultimately, different solvents will either decrease or increase the activation energy barrier; in the former case, the rate constant will increase and the rate of the reaction will accelerate, while in the latter case the rate constant will decrease and the rate of the reaction will decelerate. Hughes and Ingold52–54 were the first to present a simple qualitative solvation model for the study of solvent effects on aliphatic nucleophilic substitution and elimination reactions. The knowledge acquired from their study is encapsulated in the following set of predictive rules:30

1. When increasing the solvent polarity, the rate of the reaction will increase if the charge density of the activated complex is greater than that of the reactants; in this case, stabilization of the activated-complex by the solvent is implied.

2. When increasing the solvent polarity, the rate of the reaction will decrease if the charge density of the activated complex is lower than that of the reactants; in this case, destabi- lization of the activated-complex by the solvent is implied.

3. When changing the solvent polarity, the rate of the reaction will remain unaltered, if the charge density of the activated complex is similar to that of the reactants.

The study of solvent effects and their impact on reaction kinetics has progressed tremendously since then. Various methods, either empirical – mostly in the form of linear free-energy rela- tionships 30 – or based on theoretical chemistry, have been proposed in order to predict reaction rate constants considering solvent effects. The reader is referred to specialized textbooks30,55 for details on the empirical approaches. The most commonly applied theoretically based kinetic theory, the transition state-theory (TST), which provides a qualitative and quantitative understanding of solvent effects on reaction rate constants, is briefly reviewed next.

2.2 Conventional transition state theory

Conventional transition state theory (CTST) provides a solid theoretical framework for the idea behind the theory of Arrhenius about the mechanism of a reaction, employing the principles of statistical mechanics for the prediction of the rate constant. It was presented in 1935 almost simultaneously by Eyring50 and by Evans and Polanyi.51 Although CTST was severely criti- cized in its early stages, after more than 80 years it still remains irreplaceable thanks to its simplicity and accuracy. Within its framework, even very complex reactions may be explained qualitatively on a rather simple basis,26 while it also provides good quantitative predictions for Chapter 2. Determination of reaction kinetics: temperature and solvent eﬀects 40 large molecules, such as enzymes.56,57 The basis of transition state theory is summarized in the following paragraphs. Given an elementary bimolecular reaction

A + B → products, (2.2) where A and B are the reactants, CTST assumes that the reactants are in quasi-equilibrium with an activated complex (AB)‡, which is considered as an assemble of molecules in equilibrium with each other and with the reactants. As a result, Equation (2.2) may be written in terms of the activated complex (AB)‡ as

‡ A + B AB → products. (2.3)

The reactants pass through the activated complex configuration before reaching the final product state by following the minimum potential energy path, which is commonly referred as intrinsic reaction coordinate (IRC), and it usually corresponds to the change of a geometric parameter, e.g., bond distance or angle, along the reaction pathway. A representation of the potential energy (or total electronic energy) of a molecular system against all possible atomic configurations obtained during a reaction results in a hyper-surface in a multidimensional space, also known as potential energy surface (PES). The reactant and product configurations are both equilibrium structures and they correspond to energy minima in the PES. Similarly, the transition state is a first-order saddle point58 and it corresponds to a maximum along the reaction coordinate path and a minimum along all the other directions. A key assumption of CTST is that there are not multiple crossings along the potential energy surface, i.e., once an activated complex has been formed there is no possibility of returning to the reactants and it is hence assumed that it will proceed towards the products. Therefore, the rate constant of a reaction predicted by CTST is always an upper bound to the true rate-constant value;58 thus, this usually results to overestimated predictions. To circumvent this problem and allow the possibility of recrossing paths, a transmission coefficient κ may be introduced. This coefficient also addresses the quantum-mechanical phenomenon of tunnelling, in which molecules that do not have enough energy to surpass the activation energy barrier may “tunnel” through the barrier and form the products. Typical values of κ lie within the range 0.5-2. At lower temperatures tunnelling dominates, resulting in values of κ higher than 1, whereas at higher temperatures re-crossings happen extensively, resulting in values of κ lower than 1. Chapter 2. Determination of reaction kinetics: temperature and solvent effects 41

In an effort to overcome certain limitations and to instil generality, CTST has been expanded and refined in numerous works, resulting in a group of methods summarized under the name generalized transition state theory (GTST). Perhaps the most widely used method in this group is the variational transition state theory 59–61 (VTST). In VTST, instead of considering the transition state at a fixed maximum position in PES, as it is assumed in CTST, the notion of a dividing surface (which is allowed to vary) is introduced and the position of the surface at which the minimum reaction rate is calculated is taken to be the one of the transition state. By doing so, the effect of multiple recrossings is significantly reduced. Other treatments of GTSTs include the canonical variational state theory (CVTST), the microcanonical variational transition state theory (µVTST) and the improved canonical variational state theory (ICVTST). Interested readers are referred to specialized textbooks26,47,58 for more details.

2.3 Determination of reaction kinetics

2.3.1 Experimental techniques

Deciding upon a method to determine the rate of a reaction is a fundamental step of a kinetic investigation. Traditionally, reaction kinetics have been explored through experimental techniques and an abundance of studies may be found in the literature, where a whole spectra of different methods are applied. A few examples of experimental techniques used for reaction kinetics monitoring are presented in the following paragraphs. A large body of experimental techniques is available for monitoring reaction kinetics in the gas and liquid-phase. Whether measurements are taken in the same place that the reaction occurs or not (e.g., in a reaction vessel), the experimental techniques may be classified as in-situ or ex-situ. Ultimately, the aim is to obtain time-dependent concentration data of the species involved in the reaction. Concentration determination is possible either through direct measurements or monitoring of any chemical or physical changes happening as the reaction proceeds, for example in absorbance, pressure, pH, volume or conductivity, for which a quantitative relation with concentration may be established. Typically, in conventional kinetic investigations, ex-situ techniques are used and reactions are followed for large time-spans, e.g., hours, days, weeks or even months. A static system is employed, for example a batch reactor, in which the reactants are inserted and concentration changes are followed. Samples are withdrawn at regular time-intervals and are “quenched” before analysis takes place. Sampling errors and speed of analysis are identified as critical factors.62 In Chapter 2. Determination of reaction kinetics: temperature and solvent effects 42 modern studies, favourable ex-situ techniques employed are titration; chromatography, including gas-liquid chromatography (GLC) and high-performance liquid chromatography (HPLC); mass spectroscopy (MS); spectroscopy, including infrared Raman (IR) and visible ultraviolet spectra (UV-Vis); and spin resonance methods, including nuclear magnetic resonance (NMR) and electron spin resonance (ESR). When a reaction is studied continuously, in-situ techniques are preferred. The reaction is probed internally and any changes are monitored while the reaction is in progress. As a result, errors due to imprecision in sampling time-intervals, quenching of the reaction and loss of materials due to work-up procedures are avoided and the analysis is often fast.62 Typical monitoring time-spans extend from minutes to hours or days. Analytical techniques, including potentiometry, polarimetry, barimetry and dilatometry, as well as modified ex-situ techniques that allow in-situ monitoring, such as reaction calorimetry, IR, UV-Vis and NMR, may be employed for monitoring reaction kinetics. Once experimental concentration data have been obtained, a fundamental question that needs to be answered is how the rate of the reaction depends on the concentrations of the reacting species.26 Two main procedures are usually employed: the differential method and the integration method. The differential method is based on the direct measurement of rates through determination of the slopes of the concentration-time curves; the slopes are then analysed in terms of their dependence on the reactant concentration.49 The integration method is based on a trial-and-error procedure by choosing a provisional value for the order of the reaction, integrating the differential equation corresponding to that order and testing if the experimental time-dependent concentration data fit the resulted integrated equation.26 If not, the procedure is repeated with a different value for the order of the reaction. The differential method is superior to the integration method on this aspect, as no prior knowledge of the reaction order is required. Furthermore, in cases where the order of the reaction is a non-integer, the integration method would most likely fit the data to the integrated equation with the nearest integer-value order within the experimental error, while the differential method is able to discern possible deviations.26 If, however, the order of a reaction is well established, the integration method is expected to provide accurate rate constant and initial concentration estimates. In this thesis, in-situ NMR spectroscopy will be used for continuous monitoring of the kinetics of the reaction of interest. The integration method will be used for the analysis of the obtained concentration data as a function of time, since the order of the reaction studied is known. Chapter 2. Determination of reaction kinetics: temperature and solvent effects 43

2.3.2 Computational methods

Computational chemistry is a branch of chemistry that employs computers in solving chemical problems.58 In the last few decades it has taken centre stage as a viable and powerful approach,63 and has been heavily employed in many areas of chemical research. There is a number of reasons that may explain the rapid grow of this area; in particular, some of them are the following:63

• The progress of theoretical methodologies has reached the point that quantities of key importance, such as molecular properties and reaction energetics, may be obtained with suﬃcient accuracy to be used quantitatively.

• The outstanding advances in computer power – with increasing pace similar to no other technology – have rendered computational methods applicable to large systems consisting of several thousand atoms, which are commonly encountered in real-world applications, e.g., in drug discovery.64

• The implementation of computational methods in user-friendly software packages, requir- ing little or no prior training, has boosted signiﬁcantly their widespread use.

• Last, but not least, computational chemistry is nowadays perceived as an integral component of every chemistry-related study, acting in synergy with traditional experimental techniques, either as a complementary, a designing or an exploratory tool.

Nevertheless, it would be frivolous to assume that computational chemistry has evolved without any obstacles on its way. The very solution of its cornerstone equation, the Schrödinger equation,65 is per se a major challenge with many hurdles. An exact solution of the Schrödinger equation in terms of electron motions results in valuable information regarding the electronic energy, electron density and molecular structure of a system.58 However, the equation may be solved exactly only for one-electron systems, such as the hydrogen atom; for any other system certain simplifying approximations need to be made. Based on these approximations, one may classify the existing computational methods into four main categories: ab initio,† density- functional theory (DFT), semi-empirical and molecular mechanics (MM) methods. Major differences may be seen between them in terms of accuracy, computational cost, number of empirical parameters employed and ranges of applicability, among others.

†Ab ibitio comes from latin and it means “from the beginning”. In its modern meaning is used to signify methods that stem “from ﬁrst principles of quantum mechanics”. Chapter 2. Determination of reaction kinetics: temperature and solvent eﬀects 44

The emphasis of this thesis is on ab initio and DFT methods. Before we continue any further, we shall define a few terms which are relevant to these methods and will be regularly used from now on. In ab initio methods, the energy of a system is determined with respect to a wave function.47 Hartree-Fock (HF) theory66–68 and Møller-Plesset second-order perturbation (MP2) theory69 on one hand, and the coupled cluster (CC) theory70 on the other hand, are well-known examples of low-level and high-level ab initio methods, respectively. In DFT methods, the energy of a system is determined through functionals, i.e., functions of the electron density, which is itself a function of the spatial coordinates.47 Functionals that consider a portion of exact exchange from HF theory and exchange and correlation from other sources (either ab initio or empirical) are called hybrid functionals.47 Examples of commonly used hybrid functionals are the B3LYP,71–74 mPW1PW91,75 B1B95,76,77 and M05-2X78 functionals. A key concept used in QM calculations is the basis set; this is the mathematical representation of the (unknown) molecular orbitals of a system as a set of linear combinations of (known) functions that are called basis functions.58 For convenience, basis functions are also frequently called atomic orbital functions even though they do not correspond to the exact atomic orbitals and thus, they are not in general solutions to the atomic Schrödinger equation (due to their approximated analytical formula), even for the hydrogen-like atoms.58 Two types of basis functions are commonly used, the Slater-type orbitals (STO) and the Gaussian-type orbitals (GTO). These two types mainly differ in the functional form of the radial decay they use: in the case of STO, the radial decay is exponential in the distance of the electron from the nucleus (closely resembling the hydrogen-like atomic orbitals), whereas in the case of GTO, the radial decay is exponential in the distance of the electron from the nucleus squared.47,58 The form of the radial function determines the shape of the radial portion of the orbital which is a critical aspect in electronic structure calculations. Interested readers may consult references [47] and [58] for more information. The functional together with the basis set constitute the so-called level of theory. In the investigation of reaction kinetics, computational chemistry is making an increasingly large contribution. Santiso and Gubbins79 have presented an extensive review of a variety of different methods, along with their virtues and shortcomings, for modelling chemical reactions at electronic and atomistic levels. In the same vein, Fernández-Ramoset al.80 have focused on reviewing theoretical and computational modelling approaches for bimolecular reactions, with a special interest in cases where generally applicable kinetic methods are used. A plethora of kinetic studies that employ computational methods may be found in the references of these Chapter 2. Determination of reaction kinetics: temperature and solvent effects 45 reviews and interested readers are encouraged to resort to them for more details. Electronic-structure calculations for gas-phase thermochemistry and reaction kinetics have impressively advanced the last few years.81 For relatively small systems (approximately up to 50 electrons) there are methods, such as the CCSD(T) method,† that may predict reaction rate constants with accuracy comparable to the best modern experimental techniques; the predicted reaction rate constants may then be used for the development of complex gas-phase kinetic models.1 Higher accuracy may be achieved with the inclusion of electron correlation effects and with extrapolation in the limit of a complete basis set, i.e., a basis set with an infinite number of basis functions, although the calculations might become very expensive and time- consuming.80 For medium to large gaseous systems, less sophisticated computational methods may be employed to avoid laborious calculations, and the expected accuracy may still be quite good. DFT is usually the preferable choice as it features good level of accuracy in an affordable computational cost for medium and large reactive systems.47 MP2 on the other hand, is generally useful for non-bonded interactions in small systems, but scaling may be a significant issue as we move to larger systems.83 As no single method is expected to be ideal for all applications, one needs to establish specific criteria and make judicious choices when selecting a computational method. Predicting liquid-phase reaction rate constants to the level of accuracy required for kinetic modelling is yet to be achieved.1 Such an accomplishment requires that the multitude of solute- solvent interactions is taken into consideration, both in long and short-range. This demands either highly accurate, and thus expensive, calculations or empirical parameters to complement the predicted values. Continuous progress in theory and implementation – with concurrent progress in computer power – suggests a great future for the application of computational methods for calculations in condensed systems. The main strategies that have been presented so far in the literature to understand and model liquid-phase solute-solvent interactions are discussed in the following section.

2.4 Modelling solvent eﬀects

The very task of using computational methods to predict reaction rate constants in the liquid phase is a challenging one due to the diﬃculty of accounting for the presence of a solvent.1 Of particular interest for liquid-phase thermodynamic considerations and kinetic modelling is the

†CCSD(T) refers to coupled cluster (CC) theory with single (S) and double (D) excitations and a quasi- perturbative treatment of selected connected triple (T) excitations. 82 Chapter 2. Determination of reaction kinetics: temperature and solvent effects 46 solvation free energy, ∆G◦,solv, which refers to the energy required to transfer a molecule from the gas phase into a solution. The solvent is expected to have a major influence on the solvation free energy of both the reactants and the transition state, as well as in the determination of the Gibbs free-energy barrier between them, which subsequently will alter the rate constant of the reaction. Therefore, it is highly desirable to obtain accurate estimates for ∆G◦,solv. From a computational point of view, various methods have been proposed for calculating solvation free energies, which mainly differ in the level of detail used to describe the solvent molecules around the solute molecule. Currently, there is no known method proven to be superior to all others in capturing all solvent effects;1 each method has its own advantages and disadvantages. In the next section, a short overview of the most frequently used types of solvation models is provided.

2.4.1 Solvation models

A graphical illustration of the different approaches for representing a solvent may be seen in Fig- ure 2.2. There are approaches in which the solute molecules (shown in red) are surrounded by a large number of solvent molecules (shown in blue), both of which are treated explicitly (as shown in Figure 2.2a, b and c); in this case, solute and solvent molecules might be treated with the same model, for example, using quantum mechanics (QM) (as shown in Figure 2.2a) or molecular mechanics (MM) (as shown in Figure 2.2b) or different models (as shown in Figure 2.2c). In a diametrically opposite direction, there are approaches in which the solute molecules are treated explicitly but the solvent molecules are treated implicitly (as shown in Figure 2.2f), assuming a continuum characterized by a small number of solvent properties. In between the two extremes, there are approaches in which the solute molecules are treated explicitly and only a limited number of solvent molecules are treated explicitly, either using MM (as shown in Figure 2.2d) or QM (as shown in Figure 2.2e), while the rest are treated implicitly. Based on the above-mentioned approaches, one may classify the various types of solvation models into three categories:1 explicit, implicit and hybrid models. Each of them is briefly discussed in the following paragraphs.

Explicit solvation models

In explicit solvation models the solvent molecules around the solute molecules are treated dis- cretely. In order to acquire averaged values for the solvation free energy and other solvent Chapter 2. Determination of reaction kinetics: temperature and solvent eﬀects 47

(a) (b) (c)

(d) (e) (f)

Figure 2.2: Solvent representation by different solvation model types: from discrete to continuum models. Empty dots represent atoms or molecules calculated by quantum mechanics, full dots by molecular mechanics. Blue represents a dielectric continuum. (Adapted from Jalan et al.1) properties, sampling across all the possible system configurations is required. This may be achieved by using Monte Carlo (MC) or Molecular Dynamics (MD) algorithms.84 The molecular interactions between solute and solvent molecules in a solvated system may be modelled at various levels.1 One way is to present the solute and solvent molecules equally detailed in terms of their molecular structure. For this purpose, pure quantum-mechanical (QM) methods may be used. A characteristic example is the widely used Car-Parrinello method.85 De- spite being rigorously the most accurate, these approaches suffer from increased computational cost due to the large number of solvent molecules and the numerous MC or MD simulations needed to obtain a representative sampling across the solvent space.86 An appealing alternative is offered by pure classical molecular mechanics (MM) methods, where parametrized force fields are used to capture molecular interactions.87,88 These methods are less arduous than QM methods and thus, they may be applied to the investigation of relatively large and complex systems at reduced computational cost; sometimes even at time-spans of a few nanoseconds.89 The classical MM treatment allows for determination not only of the solvation free energy, but also of structural and dynamic information, which is difficult to obtain using other techniques.87 However, there are cases of systems that are either too large for applying QM treatment, Chapter 2. Determination of reaction kinetics: temperature and solvent effects 48 for example bio-molecular systems, or involve significant changes in the electronic state that may not be described accurately by classical MM methods, for example when bond breaking or forming takes place during a reaction.86 (A worth-noting example of a force-field developed specifically with the aim to model chemical reactions is ReaxFF, whose name stands for “reactive force field”, presented by van Duin et al.90 A recent example may be found in reference [ 91].) In these cases, in order to capture solvent effects as accurately as possible within a reasonable computational cost, quantum mechanics/molecular mechanics (QM/MM) approaches have been proposed in the literature.92–95 In QM/MM models, or otherwise called two-layer models, the solute molecules are treated in the QM level, while the rest of the system, i.e., the solvent molecules, is treated classically with a MM force field.47 This modelling scheme offers the advantage of a clear understanding of the chemical phenomena that take place, along with reduced computational cost compared with a full QM treatment. The year 2013 was a landmark year for QM/MM methods; the 2013 Nobel Prize in Chemistry was awarded to three scientists for their computational work on biological systems applying QM/MM models96 (e.g., in references [97] and [98]). Explicit models offer a comprehensive, detailed description of solvated systems. However, they are limited by the lack of elaborate classical force-field potentials that mimic satisfactorily the bulk behaviour of the solvent, especially the long-range dielectric effects.1 This is one of their main weakness, along with limitations regarding the parametrization of the force-field potentials, the accuracy of the functional used in the QM calculations and the sampling of the configura- tional phase space.87 Several groups have tried to re-parametrise force-fields using higher QM levels, resulting in the next generation of force-fields with significantly improved calculated results. Nevertheless, it is of general belief that their performance will not advance significantly unless a modification of their force-field formalism is developed.87 The predictions of explicit solvation models agree sometimes better with experimental data compared with predictions of implicit solvation models.99 However, the computational cost of explicit models still remains excessively high compared with the cost of implicit solvation models.1

Implicit solvation models

Implicit solvation models, commonly mentioned as continuum solvation models, approximate the solvent molecules to a polarizable continuum characterized by a dielectric constant. The solute molecules, which are treated in the atomic level of detail, are placed within a suitably shaped cavity inside the medium.100 The underlying approximation behind this concept is that the Chapter 2. Determination of reaction kinetics: temperature and solvent effects 49 solute-solvent interactions are independent of the solvent structure. This approximation restricts the electronic-structure calculations to the size of the solute molecules,101 and significantly reduces the computational burden, albeit at the loss of molecular-level details.24 Continuum solvation models tend to be a popular selection insofar as a fair trade-off is sought between the highest computational efficiency and an adequate level of accuracy for describing solvated systems. In the context of a continuum solvation model, the solvation process, which is the insertion of a solute from the gas phase to a solvent, may be partitioned into three steps:87 cavitation, dispersion-repulsion and electrostatics, which may be mathematically expressed as

∆G◦,solv = ∆Gcav + ∆Gvdw + ∆Gel. (2.4)

First, a cavity, of adequate size to accommodate the solute, is formed; the energy required for this task is known as the free energy of cavitation (∆Gcav). Second, dispersion-repulsion (also known as van der Waals, ∆Gvdw) forces between the solute and solvent are exerted stabilizing the system. The cavitation and dispersion-repulsion terms are usually referred together as the non- electrostatic contribution. Finally, the solute charge distribution is built in solution resulting in an electrostatic contribution term (∆Gel). This term encompasses the work required to create the solute charge distribution from the gas-phase in solution and the work required to polarize the solute charge distribution by the solvent. Decomposing the solvation process into these three steps enables the development of for- malisms of continuous solvation models that diﬀer in ﬁve aspects:58

1. How the size and shape of the cavity are deﬁned.

2. How the cavity/dispersion contribution is calculated.

3. How the charge distribution of the solute is represented.

4. How the solute is described (either with MM or QM).

5. How the dielectric medium is described.

Born was the first one to place a point charge from the vacuum into a spherical cavity in bulk solution and calculate the electronic energy distortion, neglecting any polarization effect.102 Expanding the primitive idea of Born, Onsager used a dipole inside a spherical cavity in bulk solution to describe the concept of solute polarization.103 Similarly, Kirkwood used a multipole Chapter 2. Determination of reaction kinetics: temperature and solvent effects 50 expansion inside the spherical cavity.104 These types of models describe satisfactorily the bulk solvent phase. However, they are incapable of representing interactions that are usually met in the first-solvation shell, which is made up of solvent molecules surrounding the solute cavity. In the first-solvation shell, the solvent properties are markedly different than the bulk phase properties.101 A concept devised to account for short-range solvation effects and that has found wide application is the solvent-accessible surface area (SASA), presented by Lee and Richards105 and Hermann.106 Conceptually, SASA is the area that may be engraved by the centre of a ball, of a given radius, rolling over the surface of a solute molecule. This may be used to calculate energy terms related with cavity formation and dispersive interactions. Different solvation models use different approaches for SASA.101 The Born/Onsager/Kirkood models paved the way towards the development of a very important concept, that of the mutual polarization of solute-solvent in solution. In particular, the solute charge density polarizes the dielectric medium, which in turn forms an electric field, sometimes called “reaction field”, which is further exerted back polarizing the solute charge density and so on and so forth. This iterative procedure for describing self-consistently the interaction between the solute and the solvent is known as the self-consistent reaction field (SCRF) method. One of the most frequently used family of solvation models combining the SCRF formalism with a boundary-element problem is the polarizable continuum model (PCM) presented by Miertuˇset al.107,108 In the PCM, the bulk solution is described as a polarizable dielectric medium.107 The solute molecule is enclosed into a cavity, which is constructed by overlapping spheres with atomic van der Waals radii. The surface of the cavity is partitioned in finite elements, the so-called “tesserae”, by applying the apparent surface charge (ASC) method.107 Other formulations of PCM include the dielectric PCM (D-PCM),109 the integral-equation-formalism PCM (IEF-PCM)110 and the conductor-like screening PCM (C-PCM).111 The conductor-like screening model (COSMO) is another popular option, developed by Klamt and Schüürmann.112 In order to calculate the ASC in COSMO, the surrounding medium is approximated by a conductor with an infinite dielectric constant, which is subsequently scaled by a function of the real dielectric constant. COSMO may as well be considered as a limiting case of the PCM model with an infinite dielectric constant.47 Klamt113 extended COSMO to real solvents (RS), resulting in the COSMO-RS model, which has been used successfully in many applications.114–116 In COSMO-RS, screening charge densities from COSMO are employed to model the interactions between the molecular surfaces of all the molecules in the liquid. Starting from the early 1990’s, Cramer and Truhlar developed a series of continuum solvation Chapter 2. Determination of reaction kinetics: temperature and solvent effects 51 models24,117–120 under the name of SMx, where x labels the different version. The main strategy followed in SMx models is the partitioning of the molar solvation free energy into two terms, given as ∆G◦,solv = ∆Eel + GCDS,L, (2.5) where ∆Eel is an electrostatic term,† which accounts for Gibbs free-energy changes due to electronic-structure changes, nuclear coordinate changes and long-range polarization effects and GCDS,L is a non-electrostatic term, which accounts for Gibbs free-energy changes due to short- range polarization effects and non-electrostatic effects, such as cavitation (C), dispersion (D) and solvent-related (S) structural changes.24 It is standard practice in the SMx family of solvation models to parametrize the non-electrostatic term against extensive experimental solvation databases. Due to its special relevance for the purposes of this thesis, the specific details of the SMD model24 are presented later in Section 4.1.2.

Hybrid solvation models

Hybrid solvation models represent an intermediate approach. In these, solute and solvent molecules in the first solvation shell are described explicitly, with the solvent molecules being specifically described using either a low-level QM method (super-molecule structure121) or MM methods, while the bulk solvent is treated using a continuum approach. Because of their scheme, they are also called three-layer models. Problems arising from the definition of the boundary between the different regions have been addressed in the literature.86,95,121 Hybrid models usually find application when increased accuracy is required but the size of the system prohibits the use of explicit solvation models86 or when the properties of solvent molecules in the first solvation-shell differ considerably from the ones in the bulk phase.122 However, the cost of statistical sampling required across the phase space may be enormous so their use instead of implicit solvation models is still debatable.122,123 Examples of QM/MM/continuum methods include the general liquid optimised boundary (GLOB) model124 and the solvated macromolecule boundary potential (SMBP) model125 designed for biomolecules.

2.4.2 Predictive capabilities of continuum solvation models

Evaluating the accuracy of various continuum solvation models in terms of solvation free-energy predictions is a non-trivial task. First, all models rely on the use of parameters, thus a wrong

† 24 In Marenich et al. this term is denoted as ∆GENP. Chapter 2. Determination of reaction kinetics: temperature and solvent effects 52 choice of parameters may lead to bogus results.126 Second, different models employ different training sets and levels of theory for their parametrisation; adhesion to the specific details of the computational protocol used for the parametrisation of the non-electrostatic terms of each solvation model is a prerequisite for the model’s optimal performance.126 Third, assessment of continuum solvation models by comparison with experimental data might be difficult due to the scarcity of experimental data, especially for species that are difficult to measure, for example radicals.1 The modelling solvation community has been encouraged in the past to participate in the so-called blind challenges. The idea behind a blind challenge test is to examine the predictive capabilities of available solvation models, in a set of molecules with available, but unpublished and difficult to predict, experimental solvation free-energy data. The idea was first introduced by Nicholls et al.123 and has attracted a great deal of interest among various research groups. So far only water has been considered as solvent in the challenges, such as in the SAMPL0 test,123 the SAMPL1 test,127 the SAMPL2 test128 and the Guthrie and Povar test.129 A few examples are given next regarding the performance of some commonly used continuum solvation models in computational studies and are meant to give the reader a flavour of the level of accuracy that may be currently achieved. Suitably parametrised continuum solvation models appear to achieve average errors on the solvation free-energy predictions on the order of 2.0-4.0 kJ mol−1 for small molecules and of 8.0- 13.0 kJ mol−1 for larger molecules, whereas for ions the errors are rarely below 10 kJ mol−1.1,47 For example, using the SM8 model,119 Cramer and Truhlar122 in their study reported a mean unsigned error (MUE) with experimental data of 2.5 kJ mol−1 for 940 neutral solutes. For the same set of solutes and using the COSMO-RS model Klamt et al.126 reported a MUE of 2.0 kJ mol−1. Klamt et al.126 used also the IEF-PCM MST method130 (parametrised for water, octanol, chloroform and carbon tetrachloride), resulting in a MUE of 2.7 kJ mol−1 for non- aqueous solvation and of 4.2 kJ mol−1 for aqueous solvation, but considering only the subset of solvation data that have been used to parametrise the model. In their study, Cramer and Truhlar122 also considered 332 ionic species and in this case the range of errors obtained was significantly larger: a MUE of 18 kJ mol−1 for the SM8 model, and MUE values within a range of 28.1-52.3 kJ mol−1 for the “non-SMx” models, (these are the IEF-PCM model110 implemented in Gaussian 03, the C-PCM model111 implemented in GAMESS, the PB model implemented in Jaguar and the COSMO model implemented in NWChem). The SMD model24 (where D stands for density), which succeeded the SM8 model, performs Chapter 2. Determination of reaction kinetics: temperature and solvent effects 53 similarly, although somewhat less accurately than its predecessor. Comparisons with a level of theory that both models have been parametrised, i.e., mPW1PW/6-31G(d), resulted in MUE values of 2.7 kJ mol−1 for the SMD model versus 2.3 kJ mol−1 for the SM8 model for aqueous neutral solutes and in MUE values of 2.6 kJ mol−1 for the SMD model versus 2.4 kJ mol−1 for the SM8 model for non-aqueous neutral solutes. When ionic solutes are considered, the MUE values are 18.0 kJ mol−1 for the SMD model versus 17.6 kJ mol−1 for the SM8 model. The SM12 model120 is the latest addition in the SMx family of solvation models. For this model, the reported MUE varies in the range of 2.5-3.5 kJ mol−1 for 274 aqueous neutral solutes and in the range of 2.3-2.7 kJ mol−1 for 2129 non-aqueous neutral solutes, depending on the combination of charge scheme (CM5, ESP), functional (B3LYP, mPW1PW, M06-L, M06, M06- 2X) and basis set (MG3S) implemented. For 112 aqueous ions, MUE values in the range of 11.3-15.5 kJ mol−1 were obtained. For non-aqueous ionic solutes, MUE values as high as 31 kJ mol−1 were observed, for example in solvent DMSO for 71 ionic solutes, and as low as 9.6 kJ mol−1, for example in solvent methanol for 80 ionic solutes. The authors attributed the larger errors in the predictions of ionic solvation free-energies to the large absolute energy values of the charged species. Compared with its predecessors, the SM12 model performs similarly for neutral solutes and slightly better for ionic solutes. We should mention that a distinctive advantage of the density-based SMD model is that it does not rely on any partial atomic charges scheme, as do all other SMx models that are based on the generalized Born approximation. As a result, its applicability is not limited to levels of theory with available reasonable charges;24 instead, it may be broadly applied with any electronic-structure method compatible with the IEF-PCM protocol.24 A more detailed description of the SMD model is given later in Section 4.1.2 and its fundamental equation is used to derive an expression for the liquid-phase reaction rate constant.

2.5 Modelling temperature eﬀects

Whereas solvent effects may be captured at a fairly reasonable – yet improvable – level of accuracy, very little has been done to account for temperature effects on solvation thermodynamics. Temperature effects have been seldom considered during the evolution of solvation models, although it has been early recognized as a substantial area of future work.101 As a matter of fact, until very recently solvation free-energy predictions were possible only at room temperature. Chapter 2. Determination of reaction kinetics: temperature and solvent effects 54

Incorporation of temperature effects into reactive processes relevant to industrial, environmental and pharmaceutical applications, where solvation thermodynamic quantities are needed for kinetic modelling at a range of temperatures, would be of tremendous importance. Currently, in the shortage of reliable predictive tools, temperature-dependent solvation data are mainly obtained experimentally, a task which may be quite costly and time-consuming. A number of approaches have been proposed for modelling temperature effects on solvation free energy. The two main approaches, first and second order treatment, are discussed next.

2.5.1 First order temperature dependence

The standard-state solvation free energy ∆G◦,solv at standard-state temperature T ◦ or at a given temperature T may be partitioned as131

∆G◦,solv(T ◦) = ∆H◦,solv − T ◦∆S◦,solv (2.6) or ∆G◦,solv(T ) = ∆H◦,solv − T ∆S◦,solv, (2.7) where ∆H◦,solv is the standard-state solvation enthalpy and ∆S◦,solv the standard-state solvation entropy. The major assumption posed here is that the latter terms are temperature-independent, resulting in a linear relation between ∆G◦,solv and the temperature. This is sometimes referred to as the “van’t Hoff” model. Such decomposition is always meaningful, as entropy and enthalpy are thermodynamic state functions and thus, independent of the thermodynamic path connecting two different states of the system.132 In addition, solvation enthalpy and entropy may be measured experimentally; hence, comparative studies between predicted and experimental data might provide valuable insight into solvation phenomena, enabling the re-parametrisation of currently used force-fields and the development of new ones.132 However, as Wyczalkwski et al.133 warned in their study, whereas errors on solvation free-energy predictions are on par with the ones achieved from experiments, calculations involving the decomposition of this quantity into solvation enthalpy and entropy are subject to statistical errors 10-100 times larger than the ones of solvation free-energy calculations. In the literature, several approaches have been proposed for decomposing the free energy of solvation to its constituents. Mintz et al.,134–141 developed a series of empirical correlations, based on the Abraham model formalism,142,143 for predicting enthalpy of solvation in aqueous and non-aqueous systems. In combination with the estimated value of ∆G◦,solv(T ◦) from the Chapter 2. Determination of reaction kinetics: temperature and solvent effects 55 conventional Abraham model, an estimation of ∆S◦,solv is possible, and eventually of ∆G◦,solv using Equation (2.7). Despite the method’s simplicity and good predictions of experimental values (within a range of 2.5-4.0 kJ mol−1), its applicability is limited to systems for which Abraham parameters are available and to temperature ranges close to those of the fitted experimental data. On the other hand, scaled particle theory (SPT)144–146 provides a sound theoretical framework for estimating solvation entropy changes, focusing on the concept of a cavity formation inside the solvation medium. A set of analytical expressions for the solvation free energy due to cavitation ∆Gcav, as a function of solute and solvent radii (assumed as hard spheres) and of solvent density, has been given by Pierotti.146 In this set of equations, the temperature dependence of ∆Gcav is surprisingly simple. The cavitation entropy ∆Scav may be linked to ∆Gcav via a temperature derivative as147

θ∆Gcav ∆Gcav ∆Scav = − ≈ − (2.8) θT P T neglecting other entropic eﬀects, for example due to reorganization of the solvent. This might be a reasonable assumption in most apolar, aprotic solvents, where weak solute-solvent interactions are encountered; however, it might lead to signiﬁcant errors in the presence of strong solute- solvent interactions, for example hydrogen-bonding, in solvents such as water and alcohols.148 In order to calculate ∆G◦,solv at temperature T , Ashcraft et al.,148 using Equations (2.6)-(2.8), suggested that ∆G◦,solv(T ) = ∆G◦,solv(T ◦) − (T − T ◦)∆Scav. (2.9)

Given that highly precise values are used for the solute and solvent parameters in the model of Pierotti,146 this method may be useful when fast temperature-dependent estimates of the solvation energy are needed with little experimental data available.148 Bidon-Chandal et al.149 pursued this line of modelling, i.e., decomposing the solvation free energy in a continuum solvation framework. They employed temperature-dependent cavity size and dielectric constant in order to estimate the electrostatic component of the solvation enthalpy of neutral solutes in water and octanol.

2.5.2 Second order temperature dependence

The major assumption of temperature-independent ∆H◦,solv and ∆S◦,solv terms is severely queried when temperatures far from the one assumed in the standard state are considered. Chapter 2. Determination of reaction kinetics: temperature and solvent eﬀects 56

In such cases, higher order temperature eﬀects need to be considered150 and these are most often introduced through the heat capacity of solvation (∆Cp). The majority of the proposed approaches have been developed for aqueous systems, largely because of their great scientiﬁc impact on biological systems. An example may be found in the work of Elcock and McCammon,151 who introduced a continuum solvation model for temperature-dependent hydration free-energies of amino acids at a range of biologically relevant temperatures (278-293 K). Pioneering work in the area of temperature-dependent solvation models has been carried out by Chamberlin and coworkers.152,153 In 2006, Chamberlin et al.152 introduced the continuum solvation model SM6T, in which the temperature dependence of the aqueous free energy of solvation is taken into account through a non-zero – albeit temperature-independent – standard- ◦ state heat capacity of solvation ∆Cp . In the SM6T model the solvation free energy may be written as152

T ∆G◦,solv(T ) = ∆G◦,solv(T ◦) − ∆S◦,solv(T ◦)[T − T ] + ∆C◦ (T − T ◦) − T ln . (2.10) 0 p T ◦

The SM6T model was parametrized over an extended set of 2356 experimental aqueous solvation data consisting of atoms H, C and O at temperatures ranging between 273 and 373 K. The temperature dependence of the electrostatic component of the solvation free energy was found to be negligible, indicating that the temperature dependence was largely due to the non- electrostatic component, which was modelled as being proportional to the solute SASA. A key feature of the SM6T model, elaborately discussed in the reference paper,152 is the Coulomb radii being independent of temperature and the errors introduced because of this assumption. The performance of the final model is satisfactory, reproducing aqueous free-energies with a mean unsigned error of 0.55 kJ mol−1. An extended version of the SM6T model, the SM8T model,153 includes additional solute compounds containing N, F, Cl, Br and S atoms. This model has been parametrised against 4403 aqueous solvation free-energy data of 348 species at temperatures ranging between 273 and 373 K. The authors reported a MUE of 0.33 kJ mol−1. Chamberlin and co-workers152,153 have expressed their intentions to further extend the SM6T model to more organic solvents and to near-critical and supercritical areas for aqueous solvent. Höfingerand Zerbetto154 adopted a more pragmatic approach for delivering temperature- dependent predictions for the solvation free energy on the basis of modifying an already existing continuum solvation model. For this task, the PCM decomposition scheme of the solvation free energy was employed. The analytical expressions for the cavitation term, the Chapter 2. Determination of reaction kinetics: temperature and solvent effects 57 dispersion-repulsion term and the electrostatic term were modified by introducing corresponding temperature-dependent parameters, such as cavity-formation coefficients, solute-solvent polarizabilities and dielectric constants. The model was tested with a set of compounds in aqueous solution and a comparison was made between experimental data and the estimated enthalpies and entropies of solvation. As in the study of Chamberlin et al.,152 the dominant role of the cavitation term was highlighted. An alternative way to estimate solvation thermodynamic quantities, and thus their temperature dependence, has its origin in the relation between chromatographic quantities, for example partition coefficients, and retention times.1 The temperature dependence of chromatographic quantities may be explored from either an empirical or a molecular point of view. From an empirical point of view, methods such the ones developed by Mintz and co-workers,134–141 which were discussed earlier, may be used to introduce temperature dependence on the partition coefficients. From a molecular point of view, the key idea is that the contributions to solvation free energy stem from two main sources: first, the change in internal rotations and vibrations of the solute molecule and second, the change in all the other terms, such as the cavity formation and the solute interactions with the bulk solvent.1 Fekete et al.155 and Gonzalez156 employed statistical mechanical principles to estimate the change in heat capacity ∆Cp as a result of the internal structure change of a solute upon insertion in solution. This method neglects the temperature dependence of the cavity formation term and also the temperature dependence of the solute-solvent interaction term, adopting a much different approach than the classical decomposition approach. The estimated values for ∆Cp are in poor quantitative agreement with experimental values. Nevertheless, it acts as a counter-example of what Chamberlin et al.152 and Höfingerand Zerbetto154 suggested, about the cavity formation term being the major source of temperature dependence in the solvation free-energy term.

2.5.3 Other approaches

Several other approaches attempting to account for temperature dependence in solvation processes may be found in the literature. A great number of them are focused on the hydration process and the various complications arising from the nature of the water molecule. It should be noted that, even though these methods are speciﬁc to hydration phenomena, they may as well serve as reliable starting points for a broader representation of temperature dependence in many other solvents. Characteristic examples of such methods are the work of Kinoshita and co-workers on hydration thermodynamics,157–159 the work of Sharp and co-workers160–162 Chapter 2. Determination of reaction kinetics: temperature and solvent eﬀects 58 using the Random Network Model of water, equations of state (EoS) applied to estimate various components of the solvation thermodynamics,121,163–165 the group additivity scheme166,167 and the Langevin-Dipole (LD) solvation model168 for estimation of the hydration entropy.

2.6 Summary

In this chapter, we introduced the reader to the fundamental concepts of chemical kinetics as well as to the most widely used theoretical, experimental and computational tools to explore chemical reactions. We began by giving the definitions of the central objects of study of this thesis. We then presented the theoretical framework of one of the most important kinetic theories, the well-known conventional transition state theory, whose mathematical formulation will be used in the following chapters. We briefly reviewed a number of experimental techniques which are frequently employed to determine and analyse reaction kinetics. In addition to this, we gave a short overview of various computational methods that have attracted significant scientific interest in the last few decades and have been successfully applied to the study of reaction kinetics. One of the primary focuses of this thesis is on the computational investigation of liquid- phase reaction kinetics, a task which is directly linked to the way solute-solvent interactions are modelled. Therefore, in Section 2.4, we discussed various methods that have been proposed in the literature for describing the solvent media, the so-called solvation models. Explicit solvation models were recognized as computationally expensive and time consuming approaches, albeit with accurate predictive capabilities. Nevertheless, their suitability in engineering applications where quick, qualitative answers are sought is questionable. On the other hand, implicit solvation models, thanks to the less-rigorous way of modelling the solvent, are much less computationally intense and a lot faster. If suitably parametrised to account for the non-electrostatic interactions, they may lead to reasonable predictions. They tend to be the first choice, when a reasonable level of accuracy is required at a low computational cost. Hybrid models lie in-between the explicit and implicit models, significantly reducing the computational burden imposed by the explicit consideration of the solvent while capturing the specific solute-solvent interactions in the first solvation shell. However, literature evidence shows that in some cases their application might not be the best option and a continuum solvation model might as well be sufficient.122,123 In a nutshell, there is no universal solvation model that works well in general. An informed decision based on the specific requirements of the system studied, the computing resources available and Chapter 2. Determination of reaction kinetics: temperature and solvent effects 59 the accuracy required is highly advisable before choosing a solvation model. In Section 2.5, we discussed the difficult and challenging task of incorporating temperature effects into the solvation free-energy calculations. Several research endeavours have been dedicated on this task because of its many benefits – the gain of physical insight into the thermodynamics of a solvated system, the elucidation of the potential temperature dependence of crucial parameters (e.g., dielectric constants, polarizabilities, solute geometry), the boost in predictive capabilities of the developed models and the extension of their applicability range, to name a few. Current research in the field is hindered by the sparsity of experimental data at temperatures other than 298 K and the large number of parameters involved in solvation processes. Regardless, the various approaches proposed in the literature shape our current understanding about the terms with major impact on solvation processes, which seem to be mostly the non-electrostatic terms, and suggest new paths of exploration. Chapter 3

Predicting temperature eﬀects on gas-phase reaction kinetics

3.1 Introduction

Understanding and modelling reaction kinetics are integral parts of process development. In areas such as combustion169 and atmospheric chemistry170 hydrogen abstraction reactions between the hydroxyl radical and hydrocarbons are of specific interest. These types of reactions are mechanistically simple, have been studied in detail and are well documented in the literature.171,172 Their related areas of process application, however, require kinetic models involving a large number of rate constants over broad temperature ranges, which may be challenging. Reliable sources of data may be found in databases reviewing experimental studies for the gas-phase rate constants for the reactions of the hydroxyl radical with alkanes and cycloalkanes spanning temperature ranges from approximately 180 to 2000 K.2,173–178 It is common practice to fit the experimental values to an empirical model, usually the Arrhenius equation,29 to obtain kinetic parameters. The Arrhenius model contains two parameters: the pre-exponential factor and the activation energy, which are strictly applicable within the temperature interval used in the fitting. According to the Arrhenius equation, the logarithm of the rate constant of a reaction is expected to be linear in the absolute inverse temperature. In kinetic studies where the rate constant is measured over a narrow temperature range (less than 100 K), this condition is usually verified. If the measured rate constants do not follow linearity, then one usually resorts to an expanded version of the Arrhenius equation179 that includes one extra parameter to fit the data. (The reader is encouraged to consult reference [180] for a detailed review.)

60 Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 61

Experimentally-deduced parameters in such empirical models are subject to uncertainty arising from measurement errors as well as limitations of the models. These limitations concern mainly empirical aspects, such as the nature of some formulas to be more linear than others (e.g., logarithmic plots tend to be more linear), and the number of parameters involved in the equations. Increasing the number of parameters usually results in a better fit to the experimental data, for example in the case of the generalized version of the Arrhenius equation. Yet the quality of fit is not by itself a sufficient argument for selecting a certain model over another; the parametric uncertainty as well as the model predictive capabilities must also be taken into account. Interpretability is another desirable – albeit not mandatory – feature of empirical models. The Arrhenius equation is a characteristic example of an empirical model that has historically prevailed over models with similar, or even better, fitting agreement because it contains parameters, such as the activation energy, that are meaningfully linked to the occurring chemical reaction. However, empirical models may or may not determine all chemical phenomena occurring during a reaction. For example, particularly for the class of hydrogen abstraction reactions, phenomena such as tunnelling or the strong temperature-dependence of vibrational partition functions in the presence of low-frequency bending modes181 are not captured by any empirical parameter. Quantum-mechanical (QM) methods provide a promising alternative and may, in principle, be used to determine kinetic quantities such as rate constants, tunnelling and the temperature variation of partition functions. Since they do not depend on experimental data and may in theory provide a physically meaningful analysis of the kinetics, they are frequently used to undertake systematic kinetic studies of reacting systems. A number of comprehensive benchmark computational studies exists, in which the electronic activation barrier for diverse reaction types is calculated.182–184 The activation barrier value is theoretically linked to the activation energy parameter of the Arrhenius equation.55 In general, QM methods are computationally demanding, thus their use is prohibitive in large-scale engineering applications, for example in drug design where large biomolecules are of interest or furnace design where thousands of reactions take place. Nevertheless, their use is affordable in small-size molecular systems of a few tens of atoms. For the hydrogen abstraction reactions between the hydroxyl radical and alkanes, a number of computational studies has been presented in the literature3,4,91,185–194 ranging from the basic framework of Hartree-Fock (HF) theory to very expensive composite methods, such as the popular G2 method,195 and the coupled-cluster method with triple-excitation terms, CCSD(T).196,197 Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 62

Despite the large variety of computational methods tested, there is no clear consensus on which one is the most appropriate method to study the energetics for this class of reactions, as contra- dictory examples of their performance may be found. For example, the HF methods have been known to overestimate barrier heights due to the dynamical electron correlation energy change as the reaction proceeds from the reactants to the transition state.198 Pure density functional theory (DFT) methods tend to underestimate the energy barrier due to self-interaction errors which are prominent for non-equilibrium structures (such as transition states).199–201 However, hybrid DFT methods, in which the exact HF exchange energy term varies, may appear to perform better for the computation of activation barriers due to significant error cancellations.201,202 Composite methods, such as G2 or CBS-QB3, occasionally encounter performance issues due to problems with the ground-state zero-order HF description and possibly due to differences in the transition state geometry, which is usually computed at lower level of theory than the transition state energy.203 In conjunction to the choice of method, the size and complexity of the basis set have a significant impact on the calculated activation energy value. As a result, the accuracy that may be expected from different QM levels of theory is open to critical evaluation, and hence, finding a universal level of theory that captures best the energetics of hydrogen abstraction reactions remains a challenge. Increasingly accurate electronic structure methods are constantly being developed by the computational chemistry community (e.g., W4204, HEAT205, CCSDT(Q)206) and, in the limit of a complete basis set (CBS), can be used to predict barrier heights, and hence rate constants, with sufficient accuracy to be used in a number of mechanis- tic applications81. However, these methods are currently difficult to access without significant expertise and computational resources. In this chapter we present a systematic study of the reaction between ethane and the hydroxyl radical ‡ C2H6 + ·OH [C2H5...H...OH] → ·C2H5 + H2O, Reaction 1 with a detailed investigation of temperature effects and the development of hybrid QM correlative models. We first calculate QM values for the activation energy barrier and the corresponding rate constant of the reaction using different levels of theory. Kinetic calculations are performed using conventional transition state theory (CTST)50,51 with the Wigner tunnelling correction factor.207 We then identify the level of theory that provides the most accurate value of the reaction rate constant at 298 K by comparison to experimental data and we report structural parameters and frequencies for the optimized structures. Of the levels of theory studied for this reaction, we find the hybrid DFT functional M05-2X with the cc-pV5Z basis set to be Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 63 the most accurate level of theory for this reaction, and proceed to use it to calculate reaction rate constants over a range of temperatures, from 200 to 1250 K. In the second part of this work we investigate the development of hybrid kinetic models that combine a small number of experimental reaction rate data with QM-calculated activation energy values and reaction rate constants to estimate the parameters of the Arrhenius equation and its generalized version. Such correlative expressions may be used in large-scale process modelling due to their low computational cost. The level of accuracy to which we obtain parameters for the Arrhenius equation and its generalized version determines the predictive capability of the kinetic models derived. We therefore focus on identifying reliable modelling strategies for the prediction of reaction rate constants over broad temperature ranges based on accurately determined kinetic parameters for Reaction 1.

3.2 Computational methodology

3.2.1 Quantum-mechanical calculations

All electronic structure calculations are performed using the software Gaussian 09 (release C.01).208 Various levels of theory are considered for the calculation of the activation energy barrier of the reaction. In particular, DFT functionals B3LYP,71–74 M05-2X,78 and M06-2X,209 the second-order Møller-Plesset perturbation theory MP2210 and equivalent spin projection method PMP2211,212 are selectively combined with the following basis sets: 3-21G, 6-21G, 6-31G, 6- 31G(d), 6-31G(d,p), 6-31+G(d,p), 6-311++G(d,p), 6-311G(2d,d,p), 6-311+G(2d,p), cc-pVTZ, aug-cc-pVTZ, cc-pVQZ, aug-cc-pVQZ, cc-pV5Z, and CBSB7. In addition, the composite methods CBS-QB3213,214 and W1BD215,216 are considered; no basis set specification is required in these cases. The adj2-cc-pVTZ basis set, first adjusted by Melissas and Truhlar4 to capture accurately the barrier height of the reaction of interest, is also tested with the MP2 and PMP2 methods in the present work. All vibrational modes are calculated under the harmonic approximation. Internal rotations have been identified for this reaction in previous studies4,188,189; the effect of neglecting the anharmonicity of these modes becomes critically important at higher temperatures. According to the study of Sekuˇsaket al.188, a decrease in the reaction rate constant by 6% at 1000 K is possible. Nevertheless, in order to avoid increasing the computational cost, we accept this margin of error in our calculations. For B3LYP and MP2, the vibrational frequencies as well as the zero-point energies are scaled, by factors 0.9614 and 0.9427 respectively, to compensate for the inaccuracy of the functionals.217 Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 64

Special techniques are available to describe systems involving radicals with unpaired spin electrons. Either an unrestricted approach or a restricted open-shell approach is usually followed (interested readers are referred to reference [218] for an example), depending on whether electrons of opposite spin are allowed or not to occupy different spatial orbitals. The unrestricted approach gives values greater than 0.75 for the total spin-squared operator , although 0.75 is the default value for a doublet state, due to spin contamination from higher energy states. The method allows however for more variational freedom, as the spatial restriction of opposite spin- electrons is removed, and as a result it leads to lower total energies. Lower energies are usually an indication of more stable conformations for a given molecular structure, but in this case this lowering may correspond to a more stable configuration or may be an artifact created by the unrestricted approach. On the other hand, the restricted open-shell approach results in a spin-contamination-free wave function and reasonable total energies; however, the singly occupied orbitals are not uniquely defined and the associated energies do not strictly obey Koopmans’ theorem,219 and are hence subject to physical misinterpretation. Additional schemes are also available for “projecting out” or “annihilating” the contaminant part of the unrestricted approach once the wave function has been obtained.220 This comes at the price of an increase in the CPU time requirements. Previous studies of reactions involving radicals have emphasized the need to take into account spin contamination for the accurate estimation of the activation energy barrier.4,188–191,221 In our calculations, we adopt the unrestricted approach and the spin-projection approach for the calculations with the MP2 method. The value of is monitored to ensure that the impact of spin contamination is minimal. The acceptable range is indicated by the empirical rule, presented by Young,222 according to which the effect of spin contamination is not significant if there is less than 10% difference between the values of and S(S+1).

3.2.2 Rate-constant calculations

Considering a bimolecular gas-phase reaction, given as

‡ A + B AB → products. Reaction 2 Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 65 the ideal-gas reaction rate constant kIG may be computed using conventional transition state theory50,51,223 (CTST) following

−νi ‡ ◦,IG kBT Y RT ∆ G kIG = κ exp − , (3.1) h p◦ RT i where κ is the transmission coefficient to correct for tunnelling effects, kB is the Boltzmann constant, T is the absolute temperature, h is the Planck constant, R is the ideal gas constant, ◦ p is the standard-state pressure (1 atm) and νi is the stoichiometric coefficient of species i for i = A, B (reactants) and AB‡ (transition state) with a value of -1 for each of the reactants and +1 for the transition state for Reaction 2. The term G◦,IG is the ideal-gas molar Gibbs free energy, and ∆‡ stands for the difference between the transition state and the reactants free energies, so that ∆‡G◦,IG = G◦,IG − G◦,IG − G◦,IG. (3.2) AB‡ A B

Equation (3.1) may also be formulated in terms of the corresponding partition functions of the diﬀerent species as224

−νi ‡ el,IG kBT Y RT Y 0 νi ∆ E kIG = κ q ,IG(T ) exp − , (3.3) h p◦ i RT i i where the ideal-gas molar Gibbs free-energy term has been replaced by the product of the ideal- 0,IG gas molecular partition function of each species i at temperature T , qi (T ), and the ideal-gas electronic energy Eel,IG. Tunnelling eﬀects may be described quantitatively following one-dimensional methods such as those of Wigner,207 Skodje and Truhlar,225 and Eckart,226 where the reaction path and the tunnelling path coincide. These approaches result in simple analytic expressions for the transmission coeﬃcient, and this makes them attractive in kinetic applications because of their low computational cost. Here, we use the Wigner correction factor207

1 hν‡ 2 κ = 1 + , (3.4) 24 kBT where ν‡ is the magnitude of the imaginary frequency of the transition-state structure in cm−1. In general, Wigner’s approach predicts a larger tunnelling effect for reactions having thin barriers (large ν‡) than wide barriers (small ν‡).181 In hydrogen-transfer reactions tunnelling may be important;203,227–229 in this study, the effectiveness of the Wigner expression is assessed. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 66

Although more advanced methods have been developed to address tunnelling, i.e., small curvature tunnelling230,231 (SCT) or large curvature tunnelling232 (LCT) methods, these are not considered here.

3.3 Computational investigation of the reaction kinetics

3.3.1 Activation energy barrier

The value for the activation energy at 298 K for Reaction 1 has been reported by Atkinson2 as 9.11 kJ mol−1 (uncertainty range of 8.99-9.22 kJ mol−1) deduced from experimental kinetic data. Our calculated electronic activation energy barriers at 0 K, including the zero-point vibrational energy corrections, for 37 selected combinations of methods and basis sets (levels of theory) are presented in Table 3.1. Information about the specific QM-calculated quantities at each level of theory are reported in AppendixA. As shown in Table 3.1, a broad range of values is obtained, including negative values for the case of the B3LYP functional. Negative values for the electronic activation energy barrier of Reaction 1 were also reported by Hu et al.,194 who found a value of -4.18 kJ mol−1, close to the one reported here (-4.01 kJ mol−1), using B3LYP/6-311++G(d,p). Kobayashi et al.221 reported a negative barrier height of -13.4 kJ mol−1 using B3LYP/cc-pVTZ for Reaction 1. Hybrid DFT methods have been noted to underestimate systematically the barrier heights of hydrogen abstraction reactions122,185,233–235 (sometimes by tens of kJ mol−1). In particular, relating to B3LYP, Chandra and Uchimaru236 have suggested this functional is unreliable for calculations of this class of reactions. Lynch and Truhlar202 investigated a test set of 22 reactions (including Reaction 1) and concluded that B3LYP is one of the least effective functionals for determining barrier heights. Through careful examination of DFT methods, it has been suggested that the Hartree-Fock exchange part of the calculation is responsible for this discrepancy.237 An empirical localized orbital correction model has been proposed by Hall et al.238 to redress this shortcoming of DFT methods. When applied to the B3LYP functional particularly, this empirical scheme resulted in a mean unsigned error value reduced by 8 kJ mol−1 compared with the uncorrected B3LYP calculations for a large data set of 105 barriers heights. The M05-2X and M06-2X functionals give positive activation barriers, which is encouraging, although it may also be appreciated from the table that a broad range of values is obtained for different basis sets. For the family of basis sets developed by Ditchfield et al.,239 the value of the electronic activation energy barrier found using the modest 6-31G(d) basis set is 15.26 Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 67 kJ mol−1. As we augment this basis set by adding polarization and diffuse functions the value of the electronic activation energy barrier decreases to 8.69 kJ mol−1 for M05-2X with the 6- 311+G(2d,p) basis set and to 9.32 kJ mol−1 for M06-2X with the 6-311++G(d,p) basis set. By using Dunning’s240,241 correlation-consistent basis sets (cc-pVTZ, cc-pVQZ, cc-pV5Z) and their augmented (aug-) forms with diffuse functions, the lowering of the reaction barrier height is even more pronounced. Particularly for M05-2X, we generate activation barriers that are among the lowest positive values computed in our work, ranging from the lowest overall value found with cc-pV5Z at 6.65 kJ mol−1 up to 7.12 kJ mol−1 with cc-pVQZ basis set. We also test the CBSB7213 basis set which constitutes one of the multiple steps of the CBS-QB3 method. For M05-2X and M06-2X with this basis set the values of the activation barrier are 8.79 and 8.29 kJ mol−1, respectively. The majority of M05-2X and M06-2X calculations result in activation barrier values in the vicinity of 8 kJ mol−1 (or lower), which is close to chemical accuracy (4 kJ mol−1). Reaction 1 has been included in the HTBH38/4 database182,242 as part of the test set for assessing the performance of the M05-2X and M06-2X functionals from Zhao et al.78,209 In this database, a value of 14.2 kJ mol−1 is reported as the best estimate of the classical barrier height of Reaction 1;† the best estimate value was obtained from the combination of experimental and theoretical kinetic data using the method described in the work of Lynch et al.243 Comparing this value with single-point energy calculations performed with the MG3S basis set gives a mean unsigned error (MUE) of 5.61 kJ mol−1 for the M05-2X functional and of 4.73 kJ mol−1 for the M06-2X functional. To the best of our knowledge, Reaction 1 has not been studied elsewhere using the M05-2X and M06-2X functionals, despite the existence of several studies that have highlighted the suitability of these functionals for the computation of barrier heights and their broad applicability to similar chemical systems.78,235,244

†With respect to this thesis, no direct comparison can be made with either the experimental value of the activation energy reported by Atkinson (9.11 kJ mol−1) or the best estimate value reported in the database HTBH38/4 (14.2 kJ mol−1) with the values of the activation barrier height corrected for zero-point energy contributions obtained from QM (as presented in Table 3.1), as they all correspond to different energy definitions. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 68

Table 3.1: Electronic activation energy barriers ∆‡Eel,IG, corrected for the zero-point vibrational energy, calculated for various electronic structure QM methods. Rate constants kIG are calculated using CTST (Equation (3.3)) at 298 K and compared with available experimental values kExpt. at 298 K from the literature. The uncertainty for the kExpt. values at 298 K is given within the parenthesis.

∆‡Eel,IG/kJ mol−1 kIG ×10−8/dm3 mol−1 s−1 Level of theory This work Previous studies Ref. This work B3LYP/3-21G −9.84 1661 B3LYP/6-21G −3.18 118.9 B3LYP/6-31G −2.04 56.53 B3LYP/6-31G(d) −2.01 60.71 B3LYP/6-31+G(d,p) −4.35 185.2 B3LYP/6-311++G(d,p) −4.01 -4.18 [194] 74.53 B3LYP/6-311G(2d,d,p) −6.03 179.5 M05-2X/6-31G(d) 15.26 0.0706 M05-2X/6-31+G(d,p) 9.02 0.4140 M05-2X/6-311++G(d,p) 9.23 0.3601 M05-2X/6-311+G(2d,p) 8.69 0.4775 M05-2X/CBSB7 8.79 0.3855 M05-2X/aug-cc-pVTZ 6.82 1.133 M05-2X/cc-pVQZ 7.12 0.9734 M05-2X/aug-cc-pVQZ 6.98 1.094 M05-2X/cc-pV5Z 6.65 1.556 M06-2X/6-31G(d) 13.27 0.1817 M06-2X/6-31+G(d,p) 9.01 0.3870 M06-2X/6-311++G(d,p) 9.32 0.2463 M06-2X/CBSB7 8.29 0.4685 M06-2X/aug-cc-pVTZ 7.91 0.7162 MP2/6-31G(d) 39.46 0.000008127 MP2/6-31+G(d,p) 27.74 0.0005691 MP2/6-311++G(d,p) 25.04 25.53 [194] 0.001290 MP2/6-311+G(2d,p) 23.54 5.76a/22.33b [186] 0.002588 MP2/cc-pVTZ 20.50 17.20c [3] 0.01163 MP2/adj2-cc-pVTZ 19.98 17.33 [4] 0.01827 MP2/cc-pVQZ 19.10 0.01978 PMP2/6-31G(d) 29.59 0.0004355 PMP2/6-31+G(d,p) 19.91 0.01337 PMP2/6-311++G(d,p) 17.06 17.53 [194] 0.03221 PMP2/6-311+G(2d,p) 15.87 -8.00a/13.06b [186] 0.05705 PMP2/cc-pVTZ 12.80 0.2597 PMP2/adj2-cc-pVTZ 12.30 9.75d [4] 0.4048 PMP2/cc-pVQZ 11.81 0.3732 CBS-QB3 9.45 0.3202 W1BD 11.49 0.1492 Experimental studies kExpt. ×10−8/dm3 mol−1 s−1 Ref. Greiner (1970) 2.048 (0.060) [15] Leu (1979) 1.566 (0.241) [16] Jeong et al. (1984) 1.843 (0.126) [17] Schiffman et al. (1991) 1.463 (0.072) [18] Dóbéet al. (1991,1992) 1.650 (0.241) [19,20] Sharkey and Smith (1993) 1.777 (0.084) [22] Finlayson-Pitts et al. (1993) 1.662 (0.042) [21] 1.626 (0.070) [21] Atkinson (2003)e 1.496 (0.300) [2] a Single-point energy calculation for geometries optimized at the HF/6-31G(d,p) level of theory. b Single-point energy calculation for geometries optimized at the MP2/6-31G(d,p) level of theory c Optimized at the MP2/aug-cc-pVDZ level of theory. d Single-point energy calculation for geometries optimized at the MP2/adj2-cc-pVTZ level of theory. e Recommended expression based on experimental data, evaluated at 298 K. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 69

The performance of the MP2 and PMP2 methods is tested using the same selection of basis sets. The choice of basis set is found to alter the energy barrier significantly: as expected, the more extensive the basis set used, the lower the barrier height obtained. For example, when the relatively small 6-31G(d) basis set is used, the activation energy barrier is calculated to be 39.46 kJ mol−1. The value decreases by half when using the much more extensive cc-pVQZ basis set, resulting in a value of 19.10 kJ mol−1. Similarly for the PMP2 method, the energy found decreases from 29.59 kJ mol−1 for the 6-31G basis set to 11.81 kJ mol−1 for the cc-pVQZ basis set. In comparison with MP2, the barrier heights calculated with PMP2 are on average 8.00 kJ mol−1 lower. This difference may be ascribed to the use of the projected method embedded in PMP2 to treat spin-contamination effects. The spin-contamination treatment of the projected method is distinctly reflected in the value of the activation barrier. Several studies exist in the literature in which MP2 or PMP2 methods are used for calculating the energetics of Reaction 1. Sekusak et al.186 report a single-point energy value at the MP2/6- 311+G(2d,p) level of theory (with the geometries optimized at the HF/6-31G(d,p)) level of theory) of 5.76 kJ mol−1 and a single-point energy value at the PMP2/6-311+G(2d,p) level of theory (with the geometries optimized at the HF/6-31G(d,p) level of theory) of -8.00 kJ mol−1. In the same study, when optimizing the geometries at the MP2/6-31G(d,p) level of theory the corresponding energy values reported are significantly different: 22.33 kJ mol−1 for a single-point energy calculation at the MP2/6-311+G(2d,p) level of theory and 13.06 kJ mol−1 for a single- point energy calculation at the PMP2/6-311+G(2d,p) level of theory. In later studies,188,189 Sekuˇsakand co-workers identified a weakly-bound pre-reaction van der Waals complex with a structure almost identical to the isolated reactants; the energy difference of this complex from the reactants (corrected with the zero-point energy) has been calculated to have a value of -2.51 kJ mol−1 at the MP2/6-31G(d,p) level of theory and a value of -0.84 kJ mol−1 at the G2(MP2) level of theory. Hashimoto and Iwata3 performed calculations at the MP2/aug-cc-pVDZ level of theory resulting in an activation energy barrier of 17.20 kJ mol−1. Our value using the cc-pVTZ basis set devoid of diffuse functions at the MP2 level is 20.50 kJ mol−1. Hashimoto and Iwata also calculated the energy difference of the pre-reaction complex from the reactants (corrected with the zero-point energy) to have a value of -1.00 kJ mol−1 at the MP2/aug-cc-pVDZ level of theory. An interesting basis set, employed here in conjunction with the MP2 and PMP2 methods, is the adj2-cc-pVTZ, basis set created by Melissas and Truhlar4 using Gaussian 92. It has a modified f-shell function for the oxygen atom, in order to equate the scaling factors of the formation of the O−H bond and the breaking of the C−H bond occurring during Reaction Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 70

1. We reproduce this basis set following the same modification procedure. For the MP2 and the PMP2 methods our values of 19.98 kJ mol−1 and 12.30 kJ mol−1, respectively, are on average 2.60 kJ mol−1 higher than their single-point energy values (17.33 kJ mol−1 and 9.75 kJ mol−1, respectively). The composite CBS-QB3 and W1BD methods are also evaluated here. These are computational methods which include a number of pre-defined QM calculations and they are expected to be highly reliable. Usually, one achieves a good description of reaction energetics using these methods and although they are computationally more demanding than DFT and MP2 methods, the small molecules treated here render them affordable to use. We compute barrier heights of 9.45 kJ mol−1 for CBS-QB3 and 11.49 kJ mol−1 for W1BD. These values are lower than any of those found with the MP2 and PMP2 methods and than those found with M05-2X and M06-2X functionals together with the 6-31G(d) basis set. The value of 9.45 kJ mol−1 obtained with CBS-QB3 is in good agreement with the values calculated with M05-2X/6-311++G(d,p) (9.23 kJ mol−1) and with M06-2X/6-311++G(d,p) (9.32 kJ mol−1). The slightly higher value of 11.49 kJ mol−1 with W1BD agrees best with the values derived from PMP2/cc-pVQZ (11.81 kJ mol−1) and from PMP2/adj2-cc-pVTZ (12.30 kJ mol−1). As may be appreciated by inspecting Table 3.1, the range of values for the energetics obtained from the different electronic-structure methods is remarkably broad. The values for the activation energy barrier cover a range of almost 50 kJ mol−1 (including negative values). Some general trends are also noticeable in our calculations, in particular, lower values of activation energy barriers are obtained with larger basis sets and with a better treatment of spin contamination. The activation energy barrier is the determining factor in the calculation of the rate constant (cf. Equation (3.3)), and hence, this broad variability concomitantly leads to rate-constant values spanning over eight orders of magnitude. This large variation may be ra- tionalized on the basis of the approximations and assumptions attributed to each computational method as well as the size of the basis set used in any case. The combination of the method and basis set dramatically influences the electronic energy barrier values.

3.3.2 Reaction rate constant at 298 K

The rate-constant value for Reaction 1 at 298 K recommended by Atkinson2 is 1.496×108 dm3 mol−1 s−1. This value is derived from the evaluation of a three-parameter Arrhenius-type expression ﬁtted to a large number of experimental data points over a temperature range from 180 to 1230 K. Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 71

Using our calculated activation barriers, the rate constant of Reaction 1 at 298 K is computed using CTST (Equation (3.3)) for each of the levels of theory presented. The values are given in Table 3.1 and plotted in Figure 3.1.

104 3

-1 10

s 102 -1 101 100 mol 3 10-1 -2

/dm 10

-8 10-3 10-4

x10 -5 k 10 -6 10 3-21G6-21G6-31G6-31G(d)6-31+G(d,p)6-311++G(d,p)6-311G(2d,d,p)6-31G(d)6-31+G(d,p)6-311++G(d,p)6-311+G(2d,p)CBSB7aug-cc-pVTZcc-pVQZaug-cc-pVQZcc-pV5Z 6-31G(d)6-31+G(d,p)6-311++G(d,p)CBSB7aug-cc-pVTZ6-31G(d)6-31+G(d,p)6-311++G(d,p)6-311+G(2d,p)cc-pVTZadj2-cc-pVTZcc-pVQZ 6-31G(d)6-31+G(d,p)6-311++G(d,p)6-311+G(2d,p)cc-pVTZadj2-cc-pVTZcc-pVQZ CBS-QB3 W1BD Expt.

Level of theory B3LYP M06-2X PMP2 W1BD

M05-2X MP2 CBS-QB3 Expt.

Figure 3.1: Rate constants of Reaction 1, calculated using Equation (3.3) at 298 K for various electronic structure levels of theory. Columns of the same colour correspond to the same method. For each method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The QM-calculated rate-constant values at 298 K are compared with the experimental value obtained by evaluating the recommended expression given by Atkinson2 at the same temperature. The black dashed line indicates the experimental reaction rate value so it may easily be compared to computed values.

The exponential term in the CTST expression plays a dominant role in determining the rate-constant value. Therefore, as expected given the calculated barrier heights, the rate of the reaction is considerably overestimated with the B3LYP functional, by up to three orders of magnitude. In the case of the M05-2X and M06-2X functionals, the corresponding rate constants vary from the same order of magnitude as the experimental value to an underestimate of two orders of magnitude depending on the basis set used. The rate constants predicted from the energy barriers calculated with the MP2 and PMP2 methods approach the experimental value Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 72 gradually as the size and the complexity of the basis set increase. However, PMP2 values only come within one order of magnitude of the experimental value (two orders for MP2), even with the use of more extended basis sets, such as aug-cc-pVTZ. The values computed with the CBS- QB3 and W1BD composite methods achieve a similar level of agreement. Our best match with the experimental value at 298 K is achieved with M05-2X/cc-pV5Z. In the following section, geometry and frequency calculations are reported using this level of theory.

3.3.3 Geometry and frequency calculations

The structural parameters for the reactant species C2H6 and ·OH, optimized using M05-2X/cc- pV5Z, are summarized in Table 3.2. We compare our calculated values with available equilibrium parameters estimated based on experimental data measured with spectroscopic techniques245,246 (infrared, Raman, microwave and electronic spectroscopy in the ultraviolet and visible region) and the electron diffraction method.245 As shown in Table 3.2, using the M05-2X/cc-pV5Z level of theory, we find bond lengths in good agreement with experimental values. For C2H6, our values agree best with the spectroscopic data, although for the C-C bond length our calculated value is in better agreement with the value estimated based on the electron diffraction method.

Table 3.2: Optimized bond lengths (r) and bond angles (∠) between various atoms, calculated at the M05-2X/cc-pV5Z level of theory, for reactants C2H6 and ·OH in comparison with experimental values from the literature. QM=Quantum mechanics, SP=Spectroscopic techniques, ED=Diﬀraction method.

C2H6 ·OH o Method rC-C /A˚ rC-H /A˚ ∠HCH/ Ref. r/A˚ Ref. QM 1.5239 1.0864 107.76 This work 0.9683 This work SP 1.5280 1.0877 107.31 [245] 0.9697 [246] ED 1.5240 1.0890 106.90 [245]

Following the geometry optimizations, frequency calculations are performed for the reactants. The calculated vibrational frequencies, shown in Table 3.3, are in agreement with the harmonic frequencies observed experimentally by Hansen,247 Miller,248 and Chase.249 The max- −1 imum discrepancy observed is 53 cm for the first mode for both C2H6 and ·OH. We note −1 −1 that larger differences, with a maximum of 199 cm for C2H6 (fifth mode) and 218 cm for ·OH (first mode), are observed when comparing with the fundamental frequencies reported by Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 73

Shimanouchi;250 since the calculated vibrational frequencies are obtained under the harmonic regime, such deviations are expected.

Table 3.3: Vibrational frequencies ν for the equilibrium optimized structures for the reactants

C2H6 and ·OH at the M05-2X/cc-pV5Z level of theory and from experimental studies.

ν/cm−1 Species Mode This work Expt. Harmonic Ref. Expt. Fundamental Ref.

C2H6 1 3096 3043 [247,248] 2954 [250] C2H6 2 1441 1449 [247,248] 1388 [250] C2H6 3 1021 1016 [247,248] 995 [250] C2H6 4 303 303 [247,248] 289 [250] C2H6 5 3095 3061 [247,248] 2896 [250] C2H6 6 1424 1438 [247,248] 1379 [250] C2H6 7 3142 3175 [247,248] 2969 [250] C2H6 8 1525 1552 [247,248] 1468 [250] C2H6 9 1235 1246 [247,248] 1190 [250] C2H6 10 3168 3140 [247,248] 2985 [250] C2H6 11 1528 1526 [247,248] 1469 [250] C2H6 12 829 822 [247,248] 822 [250] ·OH 1 3788 3735 [249] 3570 [249]

In Table 3.4 the geometrical parameters for the transition-state structure calculated at the M05-2X/cc-pV5Z level of theory are reported. The transition state has a staggered conformation, as shown in Figure 3.2. The breaking bond C(2)-H(4) is only 0.072 A˚ longer than the equivalent bond in the reactant C2H6, suggesting that the transition state presented here exhibits a reactant-like character. Vibrational frequency analysis of the transition-state structure reveals a unique imaginary frequency (negative eigenvalue) with a value of -848 cm−1.

3.3.4 Rate constants at diﬀerent temperatures

Using the same level of theory (M05-2X/cc-pV5Z), we perform frequency calculations for Reac- tion 1, and we compute the partition functions at 30 different temperature points from 200 to 1250 K (reported in AppendixA). To facilitate comparison with other methods, we denote these calculations as “CTST/W”, where “W” refers to the Wigner transmission coefficient. A wealth of temperature-dependent experimental kinetic data are found in the literature2,178 for this reaction. Our choice to employ M05-2X/cc-pV5Z is based on the close agreement achieved with the experimental rate constant at 298 K, together with the good agreement observed between calculated and experimental bond lengths and angles of the reactants. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 74

O(1)

H(7)

H(2) H(4)

g c b d H(3) C(2)

H(5) C(1) e a f H H(1) (6)

Figure 3.2: Optimized transition-state structure at the M05-2X/cc-pV5Z level of theory. The optimized values for the bond lengths, bond angles, and dihedral angles are reported in Table 3.4.

Table 3.4: Optimized structure of the transition state at the M05-2X/cc-pV5Z level of theory. The bond lengths between various atoms are represented by r, the bond angles are as speciﬁed in Figure 3.2, τ1 is the H(2)-C(1)-C(2)-H(1) dihedral angle, τ2 is the H(1)-C(2)-C(1)-H(3) dihedral angle, φ is the H(4)-C(2)-C(1)-H(1) dihedral angle, χ1 is the H(5)-C(2)-C(1)-H(4) dihedral angle,

χ2 is the H(4)-C(1)-C(2)-H(6) dihedral angle, ψ is the O(1)-H(4)-C(2)-C(1) dihedral angle and ω is the H(7)-O(1)-H(4)-C(2) dihedral angle.

Bond length/A˚ Bond angle/◦ Dihedral angle/◦

rC(1)−C(2) 1.5125 a 110.52 τ1 −119.84

rC(1)−H(1) 1.0885 b 111.04 τ2 119.79

rC(1)−H(2) 1.0855 c 111.05 φ −179.30

rC(1)−H(3) 1.0854 d 107.89 χ1 −117.63

rC(2)−H(4) 1.1587 e 113.25 χ2 −115.73

rC(2)−H(5) 1.0847 f 113.61 ψ −95.38

rC(1)−H(6) 1.0848 g 171.15 ω −40.57

rO(1)−H(7) 0.9661 h 96.66

rO(1)−H(4) 1.4380 Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 75

24 23 )

-1 22 s

-1 21

mol 20 3 19 /dm k 18 ln( 17 16 0 1 2 3 4 5 6 1000/(T/K)

Figure 3.3: Rate constants of Reaction 1 as a function of inverse temperature in the range 200- IG 1250 K. The red squares () correspond to the k values calculated in this work (CTST/W), the green triangles (4) to the values computed by Hashimoto and Iwata3 using CTST-ZCT, and the blue circles ( ) to the values calculated by Melissas and Truhlar4 using CVT/SCT. The black curve ( ) represents the rate expression recommended by Atkinson.2 The black diamonds ( ) indicate the experimental values5–13 at the speciﬁc temperatures for which QM calculations are3 performed. Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 76

The calculated rate-constant values, obtained evaluating Equation (3.3) at 30 different temperatures within the range 200-1250 K, are reported in Table 3.5. Of the earlier computational studies that have been dedicated to this reaction, here we select and present the findings of two studies and compare them to our results. Hashimoto and Iwata3 also employed the CTST theory but with the zero-order interpolated approximation embedded in the zero-curvature tunnelling (ZCT) method. Their calculations cover a range of low temperatures from 200 to 310 K. Melissas and Truhlar4 employed the more advanced canonical variational transition state theory (CVT) with small-curvature tunnelling (SCT) corrections considering a more extended temperature range from 200 to 3000 K. Experimental data from the literature are also listed in Table 3.5 for comparison. The experimental values are obtained using a three-parameter expression recommended by Atkinson2 evaluated at each of the temperatures of interest. The logarithmic values of calculated and experimental reaction rates are presented as a function of inverse temperature in an Arrhenius-like plot in Figure 3.3. Clear deviations from linearity are noticeable for both the experimental and the calculated values, especially for temperatures above 500 K. Large differences are observed between our calculations and those of Hashimoto and Iwata3 considering that both studies employ CTST. It is shown in Table 3.5 that, for the temperature range considered by Hashimoto and Iwata (200-310 K), we calculate significantly lower rate- constant values with an order of magnitude difference from their values. The difference is unexpected as Hashimoto and Iwata obtained an activation energy barrier value of 8.45 kJ mol−1 using the CCSD(T)/aug-cc-pVDZ level of theory, which is higher compared with our value of 6.65 kJ mol−1 at the M05-2X/cc-pV5Z level of theory (cf. Table 3.1). The use of frequencies and zero-point energies optimized at a lower level of theory (MP2/aug-cc-pVDZ) in their study and the use of a different tunnelling correction factor, namely the zero-order interpolated approximation to the zero-curvature method, could be the main contributing factors to this difference. Interestingly, despite our use of the more approximate CTST, our estimated rate constants appear to be in line, if slightly overestimated, with the predictions obtained by Melissas and Truhlar4 using CVT, except the three lowest temperature points for which we have calculations in common, 200, 210, and 225 K. Compared with experiments, our predictions of the rate constants at temperatures other than 298 K are in good agreement with the correlated values given by Atkinson2 as well as with the individual experiments5–13 over the entire temperature range. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 77

Table 3.5: Rate constants for the kinetics of Reaction 1 obtained from diﬀerent computational and experimental studies. CTST/W refers to the calculations of this work obtained at the M05-2X/cc-pV5Z level of theory using the CTST theory (Equation (3.3)) with the Wigner tunnelling factor (Equation (3.4)). CTST/SCT refers to the calculations of Hashimoto and Iwata3 obtained using the CTST theory with zero-curvature tunnelling (ZCT) correction method for the geometries and frequencies optimized at the MP2/aug-cc-pVDZ level of theory and the barrier heights obtained at the CCSD(T)/aug-cc-pVDZ level of theory. CVT/SCT refers to the calculations of Melissas and Truhlar4 obtained using the canonical-variational theory (CVT) with small-curvature tunnelling (SCT) correction method for optimized structures at the PMP2//MP2/adj2-cc-pVTZ level of theory. Atkinson’s recommended rate-constant expression2 (evaluated at the various temperatures) and individual experimental values5–13,15–22 (with the uncertainty given within the parenthesis) from the literature are also reported.

kIG ×10−8/dm3 mol−1 s−1 kExpt. ×10−8/dm3 mol−1 s−1 T/K CTST/W CTST/ZCT 3 CVT/SCT 4 Expt. 2 Expt. Ref. 200 0.474 1.855 0.564 0.296 0.273 (0.015) [6] 210 0.556 2.090 0.620 0.367 - 213 0.581 - - 0.391 0.392 (0.004) [6] 220 0.643 2.337 - 0.449 - 225 0.690 - 0.716 0.494 0.486 (0.007) [6] 230 0.737 2.608 - 0.542 0.531 (0.014) [7] 240 0.838 2.897 0.825 0.646 0.632 (0.024) [5] 250 0.945 3.210 0.897 0.762 0.753 (0.008) [6] 270 1.180 3.896 1.072 1.030 - 273 1.217 - 1.102 1.075 1.071 (0.026) [7] 280 1.307 4.276 - 1.184 1.253 (0.010) [6] 298 1.556 - 1.380 1.496 1.704 (0.117)a [15–22] 299 1.569 - - 1.512 1.482 (0.019) [7] 300 1.583 5.095 1.403 1.530 1.560 (0.048) [8] 310 1.732 5.540 - 1.724 1.778 (0.030) [6] 325 1.970 - 1.704 2.041 2.086 (0.022) [6] 327 2.002 - - 2.086 2.035 (0.022) [7] 355 2.498 - - 2.773 2.764 (0.030) [7] 396 3.342 - - 3.991 3.920 (0.163) [10] 400 3.432 - 2.910 4.124 4.643 (0.458) [9] 499 6.184 - - 8.219 9.515 (0.602) [9] 500 6.217 - 5.336 8.269 - 595 9.946 - - 13.732 13.791 (1.566) [11] 600 10.176 - 8.913 14.062 - 705 15.875 - - 21.975 20.957 (0.867) [10] 800 22.639 - 20.716 30.777 30.532 (2.047) [9] 974 39.743 - - 50.998 50.405 [12] 1000 42.888 - 40.469 54.478 - 1225 77.462 - - 89.599 92.741 (14.453) [13] 1250 82.186 - 78.890 94.056b - a Average value of experimental data at 298 K reported in Table 3.1. The recommended value is not included. b Evaluated using Atkinson’s expression 2 although it is outside the recommended temperature range (180-1230 K). Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 78

In order to be able to compare quantitatively our calculations with the experimental data of Table 3.5, we calculate the mean absolute percentage error (MAPE)

N IG Expt. 1 X k (Tj) − k (Tj) MAPE = × 100, (3.5) N kExpt.(T ) j=1 j

IG Expt. where k (Tj) is a computed rate constant at a given temperature Tj and k (Tj) is an experimental rate constant at a given temperature Tj for N temperature points. We also calculate the scale-dependent mean absolute error (MAEln) for comparisons between the logarithmic values of the rate constants, given as

N 1 X IG Expt. MAEln = ln k (Tj) − ln k (Tj) . (3.6) N j=1

Atkinson’s recommended expression provides a very accurate description of the experimental kinetic data with a MAEln of 0.036. Henceforth, comparisons are performed with these correlated rate-constant values instead of the individual experimental data. The overall agreement with the experimental rate constants over the whole temperature range of 200-1250 K is very good with a MAEln of 0.213. In order to elucidate the dominant eﬀects that impact the rate-constant calculations at diﬀerent temperatures, we split the temperature range into three domains: from 200 to 299 K, from 300 to 499 K, and from 500 to 1250

K; we report the MAEln in each domain in Table 3.6. In the low temperature range, the QM- calculated values consistently overestimate the experimental values and the resulting value for the MAEln is 0.245. These deviations are most likely caused by tunnelling, which is prominent at lower temperatures. Assuming the error in these conditions arises entirely from the transmission coefficient, this would suggest that the Wigner approach overestimates the transmission coefficient that would match calculations and experiments in this region by a factor of 1.28 on average. In the temperature range between 300 and 499 K our computational model captures the experimental values most accurately. The calculated MAEln is 0.108. This agreement supports our theoretical and computational choices (the use of the Wigner tunnelling correction factor and the harmonic oscillator approximation for calculating the vibrational partition functions) in this temperature range. At temperatures beyond 500 K and up to 1250 K the computed rate constants systematically underestimate the experimental values, resulting in a MAEln value of 0.259. The discrepancy originates from the harmonic oscillator approximation adopted in this work, which is no longer valid at such high temperatures. Accurate treatments for hindered Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 79 rotations are available,81,224,251 and they should be generally considered in order to obtain more accurate predictions in this temperature range. Melissas and Truhlar4 recognized in their study the existence of hindered rotors in the transition-state structure and quantified their influence; they reported correction factors for hindered versus harmonic approximations for temperatures between 250 to 3000 K, which become more important at higher temperatures reaching a value of 0.493 at 3000 K. We note that tunnelling effects contribute little, or not at all, to the error value in this area as they are almost negligible in this temperature range.

Table 3.6: Mean absolute percentage error (MAPE) (Equation (3.5)) and mean absolute error of the logarithm (MAEln) (Equation (3.6)) for the QM-calculated rate-constant values with CTST/W for Reaction 1 compared with the correlated values given by Atkinson’s expression2 over temperature ranges 200-1250 K, 200-299 K, 300-499 K and 500-1250 K.

T/K MAPE (%) MAEln 200-1250 22.04 0.213 200-299 29.10 0.245 300-499 9.89 0.108 500-1250 22.64 0.259

3.4 Hybrid correlative models

Quantum-mechanical methods may provide accurate predictions of the rate of the reaction of interest, as shown. However, they are inherently difficult to validate in the absence of experimental data, and they are also subject to theoretical limitations that arise from assumptions (see large deviations at high temperatures due to the harmonic approximation). There are of course highly accurate theoretical methods (including treatment of hindered rotations81,224,251) with known small uncertainties that are validated over large data sets184 and are particularly efficient in producing accurate barrier heights. However, their computational cost is prohibitively expensive for all but the smallest of molecules. Furthermore, it is not always practical to resort to such computationally demanding techniques for reactions with large-scale mechanisms. Furthermore, it is not always practical to resort to such computationally demanding techniques, especially in large-scale systems. On the other hand, experimental measurements may also be difficult, expensive, and time-consuming. Correlative techniques, which may be very accurate and computationally fast, are often preferred. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 80

With this in mind, we consider now the combination of QM calculations with experimental kinetic data to develop correlative models of the reaction kinetics. We consider first the accuracy of commonly used Arrhenius-type29,179 models when fitted to measured or computationally obtained rate constants over specified temperature ranges. The estimation of reliable, statistically significant values of the Arrhenius parameters may be challenging. Large numbers of data points are usually required due to the high degree of correlation between the activation energy and the pre-exponential factor.252,253 In this thesis, we develop a hybrid strategy for building Arrhenius-type models whose parameters are derived from both experiments and QM calculations. Through this hybrid approach, we aim to improve the accuracy of the model in predicting the reaction rate constant while decreasing the parametric uncertainty. The effectiveness of the proposed hybrid approaches for building reliable models using very few experimental data points is of special interest.

3.4.1 Methodology

The Arrhenius equation29 has been extensively used in studies of the changes of reaction rate constants with temperature due to its simplicity and its broad applicability when narrow temperature ranges are considered. Though empirical when ﬁrst formulated, the equation of Arrhenius may be related theoretically to the frequency of collisions introduced from collision theory254,255 and the height of an energy barrier for a reaction. The combination of the two terms gives the number of successful collisions that lead to a reaction.55

Arrhenius-type models may be derived by deﬁning the activation energy Ea(T ) at any temperature as55 d ln k E (T ) = a . (3.7) dT RT 2

If the activation energy is assumed to be equal to a value E˜a independent of temperature, so that

Ea(T ) = E˜a, (3.8) after integration Equation (3.7) reduces to the well-known Arrhenius equation

! E˜ k = A exp − a , (3.9) 1 1 RT or, in logarithmic form, to E˜ ln k = ln A − a , (3.10) 1 1 RT Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 81 where A1 is the so-called pre-exponential factor. Subscript “1” is used hereafter to refer to quantities obtained when assuming the activation energy is temperature-independent. Experimental or predicted rate-constant values may be used to determine A1 and Eã in a very straightforward manner. The assumption of a temperature-independent activation energy leads to a linear dependency of ln k1 on 1/T which is well known to be an oversimplification (cf. the data and results in the previous section). Alternatively, if a linear dependence is used for Ea(T ), the following expression is often postulated:26

? Ea(T ) = Ea + mRT, (3.11)

? where Ea and m are temperature-independent parameters. In this case, the rate constant is obtained from Equation (3.7) as

E? k = A T m exp − a , (3.12) 2 2 RT where an additional temperature dependence appears in the pre-exponential factor through parameter m, and the pre-exponential factor A2 is temperature-independent as before. In logarithmic form the expression is given as

E? ln k = ln A + m ln T − a . (3.13) 2 2 RT

This expression is usually referred to as the generalized Arrhenius (GA) equation.179 Subscript “2” hereafter refers to quantities obtained when assuming the activation energy is linearly depen- ? dent on temperature (Equation (3.11)). For a given reaction, parameter Ea should not typically be expected to be the same as the activation energy Eã appearing in the original Arrhenius equation. The GA expression is usually applied when data show significant deviations from linearity – either due to large experimental uncertainty, or, more often, due to extended temperature ranges being considered. For cases when the kinetic data fail to comply to the linearity dictated by the Arrhenius equation, satisfactory results are found when analysing those data in terms of the GA expression.2,256,257 It is important, however, to note that even in the case of highly accurate experimental kinetic data, there might exist several widely different parameter sets that may fit almost equally well the data when using Equation (3.12), in other words, the inherent uncertainty of estimating the GA parameters is significant. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 82

In the context of our work it is useful to relate the activation energies resulting from the Arrhenius and GA models to quantities computed via electronic structure (QM) methods. At a given temperature, the Gibbs free energy of a reaction may be written in terms of its enthalpic and entropic contributions as

∆‡G◦,IG(T ) = ∆‡H◦,IG(T ) − T ∆‡S◦,IG(T ), (3.14) where ∆‡H◦,IG(T ) is the ideal-gas enthalpy of activation and ∆‡S◦,IG(T ) is the ideal-gas entropy of activation. Substituting Equation (3.14) in Equation (3.1) results in

−νi ‡ ◦,IG ‡ ◦,IG kBT Y RT ∆ S (T ) ∆ H (T ) kIG = κ exp exp − , (3.15) h p◦ R RT i for i = A, B (reactants) and AB‡ (transition state), which may be written in a more concise form as ∆‡H◦,IG(T ) kIG = C(T ) exp − , (3.16) RT or, in logarithmic form, as ∆‡H◦,IG(T ) ln kIG = ln C(T ) − , (3.17) RT where the pre-exponential temperature-dependent term takes the form

−νi ‡ ◦,IG kBT Y RT ∆ S (T ) C(T ) = κ exp , (3.18) h p◦ R i or ‡ ◦,IG kBT X RT ∆ S (T ) ln C(T ) = ln κ − ν ln + . (3.19) h i p◦ R i Equating Equations (3.13) and (3.17), and diﬀerentiating them with respect to temperature (neglecting the temperature dependence of the tunnelling correction factor and the entropy), ? 55 results in an expression that links parameters Ea and m to the activation enthalpy:

? ‡ ◦,IG X Ea = ∆ H (T ) + (1 − m − νi)RT. (3.20) i Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 83

The activation enthalpy term is related to the electronic energy barrier, ∆‡Eel,IG, obtained from QM calculations as

‡ ◦,IG ‡ el,IG ‡ therm X ∆ H (T ) = ∆ E + ∆ E (T ) + νiRT, (3.21) i where ∆‡Etherm(T ) is a correction term for the internal thermal activation energy accounting for the eﬀects of molecular translation, rotation and vibration. Substituting Equation (3.21) in Equation (3.20) results in

? ‡ el,IG ‡ therm Ea = ∆ E + ∆ E (T ) + (1 − m)RT, (3.22) which may be further inserted into Equation (3.13) to give

∆‡Eel,IG + ∆‡Etherm(T ) k = A T m exp − − (1 − m) , (3.23) 2 2 RT or ∆‡Eel,IG + ∆‡Etherm(T ) ln k = ln A + m ln T − − (1 − m). (3.24) 2 2 RT

In Equation (3.24) quantities ∆‡Eel,IG and ∆‡Etherm are obtained from QM calculations and the remaining parameters A2 and m may be obtained by ﬁtting to available experimental reaction rate data. We refer to this model as a hybrid approach since we employ both experimental and computed data to estimate the reaction rate constant.

3.4.2 Arrhenius-type hybrid models

In this section, different combinations of experimental and computational data are used, sometimes for the entire temperature range studied and sometimes for just a few temperature points, with the aim to derive the parameters of Equations (3.10), (3.13) and (3.24) such that they reproduce the known experimental reaction rate data and such that the values of the derived parameters are statistically significant and specifically their confidence intervals are as small as possible. In Table 3.7 the different models developed in this work are presented. For each model, the equation on which the model is based is given. Whether experimental data or QM- calculated data or a combination of both are used for fitting a model’s parameters is indicated by descriptors such as “Expt.”, which refers to the use of experimental reaction rate constants, and “QM”, which refers to QM-calculated rate constants with CTST/W or activation energy Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 84 values at the M05-2X/cc-pV5Z level of theory obtained in this work. Within the parentheses following the descriptors, the temperature range over which data are taken is indicated (either the entire temperature range from 200 to 1250 K or a specific number of temperature points, in which case the points are explicitly written). We first develop model Arrhenius, which refers to the use of the Arrhenius equation (Equa- tion (3.10)), and model GA, which refers to the generalized-Arrhenius equation (Equation (3.13)), by fitting both of them to the experimental reaction rate data of Atkinson2 over the temperature range from 200 to 1250 K. Model GAQM refers to the use of the generalized-Arrhenius equation (Equation (3.13)) fitted to QM-calculated reaction rate constants as computed using CTST/W. It should be noted that model GAQM is almost entirely predictive as it is based entirely on QM calculations, with experimental data at 298 K being used only to select a level of theory. A hybrid strategy is then considered in the case of model HM A, which refers to a hybrid model ? based on the understanding that parameter Ea has the largest impact on the calculation of reaction kinetics as it is directly linked to the activation barrier height of the reaction. In this ? ‡ el,IG ‡ therm model, Ea is given by Equation (3.22), with quantities ∆ E and ∆ E obtained from

QM calculations at the M05-2X/cc-pV5Z level of theory, and parameters A2 and m obtained by fitting to experimental reaction rate data in the range from 200 to 1250 K. In developing model HM A, we take full advantage of the information provided by available experimental reaction rate data while performing only the minimum number of QM calculations possible by using, for ? the calculation of Ea, the value of the thermal correction activation energy term at 298 K only. This assumption significantly reduces the time required to develop the hybrid model. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 85 a is used while for the a 2 ? a a ˜ E m m E refer to Equation ( 3.13 ). Descriptor “Expt.” refers to experimental ? a are used. HM=Hybrid Model. E 7 and m , 2 A 1 2 A A ( 3.13 )( 3.13 ) QM(200-1250)( 3.24 ) Expt.(273, 299, 327, 355) Expt.(273, 299, 327, Expt.(273, 355) 299, 327, 355) Expt.(273, QM(200-1250) 299, 327, Expt.(273, 355) 299, 327, 355) Expt.(273, 299, 327, 355), QM(298) QM(200-1250) refer to Equation ( 3.10 ). Parameters a ˜ E 4T and 1 QM 4T A Model Equation No. Model Equation No. Calculated using Equation ( 3.22 ) at 298 K. Arrhenius ( 3.10 )GAGA Expt.(200-1250)HM AGA HM A ( 3.13 )a ( 3.24 ) Expt.(200-1250) Expt.(200-1250) - Expt.(200-1250) Expt.(200-1250) Expt.(200-1250) Expt.(200-1250) Expt.(200-1250), QM(298) Table 3.7: Data sets used toParameters estimate the parameters of the various Arrhenius-typereaction models rate developed in data. this study Descriptor forM05-2X/cc-pV5Z the “QM” level kinetics refers of ofReaction to theory. 1 . QM-calculated Withindata the reaction parenthesis are rate following fitted data the is obtained descriptors given.specific with , the temperature CTST/W Regarding range points or the or the activation experimental the experimental energy specific data, values values point(s) for of at of Talukdar the the temperature et entire at al. temperature which range the equation of Atkinson Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 86

Models GA4T and HM A4T are of particular interest here. In these, instead of using the entire temperature range, only four temperature points, taken over a narrow temperature range, are considered. More points could be used in general, but given that there are three parameters to be estimated in the generalized Arrhenius equation, four is the minimum possible number of points. By reducing the number of data points, we significantly reduce the amount of information that is needed to compute the reaction rate constants. This reduction in data increases the intrinsic uncertainty of the parameters in the models but it is a relevant scenario in many reactions for which only a few experimental data points are available or for which only a few QM computations are performed. In the development of models GA4T and HM A4T, A2 and m are determined by fitting to experimental reaction rate constants at 273, 299, 327, and 355 K; in addition, in ? model HM A4T, Ea is obtained from Equation (3.22) by carrying out QM calculations at the M05-2X/cc-pV5Z level of theory to obtain ∆‡Eel,IG and ∆‡Etherm, as in model HM A. Rate constants at the four temperature points (273, 299, 327, 355 K) are those reported by Talukdar et al.7 We choose this set of values as they are sufficiently spread above and below the 298 K temperature and are evenly distributed.

3.4.3 Results

Performance of the models

The accuracy of a model in capturing a certain data set is often not enough to render the model successful for application; the statistical significance of the parameters constituting the model is also important. In Table 3.8, the parameters for the various models and their 95% confidence intervals are listed. The corresponding MAEln values calculated by comparison with the experimental reaction rate data of Atkinson2 using the entire temperature range (200- 1250 K) are also reported in Table 3.8. In Figure 3.4, the various models developed in this study are presented and compared with the experimental reaction rate data fitted by Atkinson’s expression.2 Inspecting Table 3.8 and Figure 3.4, it may be seen that all the models perform well and capture the experimental reaction rate data. The GA model (Figure 3.4a) performs best, fitting the experimental reaction rate data over the entire temperature range studied with a MAEln value of 0.002. The negligible error value of this approach is not surprising as the model exploits the entire set of experimental reaction rate data available, and its three-parameter form allows for temperature effects to be taken into account. Use of the Arrhenius model leads to the largest error (MAEln=0.220) due to the lack of temperature dependence in the pre-exponential factor Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 87 which given the broad temperature range studied in this work is an unrealistic assumption. It QM is interesting to consider model GA now, which leads to a MAEln of 0.195. As shown in Figure 3.4b, this model results in larger deviations from the experimental reaction rate data at low (overestimating) and high (underestimating) temperature regions while it performs better in the intermediate temperature range. However, acknowledging that model GAQM is almost entirely predictive, the agreement with experimental reaction rate data for the entire temperature range from 200 to 1250 K is considered good. Model GAQM was developed using exclusively QM-calculated reaction rate data, hence when comparison is made with those QM-calculated values for the entire temperature range (200 to 1250 K) (cf. Table 3.5) the corresponding error QM value, denoted as MAEln , is low (0.090). Hybrid model HM A is in good agreement with the experimental rate constants; the calculated MAEln is 0.069. In contrast to the QM-based model GAQM, in which no experimental information is used, we find that using experimental reaction rate data to fit two of the three parameters (m and A2) lowers the corresponding error value considerably and enhances the range of applicability of the model as shown in Figure 3.4b. The fact that hybrid model HM A performs better than the classical Arrhenius model, with a MAEln value much lower than 0.220 (cf. Table 3.8), is also encouraging. 7 The GM A4T and HM A4T models developed in this study using the reduced data are considered especially successful; they lead to comparatively low MAEln values across the entire temperature range even though only four experimental data points are needed over a narrow range. In model GA4T, the level of accuracy achieved is very good (cf. Figure 3.4c). The 2 experimental reaction rate data of Atkinson are reproduced with a MAEln value of 0.021 (the second lowest error of this study). For hybrid model HM A4T, the agreement achieved with experimental reaction rate data is also very good, with a MAEln value of 0.118, a much lower value than those of the GAQM model and the classical Arrhenius model. These results highlight the benefits of combining experimental and QM-calculated information in the development of the hybrid models. Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 88 are given ? a E and ln ln a ˜ E 47 0.220 2244 0.002 87 0.195 12 0.069 18 0.021 0.118 ...... and parameters m − K . 1 10 − s − 1 10 95% CI MAPE (%) MAE 95% CI MAPE (%) MAE − 5.28-5.585.05-7.00 6 10 × mol b b 3 ? a a ˜ E E for the various Arrhenius-type models developed for the kinetics a 95% CI 95% CI is given in dm 2 A m m 2.00 1.99-2.01 4.15 4.14-4.16 0 - -1.91 1.58-2.241.39 3.00 1.32-1.451.97 11.07 5.43 0.00-33.7 10.43-11.701.15 1.83-4.17 4.19 0.75-1.54 6.02 21 0.00-86.0 19 2 4 6 6 10 3 5 10 10 10 10 × × × 10 10 , parameter × 1 × × − s -3.50 -2.33 -8.97 1 -1.96 10 10 10 -9.08 -7.41 − 10 3 − 5 − − 95% CI 95% CI 10 10 10 10 10 10 mol × × × × × × 3 1.21 8.92 1.00 2.98 1.00 1.00 3 4 5 4 6 10 10 10 10 10 10 10 1 2 × × × × × × A A is given in dm 1.04 1.08 2.45 1 (Equation ( 3.5 )) is also reported, comparing with the experimental reaction rate data for Reaction 1 , as shown in A ln 4T QM 4T Model Calculated using Equation ( 3.22 ) at 298 K. Standard confidence interval calculations leads to some negative values for the parameters. These unphysi- Arrhenius 1.54 Model GA 9.00 a cal values haveb been replaced by zero or an arbitrarily small value of 1.00 GA HM AGA HM A 5.20 . The MAE 1 − Table 3.8: Comparison of parameters andof theirReaction 95% 1 . confidence intervals Parameter (CI) Table 3.5 for the range 200-1250 K. HM=Hybrid Model. in kJ mol Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 89

Model parameters and statistical signiﬁcance

The classical Arrhenius model relies on two parameters (A1 and Eã) compared with all the ? other models considered in this work, which contain three parameters (A2, m, and Ea). As ˜ ? a result, distinctly different values are obtained for parameters A1 and A2, and Ea and Ea (cf. Table 3.8), and thus, comparisons regarding parameter values are made only between the three-parameter models. Nevertheless, for all the models derived based on the generalized m Arrhenius equation, the value of the pre-exponential term A2T is comparable with the value of parameter A1. Parameters A1 and A2 are inherently difficult to estimate, as is perceivable from the large confidence intervals obtained for these parameters in all the models derived here, even in cases where an extensive set of reaction rate data (experimental or QM-calculated) is used.

Conversely, parameter Eã in the Arrhenius model is well characterized with a comparatively narrow confidence interval, lying in a range of ±0.63 kJ mol−1 of the central value. Of the three-parameter models considered, the GA model is the one resulting in the tightest 95% confidence intervals for its three parameters. This observation is in line with the expec- tation of obtaining statistically reliable parameters when an abundance of experimental data ? is available. In this model, the values we obtain for parameters m and Ea, with narrow 95% confidence intervals of ±0.01 of the central value in each case, are identical to the values reported by Atkinson in his recommended reaction rate expression.2 Also, evaluating Equation (3.11) at −1 298 K, using the GA model parameter values, results in a value of 9.11 kJ mol for the Ea(298) −1 parameter, which is comparable with the value of 11.07 kJ mol found for Eã in the classical ? Arrhenius model. Interestingly, the values estimated for all three parameters A2, m, and Ea of QM the GA model are included within the 95% confidence intervals of models GA and GA4T. They are not included in the 95% confidence intervals of either of the hybrid models (HM A or

HM A4T), with the exception of A2 in the GA model which is included in the 95% confidence interval of A2 in the HM A4T model. Reducing the number of experimental values used for the fitting of the reaction rate data ? drastically increases the uncertainty in the estimated parameters A2, m, and Ea and as a consequence, the corresponding 95% confidence intervals broaden. This is clearly noticeable in models GA4T and HM A4T and more so in model GA4T because only experimental sources of information are used in developing this model, and so it effectively has one more fitted parameter than model HM A4T. The values for the fitted parameters for model GA4T are very similar to the parameter values found for the GA model and lie just outside the 95% confidence ? intervals. However, the 95% confidence intervals of all three parameters A2, m and Ea of Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 90

(a) 24

) 23 -1

s 22 -1 21

mol 20 3 19 /dm

k 18

ln( 17 16 0 1 2 3 4 5 6 1000/(T/K) (b) 24

) 23 -1

s 22 -1 21

mol 20 3 19 /dm

k 18

ln( 17 16 0 1 2 3 4 5 6 1000/(T/K) (c) 24

) 23 -1

s 22 -1 21

mol 20 3 19 /dm

k 18

ln( 17 16 0 1 2 3 4 5 6 1000/(T/K)

Figure 3.4: Arrhenius plots of the various models developed in this work compared with experimental reaction rate data reported by Atkinson2 for Reaction 1 over a temperature range from 200 to 1250 K, as shown in Table 3.5, illustrated here by black diamonds ( ). (a) GA model QM ( ) and Arrhenius model (··· ). (b) GA model ( ) and HM A model (···3). (c) GA4T model

( ) and HM A4T model (··· ). Chapter 3. Predicting temperature eﬀects on gas-phase reaction kinetics 91

4 model GA4T are larger by a factor of 10 (or more) than the 95% conﬁdence intervals of model

GA. In summary, the large value of the 95% confidence intervals of model GA4T suggests that this model consists of statistically unreliable parameters. By comparison, in model HM A4T, reducing the number of experimental data points has a smaller impact on the 95% confidence ? intervals of the parameters. We obtain the confidence interval of parameter Ea in the hybrid models by evaluation of Equation (3.20) for the low and high bound values of the 95% confidence interval of parameter m. The use of the QM-calculated activation energy values in parameter ? Ea together with the experimental data selected results in tighter confidence intervals. This is the key advantage of our proposition of hybrid models.

Discussion

On the basis of the preceding analysis, a number of recommendations may be made for the use of each of the models proposed in this thesis considering their suitability in terms of the level of accuracy that may be needed to capture the reaction rate data, the amount of available reaction rate data (experimental or QM-calculated) and the intrinsic uncertainty carried by the estimated parameters. When an abundance of experimental reaction rate data is available, speaking strictly in terms of accuracy, the empirical model GA leads to the best agreement with the experimental data. Our proposed hybrid model HM A appears as an appealing alternative, capturing the gas-phase kinetics of Reaction 1 very well, in particular, far better than the empirical Arrhenius model, which is conventionally used in many kinetic studies. As part of the hybrid strategy to build model HM A, performing one extra QM calculation and obtaining information relevant ? to parameter Ea has the advantage of reducing the MAEln of the rate-constant value by 0.150 compared with the conventional Arrhenius model’s MAEln value. In addition, more reliable ? estimates for parameters A2, m, and Ea with narrower 95% conﬁdence intervals are obtained; this is a key advantage of the hybrid model. The importance of the hybrid strategy is particularly highlighted when few data are available. In the scenario of considering only four experimental reaction data points at diﬀerent temperatures, again speaking strictly in terms of accuracy, the

GA4T model provides the best agreement with the experimental data; however, its estimated parameters in this case are, to a very large extent, statistically meaningless. By comparison, our hybrid model HM A4T, seems quite robust, with one of the two parameters estimated (m) especially well-defined, when fitted to four temperature points (273, 299, 327, 355 K), and very Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 92 good overall agreement (MAEln=0.118) achieved, including extrapolation to a large range of temperatures (200-1250 K).

3.5 Summary

In this chapter, the effect of temperature on the kinetics of a hydrogen abstraction reaction was studied computationally. We identified the best quantum-mechanical (QM) method for describing accurately the kinetics of this reaction and we further developed a novel hybrid approach to build better correlative models by combining QM calculations with experimental data. For the calculation of the activation energy barrier of the reaction, a thorough screening of various levels of theory was performed. Accounting for spin-contamination effects was found to improve the accuracy of the calculated activation barrier for the reaction; thus, unrestricted and spin-projected methods were used. Our calculations indicated that B3LYP fails to produce reasonable results, giving negative values for the activation barrier. The use of M05-2X and M06-2X results in positive activation barriers, in agreement with experiments, with lower values for more complex basis sets. Predicted activation barriers lie between 6.65 and 15.26 kJ mol−1 for M05-2X and between 7.91 and 13.27 kJ mol−1 for M06-2X for different basis sets. A similar pattern was observed for MP2 and PMP2, which gave gradually decreasing values of the activation barrier for increasingly complex basis sets. The lowest overall activation barrier was calculated at the M05-2X/cc-pV5Z level. For each of the levels of theory tested, the corresponding rate constants were calculated at 298 K. We used conventional transition state theory, with the Wigner207 correction factor accounting for tunnelling effects. The M05-2X/cc-pV5Z level of theory gave a rate-constant value of 1.556×108 dm3 mol−1 s−1, which is in remarkably good agreement with the experimental value 1.496×108 dm3 mol−1 s−1 obtained using a three- parameter generalized Arrhenius-type formula2 reported by Atkinson.2 Optimized structures and vibrational frequencies for the reactants at 298 K, calculated at the same level of theory, also agreed well with experimental measurements. Next, the temperature dependence of the reaction rate constant was studied in a broad temperature range between 200 and 1250 K. Good agreement with experimental reaction rate constants, obtained by evaluation of Atkinson’s expression,2 was achieved at the M05-2X/cc- pV5Z level. An overall MAEln of 0.213 was found comparing with experimental data for the whole temperature range studied. A MAEln value of 0.108 was found comparing with the Chapter 3. Predicting temperature effects on gas-phase reaction kinetics 93 experimental rate constants over an intermediate temperature range (300-499 K), in which our assumptions concerning the tunnelling effect and the harmonic approximation are reasonable and in line with our goal of securing a low computational cost. Our findings were also in line with the results of Melissas and Truhlar4 who used variational transition state theory to calculate the reaction rate constants of the same reaction. The good agreement observed with their calculations indicates that CTST may be used to describe the kinetics of the reaction of interest when combined with a highly accurate computational method, such as the M05-2X/cc-pV5Z level of theory, without the need for the more advanced variational transition state theory. New hybrid correlative models, based on a generalized Arrhenius form, have been proposed to predict reaction rates combining QM-calculated reaction rate data and activation energy values with experimental reaction rate data. We find that these hybrid models offer the best balance between model accuracy and precision of the estimated parameters (narrow confidence intervals). Particularly, hybrid model HM A4T, where only four experimental reaction rate data points at four different temperatures are considered and in which one QM calculation is ? performed at 298 K to obtain information pertinent to parameter Ea, proved a good alternative to the empirical GA4T model. The MAEln value for the HM A4T model was 0.118, compared with the corresponding value of 0.021 for the GA4T model. Moreover, two of the three parameters (m ? and Ea) included in the HM A4T model were found to be well characterized with tight confidence intervals (approximately ±0.40 and ±0.98 kJ mol−1 of the central values, respectively) compared with the ill-defined parameters found for the GA4T model. Due to its pre-exponential nature, parameter A2 is intrinsically difficult to characterize accurately in either model. Overall the methodology for developing hybrid models proposed in this work constitute a useful predictive approach that may be easily extended to other reactions. Hybrid models offer the advantage of incorporating valuable information from experiments in their parameters, while achieving good reliability (statistical significance). They require less experimental effort to develop than fully empirical models, and they may prove a particularly useful tool when studying reactions for which a limited number of experimental reaction rate data exists. Chapter 4

Kinetic investigation of a Williamson reaction in acetonitrile

In this chapter, we study the eﬀects of solvent and temperature on the rate constant of a liquid-phase reaction, from both an experimental and a computational perspective. We begin by presenting an expression for the liquid-phase reaction rate constant of a bimolecular reaction according to conventional transition state theory (CTST). We present the solvation model SMD, which is used to account for the solvent presence, and we integrate its mathematical formalism in the CTST expression. Then, we introduce the liquid-phase reaction which is selected here for investigation, a Williamson reaction, and we review previous experimental studies on this reaction. We describe the experimental methodology that we follow for monitoring the reaction in acetonitrile-d3 at various temperatures and we then proceed by presenting the main experimental results. In the remainder of the chapter, we investigate this reaction from the computational point of view. We describe the computational protocol for performing quantum- mechanical calculations and we implement it to the Williamson reaction of interest. We discuss the main computational ﬁndings in comparison with the experimental data obtained previously. The results of this chapter will have practical importance later in Chapter 5.

94 Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 95

4.1 Theoretical background for kinetic predictions in the liquid phase

Conventional transition state theory (CTST), as has been already discussed in Chapter 2 and applied in Chapter 3 for gas-phase reaction kinetics, provides a simple theoretical framework for calculating liquid-phase reaction kinetics as well. The combination of CTST with an implicit solvation model for mimicking the solvent is particularly appealing when one must balance computational cost and accuracy, and such combination has been found to be eﬀective in previous works (e.g., Furuncuo˘glu Ozaltin¨ et al.,258 Galano and Alvarez-Idaboy,259 Stanescu and Ache- nie260). In particular, the combination of CTST with the SMD continuum solvation model of Marenich et al.24 was successfully applied in the work of Struebing et al.45 for the prediction of the kinetics of a Menschutkin reaction in several solvents. There are surely other approaches available to predict reaction kinetics in the liquid phase, for example the variational transition state theory (as discussed in Section 2.2) and methods for explicit consideration of the solvent (as discussed in Section 2.4.1). However, such approaches are beyond the scope of this thesis and the reader is referred to the review of Fern´andez-Ramos et al.80 for further information. In the next sections, an expression for the rate constant of a bimolecular reaction in the liquid phase is presented according to the principles of CTST and the SMD model.

4.1.1 Liquid-phase reaction rate constant by CTST

Similarly to the gas-phase reaction rate constant kIG presented in Chapter 3, the liquid-phase reaction rate constant kL of a bimolecular liquid-phase reaction, given as

‡ A + B AB → products, (4.1) may be computed using conventional transition state theory50,51,223 (CTST) following

‡ ◦,L kBT Y νi ∆ G kL = κ c◦,L exp − , (4.2) h i RT i where κ is the transmission coefficient to correct for tunnelling effects, kB is the Boltzmann ◦,L constant, T is the absolute temperature, h is the Planck constant, R is the ideal gas constant, ci is the standard-state molar concentration for species i = A, B (reactants) and AB‡ (transition state) for Equation (4.1) and νi is the stoichiometric coefficient of species i with a value of -1 for each of the reactants and +1 for the transition state. G◦,L is the standard-state molar free Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 96 energy in the liquid phase and ∆‡ stands for the difference between the transition state and the reactants free energies, so that

∆‡G◦,L = G◦,L − G◦,L − G◦,L. (4.3) AB‡ A B

The term ∆‡G◦,L is also known as the standard-state molar Gibbs free energy of activation. Similarly to the gas-phase, in the liquid phase the Gibbs free energy of activation at a given temperature may be expressed as

∆‡G◦,L(T ) = ∆‡H◦,L(T ) − T ∆‡S◦,L(T ), (4.4) where ∆‡H◦,L(T ) is the standard-state molar enthalpy of activation in the liquid phase and ∆‡S◦,L(T ) is the standard-state molar entropy of activation in the liquid phase. Substituting Equation (4.4) in Equation (4.2) results in

‡ ◦,L ‡ ◦,L kBT Y νi ∆ S (T ) ∆ H (T ) kL = κ c◦,L exp exp − . (4.5) h i R RT i

261 ◦,L According to Ho et al., for species i at temperature T =298.15 K and concentration ci =1 −3 ◦,L mol dm , the term Gi may be related to the gas-phase standard-state free energy via

◦,L ◦,IG p→c,IG ◦,solv Gi = Gi + ∆Gi + ∆Gi , (4.6)

◦,IG where Gi is the ideal-gas molar free energy for species i at temperature T =298.15 K and ◦ p→c,IG pressure p =1 atm. The term ∆Gi accounts for the conversion from the gas-phase standard ◦ ◦,IG −3 state of T =298.15 K and p =1 atm to T =298.15 K and ci =1 mol dm and may be calculated as261 p→c,IG RT ∆Gi = RT ln ◦ . (4.7) pi

◦,solv ∆Gi is the standard-state free energy of solvation for species i accounting for the energy ◦,IG required to transfer 1 mole of species i from the ideal-gas phase at T =298.15 K and ci =1 mol −3 ◦,L −3 dm to the liquid phase at T =298.15 K and ci =1 mol dm . For a bimolecular reaction, as shown in Equation (4.1), the diﬀerence in the liquid-phase molar Gibbs free energies of the transition state and the reactants (∆‡G◦,L) is given as

∆‡G◦,L = ∆‡G◦,IG + ∆‡∆Gp→c,IG + ∆‡∆G◦,solv. (4.8) Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 97

Substituting Equation (4.8) to Equation (4.2) results in

‡ ◦,IG ‡ p→c,IG ‡ ◦,solv kBT Y νi ∆ G + ∆ ∆G + ∆ ∆G kL = κ c◦,L exp − . (4.9) h i RT i

The gas-phase Gibbs free energy of activation ∆‡G◦,IG may be expressed in terms of the corresponding partition functions of the diﬀerent species,224 as also used in Equation (3.3), and so Equation (4.9) may be written as

‡ el,IG ‡ p→c,IG ‡ ◦,solv kBT Y νi Y 0 νi ∆ E + ∆ ∆G + ∆ ∆G kL = κ c◦,L q ,IG(T ) exp − . h i i RT i i (4.10) The standard-state molar free energy of solvation ∆G◦,solv may be calculated using a solvation model. In this work, we employ the continuum solvation SMD model developed by Marenich et al.,24 which is brieﬂy described in the next section.

4.1.2 The SMD solvation model

Starting from the early 1990’s, Cramer and Truhlar developed a series of continuum solvation models24,117,118,120,122 under the name of “SMx”, where x labels the diﬀerent model version. In these models, the solvent is treated as a continuum characterized by its dielectric constant (ε). The main strategy followed in the SMx family of solvation models is the partitioning of the molar free energy of solvation into two terms as

∆G◦,solv = ∆Eel + GCDS,L = Eel,L − Eel,IG + GCDS,L, (4.11) where ∆Eel is an electrostatic term, which accounts for the change in the internal energy when transferring a solute from the gas to the liquid phase, and GCDS,L is a non-electrostatic term, which accounts for free-energy changes due to short-range electrostatic effects and non- electrostatic effects, such as cavitation (C), dispersion (D) and solvent-related (S) structural changes.24 The majority of SMx models represent the solute as a collection of partial charges placed in a cavity and rely on the generalized Born approximation47 for defining the bulk electrostatic effects. In the SMD model, the bulk electrostatic contribution is assumed to be the result of a self-consistent reaction field treatment. Therefore, unlike most of its predecessors in the SMx family, the SMD model represents the solute using the electron charge density24 and relies on the Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 98 solution of the non-homogeneous Poisson (NPE) equation according to the integral-equation- formalism protocol of the polarizable continuum model (IEF-PCM).110,262–264 This is a distinct advantage of the SMD model that allows it to be used with any electronic structure method for which the PCM algorithm is applicable.24 The SMD model is a particularly appealing option because of its well-established foundation on physical laws, its simple mathematical implementation and its low computational cost.126

The electrostatic term

According to the principles of the electrostatic theory of dielectric media,47 a medium may be described by its dielectric constant (ε), which is a scalar constant for isotropic homogeneous media and a scalar function of position for isotropic non-homogeneous media. In the case of a linear isotropic homogeneous medium, Poisson’s equation is expressed as

2 ε∇ Φ = −4πρf (4.12)

where Φ is the electric potential and ρf is the free charge density, which accounts for the charge density per unit volume (in the SMD model it relates to the solute charge density). In the case of a linear isotropic non-homogeneous medium, where the free-charge density and the electric potential depend on position r, the NPE may be applied as47

∇(ε∇Φ(r)) = −4πρf(r). (4.13)

In the context of quantum mechanics, the electrostatic contribution of the free energy of solvation may be expressed in relation to the reaction ﬁeld φ, which is equal to the total electric potential Φ minus the electric potential of the gas phase solute molecule Φ(0), resulting in24,47

e e X ∆Eel,L = hΨ|H(0) − φ|Ψi + Z φ − hΨ(0)|H(0)|Ψ(0)i, (4.14) 2 2 k k k where Ψ is the polarized solute electronic wave function in solution, H(0) is the solute electronic

Hamiltonian in the gas phase, e is the atomic unit of charge, φk is the reaction field at atom k, (0) Zk is the atomic number of atom k and Ψ is the electronic wave function in the gas phase. In the SMD model, the boundary between the solute cavity and the solvent continuum is specified as a superposition of nuclear-centered spheres with intrinsic Coulomb radii ρk, which depend solely on Zk. This boundary shapes the solvent-accessible surface area (SASA). As a result, the Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 99 reaction field at an arbitrary position r within the cavity is given as24

X qm φ(r) = , (4.15) |r − r | m m where rm is the position of an element m (also known as tessarae) of surface on the solute-solvent boundary and qm is the apparent surface charge on element m.

The cavity-dispersion-solvent structure term

The non-electrostatic component of the free energy of solvation, as shown in Equation (4.11), includes empirically any energy changes due to cavity formation, dispersion and solvent-structure eﬀects in the ﬁrst solvation shell. In the SMD model, this term is calculated as24

NA NA CDS,L X [M] X G = σkAk(R, {RZk + rs}) + σ Ak(R, {RZk + rs}), (4.16) k k

[M] where NA is the number of atoms, σk is the surface tension of atom k, σ is the molecular surface tension of atom k and Ak is the solvent-accessible surface area (SASA) of atom k, which depends on the geometry of the solute R, the atomic van der Waals radii RZk and the solvent radius rs. The atomic surface tension is given as

N XA σk =σ ˜Z + σ˜Z Z 0 Tk({Z 0 ,R 0 }), (4.17) k k k k kk k0

whereσ ˜Z is a parameter speciﬁc to the atomic number Zk of atom k, σZ Z 0 is a parameter that k k k 0 depends on the atomic numbers Zk and Zk0 of atoms k and k , respectively and Tk({Zk0 ,Rkk0 }) is a geometry-dependent function (the cutoﬀ tanh). In SMD, the atomic and molecular surface tensions may be written as functions of the bulk solvent properties and the following expressions are postulated:

[nD] [α] [β] σ˜θ =σ ˜θ nD +σ ˜θ α +σ ˜θ β (4.18) and

2 2 2 σ[M] =σ ˜[γ]γ +σ ˜[ϕ ]ϕ2 +σ ˜[ψ ]ψ2 +σ ˜[β ]β2, (4.19)

where the subscript θ corresponds to either Zk or ZkZk0 , whereas the solvent properties used are the following: nD is the refractive index at room temperature (which is by convention taken at 293 K for this quantity), α is Abraham’s hydrogen bond acidity, β is Abraham’s hydrogen bond Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 100 basicity, γ is the macroscopic surface tension at 298.15 K (in units of cal mol−1 A˚−2), φ is the aromaticity, deﬁned as the fraction of solvent atoms that are aromatic carbon atoms, and ψ is the electronegative halogenicity, deﬁned as the fraction of solvent atoms that are electronegative

[nD] [α] [β] halogen atoms (F, Cl, or Br). The coeﬃcientsσ ˜θ ,σ ˜θ andσ ˜ are empirical parameters dependent on θ, whileσ ˜[γ],σ ˜[ϕ2],σ ˜[ψ2] andσ ˜[β2] are empirical parameters independent of θ. For the training set of the SMD model,24 a diverse set of 2821 solvation energy data has been used, of which 112 energy data correspond to aqueous ionic solvation free energies, 220 to solvation free energies for 166 ions in acetonitrile, methanol and dimethyl sulfoxide, 2346 to solvation free energies of 318 neutral solutes in 91 solvents (water and 90 non-aqueous solvents) and 143 to transfer free energies for 93 neutral solutes between water and 15 organic solvents.24 The single set of parameters employed by the SMD model, namely the intrinsic atomic Coulomb radii and the atomic surface tension coeﬃcients, has been optimized over six electronic structure methods (M05-2X/MIDI!6D, M05-2X/6-31G(d), M05-2X/6-31+G(d,p), M05-2X/cc- pVTZ, B3LYP/6-31G(d) and HF/6-31G(d)).24 Although the bulk electrostatics used for the SMD parametrization are calculated using the IEF-PCM algorithm,24 the obtained parameters might also be used with other algorithms that solve the NPE, such as, for example, the COSMO and the C-PCM algorithms.111,112,265

4.1.3 Liquid-phase reaction rate constant and selectivity ratio by CTST and the SMD solvation model

A more detailed expression for the liquid-phase rate constant according to the CTST theory and the SMD solvation model may be obtained by introducing Equation (4.10) into Equation (4.9), resulting in

‡ p→c,IG ‡ el,L ‡ CDS,L kBT Y νi Y 0 νi ∆ ∆G + ∆ E + ∆ G kL = κ c◦,L q ,IG(T ) exp − . (4.20) h i i RT i i

The liquid-phase rate constant shown in Equation (4.20) may be obtained when certain information is available, namely the geometry of the reactants and transition state and the following seven solvent properties: the dielectric constant, Abraham’s hydrogen bond acidity and basicity, the refractive index, the surface tension, aromaticity and halogenicity. When competing reactions with two possible pathways are investigated, the selectivity ratio of one pathway (I) versus the other (II) may be determined by forming the ratio of the Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 101 corresponding rate constants, as

L ‡ ◦,L ‡ ◦,L ! kI κI ∆ GI − ∆ GII L = exp − . (4.21) kII κII RT

Assuming that the two different transition states stem from the same reactant configuration, Equation (4.21) may be further simplified, resulting in

L ◦,L ! kI κI ∆GII-I L = exp , (4.22) kII κII RT

◦,L where ∆GII-I corresponds to the diﬀerence in Gibbs free energy between the transition-state structures of the two alternative pathways, given as

◦,L ◦,L ◦,L ∆GII-I = GTS,II − GTS,I. (4.23)

This quantity essentially determines the route and selectivity of the reaction and is highly inﬂuenced by the solvent used. In the next section, with the aim of assessing to what extend this approach may be used to predict selectivity and kinetics in diﬀerent solvents, we focus on a liquid-phase regioselective reaction which meets the requirements for this task.

4.2 The Williamson reaction of sodium β-naphthoxide and benzyl bromide

The Williamson ether synthesis is a classic reaction in organic chemistry for preparing ethers. It was named after Alexander W. Williamson266 who first studied the formation of unsymmetrical ethers by the interaction of an alkoxide with an alkyl halide. Typically the alkoxide reacts with the alkyl halide via a bimolecular nucleophilic substitution (SN2) mechanism to form an ether. Both symmetrical and asymmetrical ethers are prepared via this reaction. Apart from its historical importance in validating the structure of ethers,267 the Williamson reaction has gained attention because it exhibits significant solvent effects that determine the selectivity of the reaction when side reactions occur simultaneously. A side reaction usually occurs when a phenoxide is used as the nucleophile, and results in the alkylation of the phenoxide at an alternative site to the oxygen site, usually a carbon site. The ratio of the O-alkylation reaction Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 102 versus the C-alkylation reaction is strongly dependent on the solvent used and, as a result, a whole spectrum of ratio values may be observed by changing the solvent medium. An interesting example of a Williamson reaction, for which significant solvents effects are observed on the selectivity, is the reaction between sodium β-naphthoxide and benzyl bromide, shown in Figure 4.1. Apart form the main ether synthesis reaction, a competing reaction also occurs as a result of the ambident character of the β-naphthoxide ion,268 shown in Figure 4.2. Due to the ionic charge being delocalized by resonance over two alternative sites, an oxygen and a carbon site, the β-naphthoxide ion is prone to bond-formation at either of the two sites.

O NaBr

O Na Br

OH NaBr

Figure 4.1: The reaction of sodium β-naphthoxide and benzyl bromide to form benzyl β-naphtyl ether (O-alkylated product) and 1-benzyl-2-naphthol (C-alkylated product). (Adapted from reference [14].)

O O

Figure 4.2: The β-naphthoxide ion possesses two potential reactive sides due to delocalization of the negative charge. (Adapted from reference [14].)

Previous studies

A small number of experimental studies14,269–272 on the reaction between sodium β-naphthoxide and benzyl bromide may be found in the literature. The impact of solvent on selectivity is investigated by performing the reaction in different solvents or solvent-mixtures and determining the final product yields. Zagorevski269,270 was the first to study the alkylation reaction of β-naphthoxide salts with substituted benzyl bromides in various solvents, such as acetone, dioxane and acetone-benzene Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 103 mixtures. In particular, Zagorevski studied the reaction in acetone at temperatures 329-331 K, in 1,4-dioxane at temperature 373 K and in the solvent-mixture acetone-benzene at temperatures 328-330 K. For acetone, the O-alkylated product was formed exclusively. For 1,4-dioxane, the alternative reaction pathway was preferred resulting in a 26% yield for the C-alkylated product, which is higher than the 23% yield obtained for the O-alkylated product. For the solvent- mixture of acetone-benzene the O-alkylation pathway again predominated with a 54% yield and a significantly smaller yield of 3% for the C-alkylated product. Building on the experimental discovery of Zagorevskii,269,270 Kornblum et al.14 extended the study into the determining role of solvent on selectivity for the reaction between sodium β-naphthoxide and benzyl bromide in a larger set of solvents. In polar aprotic solvents, such as DMF and DMSO, yields of the O-alkylated product of 97% and 95%, respectively, were obtained, with less than 1% isolated C-alkylated product in both cases. According to the authors, O-alkylation is favoured in DMF and DMSO as it involves the charge transfer from the oxygen atom to the bromide ion which is stronger than the electrostatic interaction of the bromide ion and the sodium cation. Their reasoning was based on the ability of a solvent to affect the Coulombic interactions between two point charges, expressed by the dielectric constant ε. Therefore, in aprotic solvents with relatively high dielectric constants, such as DMF and DMSO (with respective ε values of 46.45 and 36.71 at 298 K reported by Reichardt30), the sodium ion is highly solvated and thus, the electrostatic interaction with the bromide ion is minimized. Essentially, the Coulombic factor opposing the O-alkylation loses importance in a high dielectric-constant solvent. In less polar aprotic solvents, such as THF and 1,2-DME, yields of the O-alkylated product of 60% and 70%, respectively, and yields of the C-alkylated product of 36% and 22%, respectively, were obtained.14 In order to explain why the C-alkylation occurs in a larger extent in these solvents, the authors followed the same line of reasoning as before. The C-alkylation is favoured in THF and 1,2-DME as the charge transfer from the oxygen atom to the bromide ion is weaker than the electrostatic interaction of the bromide ion and the sodium cation. In aprotic solvents with relatively low dielectric constants, such as THF and 1,2-DME (with respective ε values of 7.58 and 7.20 at 298 K reported by Reichardt30), the sodium ion is less solvated and thus, the electrostatic interaction with the bromide ion is maximized. Essentially, the Coulombic factor opposing the O-alkylation gains importance in a low dielectric constant solvent. In polar protic solvents the opposite effect, of promoting the C-alkylation, was observed. For solvents such as 2,2,2-trifluoroethanol and water, high yields of the C-alkylated product of 85% Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 104 and 84%, respectively, were obtained with yields of the O-alkylated product of only 7% and 10%, respectively. The reversal in the course of the reaction for solvents 2,2,2-trifluoroethanol and water, in accordance with previous studies,273,274 was attributed to the strong hydrogen bonding capability of these solvents with the oxygen atom. As the authors explained, in protic solvents the oxygen atom will be intensely solvated, a phenomenon called “selective solvation”. As a result, the oxygen atom will be less available for nucleophilic displacement resulting in the inhibition of the O-alkylation and hence the promotion of the C-alkylation pathway.274 Even in protic solvents less capable of hydrogen-bonding than 2,2,2-trifluoroethanol and water, such as ethanol and methanol, a substantial amount of the C-alkylated product is formed, with yields of 28% and 34%, respectively. From a different angle, Akabori and Tuji271 studied the reaction between sodium β-naphthoxide and benzyl bromide with the aim of identifying the effect of two macrocyclic polyethers (benzo- 18-crown-6 and 4,7,13,16,21,24-hexaoxabicyclo[8.8.8.]hexacosane) on the course of the reaction. For their study, they considered solvents THF, benzene, water, DMF and acetonitrile. Their main findings suggested that in solvents with low dielectric constant, such as THF and benzene, where the β-naphthoxide ion and sodium cation exist in the form of an ion-pair,14 the addition of the macrocyclic polyethers promoted the O-alkylation as a consequence of the ability of the macrocyclic polyethers to dissociate the ion-pair.271 In solvents with a high dielectric constant, such as DMF and acetonitrile, the β-naphthoxide ion and the sodium cation exist in the form of free ions (or a loose ion-pair). As a result, the addition of macrocyclic polyethers had no effect upon the course of the reaction, which yielded high O-alkylation ratios. The surprisingly high O-alkylation ratios observed in water in the presence of the macrocyclic polyethers were ascribed to the role of the macrocyclic polyethers as phase-transfer catalysts, which enabled the reaction to proceed via the formation of a loose ion-pair between the β-naphthoxide ion and the sodium cation.271 Badri et al.272 investigated the selectivity of the reaction between sodium β-naphthoxide and benzyl bromide considering ionic liquids as solvents. Their venture was driven by the many advantages that the use of ionic liquids, otherwise called green solvents, has; increased safety (low volatility, low toxicity, quantitative recovery after a reaction), efficiency (good solvents, in- expensive, easy to prepare), designability (easy to functionalize) and robustness (large working temperature ranges),272,275,276 to name a few. The authors tested four different ionic liquids, Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 105 namely tetra-n-butylphosphonium bromide (n-Bu4PBr), tetra-n-methylammonium bromide (n-

Bu4NBr), 1-methyl-3-ethylimidazolin bromide (C6H11N2Br) and tetra-n-butylphosphonium chloride (n-Bu4PCl), for which they observed a similar behaviour as that of the polar aprotic solvents. Yields as high as 93-97% for the O-alkylated product and as low as 3-7% for the C-alkylated product were obtained. A summary of the percentage product yields reported in all the previous studies for this reaction at various experimental conditions is given in Table 4.1. Although the studies reviewed thus far clearly demonstrate the dominant role of solvent on the selectivity of the reaction between sodium β-naphthoxide and benzyl bromide, they lack consistency in their experimental conditions (various temperatures, solvent systems and additives used) and the reported product yields are subject to errors due to loss of material during the typical work-up procedure after the isolation of each product. In order to provide consistent and quantitative data, Ganase23 recently carried out a systematic study of the kinetics of the reaction between sodium β-naphthoxide and benzyl bromide in nine solvents (acetonitrile-d3, methanol-d4, ethanol-d6, acetone-d6, 1,2-DME, THF-d8, ethyl acetate, 1,4-dioxane, 1,4-dioxane- 23 1 d8) at 298 K. Ganase monitored the kinetics of the reaction using in situ H NMR spectrometry and determined the rate constants for the O- and C-alkylation pathways, as well as the final rate constant ratio, in each solvent studied. The values reported by Ganase23 for the rate constants kO and kC for the formation of the O- and the C-alkylated product, respectively, as well as the Expt. Expt. final rate constant ratio of kO versus kC for all the solvents studied are given in Table 4.2. The study of Ganase,23 in line with previous studies,14,269–271 confirmed the impact of solvent on the selectivity, and additionally revealed the impact of solvent on the rate constant of the reaction between sodium β-naphthoxide and benzyl bromide. As may be appreciated by Expt. Expt. inspecting Table 4.2, the rate constants kO and kC vary significantly across the different solvents studied. The final rate constant ratios reported were in reasonable agreement with product ratios from previous studies;14,269–271 nevertheless, some large discrepancies were iden- 23 tified, for example in acetonitrile-d3 (a rate constant ratio of 82:18 was reported by Ganase compared with a product ratio of 94:6 reported by Akabori et al.271) and in 1,4-dioxane (a rate constant ratio of 34:66 was reported by Ganase23 compared with a product ratio of 47:53 reported by Zagorevskii269). For the experiment conducted in 1,4-dioxane, Ganase23 argued that the disagreement with the literature value269 could be due to the different reaction temperatures at which the two experiments were conducted, highlighting the effect of temperature on the rate of the reaction. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 106

Table 4.1: Percentage product yield of the O- and the C-alkylated product (based on the isolated amounts) and the calculated percentage product O:C ratio of the reaction between sodium β- naphthoxide and benzyl bromide from available experimental studies in various experimental conditions. RT=room temperature

Alkylation Solvent ε T/K Time/h O (%) C (%) O:C (%) Ref. Watera 78.36 RT 24 10 84 11:89 [14] Water (for 5 min)a 78.36 RT 5min 8 81 9:91 [14] Water 78.36 298 24 12 73 14:86 [271] b Water+MP1 - 298 24 60 18 77:23 [271] c Water+MP2 - 298 24 77 8 91:9 [271] DMSO 46.45 RT 2h 30min 95 0 100:− [14] DMSO 46.45 313 2h 30min 95 1 99:1 [14] DMF 36.71 283-288 2h 30min 97 0 100:− [14] DMF 36.71 298 2 87 0 100:− [271] DMF 36.71 308 2h 30min 93 1 99:1 [14] b DMF+MP1 - 298 2 89 0 100:− [271] c DMF+MP2 - 298 2 90 0 100:− [271] Acetonitrile 35.94 297 42 63 4 94:6 [271] b Acetonitrile+MP1 - 297 42 60 4 94:6 [271] c Acetonitrile+MP2 - 297 42 58 3 95:5 [271] Methanola 32.66 RT 24 57 34 63:37 [14] 2,2,2-Triﬂuoroethanola 26.50 RT 24 7 85 8:92 [14] Ethanola 24.55 RT 24 52 28 65:35 [14] Acetone 20.56 329-331 1h 15min 82 0 100:− [270] Acetone+benzene - 328-330 3 54 3 95:5 [270] THFa 7.58 RT 24 60 36 63:37 [14] THFa 7.58 298 24 51 30 63:37 [271] b THF+MP1 - 298 24 65 2 97:3 [271] c THF+MP2 - 298 24 73 1 99:1 [271] 1,2-DMEa 7.20 RT 24 70 22 76:24 [14] Benzene 2.27 298 24 1 9 10:90 [271] b Benzene+MP1 - 298 24 58 12 83:17 [271] c Benzene+MP2 - 298 24 76 1 99:1 [271] Benzene-watera - RT 120 7 83 8:92 [14] Dioxane 2.21 377-379 5 23 26 47:53 [269] n-Bu4PBr - 361 2 93 7 93:7 [272] n-Bu4NBr - 360 2 97 3 97:3 [272] C6H11N2Br - 330 2 93 7 93:7 [272] n-Bu4PCl - 350 72 95 5 95:5 [272] a This yield includes any dialkylated products arising from further alkylation of the C- alkylated product. b MP1=benzo-18-crown-6. c MP2=4,7,13,16,21,24-hexaoxabicyclo[8.8.8.]hexacosane. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 107

Expt. Expt. Table 4.2: The estimated reaction rate constants kO and kC and the percentage rate Expt. Expt. constant ratio kO :kC of the reaction between sodium β-naphthoxide and benzyl bromide in nine solvents at 298 K as reported by Ganase.23 The solvents are listed in descending order of dielectric constant ε. The experiments were performed using in situ 1H NMR.

Expt. 3 −1 −1 Expt. 3 −1 −1 Expt. Expt. Solvent ε kO /dm mol s kC /dm mol s kO :kC (%) −3 −4 Acetonitrile-d3 35.94 2.757×10 6.117×10 82:18 −4 −4 Methanol-d4 32.66 5.233×10 2.917×10 64:36 −4 −4 Ethanol-d6 24.55 9.533×10 3.867×10 71:29 −3 −4 Acetone-d6 20.56 8.573×10 4.817×10 95:5 −4 −5 THF-d8 7.58 1.400×10 6.333×10 69:31 1,2-DME 7.20 4.440×10−4 8.907×10−5 83:17 Ethyl acetate 6.02 1.635×10−4 1.565×10−4 51:49 1,4-Dioxane 2.21 3.000×10−5 9.667×10−5 24:76 −5 −5 1,4-Dioxane-d8 2.21 5.000×10 9.667×10 34:66

From the preceding review, it becomes obvious that reliable kinetic measurements are of the utmost importance in order to make valid observations regarding the impact of reaction conditions on the kinetics and selectivity of our reaction of interest between sodium β-naphthoxide and benzyl bromide. In addition, large discrepancies are observed for some solvents between the literature values. In order to resolve these discrepancies, we perform experimental work for the determination of the rate constants and the selectivity ratio of the reaction between sodium β-naphthoxide and benzyl bromide in a small set of solvents and at a few temperatures. The ﬁrst solvent we select for this task, and the one we analyse in this chapter, is acetonitrile.

4.3 Experimental methodology

4.3.1 Monitoring technique

We select in situ 1H NMR spectroscopy in order to follow the kinetics of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile-d3. This monitoring technique appears the most suitable for the reaction as the two products have distinguishable aliphatic −CH2 peaks in the 1H NMR spectrum: for the O-alkylated product at 5.23 ppm and for the C-alkylated product at 4.45 ppm. For benzyl bromide, the peak we follow is an aliphatic −CH2 peak at 4.62 ppm and for sodium β-naphthoxide an aromatic −CH peak at 6.80 ppm. We use an internal standard with known concentration to quantify the concentrations of the reaction species relative to this value. As an internal standard we choose 1,3,5-trimethoxybenzene because its characteristic peaks, at 3.74 ppm and 6.10 ppm, do not overlap in the 1H NMR spectrum with Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 108 any other peak of the species involved in the reaction; as a result, the concentrations of the reaction species may be readily quantiﬁed. The instrument we use to perform the kinetic experiments is a 500 MHz Bruker CryoProbe Prodigy spectrometer with a broadband cryogenic cooler probe. This type of probe uses an open- cycle liquid nitrogen cooling system for enhanced sensitivity.277 The reduced noise contribution by the probe and electronics yields a signal-to-noise-ratio enhancement of up to a factor of 2 to 3 using the CryoProbe Prodigy technology in comparison with an equivalent room temperature probe.277 Other monitoring techniques, such as ReactIR and HPLC, were also tried in this work. Monitoring the kinetics of the reaction with ReactIR was unsuccessful as no distinct peak was identiﬁed for either of the two products. Monitoring the kinetics of the reaction with HPLC was successful; however, large errors occurred due to the calibration of the chromatography column and the detector as well as the cooling down of the reaction when it was studied at higher temperatures. As a consequence, the concentration data obtained via HPLC were discarded as unreliable.

4.3.2 Proton-exchange experiment

The acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution; thus, it is a measure of the degree of deprotonation when referring to weak acids.55 Ignoring any deviations from ideality so that all activity coeﬃcients can be set to 1, the acid − dissociation constant Ka for the equilibrium between an acid (HA) and its conjugate base (A ) and a hydrogen ion (H+), − + HA A + H (4.24) is given as55 [A−][H+] K = . (4.25) a [HA]

The value of Ka is most frequently used in the logarithmic scale, denoted as pKa, calculated as55

pKa = − log10 Ka. (4.26)

The smaller the pKa value, the stronger the acid and thus, the larger the extent of dissociation of the acid.55 Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 109

Regarding the species of the Williamson reaction of interest in this work, at 298 K the 278 pKa value reported in the literature for β-naphthol is 9.57±0.10 and for the C-alkylated product279 is 9.60±0.50. The error is larger on the latter value as it was calculated from literature references incorporating substituent effects. However, given that β-naphthol is not significantly more acidic, reprotonation by the C-alkylated product could take place. The afore-mentioned literature findings suggest that an exchange of protons takes place between sodium β-naphthoxide and the C-alkylated product. Therefore, we design an experiment in order to test this hypothesis. We prepare four separate samples: one pure sample of sodium β-naphthoxide, one pure sample of β-naphthol, one pure sample of the C-alkylated product and a mixture of sodium β-naphthoxide and the C-alkylated product at 1:1 ratio. Subsequently, we dissolve all four samples in DMSO, which is used because of its large pKa value (pKa=35 reported by Matthews et al.280) to guarantee that proton-exchange will not happen between the solvent and any of the molecules tested. For the various molecules, we monitor the aromatic carbon linked with the oxygen atom, as labelled in Figure 4.3, using 13C NMR spectroscopy. The external magnetic field perceived by the carbon nucleus changes as a function of the electronegativity of the atoms attached to it.281 Therefore, a difference in the chemical shift is anticipated for the aromatic carbon (C2) followed in sodium β-naphthoxide, for the aromatic carbon (C2) followed in β-naphthol and for the aromatic carbon (C12) followed in the C-alkylated product. Indeed, for the pure samples, the chemical shift appears at 170.7 ppm for the aromatic carbon

(C2) of sodium β-naphthoxide, at 155.0 ppm for the aromatic carbon (C2) of β-naphthol and at

153.0 ppm for the aromatic carbon (C12) of the C-alkylated product. For the mixture sample, the chemical shift appears at 163.6 ppm for the aromatic carbon (C2) of β-naphthoxide (lower than in the pure sample) and at 162.6 ppm for the aromatic carbon (C12) of the C-alkylated product (higher than in the pure sample).

(a) (c) O Na (b) OH OH 2 2 12

Figure 4.3: Labelled aromatic carbon C2 for (a) sodium β-naphthoxide and (b) β-naphthol, 13 and (c) aromatic carbon C12 for the C-alkylated product, for monitoring using C NMR spectroscopy. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 110

The large diﬀerence in the chemical shift values obtained for the aromatic carbon C2 between sodium β-naphthoxide and β-naphthol is due to the more electronegative oxygen atom (O−) in β-naphthoxide, compared with the less electronegative hydroxy (-OH) group in β-naphthol, in the vicinity of the carbon nucleus. Conversely, the chemical shift values of the aromatic carbons

C2 for β-naphthol and C12 for the C-alkylated product (155.0 ppm and 153.0 ppm, respectively) are very close (as they are both attached to an -OH group). The displacement of the chemical shifts of the aromatic carbons C2 and C12 in the mixture indicates that the electronegativity of the two carbon nuclei changes due to the different atoms attached to the neighbouring oxygen atom in sodium β-naphthoxide and the C-alkylated product. The average of the chemical shift values for the aromatic carbon of β-naphthtol (155.0 ppm) and sodium β-naphthoxide (170.7 ppm), gives approximately the value found for the chemical shift of the aromatic carbon C2 for sodium β-naphthoxide in the mixture sample (163.6 ppm). Therefore, it may be deduced that the chemical shift values of the aromatic carbon nuclei C2 for sodium β-naphthoxide at 163.6 ppm and C12 for the C-alkylated product at 162.6 ppm in the mixture sample correspond to an average of each of the two molecules in their protonated and deprototonated forms. Detection of the proton-exchange between the protonated and deprotonated forms for either sodium β- naphthoxide or the C-alkylated product using 1H NMR would be very difficult, as the proton- exchange phenomenon occurs very fast, and the resulting peaks for the -OH groups of β-naphthol and the C-alkylated product are indistinguishable. As a result of the proton-exchange experiment, an equilibrium is postulated between the protonated and deptoronated forms of sodium β-naphthoxide and the C-alkylated product, as shown in Figure 4.4. Once the C-alkylated product is produced from the reaction between sodium β-naphthoxide and benzyl bromide, protonation of sodium β-naphthoxide starts taking place, establishing an equilibrium between the two molecules. The proton-exchange phenomenon has not been identified in any of the previous studies.14,23,269–272 In this work, the equilibrium between protonated and deprotonated forms of sodium β-naphthoxide and the C-alkylated product is further considered when analysing the obtained reaction kinetic data. A kinetic model, which captures both the reaction between sodium β-naphthoxide and benzyl bromide and the proton-exchange equilibrium, is developed in Section 4.3.3. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 111

O Na OH OH O Na

Figure 4.4: The equilibrium between the protonated and deprotonated forms of sodium β- naphthoxide and the C-alkylated product.

4.3.3 Methodology for kinetic experiments

In this work, we build the experimental methodology based on the one developed by Ganase.23 As a result, preliminary tests, such as solubility and reactivity tests, are omitted and the reader is referred to the original study of Ganase23 for details on these tests. Nevertheless, when additional actions are taken, these are clearly explained here. The main steps of the experimental methodology followed in this work are as follows:

1. First, we select three diﬀerent temperatures (298, 313 and 323 K) to perform the kinetic

experiment in acetonitrile-d3. The highest temperature is selected to be lower than the 30 boiling point of acetonitrile-d3 (354.75 K).

2. Second, we determine the initial concentration of the reactants. We use a ratio of 1:1 for the two reactants and we select a relatively low value for their initial concentrations (0.09 M) in order to avoid problems with solubility. We use a solution of known concentration of

the internal standard in acetonitrile-d3. Prior to each experiment, we weigh the amounts of the two reactants (cf. Table 4.3) using an electronic scale. We then dissolve sodium

β-naphthoxide into the solution of the internal standard and acetonitrile-d3 in a vial and transfer the content of the vial to an NMR tube with a syringe. Subsequently, we add benzyl bromide into the NMR tube to initiate the reaction, and we transfer the tube into the NMR instrument which is already tuned at a selected temperature.

3. After the insertion of the sample into the NMR instrument, the ﬁrst 1H NMR spectrum is obtained in approximately 3 minutes, which is the time needed for the probe to be tuned and for the sample to be shimmed. After this time period, we collect data every six minutes for the ﬁrst one hour and every 20 minutes until the end-time of monitoring the experiment. We monitor the experiment at 298 K for approximately four hours and the experiments at temperatures 313 K and 323 K for approximately two hours. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 112

4. The raw peak values are integrated and the known concentration of the internal standard is used to convert the peak areas into concentration units. The peaks that are monitored are the following:

(a) For 1,3,5-trimethoxybenzene, an aromatic (−CH)3 peak at 6.10 ppm and a methyl

(−CH3)3 peak at 3.74 ppm. (b) For sodium β-naphthoxide, an aromatic −CH peak at ∼ 6.79-7.05 ppm (where possible).

(d) For the O-alkylated product, an aliphatic −CH2 peak at ∼ 5.19-5.23 ppm.

(e) For the C-alkylated product, an aliphatic −CH2 peak at ∼ 4.36-4.47 ppm.

Monitoring the consumption of sodium β-naphthoxide in the reaction is feasible only for a limited number of spectra at the beginning of each experiment, as the selected aromatic −CH peak overlaps with other peaks in the aromatic area and disappears. For each experiment at a diﬀerent temperature, the number of spectra integrated for sodium β- naphthoxide is 8 spectra at 298 K (45 minutes), 3 spectra at 313 K (15 minutes) and 1 spectrum at 323 K (3 minutes). The number of integrated spectra for sodium β-naphthoxide is small compared to the length of the kinetic experiments overall. However, it is appreciated that having concentration data for one of the reactants, especially in the beginning of each experiment, is crucial in the accurate estimation of the initial concentration of that reactant as well as in the determination of the reaction rate constants for the formation of the two products. Hence, we decide to retain the concentration data collected for sodium β-naphthoxide for further use in the kinetic analysis.

5. Two diﬀerent kinetic models are developed in this chapter. The ﬁrst model developed is the “2-reaction model”, which refers to the model that captures the kinetics of the reaction between sodium β-naphthoxide and benzyl bromide leading to the formation of the O- and C-alkylated products, shown in Figure 4.1. The reactions that constitute the 2-reaction model may be written, in a concise form, as

k 1 + 2 −→O 3 (4.27)

k 1 + 2 −→C 4, (4.28) Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 113

where 1 corresponds to sodium β-naphthoxide, 2 to benzyl bromide, 3 to the O-alkylated

product, 4 to the C-alkylated product, kO is the rate constant of the reaction leading to the

formation of the O-alkylated product and kC is the rate constant of the reaction leading to the formation of the C-alkylated product. The reaction between sodium β-naphthoxide and benzyl bromide follows a ﬁrst-order rate law for each of the reactants and a second-

order rate law overall. The rates rO and rC of the reactions shown in Equations (4.27) and (4.28), respectively, may be written as

rO = kO · [1] · [2] (4.29)

rC = kC · [1] · [2], (4.30)

where the numbers placed inside square brackets denote molar concentrations.282 Assum- ing a constant volume, perfect mixing and isothermal conditions inside the NMR tube,282 the equations of the 2-reaction model for the overall rate of the reaction may be written, in terms of the change in concentration for each component, as

d[1] = −(k + k ) · [1] · [2] (4.31) dt O C

d[2] = −(k + k ) · [1] · [2] (4.32) dt O C d[3] = k · [1] · [2] (4.33) dt O d[4] = k · [1] · [2] (4.34) dt C

[1]t=0 = [1]0, [2]t=0 = [2]0, [3]t=0 = 0, [4]t=0 = 0, (4.35)

where [i]t=0 is the initial concentration of component i =1, 2, 3 and 4, as speciﬁed in Equations (4.27) and (4.28) at time zero.

The second model developed in this work is the “proton-exchange model”, which, besides capturing the kinetics of the reaction between sodium β-naphthoxide and benzyl bromide, also includes the equilibrium between the protonated and deprotonated forms of sodium β-naphthoxide and the C-alkylated product, shown in Figure 4.4. The reactions that constitute the proton-exchange model may be written, in a concise form, as

k 1 + 2 −→O 3 (4.36) Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 114

k 1 + 2 −→C 4 (4.37)

k 1 + 4 −→f 5 + 6 (4.38)

5 + 6 −→kr 1 + 4, (4.39)

where 5 corresponds to β-naphthol, 6 to the deprotonated form of the C-alkylated product,

kf is the rate constant for the forward reaction towards β-naphthol and the deprotonated

form of the C-alkylated product and kr is the rate constant for the reverse reaction towards sodium β-naphthoxide and the C-alkylated product. Assuming a single-step mechanism

for the proton-exchange equilibrium, the rates rf and rr of the reactions showing in Equa- tions (4.38) and (4.39), respectively, may be written as

rf = kf · [1] · [4] (4.40)

rr = kr · [5] · [6]. (4.41)

The assumptions of perfect mixing and isothermal conditions inside the NMR tube282 are also retained here, and the equations for the proton-exchange model may be written as

d[1] = −(k + k ) · [1] · [2] − k · [1] · [4] + k · [5] · [6] (4.42) dt O C f r

d[2] = −(k + k ) · [1] · [2] (4.43) dt O C d[3] = k · [1] · [2] (4.44) dt O d[4] = k · [1] · [2] − k · [1] · [4] + k · [5] · [6] (4.45) dt C f r d[5] = k · [1] · [4] − k · [5] · [6] (4.46) dt f r

kf Keq = (4.47) kr [5] = [6] (4.48)

[Sum1] = [1] + [5] (4.49)

[Sum2] = [4] + [6] (4.50)

[1]t=0 = [1]0, [2]t=0 = [2]0, [3]t=0 = 0, [4]t=0 = 0, [5]t=0 = 0, (4.51) Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 115

where Keq is the equilibrium constant, [Sum1] (referred here as “total naphthol”) is the sum of the concentrations of the protonated ([5]) and deprotonated ([1]) forms of sodium

β-naphthoxide, [Sum2] (referred here as “total C-alkylated product”) is the sum of concentrations of the protonated ([4]) and deprotonated ([6]) forms of the C-alkylated product

and [5]t=0 is the initial concentration of β-naphthol at time zero.

The two kinetic models developed in this work are implemented in gPROMS.283 Con- centration data obtained from integration of the 1H NMR spectra for the reactants and

products of the reaction are used to estimate the rate constants kO and kC of the 2-reaction

model and, additionally, the equilibrium constant Keq of the proton-exchange model. In

both models, the initial concentration values for the reactants at time zero, [1]0 and [2]0, are also estimated. By doing so, we may identify any signiﬁcant diﬀerences from the weighted values which may be due to errors in weighting the amounts of the reactants (due to the accuracy of the scale) or due to errors when transferring the solution from the vial to the NMR tube via a syringe (due to some material left either in the vial or in the syringe).

4.4 Experimental results

4.4.1 Kinetic experiments at 298, 313 and 323 K

In this section, we present and discuss the results of the kinetic experiments for the reaction between sodium β-naphthoxide and benzyl bromide performed in acetonitrile-d3 at three diﬀer- ent temperatures. We ﬁrst present the results obtained by implementing the 2-reaction model in gPROMS, and then the results obtained by implementing the proton-exchange model in gPROMS only at 298 K for comparison. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 116

2-reaction model

The experimental concentration proﬁles for the components of the reaction between sodium

β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K are shown (with symbols) in Figures 4.5, 4.6 and 4.7, respectively. The molecules monitored in this reaction are the following: the reactants, sodium β-naphthoxide and benzyl bromide, and the products, the O-alkylated product and the C-alkylated product. The concentration of each reaction component is represented by a different symbol and colour. The curves of the corresponding colours represent the calculated concentration values for each reaction component obtained by implementing the 2-reaction model in gPROMS. For each experiment at a different Expt. Expt. temperature, the estimated reaction rate constants kO and kC are presented in Table 4.4. The sensitivity of each estimated parameter is summarized in its 95% confidence intervals, which are also reported in the table. The 95% confidence intervals are estimated on the basis of a constant variance model assuming an error of 0.01 dm3 mol−1 for the concentration values. Superscript “Expt.” in the estimated parameters refers to the use of experimental concentration data to fit those parameters. In addition, the estimated initial concentration values for the reactants at time zero, [1]0 and [2]0, along with their 95% confidence intervals, are presented in Table 4.3. As may be seen in Figures 4.5, 4.6 and 4.7, the fitting of the 2-reaction model to the experimental concentration data of the reaction components is reasonably good for all the three temperature-dependent experiments. The shortage of experimental measurements for the concentration of sodium β-naphthoxide results in the underestimation of the predicted concentration values (the blue curves shown in Figures 4.5, 4.6 and 4.7) in all three experiments. The predictions of the 2-reaction model deviate slightly from the experimental data and these deviations could possibly indicate the existence of a reversible reaction and potentially an equilibrium between reactants and products. The existence of such an equilibrium would justify these deviations from the experimental data (underestimation of reactants concentrations and overestimation of the products concentrations). In the study of Ganase,23 the reversibility of the reaction was tested by putting together the two products in the solvent and no formation of the reactants was observed. An alternative source of the model deviations from the experimental data could be the impact of neglecting the formation of the double C-alkylated product from further alkylation of the C-alklylated product (which will be thoroughly discussed in Chapter 5). Nevertheless, the impact of this source is expected to be minor for the specific solvent investigated here given the small amount of the C-alkylated product resulted. The small differences Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 117

0.1 Sodium β-naphthoxide Benzyl bromide O-alkylated product 0.08 C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 2000 4000 6000 8000 10000 12000 14000 16000 t/sec

Figure 4.5: Experimental concentration data as a function of time in acetonitrile-d3 at 298 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS. between the estimated and weighted initial concentrations of the reactants may be either due to the error in the weighted values because of the loss of small reactant amounts (weighted separately in vials) when transferring them with a syringe to the NMR tube to initiate the reaction; or due to inaccuracies in the volume used, which would also lead to inaccuracies in the concentrations. For both of the reactants, the initial concentration values are underestimated when compared with the weighted values, as shown in Table 4.3. The estimated 95% conﬁdence intervals lie (on average for the three experiments) on the following ranges from the estimated initial concentration values: ±0.007 mol dm−3 for sodium β-naphthoxide and ±0.008 mol dm−3 for benzyl bromide. Expt. Expt. As may be seen in Table 4.4, a value of 95:5% is obtained for kO :kC for the experiment at 298 K. This value slightly increases to 96:4% as the temperature increases to 313 and 323 K. Based on these values, a clear preference of the selectivity of the reaction towards the O-alkylated product is demonstrated, for the temperature range studied. This outcome is in accordance with previous studies14,23,270,271 in which the reaction between sodium β-naphthoxide and benzyl bromide was investigated in polar aprotic solvents; in particular, the percentage rate constant Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 118

0.1 Sodium β-naphthoxide Benzyl bromide O-alkylated product 0.08 C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 t/sec

Figure 4.6: Experimental concentration data as a function of time in acetonitrile-d3 at 313 K measured with in situ 1H NMR spectroscopy. The different symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS. ratio value of 95:5 at 298 K obtained in this work is in excellent agreement with the value of 94:6 at 298 K reported by Akabori and Tuji271 for the reaction in acetonitrile (cf. Table 4.1). On the other hand, appreciable difference is observed when comparison is made with the value of 82:18 reported by Ganase;23 the reason for this discrepancy is explained later in this section. Expt. The estimated values for the reaction rate constant kO , reported in Table 4.4, are always Expt. one order of magnitude larger than the estimated values for the reaction rate constant kC , indicating the much faster reaction yielding the O- over the C-alkylated product in acetonitrile- d3, independently of the temperatures studied in this work. As the temperature increases, the values of both rate constants drastically increase by an order of magnitude, suggesting a substantial increase in the rates of both reactions. For the three experiments performed at different temperatures, the estimated 95% confidence intervals are of the same order of magnitude as the estimated rate-constant values; hence, the intrinsic uncertainty of the estimated rate-constant values is appreciable. Of the three experiments performed in acetonitrile-d3, the experiment at temperature 323 K is the one with the largest 95% confidence intervals for the estimated rate-constant values. At 323 K the reaction is very fast and the experiment is completed in Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 119

0.1 Sodium β-naphthoxide Benzyl bromide O-alkylated product 0.08 C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 t/sec

Figure 4.7: Experimental concentration data as a function of time in acetonitrile-d3 at 323 K measured with in situ 1H NMR spectroscopy. The different symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS. approximately 70 minutes, as sodium β-naphthoxide is consumed. The rate of a reaction is dominated by the first stages, when the reactant concentrations are at their maximum, and the rate decreases as the reactants are consumed. Therefore, the rate constant of a reaction, being the slope of a concentration-versus-time plot, may be defined accurately based on a large number of concentration measurements collected for as long as the concentration is changing with time. For the experiment at 323 K, the small number of 1H NMR spectra collected before the rate of the reaction dropped almost to zero, deteriorates appreciably the level of precision for the estimated rate-constant values; as a consequence, the corresponding 95% confidence intervals broaden. The constant variance model adopted in this work implies that the true parameter values may be found within a symmetric confidence region; however, this is an approximation and the true variance of measurement error is unspecified. As a consequence, negative values are obtained as lower bounds for the confidence intervals of the C-alkylation rate constant estimates. Since no physical meaning may be attributed to a negative rate constant, we restrain the lower bounds of the relevant confidence intervals to a zero value. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 120

We emphasize again that the confidence intervals are calculated based on an assumed error value and they are only indicative of the uncertainty involved in the estimated parameters. The best way to properly calculate confidence intervals is by using a probability-based sampling method, where each sample has a known probability of being selected and as a result, the like- lihood of a sample being representative of the population is improved.284 For a non-probability sampling method, no information is available about the sampling distribution of means and presumably the calculated confidence intervals lack statistical meaning. Yet, it is of usual practice for confidence intervals to be calculated with non-probability samples.284 In comparison with the rate-constant values reported by Ganase23 (cf. Table 4.2), who also studied the kinetics of the reaction in acetonitrile-d3 at 298 K, the rate-constant values obtained in this work are very different. These large differences led to a major review of the results obtained in the previous study. A thorough analysis of the raw 1H NMR spectra obtained from Ganase23 revealed an incorrect integration of the peak corresponding to the C-alkylated product, which is integrated together with an adjacent peak; as a result, a higher ratio value for the C- alkylated product is obtained. In this study, the adjacent peak is also present (at 4.50 ppm), but it does not interfere with the peak of the C-alkylated product (∼ 4.36-4.47 ppm); thus, it is possible to integrate the peak for the C-alkylated product alone. The higher resolution of the 500MHz NMR instrument used in this work (as opposed to a 400MHz NMR instrument used in the previous work) could explain why the two peaks are separated in the spectra collected.

(Although it is postulated that the unidentified peak observed at 4.50 ppm in acetonitrile-d3 originates from the O-alkylated product, further investigation is required to fully characterize 23 it.) In light of this finding, the rate-constant values reported by Ganase in acetonitrile-d3 are discarded as inaccurate and no further comparison is made in this work with respect to those values. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 121 3 (%) Expt. C k 95:5 96:4 96:4 : Expt. O 95% CI k 4 3 3 − − − 3 10 10 10 − × × × 95% CI /mol dm e 0 [2] [0.000-5.691] [0.000-9.215] [0.000-17.27] -naphthoxide and benzyl bromide in acetonitrile-d β 3 − 1 − s is also reported. 1 3 3 4 − − − − Expt. C /mol dm 10 10 10 mol k w 0 : × × × 3 -naphthoxide ([1]) and benzyl bromide ([2]), for the reaction studied Expt. O /dm β k 3.390 1.656 7.194 Expt. C k 2 2 2 95% CI [2] − − − , with their 95% confidence intervals (CI), obtained by fitting the 2-reaction model 10 10 10 × × × Expt. C 3 k − and 95% CI -naphthoxide Benzyl bromide Expt. /mol dm O β e 0 k [2.538-6.508] [3.617-12.77] [0.976-1.620] [1] 1 − 3 s − Sodium 1 2 2 2 − − − − 10 10 10 mol × × × 3 /mol dm w 0 /dm Expt. O k 298313323 0.090 0.090 0.090 0.073 0.065 0.064 [0.066-0.080] [0.057-0.073] [0.058-0.070] 0.090 0.090 0.090 0.079 0.079 0.079 [0.072-0.087] [0.071-0.088] [0.072-0.086] at three different temperatures. The reported concentration values correspond to the weighted amounts, indicated by the T/K [1] 3 313323 4.523 8.194 298 1.298 T/K Table 4.3: Initial concentrations atin time acetonitrile-d zero of thesuperscript reactants, “w”, sodium and the estimated values, indicatedthe by 2-reaction the model superscript “e”, in with gPROMS. their 95% confidence intervals (CI), obtained by implementing Table 4.4: Estimated reaction rate constants to experimental concentration data forat the three components different of temperatures. the The reaction percentage between rate sodium constant ratio of Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 122

Proton-exchange model

A similar analysis of the obtained experimental kinetic data is performed by implementing the proton-exchange model in gPROMS. Only the experimental concentration profiles at 298 K (for which experiment we have the maximum information we can use to specify the quantity [Sum1]) are analysed and presented here and the estimated results are compared with the findings of the 2-reaction model. In the proton-exchange model, in addition to the reaction between sodium β-naphthoxide and benzyl bromide, the equilibrium between the protonated and deprotonated forms of sodium β-naphthoxide and the C-alkylated product, as shown in Figure 4.4, is also considered. As a result, the experimental concentration profile for sodium β-naphthoxide is equal to the sum of the concentrations of its protonated and deprotonated forms, referred to in this model as “total naphthol”. (As discussed in Section 4.3.2, according to the postulated equilibrium, the aromatic −CH peak in the 1H NMR spectrum assigned to sodium naphthoxide is an average peak of its protonated and deprotonated forms.) Similarly, the experimental concentration profile for the C-alkylated product is equal to the sum of the concentrations of its protonated and deprotonated forms, referred to in this model as “total C-alkylated product”. The experimental concentration profiles of the various components followed in the reaction (based on the postulates of the proton-exchange model) are shown in Figure 4.8. The concentration of each reaction component is represented by a different symbol and colour. The curves of the corresponding colour represent the calculated concentration values for each reaction component obtained by implementing the proton-exhcange model in gPROMS. The estimated reaction Expt. Expt. rate constants kO and kC , and the equilibrium constant Keq, together with their 95% confidence intervals, are presented in Table 4.6. Additionally, the estimated initial concentration values for the reactants at time zero, [1]0 and [2]0, and their 95% confidence intervals, are presented in Table 4.5. As shown in Figure 4.8, the quality of fitting the proton-exchange model to the experimental concentration data of the various components is reasonably good; however, no significant improvement is observed compared with the fitting of the 2-reaction model. The initial concentration values for sodium β-naphthoxide and benzyl bromide are slightly underestimated, as shown in Table 4.5 compared with the weighted values. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 123 (%) -naphthoxide and the Expt. C β k 93:7 : ), for the reaction studied 0 Expt. O 95% CI k 3 − 3 10 − , with their 95% confidence intervals × Expt. eq K 95% CI /mol dm e 0 [2] [0.000-3.489] ) and benzyl bromide ([2] 0 3 − 1 − s 1 is also reported. 3 − − /mol dm 10 mol Expt. C w 0 × k 3 : -naphthoxide ([1] /dm Expt. 1.113 O β k Expt. C k , and the equilibrium constant 2 95% CI [2] − 4 Expt. C k 10 10 × × 3 − and 95% CI 95% CI Expt. O k -naphthoxide Benzyl bromide /mol dm β e 0 [0.000-4.281] [1] 1 − 3 s − Sodium 1 2 − − 10 mol × 3 Expt. eq /mol dm K w 0 /dm Expt. O k 298 0.090 0.077 [0.067-0.087] 0.090 0.081 [0.073-0.089] T/K [1] at 298 K. The reported concentration values correspond to the weighted amounts, indicated by the superscript “w”, and the 3 298298 1.388 102 [0.00-1.42] T/K T/K -naphthoxide and benzyl bromide and the equilibrium between the protonated and deprotonated forms of sodium Table 4.5: Initial concentrations at time zero of the reactants, sodium Table 4.6: Estimated reaction rate constants in acetonitrile-d estimated values, with their 95%the confidence superscript intervals “e”. (CI), obtained by implementing the proton-exchange model in gPROMS, indicated by (CI), obtained by fitting theβ proton-exchange model to experimental concentration data for the components of the reaction between sodium C-alkylated product at 298 K. The percentage rate constant ratio of Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 124

0.1 Total Naphthol Benzyl bromide O-alkylated product 0.08 Total C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 2000 4000 6000 8000 10000 12000 14000 16000 t/sec

Figure 4.8: Experimental concentration data as a function of time in acetonitrile-d3 at 298 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.42)-(4.51) for the proton-exchange model in gPROMS.

Expt. Expt. In general, the estimated values for the reaction rate constants kO and kC , as shown in Table 4.6, as well as the initial concentrations of the reactants, as shown in Table 4.5, are similar to the corresponding values obtained for the 2-reaction model. Also, the percentage rate constant ratio value of 93:7 obtained for the proton-exchange model is in excellent agreement with the value obtained for the 2-reaction model (95:5) and with the percentage product ratio value reported by Akabori and Tuji271 (94:6). However, it is clear that the 95% confidence intervals for all the quantities estimated using the proton-exchange model are subject to significant uncertainty, especially in the case of the equilibrium constant: the value is statistically insignificant. The broadness of the 95% confidence intervals obtained in the proton-exchange model is due to the way the model is defined and the amount of data used to fit the model. The proton-exchange model constitutes a more complete representation of the kinetic and thermodynamic phenomena involved in the reaction system, in comparison with the 2-reaction model; however, its implementation in the experimental concentration data creates an underspecified problem. An unspecified problem occurs when the amount of information provided is insufficient to determine all the unknown parameters and a large space of possible solutions is available; Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 125 as a consequence, the estimated parameters are intrinsically ill-defined. Specifying two of the four experimental data sets (the total naphthol and the total C-alkylated product) as sums of two other unknown quantities, poses significant limitations in the performance of the parameter estimation algorithm. Consequently, the values of the estimated parameters carry significant uncertainty, which is depicted by their large confidence intervals. By inspecting Tables 4.6 and 4.5, it is clear that this is exactly the case for the estimated parameters of the proton-exchange model. By contrast, the 2-reaction model, although physically less representative of the kinetic and thermodynamic phenomena of the reaction, achieves a higher level of confidence for the estimated parameters; the corresponding 95% confidence intervals are considerably narrower and the parameters well-defined. Furthermore, the similarity between the values of the rate constants estimated from both the 2-reaction and the proton-exchange model suggests a minor impact of the equilibrium on the kinetics of the system. This is true because of the small amount of the C-alkylated product produced in acetonitrile-d3, which allows us to neglect the equilibrium without introducing a significant error in our kinetic analysis. In light of the above, the results from the 2-reaction model, reasonably accurate and more well-specified, are used henceforth to explore further the effect of temperature on the reaction rate constants and to calculate temperature-dependent thermodynamic quantities. This is the goal of the next section.

4.4.2 Thermodynamic analysis based on experimental data

With the aim to illustrate the eﬀect of temperature on the rate constants, the estimated rate constants are interpreted according to two equations. First, we use the Arrhenius equation, shown in Equation (3.10), to plot ln k versus 1/T , where the slope of the plot corresponds to

E˜a/R and the y-intercept to ln A1. An alternative analysis may be done in terms of the Eyring equation, given as ‡ ◦,L kBT Y νi ∆ G kL = c◦,L exp − , (4.52) h i RT i or, by analysing the Gibbs free energy of activation using Equation (4.4), as

‡ ◦,L ‡ ◦,L kBT Y νi ∆ S ∆ H kL = c◦,L exp exp − , (4.53) h i R RT i Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 126 which may be written in a logarithmic form, as

L ‡ ◦,L ‡ ◦,L ! k ∆ H ∆ S kB Y νi ln = − + + ln c◦,L . (4.54) T RT R h i i

We then plot ln(k/T ) versus 1/T , where the slope of the plot corresponds to −∆‡H◦,L/R ‡ ◦,L ◦,L −3 and the y-intercept to ∆ S /R + ln(kB/h), for concentration ci =1 mol dm . In order to compare with theoretical estimates, the Gibbs free energy of activation may be computed using Equation (4.4) for any temperature included within the range for which experimental rate constants have been obtained. The underlying assumption here is that the enthalpy and entropy of activation are temperature-independent over a relatively narrow temperature range. Extrapolating to a temperature value outside this range might result in significant errors due to violation of this assumption. Equation (4.52) is an expression frequently used for the analysis of temperature-dependent experimental data. It represents the phenomenological formulation of CTST, as shown in Equa- tion (4.2), in which the transmission coefficient κ has been omitted or assumed to be the unity. These two equations may, under certain circumstances, be equivalent.285–287 The Eyring equation is usually simpler to apply and the main reasons for considering the CTST expression relate to the interest of conveying physical insight with respect to quantum effects (such as recrossing and tunnelling effects) that are expressed through the transmission coefficient. Even though the exact form of the transmission coefficient was derived by Kuppermann,287 usually a more pragmatic and simple model is applied. In any case, as Pu et al.288 pointed out, the validity of CTST should not be doubted; instead, a more fruitful discussion may be about the accuracy of the corresponding CTST expression considering a specific value or a specific model for the transmission coefficient.288 In this work, when fitting the Eyring equation to experimental data, we assume a value of one for the transmission coefficient. This is a reasonable assumption as no evidence of the presence of tunnelling effects is reported in the literature for this type of reaction. Even more so, tunnelling effects are usually more prominent at lower temperatures; the range of temperatures studied here hardly justifies the necessity of accounting for tunnelling effects. The validity of this assumption will also be discussed in a later section, where values for the transmission coefficient are calculated by considering the Wigner scheme.207 The experimental rate constants for both alkylation pathways at various temperatures are presented in Arrhenius-like and Eyring-like plots, in Figures 4.9 and 4.10, respectively. The Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 127 quality of fit (R2) and the linear equation derived in each case are also given in the graphs. For the temperature range studied, from 298 to 323 K, the experimental rate constants for the O- and the C-alkylation pathways strictly follow linearity, when fitted as functions of inverse temperature to both the Arrhenius and the Eyring equations (R2> 0.99 in Figures 4.9 and 4.10). This is because of the narrow temperature range studied and the small number of performed experiments within this range.

0 O-alkylation -2 C-alkylation

-4 y=-7158.8x + 19.705 R2=0.9957 -6 y=-5892.6x + 12.505 2 Expt. -8 R =0.9914 k

ln -10

-12

-14

-16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 4.9: Arrhenius plot of the logarithm of experimental reaction rate constants as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K.

The estimated parameters of the Arrhenius equation and the Eyring equation are listed, Expt. respectively, in Tables 4.7 and 4.8. As shown in Table 4.7, the ratio of A1 for the O- versus 3 Expt. −1 the C-alkylation has a value of 10 . For Eã , large values (around 50-60 kJ mol ) are obtained for the O- and the C-alkylation; thus, a fairly large energy barrier is suggested for the Expt. reactants in order to reach either of the two products. However, the difference in Eã between the two pathways is only 10 kJ mol−1. When analysing the rate constants in terms of the Eyring equation, as shown in Table 4.8, similar observations may be made: the values for the obtained thermodynamic parameters for the two alkylation pathways are quite large but the relative differences between them are fairly small. (When multiplied by temperature, the values for the estimated parameter ∆‡S◦,Expt., Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 128

0 O-alkylation -2 C-alkylation

-4

/T) -6 y=-6849.1x + 12.969 -8 2

Expt. R =0.9953

(k -10

ln y=-5582.9x + 5.7692 2 -12 R =0.9905

-14

-16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 4.10: Eyring plot of the logarithm of experimental reaction rate constants over temperature as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium

β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K.

Table 4.7: Estimated parameters of the Arrhenius equation (Equation (3.10)) for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K.

Expt. 3 −1 −1 ˜Expt. −1 A1 /dm mol s Ea /kJ mol O-alkylation 3.61×108 59.52 C-alkylation 2.70×105 48.99

Table 4.8: Estimated thermodynamic parameters of the Eyring equation (Equations (4.54) and

(4.4)) for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile-d3 at temperatures 298, 313 and 323 K.

∆‡H◦,Expt./kJ mol−1 ∆‡S◦,Expt./J K−1 mol−1 ∆‡G◦,Expt.(298)/kJ mol−1 O-alkylation 56.95 −89.72 83.68 C-alkylation 46.42 −149.58 90.99 ◦,Expt. −1 ◦,Expt. −1 −1 ◦,Expt. −1 ∆HC−O /kJ mol ∆SC−O /J K mol ∆GC−O (298)/kJ mol Difference C-O −10.53 −59.86 7.31 Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 129 although small in absolute numbers, contribute to the calculation of parameter ∆‡G◦,Expt.(298) to the same order of magnitude as parameter ∆‡H◦,Expt..) Of particular interest here is the ◦,L quantity ∆GC−O(T ), which is the standard-state Gibbs free energy difference between the C- and the O-alkylation transition-state structures at a given temperature. This quantity dictates the selectivity of the reaction at a given temperature by determining the final rate constant ◦,L ratio, which is proportional to the exponential of ∆GC−O(T ) over RT . From the experiments ◦,Expt. −1 performed in acetonitrile-d3, the value obtained at 298 K for ∆GC−O is 7.31 kJ mol (cf. Ta- ble 4.8). Given the large values for ∆‡G◦,Expt.(298) estimated for the two alkylation pathways, the difference between them is fairly small; however, this small energy difference gives rise to Expt. Expt. a ratio value for kO :kC of 95:5. Thus, it is apparent the major impact that parameter ◦,L ∆GC−O(T ) has on the determination of the selectivity of a reaction at a certain temperature and the key role of the solvent in altering this parameter.

4.5 Computational methodology

In this section, we investigate the Williamson reaction between sodium β-naphthoxide and benzyl bromide from a computational point of view. We aim for a thorough assessment of the predictive capabilities of various quantum-mechanical (QM) methods combined with the CTST theory and the SMD model for the kinetics and selectivity of the reaction. The eﬀect of the selection of the level of theory (method and basis set) on the QM predictions is scrutinized. Our experimental

ﬁndings for the rate constants and selectivity ratios using solvent acetonitrile-d3 may be used to make comparisons with the predicted values. To the best of our knowledge, the reaction between sodium β-naphthoxide and benzyl bromide has not been previously studied computationally.

Computational details

Throughout this work all calculations are performed using the software Gaussian 09 (release C.01).208 The SMD model is used to account for the presence of solvent as a continuum. The macroscopic descriptors required by SMD to describe a solvent medium are available in Gaussian 09 for a number of solvents; acetonitrile is one of them. We note that acetonitrile is one of the 90 non-aqueous solvents for which solvation data have been used for the parameterization of the SMD model. Specifically, solvation free energies for seven neutral solutes in acetonitrile were used for the parametrization. In addition, solvation free energies for 69 ionic solutes in Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 130 acetonitrile have been used. Of the components that are involved in this work, none is included in the lists of 7 neutral or 69 ionic solutes selected. The effect of the level of theory on the calculation of the selectivity and the kinetics of the reaction is explored here by considering a number of different DFT functionals. Specifically, we consider the hybrid functionals B3LYP,71–74 M05-2X78 and M06-2X209 and the long-range corrected functional with empirical dispersion wB97xD,289 with a number of basis sets, namely 6-31G(d), 6-31+G(d), 6-31+G(d,p), 6-311+G(d,p), 6-311++G(d,p), 6-311G(2d,d,p), cc-pVDZ and cc-pVTZ, in 18 different combinations. No scaling factors are considered when calculating the vibrational frequencies and zero-point energies, except in the calculations using the B3LYP/6-31+G(d) level of theory, where a scale factor of 0.9614 is applied.217 For each level of theory tested, the geometries of the reactants and the two transition states corresponding to the two alkylation pathways (O and C) are optimized, first in the gas phase and then in the liquid phase, i.e., the solute geometry is optimized in the field generated by the dielectric constant of the solvent. In order to guarantee that a minimum (in the case of the reactants) or a first order saddle point (in the case of a transition state) has been found, we use the keyword “calcall”, which demands the calculation of analytic second derivatives (also known as force constants) at every point of the optimization; as a result, the stationary point obtained may be fully characterized. The optimizations are completed using the keyword “ultrafine” for specifying the integration grid during the numerical integrations.† The keyword “vtight” is used for requesting the highest accuracy for the convergence criteria. The tightest convergence criteria used by Gaussian 09 are the following:290

1. The maximum component of the force must be below the cutoﬀ value of 2×10−6.

2. The root-mean-square of the forces must be below the cutoﬀ tolerance value of 1×10−6.

3. The calculated displacement for the next step must be below the cutoﬀ value of 6×10−6.

4. The root-mean-square of the displacement must be below the cutoﬀ value of 4×10−6.

We calculate the reaction rate constant for each alkylation pathway using Equation (4.20), in which the value for the transmission coeﬃcient κ is given from the Wigner207 tunnelling expression (Equation (3.4)). For the imaginary frequency required in the Wigner expression we use the value obtained after a liquid-phase optimization. The selectivity of the reaction towards

†The keyword “ultraﬁne” requests a pruned (99,590) grid. This type of grid has been optimized to use the minimal number of points required to achieve a certain level of accuracy. 290 Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 131 the O- or the C-alkylated product may be determined using Equation (4.2), as

! kQM κQM ∆‡G◦,QM − ∆‡G◦,QM O = O exp − O C , (4.55) QM QM RT kC κC where the subscripts “O” and “C” indicate the O- and C-alkylation pathways, respectively and the superscript “QM” refers to the use of QM methods to determine these quantities. Adopting the assumption that the two different transition states originate from the same reactant configuration, Equation (4.55) may be further simplified, resulting in

! kQM κQM ∆G◦,QM O = O exp C-O , (4.56) QM QM RT kC κC

◦,QM where ∆GC-O corresponds to the QM-calculated diﬀerence in Gibbs free energy between the C- and the O-alkylation transition-state structures, given as

◦,QM ◦,QM ◦,QM ∆GII-I = GTS,II − GTS,I . (4.57)

The impact of the solvent on this quantity is subject to more detailed investigation in our work.

4.6 Computational results

4.6.1 Structure search for stable conformations of sodium β-naphthoxide

The sodium cation is electrostatically retained in the proximity of the oxygen atom of β- naphthoxide; however, other energetically favoured configurations might also exist. In order to ascertain the minimum electronic energy configuration of sodium β-naphthoxide, we perform a potential energy surface (PES) scan calculation. A relaxed PES scan calculation consists of a series of geometry optimizations in which a structural parameter of interest, such as a bond length or a torsion angle, is fixed to a specific range of values while all the other structural parameters are allowed to vary to their favourable position.290 An alternative type of rigid PES scan calculation is also possible, where single-point calculations are performed at a fixed geometry. A relaxed PES scan allows for more degrees of freedom and thus more flexibility in the geometry of the molecule, which results in a more realistic analysis of the dependence of the energy on the varying parameter. In this work we opt for the relaxed PES scan calculation. By scanning the PES along the distance between sodium cation and the oxygen atom of β-naphthoxide, we obtain the electronic energy of the optimized geometries in vacuum and in Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 132

−622.650 Vacuum

−622.700 −1

−622.750

−622.800

/a.u. Particle −622.850 el, IG

E −622.900

−622.950 0.00 2.00 4.00 6.00 8.00 10.00 r/Å

Figure 4.11: Electronic energy (Eel,IG) as a function of the sodium-oxygen distance (r) in sodium β-naphthoxide in vacuum performing a relaxed potential energy surface scan at the B3LYP/6- 31+G(d) level of theory. acetonitrile, as shown in Figures 4.11 and 4.12, respectively. The effect of the level of theory on the energetics of the scan calculation is not of interest here, so we arbitrarily select the B3LYP/6- 31+G(d) level of theory to illustrate the scan results. By inspecting Figure 4.11, we may easily determine in vacuum the distance that corresponds to an equilibrium structure (req) and which has the lowest energy (minimum) for the region of the PES scanned; at the B3LYP/6-31+G(d) level of theory, a value of req=1.984 A˚ is obtained. At this configuration, the sodium cation interacts with the oxygen atom of β-naphthoxide forming an ion-pair. As the sodium cation moves away from the oxygen atom of β-naphthoxide, the electronic energy increases until it reaches a plateau value where apparently the two atoms are completely dissociated. This energy value of the pair is the same as the energy of the two ions optimized separately. In the presence of solvent acetonitrile, the distance between the sodium cation and the oxygen atom of β-naphthoxide in the equilibrium structure is req=2.067 A,˚ which is longer than the equivalent distance of the equilibrium structure in vacuum. Furthermore, the final configuration in acetonitrile has lower energy than in vacuum, which is the result of the presence of the solvent that decreases the electrostatic interactions and stabilizes the molecule in lower energy levels. Similarly to the results in vacuum, as the distance of the sodium-oxygen atoms in acetonitrile increases, the electronic energy increases towards a plateau value where the two atoms are remote Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 133

−622.650 Acetonitrile Vacuum −622.700 −1 −622.750

−622.800

−622.850 /a.u. Particle el E −622.900

−622.950 0.00 2.00 4.00 6.00 8.00 10.00 r/Å

Figure 4.12: Electronic energy (Eel) as a function of the sodium-oxygen distance (r) in sodium β- naphthoxide in acetonitrile performing a relaxed potential energy surface scan at the B3LYP/6- 31+G(d) level of theory. The results from the calculations in vacuum are shown for comparison. from each other. Thereafter, we model sodium β-naphthoxide in the form of an ion-pair, as it is most likely to be encountered, both in the gas and liquid phase.

4.6.2 Transition-state structures for the O- and C-alkylation

The task of locating the transition state for a specific reaction may be rather challenging; a good initial guess is usually highly desirable in order to avoid laborious and time consuming calculations.63 For the reaction between sodium β-naphthoxide and benzyl bromide, Kornblum et al.14 provided drawings of probable transition state configurations for the O- and C-alkylation pathways. For the O-alkylation pathway, it was suggested that the reaction proceeds via a transition-state structure in which the oxygen atom is aligned with the carbon and bromine atoms of benzyl bromide, so that the sodium cation is relatively remote from the bromine atom.14 On the contrary, for the C-alkylation a non-linear arrangement was postulated where the departing bromine atom is relatively close to the sodium cation, which is found in-between the bromine atom and the oxygen atom.14 In the absence of any other relevant information, these drawings are used in our investigation as guidelines for locating the two transition states. For illustrative purposes, the structures optimized at the B3LYP/6-31+G(d) level of theory are shown in the figures of this section. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 134

O C(1)

C(2) C(3) Br

177.92o

Dihedral angle: o C(1)-C(2)-C(3)-Br=-180.00

Figure 4.13: A transition-state structure (TS 1) for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 135

In vacuum, a computational search of the transition state for the O-alkylation pathway results in two candidate structures, each of them having one negative eigenvalue (imaginary frequency) associated with the Hessian matrix. In order to ensure that each of the two candidate structures links the reactants and products corresponding to the O-alkylation reaction, we additionally perform an intrinsic reaction coordinate (IRC) type of calculation. In the IRC calculation, we start from the optimized transition-state structure moving downhill along the reaction path towards the direction of the reactants and the products.290 In either direction, the IRC calculation terminates successfully, albeit the energies of the end-point structures are higher than the energies obtained when the two reactants, as well as the two products, are optimized separately. As it is common for the IRC algorithm to stop prematurely before reaching the minimum on either side, further optimization of the end-point structures found is highly recommended.290 Indeed, by following this strategy we obtain the desired reactant and product configurations. For the O-alkylation pathway, the difference between the two transition-state structures is the orientation of the plane of the β-naphthoxide ion with respect to the plane defined by the bromine atom (Br) and the carbon atom (C3) from which the bromine atom departs. In one transition-state configuration (TS 1) the plane of the β-naphthoxide ion lies in the same plane as ◦ the departing bromine atom Br and the carbon atom C3 (with a dihedral angle of -180.00 ), as shown in Figure 4.13, whereas in the other transition-state configuration (TS 2) the plane of the β-naphthoxide ion is nearly perpendicular to the plane of the bromine atom Br and the carbon ◦ atom C3 (with a dihedral angle of 76.40 ), as shown in Figure 4.14. TS 1 resembles closely the drawing of Kornblum et al.;14 however, TS 2 has an electronic energy value of -3465.54818 a.u., which is slightly lower than the electronic energy value of TS 1 of -3465.54748 a.u., and thus corresponds to a more stable configuration. The most favourable proceeding pathway in CTST for a reaction is expected to be the one corresponding to the lowest energy transition- state structure.26 As a result, we assume hereafter that the O-alkylation reaction proceeds in vacuum solely via TS 2. (We make the same statement also for the liquid-phase after calculating the energies for both TS 1 and TS 2 in the liquid phase and finding that the energy of TS 2 is lower than TS 1, as discussed in a forthcoming paragraph.) The structural parameters for the optimized geometries at the B3LYP/6-31+G(d) level of theory in vacuum of both transition states (TS 1 and TS 2) of the O-alkylation pathway may be found in AppendixB. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 136

C(2)

C(3)

Br C (1) 178.60o

Dihedral angle: o C(1)-C(2)-C(3)-Br=76.40

Figure 4.14: An alternative transition-state structure (TS 2), energetically more favoured, for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 137

In a similar way, the computational search for the C-alkylation pathway results in one candidate transition state with one imaginary frequency, which is further verified by performing an IRC calculation. The transition-state structure that we postulate for the C-alkylation pathway 14 is shown in Figure 4.15. As suggested by Kornblum et al., the carbon atom C3 from which the bromine atom Br departs is above the plane of the β-naphthoxide ion, which attacks rearward the departing bromine atom, as demanded by the SN2 mechanism. The C-alkylation proceeds via a quasi six-atom transition state and the only difference between the transition state we identify here and the one suggested by Kornblum et al.14 is the sodium cation stabilizing the benzyl ring rather than being close to the bromide ion. The structural parameters for the optimized geometry at the B3LYP/6-31+G(d) level of theory in vacuum of the transition states of the C-alkylation pathway may be found in AppendixB. For the O-alkylation pathway, the two transition state configurations found in vacuum appear also in the liquid phase, with TS 2 being again lower in energy (with an electronic energy value of -3465.59924 a.u.) than TS 1 (with an electronic energy value of -3465.59908 a.u.). Structural changes are observed in the optimized geometries of the transition-state structures of both pathways from the gas phase to the liquid phase. The structural parameters for the optimized structures at the B3LYP/6-31+G(d) level of theory of the transition-state structures of both alkylation pathways in acetonitrile are given in AppendixB. In order to use a quantitative measurement of the conformational change of the transition- state structures observed in vacuum and in the liquid phase for both alkylation pathways, we employ the iterative closest point (ICP) algorithm. The ICP algorithm291–293 aligns two point sets by keeping fixed one of the two (which is called “target”) and transforming the other (which is called “source”) in order to best match the target. The algorithm minimizes iteratively the distance of the coordinates of the source from the coordinates of the target, reporting the root- mean-square error (RMSE) between the aligned source and the target. A more detailed overview of how the algorithm works may be found in the references provided. In our study, we employ the ICP algorithm embedded in MATLAB to align the transition-state structures in vacuum and in acetonitrile for both alkylation pathways. We set as target point set the optimized Cartesian coordinates of the transition-state structures in vacuum and as source point set the optimized Cartesian coordinates of the transition-state structure in the liquid phase (acetonitrile) for both alkylation pathways. For the O-alkylation pathway, we only focus on the TS 2 conformation. In the comparison of the structures obtained in vacuum and in acetonitrile, the algorithm results Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 138

C(2)

C(1) C(3)

165.42o

Dihedral angle: o C(1)-C(2)-C(3)-Br=152.18

Figure 4.15: A transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 139 in RMSE values of the O-alkylation transition-state structure TS 2 of 0.9571 A˚ and of the C- alkylation transition-state structure of 0.1128 A.˚ These structural diﬀerences are expected as the dielectric constant of acetonitrile (ε=35.94) is very diﬀerent from the dielectric constant of vacuum (ε=1.00).

4.6.3 Selectivity and reaction kinetics at 298 K

In this section, the focus is on calculating the Gibbs free-energy difference between the two ◦,QM QM QM transition-state structures (∆GC-O ), the selectivity ratio (kO /kC ) and the rate constants for the two pathways (kQM) of the reaction in acetonitrile at 298 K. For the different levels of theory, the predicted values are listed in Table 4.9 and plotted in Figures 4.16-4.18. The superscript “QM” refers to the use of QM methods to determine these quantities. The corresponding experimental rate-constant values and selectivity ratios obtained in this work at 298 K are also presented in the table and in the figures (with dashed lines) for comparison. For quantitative comparisons between the QM-calculated and experimental quantities, we calculate the absolute percentage error (APE) using the following formulas:

◦,QM ◦,Expt. ∆GC-O − ∆GC-O APE ◦ = × 100, (4.58) ∆GC-O,i ◦,Expt. ∆GC-O

kQM/kQM − kExpt./kExpt. APE = O C O C × 100, (4.59) kO/kC,i Expt. Expt. kO /kC

kQM − kExpt. APE = O O × 100, (4.60) kO,i Expt. kO

kQM − kExpt. APE = C C × 100, (4.61) kC,i Expt. kC

! 1 kQM − kExpt. kQM − kExpt. APE = O O + C C × 100, (4.62) k,i 2 Expt. Expt. kO kC where i is an experiment deﬁned by a speciﬁc combination of temperature and solvent. Fur- thermore, we calculate the mean absolute percentage error of a predicted quantity P (MAPEP) given as N 1 Xe MAPE = APE , (4.63) P N Pi e i=1 ◦ where P is either ∆GC-O, kO/kC or k, and Ne is the number of experiments. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 140

◦,QM The predictions for ∆GC-O in acetonitrile at 298 K for the various levels of theory span a narrow range of 10.5 kJ mol−1, as shown in Figure 4.16. Compared with the experimental value (7.31 kJ mol−1), the QM predictions are generally in very good agreement. The B3LYP functional, independently of the basis set used, slightly overestimates the experimental value by approximately 2.00 kJ mol−1. This energy difference is below chemical accuracy, which is usually taken to be 4.00 kJ mol−1. The wB97xD/6-311++G(d,p) level of theory is the only one ◦,QM that gives a (marginally) negative value for ∆GC-O . For the M05-2X and M06-2X functionals, ◦,QM −1 the computed values for ∆GC-O range from 0.47 to 6.67 kJ mol . Among all levels of theory, the closest prediction to the experimental value is achieved by M05-2X/6-31G(d) with a value of 6.67 kJ mol−1. This finding is consistent with the findings of Marenich et al.,24 who reported with M05-2X/6-31G(d) a mean unsigned error (MUE) for non-aqueous solvation free energies of 2.68 kJ mol−1 on a test set of 2072 neutral solutes. This error value was the lowest among all the other levels of theory used in the parametrisation of the SMD model. Particularly for acetonitrile, solvation free-energy calculations at the M05-2X/6-31G(d) level of theory resulted in a MUE of 2.64 kJ mol−1, based on a sample of 7 neutral solutes. The error value increased to 24.70 kJ mol−1, when a sample of 69 ionic solutes was considered. Despite the large sample of ionic solutes used, the error value is appreciable and gives the reader an indication of the level of accuracy expected in the solvation free-energy calculations with the SMD model when ionic (charged) species are involved. It has been recently shown that these errors may be reduced by adding a small number of explicit solvent molecules and averaging conformationally.120,294,295 For the transmission coefficient κ, which is calculated using the Wigner scheme,207 we obtain values ranging from 1.11 to 1.30 for the O-alkylation, and from 1.08 to 1.28 for the C-alkylation QM QM pathway. The ratio κO /κC lies in the range from 0.99 to 1.04. These values justify our earlier assumption to consider the transmission coefficient of the two pathways to be equal to unity when fitting the Eyring equation to the experimental rate-constant values. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 141

15.0

-1 10.0

5.0 /kJ mol

º | C-O G

∆ 0.0

-5.0 6-31G(d)6-31+G(d)6-311+G(d,p)6-311++G(d,p)6-311G(2d,d,p)cc-pVDZcc-pVTZ 6-31G(d)6-31+G(d)6-31+G(d,p)6-311++G(d,p)cc-pVTZ 6-31+G(d)6-311++G(d,p)cc-pVDZcc-pVTZ 6-31+G(d)6-311++G(d,p)

B3LYP M05-2X M06-2X wB97xD Level of theory QM Expt.

Figure 4.16: Gibbs free-energy diﬀerence between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at 298 K obtained for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 2-reaction model (cf. Table 4.8). Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 142 1 1 − s − s 1 1 − − mol 5 4 5 5 4 6 6 2 1 2 3 2 3 4 5 1 2 4 mol 3 − − − − − − − − − 0 − − − − − − − − − 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 ) and the rate constants of the /dm × × × × × × × × × × × × × × × × × × × /dm QM C QM Expt. /k C C k k 3.313 1.878 3.646 3.577 1.080 2.233 2.029 8.762 8.508 1.636 5.558 1.095 4.943 3.952 2.858 6.105 2.052 1.584 7.194 QM O k 1 1 − s − s 1 1 − − mol 3 3 3 3 4 4 3 1 3 3 4 1 2 3 1 2 mol 3 − − − − − − 0 0 0 − − − − − − − − − − 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 /dm × × × × × × × × × × × × × × × × × × × /dm QM Expt. O O k k ), the selectivity ratio ( QM , ◦ C-O QM Expt. C C 080477 9.970 11 1.241 97 1.137 58 3.575 41 1.540 20 1.128 89 1.350 3.571 98 3.096 2426 7.641 56 1.108 93 5.000 33 1.304 01 1.790 4.785 1.594 83 1.750 97 1.094 04 1.298 G ...... /k /k QM O k Expt. O k 1 1 − − 7967516246 53 88 34 67 31 45 33 48 68 55 71 15 9347 4 68 1 5711 6 06 2 1 4 2 2 1 80 52 57 1 31 18 ...... /kJ mol /kJ mol 0 O − O QM − , Expt. − , ◦ C ◦ C G G ∆ ) of the reaction in acetonitrile at 298 K at all the levels of theory tested. The corresponding experimental QM k B3LYP/6-31+G(d)B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p)B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 8 M05-2X/6-31G(d) 9 8 M05-2X/6-31+G(d) 8 M05-2X/6-31+G(d,p)M05-2X/cc-pVTZ 10 M06-2X/6-31+G(d) 9 M06-2X/6-311++G(d,p) 6 3 M06-2X/cc-pVDZ 1 M06-2X/cc-pVTZwB97xD/6-31+G(d)wB97xD/6-311++G(d,p) 0 4 1 3 2 2 Level of theoryB3LYP/6-31G(d) ∆ 9 M05-2X/6-311++G(d,p) 1 Expt. 7 O- and C-alkylation pathways ( values obtained in this work are also reported for comparison. Table 4.9: QM-calculated values for the Gibbs free-energy diﬀerence (∆ Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 143

QM QM The calculated values for kO /kC (using Equation (4.56)) are shown in Figure 4.17. The purpose of the black line shown in the figure at a value of 1.0, which is translated to a percentage ratio of 50:50, is to facilitate the distinction between the QM methods that predict a preferred route of the reaction towards the O-alkylated product (values above the line) or towards the C- alkylated product (values below the line). The fact that in acetonitrile all levels of theory predict the established selectivity preference towards the O-alkylated product at 298 K (based on ample experimental evidence of this and previous works14,23) is noteworthy. The experimental value obtained in this work for the selectivity ratio is 18.04, which is translated to a percentage ratio of 95:5. For the various basis sets combined with the B3LYP functional, we obtain selectivity ratio values from 31.77 to 68.96, which correspond to percentage selectivity ratio values from 97:3 to 99:1, respectively, favouring the O-alkylated product. The percentage selectivity ratio values computed with the B3LYP functional with all basis sets agree very well with the experimental ◦,QM value. As expected from the value found for ∆GC-O , the M05-2X/6-31G(d) level of theory gives the closest agreement to experimental selectivity ratio with a value of 15.41, corresponding to a percentage value of 94:6. Using Equation (4.20), we calculate kQM of the reaction in acetonitrile at 298 K for each of the levels of theory tested in this work and we plot the results in Figure 4.18. A range of values is obtained for the various combinations of electronic structure methods and basis sets. While the gas-phase geometries ought to be more accurate with larger basis sets, the liquid-phase energies may be more accurate with the basis sets used in parametrization. For example, among the different basis sets tested with the B3LYP functional, one would expect better performance with basis sets for which SMD has been parametrized, e.g., with B3LYP and the small basis set 6-31G(d). Nevertheless, the inclusion of a diffuse function (i.e., the + in the basis set) significantly improves the accuracy of the prediction for the absolute rate-constant values. This improvement could be explained due to the presence of bromide anion, which is large and whose weakly bound electron requires more flexibility;47 as a result, the addition of a diffuse function may prove beneficial. On the other hand, a more extensive splitting of the valence shell (number of 1’s after the 3) reduces the accuracy of the prediction. When combined with the Dunning’s correlation-consistent basis sets240,241 (cc-pVDZ and cc-pVTZ), B3LYP predicts considerably lower values for the reaction rate constants of both pathways. Regarding the effect of the basis set on the absolute rate-constant values, the same increasing or decreasing pattern is observed for the other functionals tested in this work, namely M05-2X, M06-2X and wB97xD. However, because different functionals overestimate or underestimate Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 144

100.0

10.0 C k / O k 1.0

0.1 6-31G(d)6-31+G(d)6-311+G(d,p)6-311++G(d,p)6-311G(2d,d,p)cc-pVDZcc-pVTZ 6-31G(d)6-31+G(d)6-31+G(d,p)6-311++G(d,p)cc-pVTZ 6-31+G(d)6-311++G(d,p)cc-pVDZcc-pVTZ 6-31+G(d)6-311++G(d,p)

B3LYP M05-2X M06-2X wB97xD Level of theory QM Expt.

Figure 4.17: Selectivity ratio for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at 298 K calculated using Equation (4.56) for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 2-reaction model (cf. Table 4.4). Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 145

101 100 -1 -1 s 10 -1 10-2

mol -3

3 10 10-4 /dm k 10-5 -6 10 6-31G(d)6-31+G(d)6-311+G(d,p)6-311++G(d,p)6-311G(2d,d,p)cc-pVDZcc-pVTZ 6-31G(d)6-31+G(d)6-31+G(d,p)6-311++G(d,p)cc-pVTZ 6-31+G(d)6-311++G(d,p)cc-pVDZcc-pVTZ 6-31+G(d)6-311++G(d,p)

B3LYP M05-2X M06-2X wB97xD Level of theory

O-alkylation QM O-alkylation Expt. C-alkylation QM C-alkylation Expt.

Figure 4.18: Rate constants for the O- and C-alkylation pathways of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at 298 K calculated using the CTST theory (Equation (4.20)) for various levels of theory and the SMD solvation model implemented in Gaussian 09. The columns of the same colour correspond to the same QM method. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The dashed lines indicate the experimental values at 298 K from the 2-reaction model (cf. Table 4.4). Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 146 the absolute rate-constant values, the agreement is improved or reduced inconsistently. For example, inclusion of a diﬀuse function on the 6-31G(d) basis set for M05-2X, increases the rate-constant values (as shown before for B3LYP), but since the M05-2X with 6-31G(d) overestimates the computed values, this increase leads to a worse agreement. Hence, the importance of selecting an appropriate functional for accurate absolute kinetic predictions is highlighted here. Overall, the closest agreement with experimental rate-constant values for both alkylation pathways is achieved by M05-2X/cc-pVTZ, with an overall APEk value of 46.67%. This level of theory matches best the experimental rate-constant value for the C-alkylation pathway (with

APEkC =52.22%) and it is the third best option for matching the rate-constant value of the

O-alkylated pathway (with APEkO =41.13%). It is worth noting that, for implicit solvation models, greater predictive accuracy is expected when the level of theory used for predictions is consistent with the level of theory used for parametrizing the solvation model against experimental solvation energies at a certain temperature (usually 298 K).261 In particular for the SMD model, for which the set of parameters that the model uses (intrinsic atomic Coulomb radii and atomic surface tension coefficients) is obtained as an average fitting of six levels of theory, an average performance is expected across these levels of theory. Moreover, variations might occur depending on whether relative or absolute values are investigated. When relative values are considered (as for example in this study, the free energies of the two transition states for the calculation of the selectivity ratio), the errors arising from empirical non-electrostatic terms, or from the calculation of bulk electrostatics due to the assumption of the solvent as continuum, are likely to cancel out and improve the accuracy of the predictions. On the contrary, when absolute values are sought (as for example in this study, the free energies of solvation for the calculation of the rate constant for each of the two pathways), the predictions are subject to limitations due to the parametrization of the empirical GCDS term with a certain amount of solvation energy data. In this case, the inadequacy of the empirical term GCDS to capture the short-range non-electrostatic effects may be compensated by the increasingly accurate prediction of the bulk electrostatic interactions (which is again subject to the formalism and parametrization of the functional296), which is more likely to be achieved for a small system with the use of sophisticated methods for including electron correlation and extensive basis set.58 This is the reason why different levels of theory are identified as good options for describing either relative energy differences or absolute energies (in our study, the selectivity ratio or the kinetics of the reaction). For example, M05-2X/6-31G(d), which has been identified in acetonitrile as the best level of theory for capturing the selectivity ratio of the Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 147 reaction at 298 K, fails to capture absolute rate-constant values. There are, of course, levels of theory that perform reasonably well for both reaction metrics, such as B3LYP/6-31+G(d) and M05-2X/cc-pVTZ. Interestingly, the latter is one of the levels of theory that has been used to parametrize SMD and the former is very similar to B3LYP/6-31G(d), which has also been used to parametrize SMD. For the best computational options identified here for capturing the selectivity (M05-2X/6- 31G(d)) and the kinetics (M05-2X/cc-pVTZ) of the reaction in acetonitrile, we report the absolute percentage error (APE) values in Table 4.10. The obtained APE values emphasize the fact that our choice of level of theory depends on the reaction metrics we aim to predict, the level of accuracy we want to achieve and the computational resources we have available.

Table 4.10: Absolute percentage error (APE) for the QM-calculated values for the Gibbs free- ◦,QM QM QM energy diﬀerence (∆GC-O ), the selectivity ratio (kO /kC ) and the rate constants of the two alkylation pathways (kQM) of the reaction in acetonitrile at 298 K at the M05-2X/6-31G(d) and M05-2X/cc-pVTZ levels of theory.

M05-2X/6-31G(d) M05-2X/cc-pVTZ

APE ◦ (%) APE (%) APE (%) ∆GC-O kO/kC k 8.74 14.60 46.67

4.6.4 Thermodynamic analysis based on QM-calculated data

Using the level of theory identified previously as the best option for capturing the kinetics (M05-2X/cc-pVTZ) of the Williamson reaction in acetonitrile, we perform additional QM calculations at temperatures 313 and 323 K for which we have available experimental data. The QM-calculated rate constants, obtained using Equation (4.20), are listed in Table 4.11. Similarly to the analysis we followed for the experimental rate constants, we plot the QM-calculated rate- constant values in terms of the Arrhenius (Equation (3.10)) and the Eyring (Equation (4.52)) equations, as shown in Figures 4.20 and 4.19. In the previous section, the experimentally- obtained rate constants are fitted to the Eyring equation (cf. Equation (4.52)) with the assumption of a unitary transmission coefficient. In order to be able to compare with these values, we also assume a unitary transmission coefficient for the Eyring equation when fitted to the QM- calculated rate-constant values. The quality of fit (R2) and the linear equation derived for each pathway are given in the graphs. The experimental values are also plotted in the figures (black Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 148 symbols) for comparison. The obtained parameters of the Arrhenius and the Eyring equations are listed, respectively, in Tables 4.13 and 4.14.

Table 4.11: QM-calculated rate constants for the O- and the C-alkylation pathways (kQM) of the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at the M05- 2X/cc-pVTZ level of theory using the CTST theory and the SMD model (Equation (4.20)) at Expt. Expt. temperatures 298, 313 and 323 K. The percentage rate constant ratio of kO :kC is also reported.

QM 3 −1 −1 QM 3 −1 −1 Expt. Expt. T/K kO /dm mol s kC /dm mol s kO :kC (%) 298 7.641×10−3 1.095×10−3 87:13 313 1.769×10−2 2.468×10−3 88:12 323 2.993×10−2 4.107×10−3 88:12

The performance of the CTST theory and the SMD model with M05-2X/cc-pVTZ may be appreciated by inspecting Figures 4.19 and 4.20, where the QM-calculated rate constants are in good agreement with the experimental values at the temperature range between 298 and 323 K. In particular, for the C-alkylation pathway the computed rate constants are within the 95% conﬁdence intervals of the experimental values, whereas for the O-alkylation pathway the computed rate constants are systematically below the lower bound of the experimental values for all the three temperatures investigated (cf. Table 4.4). For the three temperatures that calculations and experiments are performed with M05-2X/cc-pVTZ, the MAPEk value is

47.98%. As shown in Table 4.12, the lowest APEk value is obtained at temperature 323 K, which is the highest temperature studied. This result is unorthodox given that the SMD model has been parametrized at 298 K and therefore, we would expect that extrapolation to higher (or lower) temperatures would result in signiﬁcant errors (as it is observed for example at 313 K).

Table 4.12: Absolute percentage error (APE) and mean absolute percentage error (MAPE) values for the QM-calculated rate constants of the two pathways of the reaction (kQM) at the M05-2X/cc-pVTZ level of theory compared with the experimental values in acetonitrile at temperatures 298, 313 and 323 K.

T/K MAPEk (%) 298-328 47.98

T/K APEk (%) 298 46.67 313 54.95 323 42.31 Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 149

0 y=-5252.8x + 12.751 O-alkylation QM R2=1 C-alkylation QM -2 O-alkylation Expt. C-alkylation Expt. -4

-6 k -8 y=-5085.6x + 10.248 2 ln R =1 -10

-12

-14

-16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 4.19: Arrhenius plot of the logarithm of the QM-calculated reaction rate constants at the M05-2X/cc-pVTZ level of theory as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K. The experimental rate constants, illustrated by black symbols, are shown for comparison. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 150

0 O-alkylation QM C-alkylation QM -2 O-alkylation Expt. C-alkylation Expt. -4

-6 y=-4943.1x + 6.015 R2=1 -8 (k/T)

ln -10

-12

-14 y=-4775.9x + 3.511 R2=1 -16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 4.20: Eyring plot of the logarithm of the QM-calculated reaction rate constants at the M05-2X/cc-pVTZ level of theory over temperature as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K. The experimental rate constants, illustrated by black symbols, are shown for comparison. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 151

By comparison of Tables 4.13 and 4.7, the set of parameters (A1 and Eã) obtained when fitting the Arrhenius equation to QM-calculated and to experimental rate constants are distinctly different. On the contrary, the values of parameter ∆‡G◦(298) obtained when fitted to QM- calculated (cf. Table 4.14) and to experimental rate constants (cf. Table 4.8) are in excellent agreement, with a variation of no more than 1.50 kJ mol−1. Slightly larger variations, with a maximum value of 15.00 kJ mol−1, are observed between the individual components (∆‡H◦ and ∆‡S◦).

Table 4.13: Estimated parameters obtained by ﬁtting the Arrhenius equation (Equation (3.10)) to QM-calculated rate constants at the M05-2X/cc-pVTZ level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K.

QM 3 −1 −1 ˜QM −1 A1 /dm mol s Ea /kJ mol O-alkylation 3.45×105 43.67 C-alkylation 2.82×104 42.28

Table 4.14: Estimated thermodynamic parameters obtained by ﬁtting the Eyring equation (Equation (4.2)) to QM-calculated rate constants at the M05-2X/cc-pVTZ level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in acetonitrile at temperatures 298, 313 and 323 K.

∆‡H◦,QM/kJ mol−1 ∆‡S◦,QM/J K−1 mol−1 ∆‡G◦,QM(298)/kJ mol−1 O-alkylation 41.10 −147.54 85.07 C-alkylation 39.71 −168.38 89.88 ◦,QM −1 ◦,QM −1 −1 ◦,QM −1 ∆HC−O /kJ mol ∆SC−O /J K mol ∆GC−O (298)/kJ mol Diﬀerence C-O −1.39 −20.82 4.81

Although SMD is parametrized only at 298 K, in our study we use it to predict solvation free energies at higher temperatures (313 and 323 K). Considering the reasonably good agreement achieved with experimental data, extrapolation at higher temperatures does not seem to dramatically impact the rate-constant predictions. This may be attributed to two reasons: first, the narrow temperature range investigated, which allows for minimum error in the SMD predictions for the solvation energies; and second, the use of a computationally efficient level of theory (M05-2X/cc-pVTZ), whose accurate calculations for the bulk electrostatics counterbalance the deficiencies of the empirically parametrized GCDS term of the SMD model. Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 152

4.7 Summary

In this chapter, we focused on the prediction of solvent and temperature effects on the selectivity and kinetics of a liquid-phase reaction. First, we presented a mathematical expression for calculating the liquid-phase rate constant of a bimolecular reaction according to conventional transition state theory (CTST). The effect of solvent was considered by using the continuum solvation SMD model, in which the solvent is described as a polarizable continuum medium characterized by seven solvent descriptors. We presented the mathematical formalism of the SMD model, which then we introduced to the CTST expression. In addition, we showed that, at a given temperature, the determining factor for the selectivity preference of a reaction with two possible pathways (I and II) is the free-energy difference between the two transition states ◦ leading to the two final products (∆GII−I(T )). In order to implement the proposed methodology, we selected a Williamson reaction between sodium β-naphthoxide and benzyl bromide, for which significant solvent effects on selectivity are reported in the literature. The choice of solvent significantly impacts the selectivity of this reaction for alkylation at either oxygen or carbon sites of sodium β-naphthoxide leading to the O- or C-alkylated products, and determines the final rate constant ratio. In the first part of our study, we monitored the kinetics of the reaction in acetonitrile-d3 at temperatures 298, 313 and 323 K using in situ 1H NMR spectrometry. In the preliminary experimental investigation of the reaction, we identified an underlying phenomenon of a proton-exchange taking place between one of the reactants (sodium β-naphthoxide) and one of the products (C-alkylated product). As a result, an equilibrium between protonated and deprotonated species of these two molecules was established. We developed and implemented in gPROMS a 2-reaction model, which captures the two main pathways of the Williamson reaction. Based on this model, we obtained the reaction rate constants for the O- and the C-alkylation pathways at 298, 313 and 323 K. The percentage Expt. Expt. rate constant ratio of kO :kC at 298 K has a value of 95:5; this ratio value is in excellent agreement with the product ratio of 94:6 found in a previous experimental study.271 The level of accuracy in which the model parameters were specified is reasonable, based on the estimated values of the 95% confidence intervals. We also developed and implemented in gPROMS a second model (the proton-exchange model) which captures the proton-exchange equilibrium; however, even though the obtained kinetic parameters for this model at 298 K were similar to the ones obtained from the 2-reaction model, they suffered from large uncertainty (shown by the wide Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 153

95% confidence intervals); thus, this model was not further investigated. We analysed the reaction rate constants at 298, 313 and 323 K in terms of the Arrhenius and Eyring equations. Due to the narrow temperature range studied, straight lines for the dependence of the logarithmic values of the rate constants on the inverse temperature were obtained in both cases. The large values obtained for the activation energy parameter (Eã) for both pathways (59.52 kJ mol−1 for the O-alkylation and 48.99 kJ mol−1 for the C-alkylation) suggested significant activation barriers for the reactants to overcome. By fitting to experimental rate con- −1 ◦,Expt. stants in acetonitrile-d3 at 298 K, we obtained a value of 7.31 kJ mol for parameter ∆GC−O , which shows the selectivity preference of the reaction towards the O-alkylation pathway. In the second part of our study, we investigated the Williamson reaction from a computational perspective. We performed a thorough computational analysis using various DFT functionals and basis sets (levels of theory) in 18 different combinations. As this reaction has not been studied computationally before, we performed geometry and frequency calculations, both in vacuum and in acetonitrile, in order to obtain optimized structures for the reactants and the transition states of the two pathways. For illustrative purposes, we reported all the computational results at the B3LYP/6-31+G(d) level of theory. Our QM findings suggested that sodium β-naphthoxide reacts in the form of an ion-pair, where the minimum energy conformation was found. Two different transition-state structures were identified for the O-alkylation pathway (of which the one with the lowest energy was selected for further use) and one for the C-alkylation pathway. Subsequently, we applied the expression of the CTST theory with the SMD model ◦,QM QM QM QM to predict the selectivity (∆GC−O and kO /kC ) and the kinetics (k ) of the reaction in acetonitrile-d3 at 298 K with all levels of theory tested. We discussed the performance of each combination of DFT functional with the various basis sets. The role of the functional was recognized as the most important aspect in these calculations. The parametrization of the SMD model in the performance of the various levels of theory is also of key importance. In comparison with experiments at 298 K, all the levels of theory tested predicted successfully the selectivity preference towards the O-alkylated product. M05-2X/6-31G(d) achieved the best performance −1 ◦,QM based on selectivity criteria at 298 K, with a value of 6.67 kJ mol for ∆G (APE ◦ C−O ∆GC-O QM QM of 8.74%) and a value of 15.41 for the kO /kC (APEkO/kC of 14.60%). M05-2X/cc-pVTZ achieved the best performance based on kinetics criteria, capturing the rate constants of the two QM alkylation pathways k with a APEk value of 46.67%. We extended our QM calculations to higher temperatures (313 and 323 K) using the M05- 2X/cc-pVTZ level of theory, for which we calculated rate constants in reasonable agreement Chapter 4. Kinetic investigation of a Williamson reaction in acetonitrile 154 with the corresponding experimental values. For the three temperatures investigated (298,

313 and 323 K), we obtained a MAPEk value of 47.98%. The temperature-dependent data were subsequently analysed in terms of the Arrhenius and Eyring equation. Excellent agreement (with a difference of less than 1.50 kJ mol−1) was achieved for the individual values of ∆‡G◦,QM(298) for the two alkylation pathways when fitted to experimental values and to QM-calculated values. Based on these deviations, we deduced that, in this case, extending the use of the SMD solvation model to temperatures higher than the parametrization temperature of the empirical term GCDS does not reduce significantly the accuracy of the calculations. Overall, very encouraging results were obtained in this chapter for the effects of the solvent medium and temperature by investigating experimentally and computationally a relatively simple reaction, the Williamson reaction between sodium β-naphthoxide and benzyl bromide, in a polar aprotic solvent. These results create incentive to further investigate the reaction focusing on a different solvent medium, such as a polar protic solvent. This will be the central subject of study of Chapter 5. Chapter 5

Kinetic investigation of a Williamson reaction in methanol

In this chapter, we further investigate the Williamson reaction between sodium β-naphthoxide and benzyl bromide (introduced in Chapter4) in solvent methanol and at various temperatures. We experimentally monitor the reaction in methanol and we present the key ﬁndings of this study. We proceed by investigating the kinetics and selectivity of the reaction computationally by performing quantum-mechanical DFT calculations. The solvation model SMD is also implemented here to account for the presence of solvent. We use the computational results to make comparisons with the experimental data and to draw conclusions regarding the predictive capabilities of the computational methodology applied. Before concluding this chapter, we assess the performance of the various levels of theory for predicting accurately selectivity and kinetics metrics for the two solvents in which the reaction of interest has been studied (acetonitrile and methanol). The ﬁndings of this analysis will be used later in Chapter6.

5.1 Experimental methodology

5.1.1 Monitoring technique

For reasons explained in the previous chapter, we select in situ 1H NMR spectroscopy in order to monitor the kinetics of the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4. The same instrument (500 MHz Bruker CryoProbe Prodigy spectrometer) is used to perform the kinetic experiments. As an internal standard, we use tetramethylsilane (TMS), which is the default primary reference component included in methanol-d4.

155 Chapter 5. Kinetic investigation of a Williamson reaction in methanol 156

5.1.2 Reaction of benzyl bromide with methanol-d4

A literature survey evinces that benzyl bromide reacts with methanol yielding benzyl methyl ether.297 Although in the study of Ganase23 reactivity tests were performed and no such reaction was identiﬁed, in our work an experiment is designed to conﬁrm the existence of such a reaction. An analysis is performed using heteronuclear single quantum correlation (HSQC) NMR spectroscopy.298 An 1H–13C HSQC experiment is a highly sensitive two-dimensional (2D) experiment, which provides information regarding the correlation of the protons with the carbon atoms they are directly linked to.281 The spectrum obtained from an 1H–13C HSQC experiment constitutes a map of correlations, in which each peak corresponds to a proton-carbon coupling of attached nuclei;281 therefore, structural assignment may be considerably facilitated.

We prepare one sample of benzyl bromide at a concentration of 0.09 M in methanol-d4, which we then analyse through a 1H–13C HSQC experiment. The chemical shift of the peak 1 appearing in the H NMR spectrum at 4.55 ppm is assigned to the aliphatic −CH2 group of benzyl bromide, with a corresponding chemical shift for the aliphatic carbon appearing in the 13C NMR spectrum at 32.63 ppm. A second peak appears along the proton axis, close to the 1 peak of the aliphatic −CH2 group of benzyl bromide, with a chemical shift in the H NMR spectrum at 4.44 ppm and a corresponding peak at the carbon axis, with a chemical shift in the 13C NMR spectrum at 74.18 ppm. Based on literature references,299,300 the proton and carbon chemical shifts of the second peak are assigned to the aliphatic −CH2 group of benzyl methyl ether. In order to further validate our ﬁndings, we also analyse the initial sample of benzyl bromide in methanol-d4 by performing a 2D heteronuclear multiple bond correlation (HMBC) experiment. This experiment captures the correlations between carbon and proton atoms over longer distances, while suppresses the direct-bond correlations.281 For our sample, the HMBC experiment results in a cross-peak with a chemical shift in the 1H NMR spectrum at 4.44 ppm and a chemical shift in the 13C NMR spectrum at 56.09 ppm. The carbon chemical shift value of the cross-peak is consistent with the literature value for the carbon chemical shift of the −CD3 group of benzyl methyl ether.300 As a result, the observed cross-peak illustrates the existence of correlation between the protons of the aliphatic −CH2 group with the carbon from the −CD3 group. (The use of deuterated solvent (methanol-d4) in our experiment, results in a deuterated 1 −CD3 group which does not give a proton chemical shift in the H NMR spectrum.) Chapter 5. Kinetic investigation of a Williamson reaction in methanol 157

The results of the HSQC and HMBC experiments (summarized in Tables 5.1 and 5.2) con-

ﬁrm the existence of the side-reaction between benzyl bromide and methanol-d4, as shown in Figure 5.1. The side-reaction is taken into account when analysing the experimental kinetic data by developing a kinetic model that captures both the kinetics of the side-reaction and the kinetics of the two pathways of the reaction between sodium β-naphthoxide and benzyl bromide.

1 13 Table 5.1: H NMR and C NMR chemical shifts for the aliphatic −CH2 group in benzyl bromide and in benzyl methyl ether resulted from a 2D HSQC experiment.

Chemical shift Component 1H NMR/ppm 13C NMR/ppm Benzyl bromide 4.55 32.36 Benzyl methyl ether 4.44 74.18

1 Table 5.2: Heteronuclear chemical shift correlation between the aliphatic −CH2 group ( H NMR) 13 and the −CD3 group ( C NMR) in benzyl methyl ether obtained from a 2D HMBC experiment.

Chemical shift Component 1H NMR/ppm 13C NMR/ppm Benzyl methyl ether 4.44 56.09

CD3 Br O D3C OH HBr

Figure 5.1: The reaction of benzyl bromide and methanol-d4, resulting in the formation of benzyl methyl ether.

5.1.3 Reaction for the formation of the double C-alkylated product

In the study of Kornblum et al.,14 it was mentioned that, additionally to the two main O- and C-alkylated products that occur from the reaction between sodium β-naphthoxide and benzyl bromide, a third by-product occurs as well. Further alkylation of the deprotonated form of the C-alkylated product, which reacts with benzyl bromide, leads to the formation of 1,1-dibenzyl-2- (1H)-naphthalenone (double C-alkylated product), as shown in Figure 5.2. The protonated form of the C-alkylated product is found to be incapable of further alkylation. This is conﬁrmed by performing an experiment where the C-alkylated product with benzyl bromide are left within a vial for 24 hours (in the absence of any deprotonating agent, e.g., a base). No sign of the double Chapter 5. Kinetic investigation of a Williamson reaction in methanol 158

C-alkylated product is detected in the 1H NMR spectrum obtained after this period of time. This ﬁnding is also consistent with the fact that naphthol, which is the protonated form of β- naphthoxide, is also incapable of reacting with benzyl bromide; the two species (naphthol and the C-alkylated product) have very similar chemical structure, and thus similar chemical reactivity. Depending on the solvent used, which impacts the amount of C-alkylated product yielded, the amount of double C-alkylated product may be signiﬁcant. Kornblum et al.14 studied the reaction between sodium β-naphthoxide and benzyl bromide in ethanol, for which they reported yields of the C-alkylated product of 24% and of the double C-alkylated product of 4%; the latter amount was then included in the yield of the C-alkylated product. The amount of double C-alkylated product obtained in ethanol is appreciable and is expected to be even higher when the reaction takes place in methanol given that a higher amount of the C-alkylated product is also expected in this solvent. For these reasons, the additional alkylation reaction leading to the double C- alkylated product is considered in the development of an overall kinetic model for the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4.

O Na O Br NaBr

Figure 5.2: The reaction of benzyl bromide with the C-alkylated product (deprotonated), resulting in the formation of 1,1-dibenzyl-2-(1H)-naphthalenone (double C-alkylated product).

5.1.4 Methodology for kinetic experiments

We follow the same experimental methodology developed for acetonitrile-d3 in the previous chapter (cf. Section 4.3.3) to study the kinetics of the Williamson reaction between sodium

β-naphthoxide and benzyl bromide in methanol-d4. The speciﬁc details of the experiments performed in methanol-d4 are as follows:

1. First, we select three diﬀerent temperatures (298, 313 and 328 K) to perform kinetic

experiments in methanol-d4. The highest temperature is selected to be lower than the boiling point of methanol (337.8 K).30

2. Second, we determine the initial concentration of the reactants. We use a 1:1 ratio for the two reactants and we select relatively low values for their initial concentrations (0.09 M) in Chapter 5. Kinetic investigation of a Williamson reaction in methanol 159

order to avoid problems with solubility. Prior to each experiment, we weigh the amounts of the two reactants (cf. Table 5.3) using an electronic scale. We then dissolve sodium

β-naphthoxide into methanol-d4 (in which the internal standard is already included) in a vial and transfer the content of the vial to an NMR tube with a syringe. Subsequently, we add benzyl bromide into the NMR tube to initiate the reaction, and we transfer the tube into the NMR instrument which is already tuned at a selected temperature.

3. After the insertion of the sample into the NMR instrument, the first 1H NMR spectrum is obtained in approximately 3 minutes, which is the time needed for the probe to be tuned and for the sample to be shimmed. After this time period, we collect data every six minutes for the first one hour and every 15-20 minutes until the end-time of monitoring the experiment. We monitor the experiment at 298 K for approximately four hours, the experiment at 313 K for approximately five hours and the experiment at 328 K for approximately two hours.

4. The raw peak values are integrated and the known concentration of the internal standard is used to convert the peak areas into concentration units. The peaks that are monitored are the following:

(a) For TMS, an aliphatic peak (−CH3)4 at 0.00 ppm. (b) For sodium β-naphthoxide, an aromatic −CH peak at ∼ 6.86-7.05 ppm (where possible).

(d) For the O-alkylated product, an aliphatic −CH2 peak at ∼ 5.14-5.22 ppm.

(e) For the C-alkylated product, an aliphatic −CH2 peak at ∼ 4.40-4.44 ppm (where possible).

(f) For the double C-alkylated product, a benzyl (−CH2)2 peak at ∼ 6.61-6.67 ppm.

(g) For benzyl methyl ether, an aliphatic −CH2 peak (where possible) at ∼ 4.41-4.44 ppm.

The reaction in methanol-d4 is slower than the reaction in acetonitrile-d3; therefore, mon-

itoring the consumption of sodium β-naphthoxide in methanol-d4 is feasible for a longer period, and thus a larger number of spectra is obtained. For each experiment at a diﬀerent temperature, the number of spectra integrated for sodium β-naphthoxide is 20 spectra at Chapter 5. Kinetic investigation of a Williamson reaction in methanol 160

298 K (4 hours), 16 spectra at 313 K (2 hours and 20 minutes) and 9 spectra at 328 K (47 minutes).

The aliphatic −CH2 peaks of the C-alkylated product and benzyl methyl ether appear at neighbouring chemical shifts in the 1H NMR spectrum and in some spectra the two peaks are entirely merged. We only integrate the spectra in which the two peaks are distinguishable; therefore, the number of integrated spectra is 10 spectra at 298 K, 16 spectra at 313 K and 9 spectra at 328 K. Although we distinguish the two peaks at various time moments, especially at 313 K, they always remain very close and hence, the integrated peak area value of each one of them involves some amount of uncertainty.

5. The 2-reaction model, developed in Chapter 4, does not provide an accurate representa-

tion of the reaction system in methanol-d4, as it neglects the side-reaction between benzyl

bromide and methanol-d4 and the reaction for the formation of the double C-alkylated product. In addition, when the reaction between sodium β-naphthoxide and benzyl bro-

mide takes place in methanol-d4, the amount of C-alkylated product produced is considerable. In this case, the equilibrium established in Chapter 4 (cf. Section 4.3.3) between the protonated and deprotonated forms of sodium β-naphthoxide and the C-alkylated product should not be neglected, as it has a considerable impact on the reaction system. There- fore, we develop a new model, which is referred to as “4-reaction proton-exchange model”, which constitutes a more accurate representation of the kinetic and thermodynamic phenomena occurring in the reaction system. An overall scheme of the reaction system when

the reaction takes place in methanol-d4 may be found in AppendixC.

In the 4-reaction proton-exchange model, additionally to the kinetics of the two reaction pathways between sodium β-naphthoxide and benzyl bromide (as shown in Figure 4.1), the

kinetics of the side-reaction of benzyl bromide with methanol-d4 (as shown in Figure 5.1), the double alkylation reaction (as shown in Figure 5.2) and the equilibrium between the protonated and deprotonated forms of sodium β-naphthoxide and the C-alkylated product (as shown in Figure 4.4) are also considered. The reactions that constitute the 4-reaction proton-exchange model may be written, in a concise form, as

k 1 + 2 −→O 3 (5.1)

k 1 + 2 −→C 4 (5.2) Chapter 5. Kinetic investigation of a Williamson reaction in methanol 161

k 1 + 4 −→f 5 + 6 (5.3)

5 + 6 −→kr 1 + 4 (5.4)

k 2 + 6 −−→DC 7 (5.5)

k 2 + 8 −−−→BME 9 (5.6)

where 1 corresponds to sodium β-naphthoxide, 2 to benzyl bromide, 3 to the O-alkylated product, 4 to the C-alkylated product, 5 to β-naphthol, 6 to the deprotonated form of

the C-alkylated product, 7 corresponds to the double C-alkylated product, kDC is the rate constant of the reaction leading to the formation of the double C-alkylated product, 8 to

methanol-d4 (solvent), 9 to benzyl methyl ether, kO is the rate constant of the reaction

leading to the formation of the O-alkylated product, kC is the rate constant of the re-

action leading to the formation of the C-alkylated product, kBME is the rate constant of

the reaction leading to the formation of benzyl methyl ether, kf is the rate constant for the forward reaction towards β-naphthol and the deprotonated form of the C-alkylated

product, kr is the rate constant for the reverse reaction towards sodium β-naphthoxide

and the C-alkylated product, Keq is the equilibrium constant.

In addition to the rates speciﬁed in the 2-reaction model (cf. Section 4.3.3), two extra

equations are introduced for the rate rDC of the reaction between benzyl bromide and the deprotonated form of the C-alkylated product, as shown in Equation (5.5) and the rate

rBME of the reaction between benzyl bromide and methanol-d4, as shown in Equation (5.6), which may be written as

rDC = kDC · [2] · [4] (5.7)

rBME = kBME · [2] · [8], (5.8)

where the numbers placed inside square brackets denote molar concentrations.282 Assum- ing a constant volume, perfect mixing and isothermal conditions inside the NMR tube,282 the equations of the 4-reaction proton-exchange model for the overall rate of the reaction may be written, in terms of the change in concentration for each component, as

d[1] = −(k + k ) · [1] · [2] − k · [1] · [4] + k · [5] · [6] (5.9) dt O C f r

d[2] = −(k + k ) · [1] · [2] − k · [2] · [8] − k · [2] · [6] (5.10) dt O C BME DC Chapter 5. Kinetic investigation of a Williamson reaction in methanol 162

d[3] = k · [1] · [2] (5.11) dt O d[4] = k · [1] · [2] − k · [1] · [4] + k · [5] · [6] (5.12) dt C f r d[5] = k · [1] · [4] − k · [5] · [6] (5.13) dt f r d[7] = k · [2] · [6] (5.14) dt DC d[8] = −k · [2] · [8] (5.15) dt BME d[9] = k · [2] · [8] (5.16) dt BME

kf Keq = (5.17) kr [5] = [6] + [7] (5.18)

[Sum1] = [1] + [5] (5.19)

[Sum2] = [4] + [6] (5.20)

[1]t=0 = [1]0, [2]t=0 = [2]0, [3]t=0 = 0, [4]t=0 = 0, [5]t=0 = 0, (5.21) [7]t=0 = 0, [8]t=0 = [8]0, [9]t=0 = 0

[Sum1] (referred here as “total naphthol”) is the sum of the concentrations of the proto-

nated ([5]) and deprotonated ([1]) forms of sodium β-naphthoxide, [Sum2] (referred here as “total C-alkylated product”) is the sum of concentrations of the protonated ([4]) and

deprotonated ([6]) forms of the C-alkylated product and [i]t=0 is the initial concentration of component i =1, 2, 3, 4, 5, 6, 7, 8 and 9, as speciﬁed in Equations (5.1)-(5.6) at time zero.

The concentration of methanol-d4 at time zero, [8]0, may be assumed to be roughly equal

to the molar density of methanol-d4 at the speciﬁc temperature of the experiment. As this

value is signiﬁcantly larger than the value of the rate constant kBME and it is practically

invariant throughout an experiment, we may reasonably assume that the term kBME · [8]

remains constant for an experiment at a given temperature. The density of methanol-d4 at 298 K has a value of 888.11 Kg m−3,301 which results in a molar density of 24.62 mol dm−3. However, since no temperature-dependent data are reported in the literature for

the density of methanol-d4 at temperatures 313 K and 328 K, we report the value of the

multiplication of kBME and [8]. As the 4-reaction proton-exchange model is posed, the ratio of rate constants of the O-alkylated product over the C-alkylated product is equal Chapter 5. Kinetic investigation of a Williamson reaction in methanol 163

to the ratio of concentrations of the O-alkylated product over the sum of the C-alkylated product and the double C-alkylated product, given as

k [3] O = , (5.22) kC [4] + [7]

which may be obtained by integrating Equations (5.11), (5.12) and (5.14) and assuming a zero value for the initial concentrations.

The 4-reaction proton-exchange model is implemented in gPROMS.283 We use the concentration data obtained from integration of the 1H NMR spectra for the reactants and

products of the reaction to estimate the rate constants kO, kC, kDC, the term kBME · [8],

and the equilibrium constant Keq. For reasons explained in Section 4.3.3, the initial con-

centration values for the reactants at time zero, [1]0 and [2]0, are also estimated.

For the parameter estimation, an error of 0.01 dm3 mol−1 is assumed for the concentration values, following a constant variance model. This error value encapsulates all the uncertainty of the experimental measurements. It is of course likely that higher errors may be associated with lower concentrations, since their measurements are generally more diﬃcult and less accurate, and also associated with the integration of the overlapping peaks of the C-alkylated product and benzyl methyl ether. Another source of uncertainty emanates from the use of TMS, which is highly volatile and thus its concentration is susceptible to temperature variations. Nevertheless, the assumption of 0.01 dm3 mol−1 error for all concentration measurements is considered adequate.

5.2 Experimental results

5.2.1 Kinetic experiments at 298, 313 and 328 K

In this section, we present and discuss the results of the kinetic experiments for the reaction between sodium β-naphthoxide and benzyl bromide performed in methanol-d4 at three diﬀerent temperatures and the results obtained from the kinetic analysis performed by implementing the 4-reaction proton-exchange model in gPROMS.

4-reaction proton-exchange model

The experimental concentration proﬁles for the components of the reaction between sodium β- naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K are shown Chapter 5. Kinetic investigation of a Williamson reaction in methanol 164

0.1 Total Naphthol Benzyl bromide O-alkylated product 0.08 Total C-alkylated product Double C-alkylated product Benzyl methyl ether -3 0.06

0.04 /mol dm c

0.02

0 0 2000 4000 6000 8000 10000 12000 14000 16000 t/sec

Figure 5.3: Experimental concentration data as a function of time in methanol-d4 at 298 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (5.9)-(5.21) for the 4-reaction proton-exchange model in gPROMS.

(with symbols) in Figures 5.3, 5.4 and 5.5, respectively. The molecules monitored in this reaction are the following: the reactants: sodium β-naphthoxide and benzyl bromide; the main products: the O-alkylated product and the C-alkylated product; the by-product: the double C-alkylated product; and the side-product: benzyl methyl ether. The concentration of each reaction component is represented by a different symbol and colour. The curves of the corresponding colours represent the calculated concentration values for each reaction component obtained by implementing the 4-reaction proton-exchange model in gPROMS. For each experiment at a different Expt. Expt. Expt. Expt. temperature, the estimated reaction rate constants kO , kC , kDC , the term kBME ·[8], and Expt. the equilibrium constant Keq , together with their 95% confidence intervals, are presented in Table 5.4. The 95% confidence intervals are estimated on the basis of a constant variance model assuming an error of 0.01 dm3 mol−1 for the concentration values. The superscript “Expt.” refers to the use of experimental concentration data to fit those parameters. Additionally, the estimated initial concentration values for the reactants at time zero, [1]0 and [2]0, and their 95% confidence intervals, are presented in Table 5.3. By inspecting Figures 5.3, 5.4 and 5.5, it may be seen that the fitting agreement of the Chapter 5. Kinetic investigation of a Williamson reaction in methanol 165

0.1 Total Naphthol Benzyl bromide O-alkylated product 0.08 Total C-alkylated product Double C-alkylated product Benzyl methyl ether -3 0.06

0.04 /mol dm c

0.02

0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 t/sec

Figure 5.4: Experimental concentration data as a function of time in methanol-d4 at 313 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (5.9)-(5.21) for the 4-reaction proton-exchange model in gPROMS.

4-reaction proton-exchange model to the experimental concentration data of the various components is very good at lower temperatures, 298 and 313 K, and reasonably good at 328 K. In particular, at 298 K, despite the small number of spectra (10 spectra) integrated with the separated peaks of the C-alkylated product and benzyl methyl ether, the model captures very well the kinetics of the system. At 313 K, where the peaks of the C-alkylated product and benzyl methyl ether are mostly separated and hence an adequate number of integrated spectra (16 spectra) is obtained, the fitting agreement achieved with the 4-reaction proton-exchange model is also quite good. At 328 K, the peaks of the C-alkylated product and benzyl methyl ether overlap throughout the whole time-length of the monitored experiment, and they are barely discernible from each other. As a consequence, separate integration of the two peaks is difficult (9 spectra obtained) and the resulting concentration values involve some amount of uncertainty. Moreover, the reduced number of available data has an impact on the quality of the fitting at that temperature (cf. Figure 5.5). The increase of temperature also contributes to the increase of noise in the NMR measurements, resulting in a low signal-to-noise-ratio; nevertheless, because of the highly sensitive spectrometer used, this source of error is almost eliminated. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 166

0.12 Total Naphthol Benzyl bromide 0.1 O-alkylated product Total C-alkylated product Double C-alkylated product 0.08 Benzyl methyl ether -3

0.06 /mol dm

c 0.04

0.02

0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 t/sec

Figure 5.5: Experimental concentration data as a function of time in methanol-d4 at 328 K measured with in situ 1H NMR spectroscopy. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (5.9)-(5.21) for the 4-reaction proton-exchange model in gPROMS.

As shown in Table 5.3, for the kinetic experiments at 298 and 313 K the estimated initial concentration values for the reactants are in good agreement with the weighted values (although they are slightly overestimated for the experiment at 313 K). For both experiments, the corresponding 95% confidence intervals include the weighted values (with the exception of the weighted value of sodium β-naphthoxide at 313 K). The estimated values of the reactants lie within ranges for sodium β-naphthoxide of ±0.007 mol dm−3 and ±0.008 mol dm−3, and for benzyl bromide within ranges of ±0.008 mol dm−3 and ±0.010 mol dm−3 at temperatures 298 and 313 K, respectively. For the kinetic experiment at temperature 328 K, the initial concentration values of the reactants are overestimated compared with the weighted values most probably due to TMS losses that lead to higher concentration values for the reaction components. The corresponding 95% confidence intervals lie from the estimated values of the reactants within ranges of ±0.012 mol dm−3 for sodium β-naphthoxide and of ±0.013 mol dm−3 for benzyl bromide; yet, they do not include the corresponding weighted values. The broader confidence intervals obtained for the experiment at 328 K, compared with the corresponding values obtained for the experiments at 298 K and 313 K, may be attributed most likely to the limited number of concentration data Chapter 5. Kinetic investigation of a Williamson reaction in methanol 167 obtained for sodium β-naphthoxide (9 spectra integrated) and for the C-alkylated product and benzyl methyl ether (9 spectra integrated). As may be deduced by comparing Tables 4.4 and 5.4, the reaction between sodium β- naphthoxide and benzyl bromide proceeds much slower in a polar protic solvent, such as methanol- d4, than in a polar aprotic solvent, such as acetonitrile-d3. The estimated values for the reaction Expt. Expt. rate constants kO and kC in methanol-d4 differ by half to one order of magnitude at each Expt. of the three temperatures studied. According to the obtained values of the term kBME · [8], the value for the rate constant of the side-reaction between benzyl bromide and methanol-d4 is ex- Expt. pected to be always at least two orders of magnitude lower than kO and one and a half orders Expt. of magnitude lower than kC . Although the rate of the side-reaction is slow, as suggested from Expt. the small value obtained for the term kBME · [8], it should still be taken into account to avoid overestimating the C-alkylation product concentration. As the temperature increases, the rates of all four reactions increase, as suggested by the drastic increase observed in the corresponding rate constants. For the three experiments performed at different temperatures, the estimated 95% confidence intervals are of the same order of magnitude with the estimated rate-constant values; hence, we may assume that the intrinsic uncertainty involved in these values is considerable. This is more obvious in the 95% confidence Expt. Expt. intervals of the values of kC and kBME · [8], which rely upon a small number of measured concentration data with a large degree of uncertainty (e.g., due to TMS variations, peak overlap- Expt. ping), and of the value of kDC , which relies on fairly low measured concentrations compared to the concentrations of the other reaction species. The equilibrium constant Keq is also inherently difficult to be accurately determined due to the way the model is built and the concentration data of the species we are able to follow in our kinetic experiments (cf. specific details of the proton-exchange model in Section 4.3.3). Furthermore, the equilibrium constant will be greatly affected by the initial concentrations used for the reactants (e.g., if one species is in excess) and we expect it to be shifted in either direction depending on the concentrations of the reaction components at a certain time. Of the three experiments performed in methanol-d4, the experiment at temperature 328 K is the one with the widest 95% confidence intervals, and so it has the largest uncertainty in the estimated rate constants. The same reasons discussed before for the large 95% confidence intervals obtained for the initial concentrations of the two reactants at 328 K are also applicable here. Expt. Expt. As shown in Table 5.4, percentage values of the ratio kO :kC of 74:26 and 73:27 are obtained for the experiments at 298 and 313 K, respectively. The ratio value increases to 80:20 Chapter 5. Kinetic investigation of a Williamson reaction in methanol 168 for the experiment at 328 K; this value should be carefully evaluated given the uncertainty of the concentration measurements obtained. Despite the inherent uncertainty due to low concentration values, the value of the rate constant for the formation of the double C-alkylated product is comparable to the value of the rate constant for the formation of the C-alkylation product. This result validates our decision to include the kinetics of the additional alkylation reaction to the overall kinetic model developed. According to the percentage rate constant ratio values obtained in this work, a selectivity preference of the reaction towards the O-alkylated product is demonstrated at the temperature range studied, with a substantial percentage of the C-alkylated product and the double C-alkylated product also formed. This outcome is in line with previous studies14,23,271 in which the reaction between sodium β-naphthoxide and benzyl bromide was investigated in polar protic solvents. Depending on the hydrogen-bond capability of the solvent used, the percentage of the C-alkylated product yielded is significant or even the dominant product of the reaction, in cases of strong hydrogen bonding solvents like water and 2,2,2-trifluoroethanol. In particular, the reaction has been investigated at 298 K by Kornblum et al.,14 who reported product ratio values of the O-versus the C-alkylated product of 64:36 in methanol; and 23 Expt. Expt. by Ganase, who reported rate constant ratio values of kO :kC of 63:37 in methanol-d4 (cf. Table 4.1). With respect to the study of Kornblum et al.,14 it was mentioned that any double C-alkylated product yielded from further alkylation of the C-alkylated product was included in the yield of the C-alkylated product; as a result, their product ratio value is still comparable with our rate constant ratio value according to Equation (5.22). Although the specific details of the experiment in methanol were not mentioned, the experiment was performed similarly to the one in ethanol for which a 28% yield was obtained for the C-alkylated product and a 4% yield for the double C-alkylated product. Given that in methanol a higher yield of 34% was obtained for the C-alkylated product, a higher percentage of double C-alkylated product is expected to occur. The difference observed with our value and the value reported by Kornblum may be attributed partly to the loss of product material in their study during the work-up procedure. With respect to the study of Ganase,23 the primary data from the 1H NMR spectra have been collected and reviewed. It was found that a clear distinction between the C-alkylated product peak and the benzyl methyl ether peak was not possible, as the two peaks overlap almost throughout the whole length of the experiment. We may speculate that this is due to the low sensitivity of the particular probe type (a room temperature probe) used to perform the experiments, which subsequently concealed the existence of the side-reaction of benzyl bromide with Chapter 5. Kinetic investigation of a Williamson reaction in methanol 169

23 methanol-d4. As a result, in the original study of Ganase the total integrated peak area was assigned to the C-alkylated product. Furthermore, the additional alkylation reaction between the C-alkylated product and benzyl bromide was not taken into consideration in the kinetic modelling of the reaction. In light of these ascertainments, we conclude that the rate-constant 23 values reported by Ganase in methanol-d4 are not representative of the kinetic phenomena occurring in this solvent. The rate-constant values of that study are comparable only with the ones obtained in our study from the 2-reaction model, for which the same integration strategy is applied: the peaks of the C-alkylated product and benzyl methyl ether are integrated together and the total value is assigned to the C-alkylated product. (Indeed, the estimated rate Expt. Expt. constants and final rate constant ratio of kO :kC obtained from the 2-reaction model are in close agreement with the values of Ganase23 and the reader is referred to AppendixD for more details.) No comparison should be made between the rate-constant values reported in the study of Ganase23 and the values derived from the 4-reaction proton-exchange model in this work. Regarding the overlapping peaks of the C-alkylated product and benzyl methyl ether, we note that in our work this obstacle is partly circumvented by the use of the enhanced sensitivity CryoProbe Prodigy broadband probe, which allows for high resolution in the 1H NMR spectra and a clearer distinction between the two peaks. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 170 6 5 5 10 10 10 × × × , with their 95% (%) -naphthoxide and eq β K 95% CI Expt. C k : 80:20 73:27 74:26 ), for the reaction studied 0 Expt. O 95% CI k Expt. eq 914 [0.000-9.156] 1058 [0.000-10.69] 829.9 [0.000-10.91] K 3 − 3 3 4 6 5 5 − − − − − − 10 10 10 10 10 10 × × × × × × /mol dm e 0 [2] 95% CI 95% CI ) and benzyl bromide ([2] 3 0 − is also reported. [8], and the equilibrium constant [0.000-16.28] [0.000-2.621] [0.000-7.212] [0.000-18.58] [0.795-3.639] [3.387-15.41] · 1 Expt. C − Expt. k BME s : /mol dm k 1 1 w 0 3 3 4 6 5 5 − − − − − − − − Expt. O k 10 10 10 10 10 10 mol [8]/s -naphthoxide ([1] × × × × × × · 3 β , the term /dm Expt. 3.111 1.087 2.702 6.548 2.217 9.397 BME k Expt. DC 95% CI [2] Expt. C k k , 4 4 4 2 3 4 3 − − − Expt. C − − − − k 10 10 10 10 10 10 , × × × × × × Expt. O -naphthoxide Benzyl bromide k /mol dm β e 0 95% CI 95% CI [1] and the equilibrium between the protonated and deprotonated forms of sodium 4 [0.00-6.290] [0.00-6.785] [0.00-18.34] 3 [0.000-42.85] [0.000-27.38] [0.000-87.60] − Sodium 1 1 − − s s 1 1 2 3 4 4 4 4 -naphthoxide and benzyl bromide, the reaction for the formation of the double C-alkylated product, the reaction − − − − − − − − β /mol dm w 0 10 10 10 10 10 10 mol mol × × × × × × 3 3 /dm /dm 313328 0.090 0.090 0.100 0.113 [0.092-0.108] [0.102-0.125] 0.090 0.090 0.099 0.106 [0.089-0.109] [0.093-0.119] 298 0.090 0.087 [0.080-0.094] 0.090 0.091 [0.083-0.100] T/K [1] at 298 K. The reported concentration values correspond to the weighted amounts, indicated by the superscript “w”, and Expt. Expt. O DC k k 4 328 1.209 313 2.957 313328 4.840 4.825 298 7.581 298 2.921 T/K T/K Table 5.3: Initial concentrations at time zero of the reactants, sodium Table 5.4: Estimated reactionconfidence rate intervals (CI), constants obtained by fittingthe the reaction between 4-reaction sodium proton-exchange model tobetween benzyl experimental concentration bromide data and for methanol-d thethe components C-alkylated of product at 298 K. The percentage rate constant ratio of in methanol-d the estimated values, indicatedproton-exchange by model the in superscript gPROMS. “e”, with their 95% confidence intervals (CI), obtained by implementing the 4-reaction Chapter 5. Kinetic investigation of a Williamson reaction in methanol 171

5.2.2 Thermodynamic analysis based on experimental data

The experimental rate-constant values, obtained using the 4-reaction proton-exchange model, are plotted as a function of the inverse temperature in an Arrhenius-like plot (using Equation (3.10)) and in an Eyring-like plot (using Equation (4.52)); the results may be seen in Figures 5.6 and 5.7. The quality of ﬁt (R2) and the linear equation derived for each pathway are given in the ﬁgures. The estimated parameters of the Arrhenius and Eyring equations are listed, respectively, in Tables 5.5 and 5.6. 0 O-alkylation -2 C-alkylation y=-9013.8x + 23.034 R2=0.9986 -4

-6

Expt. y=-7972.8x + 18.573 -8 2 k R =0.9972

ln -10

-12

-14

-16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 5.6: Arrhenius plot of experimental reaction rate constants as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K.

For the temperature range studied (298-328 K), the fitted rate constants for the two pathways obey linearity, when fitted as functions of inverse temperature to both the Arrhenius and the Eyring equations (R2 0.99 in Figures 5.6 and 5.7). This may be explained because of the narrow temperature range studied and the small number of experiments performed. Expt. For the pre-exponential factor A1 of the Arrhenius equation, a difference of a factor of 86.6 was found for the ratio of the values corresponding to the O- versus the C-alkylation. For Expt. the activation energy parameter Eã , the large values obtained for the O-alkylation pathway (74.94 kJ mol−1) and the C-alkylation pathway (66.29 kJ mol−1) explain the slow reaction rates in methanol-d4; these are much slower than the rates observed in acetonitrile-d3 for which Chapter 5. Kinetic investigation of a Williamson reaction in methanol 172 we find activation energy barriers of approximately 18-20 kJ mol−1 less for the two alkylation pathways (cf. Table 4.7).

0 O-alkylation -2 C-alkylation

-4

/T) -6

-8 y=-8701.4x + 16.289 Expt. R2=0.9985 (k -10 ln -12 y=-7660.4x + 11.828 -14 R2=0.9970 -16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 5.7: Eyring plot of experimental reaction rate constants as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K.

Analysing the rate constants in terms of the Eyring equation, similar observations may be made: the values for the obtained thermodynamic parameters for the two alkylation pathways, as presented in Table 5.6, are quite large; however, the relative differences between them are fairly small. (The values of the estimated parameter ∆‡S◦,Expt., despite being small in absolute numbers, when multiplied by the temperature contribute to the calculation of ∆‡G◦,Expt.(298) in the same order of magnitude as ∆‡H◦,Expt..) As discussed before, we are particularly interested about monitoring the difference in ∆‡G◦,Expt. at 298 K between the two alkylation pathways. ◦,Expt. −1 At 298 K in methanol-d4, ∆GC−O has a value of 2.40 kJ mol . Even though the transition state of the C-alkylated product is higher in energy than the transition state of the O-alkylated product, the difference between them is not unsurpassable. As a result, a considerable number of molecules reach the higher-energy transition state of the C-alkylated pathway, in addition to the larger number of molecules that reach the lower-energy transition state of the O-alkylation pathway, leading to the major O-alkylated product in methanol. The decisive role of the solvent in promoting the one against the other pathway by stabilizing the corresponding transition states Chapter 5. Kinetic investigation of a Williamson reaction in methanol 173 in different energy levels is highlighted here. In methanol-d4, the small free-energy difference of only 2.40 kJ mol−1 between the two transition states results in a percentage rate constant ratio Expt. Expt. ◦,L of 74:26 of kO :kC . The key role of parameter ∆GC−O(T ) on determining the selectivity of a reaction and the impact of solvent on altering this parameter are demonstrated here.

Table 5.5: Estimated parameters of the Arrhenius equation (Equation (3.10)) for the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K.

Expt. 3 −1 −1 ˜Expt. −1 A1 /dm mol s Ea /kJ mol O-alkylation 1.01×1010 74.94 C-alkylation 1.16×108 66.29

Table 5.6: Estimated thermodynamic parameters of the Eyring equation (Equations (4.54) and

(4.4)) for the reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K.

∆‡H◦,Expt./kJ mol−1 ∆‡S◦,Expt./J K−1 mol−1 ∆‡G◦,Expt.(298)/kJ mol−1 O-alkylation 72.35 −62.12 90.86 C-alkylation 63.69 −99.21 93.26 ◦,Expt. −1 ◦,Expt. −1 −1 ◦,Expt. −1 ∆HC−O /kJ mol ∆SC−O /J K mol ∆GC−O (298)/kJ mol Diﬀerence C-O −8.65 −37.09 2.40

5.3 Computational methodology

In this section, we perform a systematic computational investigation of the reaction between sodium β-naphthoxide and benzyl bromide in methanol using quantum-mechanical calculations. We follow the same computational methodology described in the previous chapter for acetonitrile. The same set of levels of theory (functionals B3LYP, M05-2X, M06-2X and wB97xD with basis sets 6-31G(d), 6-31+G(d), 6-31+G(d,p), 6-311+G(d,p), 6-311++G(d,p), 6-311G(2d,d,p), cc-pVDZ and cc-pVTZ in 18 diﬀerent combinations) is assessed. For each level of theory assessed, the structures of the reactants and the two transition states are optimized anew in methanol. (The root-mean-square error values for the deviation of the optimized transition-state structures for both alkylation pathways in methanol from the corresponding structures in vacuum are given in AppendixB.) In order to account for the presence of solvent, we employ the solvation model SMD.24 Methanol is included in the list of solvents for which the SMD model has embedded the Chapter 5. Kinetic investigation of a Williamson reaction in methanol 174 seven macroscopic descriptors† required in order to describe the solvent medium. Nevertheless, we mention here that methanol is not one of the 90 non-aqueous solvents for which solvation data for neutral solutes have been used in the training set of the SMD model. Yet, solvation free energies for 80 ionic solutes in methanol have been used, with a MUE of 8.79 kJ mol−1, as reported by Marenich et al.24 However, none of the components involved in this reaction is among these 80 ionic solutes.

5.4 Computational results

5.4.1 Structure search for stable conformations of sodium β-naphthoxide

In order to identify the electrostatically most-favourable conﬁguration between the sodium cation and the oxygen atom of sodium β-naphthoxide in methanol, we perform a relaxed potential energy surface (PES) scan type calculation. This type of calculation has already been discussed in the previous chapter, when the electrostatic interactions between the sodium cation and the negatively charged oxygen atom are investigated in acetonitrile. By scanning the PES, the electronic energy of the optimized geometries of sodium β-naphthoxide in methanol is obtained as a function of the distance between the sodium cation and the oxygen atom of β-naphthoxide, as shown in Figure 5.8. The B3LYP/6-31+G(d) level of theory is an arbitrary selection to illustrate the electrostatic interactions in the sodium β-naphthoxide molecule. By inspecting Figure 5.8, we see that the minimum energy for the region of the PES scanned corresponds to an equilibrium distance with a value of req=2.104 A.˚ At this distance, the sodium cation is bound electrostatically to the negatively charged oxygen atom forming an ion-pair. At distances smaller than req, the energy increases massively as the two atoms are forced to come closer and the repulsive forces are maximized. At distances larger than req, the electronic energy increases until it reaches a plateau, where apparently the sodium cation and the oxygen atom are completely dissociated and the energy at the B3LYP/6-31+G(d) level of theory appears to have converged.

Comparing to the analogous calculation performed in acetonitrile, the value of req increases as the dielectric constant of the solvent decreases, i.e., req,Acetonitrile < req,Methanol for εAcetonitrile

> εMethanol. In general, the ion-pair is stabilized at larger distances as we change the medium from vacuum to either acetonitrile or methanol.

†The required descriptors are the dielectric constant (ε), the Abraham’s hydrogen bond acidity (α) and basicity (β), the refractive index (nD), the bulk surface tension at 298 K (γ), the aromaticity (φ) and the electronegative halogenicity (ψ). Chapter 5. Kinetic investigation of a Williamson reaction in methanol 175

−622.650 Methanol Acetonitrile −622.700 Vacuum −1 −622.750

−622.800

−622.850 /a.u. Particle el E −622.900

−622.950 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 r/Å

Figure 5.8: Electronic energy (Eel) as a function of the distance (r) of the sodium-oxygen atoms in sodium β-naphthoxide in methanol performing a relaxed potential energy surface scan at the B3LYP/6-31+G(d) level of theory. The results from the calculations in vacuum and acetonitrile are shown for comparison.

5.4.2 Selectivity and reaction kinetics at 298 K

The QM-calculated values for the Gibbs free-energy difference between the two alkylation pathways, the selectivity ratio and the rate constants of the two pathways for the reaction in methanol at 298 K for the various levels of theory are listed in Table 5.7 and plotted in Figures 5.9-5.11. Superscript “QM” refers to the use of QM methods to determine those quantities. The corresponding experimental values obtained in this work at 298 K are also presented in the table and in the figures (dashed lines) for comparison. ◦,QM As shown in Table 5.9 and Figure 5.9, the predictions for ∆GC-O in methanol at 298 K for the various levels of theory span over a range of 11.72 kJ mol−1 including also negative values. ◦,L According to the definition of ∆GC-O a negative value indicates that the predicted selectivity is favoured towards the C-alkylated product. This contradicts the experimental evidence suggesting that, even though the amount of C-alkylated product yielded is significant when the reaction is performed in methanol, the O-alkylated product is still the dominant product. B3LYP is the ◦,QM only method that gives consistently positive ∆GC-O values. M05-2X gives positive value only in combination with the 6-31G(d) basis set and negative values with all the other basis sets. Combined with each of the possible basis sets, M06-2X and wB97xD give consistently negative Chapter 5. Kinetic investigation of a Williamson reaction in methanol 176

◦,QM predictions for ∆GC-O . Among all levels of theory tested, the closest prediction to the experimental value (2.40 kJ mol−1) is achieved by M05-2X/6-31G(d) with a value of 1.93 kJ mol−1. This is consistent with what Marenich et al.24 reported about M05-2X/6-31G(d) being the level of theory with the lowest MUE value (MUE=8.79 kJ mol−1) for ionic solvation free energies among all other levels of theory used in the parametrization of the SMD model. ◦,L The accurate prediction of ∆GC-O reduces to accurate calculation of the gas-phase Gibbs free- energy difference and the solvation free-energy difference between the C- and the O-alkylation transition-state structures (cf. Equations (4.55) and (4.6)). The solvation free-energy calculation relies upon the solvation model applied. We restate here the view of Klamt et al.,126 seconded also by Ho et al.,261 regarding the predictive capabilities of continuum solvation models: their performance is expected to be optimal when they are applied in a manner consistent with their parametrization. In a nutshell, optimal performance entails a rigorous application of the continuum solvation models using the levels of theory for which the model’s parameters (atomic radii) and the non-electrostatic terms have been optimized in order to fit a number of experimental solvation energies (which consist the model’s training set) at a specific temperature. If any of the above conditions is not followed, then significant deviations from optimal behaviour may be expected. In our work, the majority of the levels of theory tested are not capable of capturing the difference between the liquid-phase free energies of the two transition-state structures in methanol. The poor prediction of such fine free-energy difference (2.40 kJ mol−1) is anticipated due to the shortage of experimental solvation free-energy data for this solvent in the training set of SMD. The specific solvent-solute interactions are not captured accurately, which possibly explains the negative values obtained for several levels of theory. With respect to the parametrization of the SMD model, a good example of level of theory is M05-2X/6-31G(d), which is one of the six electronic methods for which the single set of parameters of SMD has been optimized. M05- 2X/6-31G(d) performs best in terms of capturing the experimental Gibbs free-energy difference and selectivity ratio; its good performance is mostly related to the aforementioned reason given by Klamt et al.126 We may also argue that its performance is likely to benefit from fortuitous cancellation of errors due to the relative energy differences considered. With respect to the transmission coefficient κ, estimated using the Wigner scheme,207 we obtain values ranging from 1.12 to 1.31 for the O-alkylation and from 1.09 to 1.29 for the C- QM QM alkylation pathway. The ratio κO /κC is in the range 1.00-1.05. Therefore, our assumption for considering a transmission coefficient of unity in the Eyring equation is reasonable. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 177 1 1 − s − s 1 4 5 5 5 6 6 2 1 2 4 2 3 4 5 2 3 1 − 5 4 0 − − − − − − − − − − − − − − − − − − − 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 mol 10 10 × mol × × × × × × × × × × × × × × × × 3 × × 3 ) and the rate constants of the 1.109 /dm 1.172 2.932 2.866 9.402 1.187 1.419 5.691 5.602 4.950 8.336 2.520 3.152 2.108 4.266 8.904 7.879 2.174 2.702 /dm QM C QM Expt. /k C C k k QM O k 1 1 − s − s 1 4 4 4 4 4 4 5 5 1 1 1 2 4 3 4 4 5 2 3 1 − − − − − − − − − − − − − − − − − − − − − 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 mol mol × × × × × × × × × × × × × × × × × × × 3 3 /dm /dm QM Expt. O O k k ), the selectivity ratio ( QM , ◦ C-O QM Expt. C C 00 2.173 80 7.581 51212185 2.866 30 2.289 16 1.062 95 7.165 52 7.617 24 5.815 18 2.003 2.208 2.168 1.433 92752319 9.287 06 1.687 83 1.499 27 6.763 2.144 1.679 1.292 G ...... /k /k QM O k Expt. O k 1 1 − − 67 10 40 2 760193480359 0 21 0 72 0 50 0 28 0 0 0 0 0 0 092703831405 7 93 5 5 7 18 11 2 ...... /kJ mol /kJ mol 1 4 3 0 3 4 0 1 3 4 O − − − − − − − − − − O QM − , Expt. − , ◦ C ◦ C G G ∆ ) of the reaction in methanol at 298 K at all the levels of theory tested. The corresponding experimental QM k Level of theoryB3LYP/6-31G(d) ∆ 5 Expt. 2 B3LYP/6-31+G(d)B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p)B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 4 M05-2X/6-31G(d) 5 4 M05-2X/6-31+G(d) 4 M05-2X/6-31+G(d,p) M05-2X/6-311++G(d,p) M05-2X/cc-pVTZ 7 M06-2X/6-31+G(d) 6 M06-2X/6-311++G(d,p) 1 M06-2X/cc-pVDZ M06-2X/cc-pVTZ wB97xD/6-31+G(d) wB97xD/6-311++G(d,p) O- and C-alkylation pathways ( values obtained in this work are also reported for comparison. Table 5.7: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 5. Kinetic investigation of a Williamson reaction in methanol 178

◦,QM As expected from the range of values obtained for ∆GC-O , the QM-calculated selectivity ratios for the various levels of theory are quite diverse, as shown in Figure 5.10. The black line at a ratio value of 1.0, which corresponds to a percentage rate constant ratio of 50:50, helps the reader to easily distinguish between the methods that favour the O-alkylation pathway (values above the line) and those that favour the C-alkylation pathway (values below the line). As it may be seen, B3LYP with all basis set combinations and M05-2X with 6-31G(d) deliver selectivity ratios in accordance with the experimental evidence of the O-alkylated product prevailing in methanol. All the other levels of theory are in antithesis with the experimental value, predicting an overturn in the route of the reaction towards the C-alkylated product. The experimental value for the rate constant ratio is 2.80, which is translated to a percentage rate constant ratio of 74:26. For the various basis sets combined with the B3LYP functional, reasonable agreement with the experiment is achieved with ratio values, as presented in Table 5.7, ranging from 5.23 to 18.06, which are translated to percentage rate constant ratio values from 84:16 to 95:5, favouring the O-alkylated product. M05-2X/6-31G(d) gives the best agreement with a rate constant ratio value of 2.27, which corresponds to a percentage rate constant ratio value of 69:31. The calculated values for the rate constants of the O- and C-alkylation pathway in methanol at 298 K, obtained using Equation (4.20), for all the levels of theory tested are plotted in Fig- ure 5.11. The best agreement with experimental values is achieved by B3LYP/6-311G(2d,d,p),† which employs a rather extensive and computationally arduous basis set. Calculations with this basis set may become staggeringly demanding in terms of computational resources, as the number of atoms in the system increases. Compared with the experimental rate constants, for

B3LYP/6-311G(2d,d,p) the APEk is 37.99%. This level of theory matches the experimental rate constant of the O-alkylation pathway with an APEkO value of 10.79% (which is the second best after M05-2X/cc-pVTZ with APEkO =5.49%) and the C-alkylation pathway with an

APEkO value of 65.20%. The second best overall agreement with experiments, with a correct prediction for the selectivity preference of the reaction according to experimental evidence, is achieved by B3LYP/6-31+G(d). Augmenting the small basis set 6-31G(d), which is used in the SMD parametrization, with a diﬀuse function (+) to account for the presence of the bromide an- 47 ion, improves the performance of B3LYP/6-31+G(d) and results in an APEk value of 39.56%. Further addition of diﬀuse functions deteriorates the predictions, as it leads to underestimation of the rate constants of both the O- and the C-alkylation pathway. The combination of B3LYP

†The nomenclature for 6-311G(2d,d,p) indicates that it is a split-valence triple-zeta basis set with two additional d polarization functions on second row atoms, one d function on ﬁrst row atoms and one p function on hydrogens. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 179

15.0

-1 10.0

5.0 /kJ mol

º | C-O G

∆ 0.0

-5.0 6-31G(d)6-31+G(d)6-311+G(d,p)6-311++G(d,p)6-311G(2d,d,p)cc-pVDZcc-pVTZ 6-31G(d)6-31+G(d)6-31+G(d,p)6-311++G(d,p)cc-pVTZ 6-31+G(d)6-311++G(d,p)cc-pVDZcc-pVTZ 6-31+G(d)6-311++G(d,p)

B3LYP M05-2X M06-2X wB97xD Level of theory QM Expt.

Figure 5.9: Gibbs free-energy diﬀerence between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in methanol at 298 K obtained for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 4-reaction proton-exchange model (cf. Table 5.4). Chapter 5. Kinetic investigation of a Williamson reaction in methanol 180

100.0

10.0 C k / O k 1.0

0.1 6-31G(d)6-31+G(d)6-311+G(d,p)6-311++G(d,p)6-311G(2d,d,p)cc-pVDZcc-pVTZ 6-31G(d)6-31+G(d)6-31+G(d,p)6-311++G(d,p)cc-pVTZ 6-31+G(d)6-311++G(d,p)cc-pVDZcc-pVTZ 6-31+G(d)6-311++G(d,p)

B3LYP M05-2X M06-2X wB97xD Level of theory QM Expt.

Figure 5.10: Selectivity ratio for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at 298 K calculated using Equation (4.56) for various levels of theory and the SMD solvation model implemented in Gaussian 09. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The red dashed line indicates the experimental value at 298 K from the 4-reaction proton-exchange model (cf. Table 5.4). Chapter 5. Kinetic investigation of a Williamson reaction in methanol 181

101 100 -1 -1 s 10 -1 10-2

mol -3

B3LYP M05-2X M06-2X wB97xD Level of theory

O-alkylation QM O-alkylation Expt. C-alkylation QM C-alkylation Expt.

Figure 5.11: Rate constants for the O- and C-alkylation pathways of the reaction between sodium β-naphthoxide and benzyl bromide in methanol at 298 K calculated using CTST theory (Equation (4.20)) for various levels of theory and the SMD solvation model implemented in Gaussian 09. Columns of the same colour correspond to the same QM method. For each QM method, the basis sets assessed are ordered from left to right in ascending order of size and complexity. The dashed lines indicate the experimental values at 298 K from the 4-reaction proton-exchange model (cf. Table 5.4). Chapter 5. Kinetic investigation of a Williamson reaction in methanol 182 with Dunning’s basis sets,240,241 such as cc-pVDZ and cc-pVTZ, leads to a dramatic underestimation of both pathways and especially the C-alkylation pathway. On the contrary, the M05-2X functional performs best when combined with the cc-pVTZ basis set. This level of theory is also among the ones used to parametrize SMD and at the same time, its extensive form allows for accurate computation of the bulk electrostatic interactions. A similar behaviour is observed for the M06-2X functional, which in combination with the Dunning-type basis sets gives the most promising results. Albeit the correct selectivity is not captured, M06-2X/cc-pVDZ results in good agreement with the absolute experimental rate constants for both alkylation pathways, with an APEk value of 47.78%. In Table 5.8 we report the absolute percentage error (APE) values for the two best computational options identiﬁed, M05-2X/6-31G(d) and B3LYP/6-311G(2d,d,p), for capturing the selectivity and the kinetics of the reaction, respectively, in methanol. The obtained error values reinforce our position, based also on our ﬁndings of the previous chapter, that in order to achieve accurate absolute energy predictions in the liquid phase, usually higher levels of theory are more advantageous, whereas for relative energy predictions usually parametrized levels of theory for the solvation model are preferable.

Table 5.8: Absolute percentage error (APE) for the QM-calculated values for the Gibbs free- ◦,QM QM QM energy diﬀerence (∆GC-O ), the selectivity ratio (kO /kC ) and the rate constants of the two alkylation pathways (kQM) of the reaction in methanol at 298 K at the M05-2X/6-31G(d) and B3LYP/6-311G(2d,d,p) levels of theory.

M05-2X/6-31G(d) B3LYP/6-311G(2d,d,p)

APE ◦ (%) APE (%) APE (%) ∆GC-O kO/kC k 19.59 19.08 37.99

5.4.3 Thermodynamic analysis based on QM-calculated data

Using the level of theory (B3LYP/6-311G(2d,d,p)) identified before as the best option for capturing the rate constants of the two alkylation pathways of the reaction in methanol, we perform additional QM calculations at temperatures 313 and 328 K for which we have available experimental data. The QM-calculated rate constants, obtained using Equation (4.20), are listed in Table 5.9 and plotted in Figures 5.12 and 5.13 in an Arrhenius and in an Eyring format, respectively. The quality of fit (R2) and the linear equation derived are given in the graphs. For comparison, the experimental rate constants are also plotted in the figures (black symbols). Chapter 5. Kinetic investigation of a Williamson reaction in methanol 183

The obtained parameters of the Arrhenius and the Eyring equations are listed, respectively, in Tables 5.11 and 5.12.

Table 5.9: QM-calculated rate constants for the O- and the C-alkylation pathways (kQM) of the reaction between sodium β-naphthoxide and benzyl bromide in methanol at the B3LYP/6- 311G(2d,d,p) level of theory using CTST and the SMD model at temperatures 298, 313 and 328 QM QM K. The percentage rate constant ratio of kO :kC is also reported.

QM 3 −1 −1 QM 3 −1 −1 QM QM T/K kO /dm mol s kC /dm mol s kO :kC (%) 298 6.763×10−4 9.402×10−5 88:12 313 1.892×10−3 2.575×10−4 88:12 328 4.900×10−3 6.542×10−4 88:12

The CTST theory with the B3LYP/6-311G(2d,d,p) level of theory and the SMD model perform reasonably well predicting the rate constants of the two alkylation pathways over the range from 298 to 328 K, as may be appreciated by inspecting Figures 5.12 and 5.13. In particular, in comparison with the experimental rate constants at 298 K, the QM-calculated rate constant for the O-alkylation pathway is in very good agreement, while the one for the C-alkylation pathway is underestimated but within the estimated 95% conﬁdence intervals (cf. Table 5.4). At higher temperatures, the deviations from the experimental values become larger and our computed rate constants for both alkylation pathways are systematically underestimated compared with the experimental values. For the three temperatures at which calculations and experiments are performed, the MAPEk for the computed values is 54.46%. As shown in Table 5.10, as we extrapolate to higher temperatures, the error values increase. This trend is expected given that the SMD model has been parametrized at 298 K and so larger deviations will occur as we deviate from this temperature.

Table 5.10: Absolute percentage error (APE) and mean absolute percentage error (MAPE) values for the QM-calculated rate constants (kQM) of the two pathways of the reaction at the B3LYP/6-311G(2d,d,p) level of theory compared with the experimental values in methanol at temperatures 298, 313 and 328 K.

T/K MAPEk (%) 298-328 54.46

T/K APEk (%) 298 37.99 313 56.17 328 69.22 Chapter 5. Kinetic investigation of a Williamson reaction in methanol 184

When Table 5.11 is contrasted against Table 5.5, large differences are observed between the set of parameters found when fitting the Arrhenius equation to QM-calculated and to experimental rate constants. Similarly, by contrasting Tables 5.12 and 5.6, significant differences are observed in the values of activation enthalpies and entropies when fitted to QM-calculated and to experimental rate constants. Nevertheless, it may be seen that the liquid-phase Gibbs free energies at 298 K found when fitted to QM-calculated and to experimental rate constants are in excellent agreement, with a maximum variation of 2.90 kJ mol−1 in the case of the C-alkylation ◦,QM −1 pathway. At 298 K, the value of ∆GC−O (4.89 kJ mol ) specified by fitting the Eyring equation to the QM-calculated values agrees very well with the equivalent QM-calculated value (4.83 kJ mol−1) obtained using Gaussian (cf. Table 5.7). This is an additional verification of the small impact that the transmission coefficient has on this particular reaction and the validity of using a value of 1 in the Eyring equation (cf. section 4.4.2).

0 O-alkylation QM C-alkylation QM -2 y=-6450.9x + 14.346 O-alkylation Expt. R2=1 C-alkylation Expt. -4

-6

k -8 ln y=-6319.6x + 11.932 -10 R2=1

-12

-14

-16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 5.12: Arrhenius plot of the logarithm of QM-calculated reaction rate constants at the B3LYP/6-311G(2d,d,p) level of theory as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C-alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K. The experimental rate constants, illustrated by black symbols, are shown for comparison. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 185

0 O-alkylation QM C-alkylation QM -2 O-alkylation Expt. C-alkylation Expt. -4

-6

-8 y=-6138.5x + 7.6004 (k/T) R2=1

ln -10

-12

-14 y=-6007.2x + 5.1868 R2=1 -16 0.0030 0.0031 0.0032 0.0033 0.0034 1/T

Figure 5.13: Eyring plot of the logarithm of QM-calculated reaction rate constants at the B3LYP/6-311G(2d,d,p) level of theory over temperature as a function of inverse temperature. The diﬀerent colours and symbols, as labelled in the legend, correspond to the O- and C- alkylation pathways for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K. The experimental rate constants, illustrated by black symbols, are shown for comparison.

Table 5.11: Estimated parameters obtained by ﬁtting the Arrhenius equation (Equation (3.10)) to QM-calculated rate constants at the B3LYP/6-311G(2d,d,p) level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K.

QM 3 −1 −1 ˜QM −1 A1 /dm mol s Ea /kJ mol O-alkylation 1.70×106 53.64 C-alkylation 1.52×105 52.54 Chapter 5. Kinetic investigation of a Williamson reaction in methanol 186

Table 5.12: Estimated thermodynamic parameters obtained by ﬁtting the Eyring equation (Equation (4.2)) to QM-calculated rate constants at the M05-2X/cc-pVTZ level of theory for the reaction between sodium β-naphthoxide and benzyl bromide in methanol at temperatures 298, 313 and 328 K.

∆‡H◦,QM/kJ mol−1 ∆‡S◦,QM/J K−1 mol−1 ∆‡G◦,QM(298)/kJ mol−1 O-alkylation 51.04 −134.36 91.08 C-alkylation 49.95 −154.43 95.96 ◦,QM −1 ◦,QM −1 −1 ◦,QM −1 ∆HC−O /kJ mol ∆SC−O /J K mol ∆GC−O (298)/kJ mol Diﬀerence C-O −1.09 −20.07 4.89

As Ho et al. highlighted in their comment,261 the importance of implementing the solvation model at the temperature used for its parametrization should not be neglected. When temperatures other than the parametrization temperature of a continuum solvation model need to be considered, for example as it is usually necessary in order to partition the liquid-phase Gibbs free energy in its enthalpic and entropic contributions, more advanced models, such as SM6T,152 SM8T153 or COSMO-RS,302 in which the temperature dependence is explicitly introduced into the solvation energy, should be applied. Our computational investigation in methanol extends from 298 K to 328 K. Albeit SMD is parametrized only at 298 K, extrapolation at higher temperatures does not aggravate im- mensely the liquid-phase Gibbs free-energy predictions. This could be explained by the narrow temperature range investigated, which allows for minimum error in the SMD predictions for the solvation free energies, and also by the use of a computationally eﬃcient level of theory (B3LYP/6-311G(2d,d,p)), whose accurate calculations for the bulk electrostatics counterbalance the deﬁciencies of the empirically parametrized GCDS term in the SMD model.

5.5 Performance of levels of theory for acetonitrile and methanol

A thorough experimental and computational analysis of the Williamson reaction in two separate solvents, acetonitrile and methanol, has been conducted and presented in this and the previous chapters. The most accurate levels of theory capturing the selectivity and the kinetics for each solvent have been identiﬁed. When the reaction is considered in both solvents, it would be most useful to elicit one level of theory that performs on average best for selectivity predictions and one level of theory that performs on average best for kinetics prediction; this is the goal of this section. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 187

To begin with, we calculate the MAPE values using the formula shown in Equation (4.63) in ◦,QM QM QM QM order to perform an error analysis for the QM-calculated quantities ∆GC-O , kO /kC and k in acetonitrile and methanol at temperature 298 K. From the error analysis, we see that very few levels of theory give reasonable error values for both metrics (selectivity and kinetics). As fas as capturing selectivity metrics is concerned, M05-2X/6-31G(d) is by far the most successful level of theory, with a MAPE ◦ value of 14.17% and a MAPE value of 16.84%. This ∆GC-O kO/kC outcome is expected according to the error values obtained for the two individual solvents. The other levels of theory give MAPE ◦ values in the range between 42.22% and 192.40%, and ∆GC-O

MAPEkO/kC values in the range between 65.34% and 412.9%. In the case of B3LYP, among all the basis sets considered, given the diversity in size and amount of computational eﬀort required, only B3LYP/6-311++G(d,p) seems to capture selectivity metrics in an acceptable way

(MAPE ◦ =42.22% and MAPE =81.24%). Even though augmenting the simple basis ∆GC-O kO/kC set 6-31G(d) decreases the error in all metrics, the absolute error values are still large. The same trend of decreasing error with extending basis set is observed when B3LYP is combined with the Dunning’s basis sets (cc-pVDZ and cc-pVTZ); however, the overall agreement with experimental values is worse than when the 6-31G(d) family of basis sets is used. Nevertheless, all combinations of the B3LYP functional with basis sets capture the correct selectivity preference for methanol. The functionals M05-2X, M06-2X and wB97xD in various combinations with the basis set tested perform slightly better than B3LYP for selectivity prediction, giving MAPE ◦ ∆GC-O values between 16.83% and 186.76%, and MAPEkO/kC values between 19.70% and 93.78%. However, it should be acknowledged that all these levels of theory, except for M05-2X/6-31G(d), are unsuccessful in capturing the correct selectivity preference for methanol. When accurate prediction of the two absolute reaction rate constants is sought, B3LYP out- performs the rest of electronic structure methods (of course, a few exceptions are also present).

The lowest overall MAPEk value, that is 44.05%, is obtained by using the B3LYP functional with the small basis set 6-31+G(d). As it has been already discussed in Section 4.6.3, the advantages of employing the 6-31+G(d) basis set are first, that it is only by one diffuse function (+) larger than the 6-31G(d) basis set, which has been used in the parametrization of the SMD model, thus reaps all the pertinent benefits; and second, that the extra diffuse function allows for better treatment of the large bromide atom. Good absolute rate-constant predictions are achieved also from B3LYP/6-311G(2d,d,p), with a MAPEk value of 58.36%, which is the second best overall error value. We remind the reader that B3LYP/6-311G(2d,d,p) has been identified Chapter 5. Kinetic investigation of a Williamson reaction in methanol 188 as the best option for capturing the kinetics in solvent methanol (cf. Figure 5.11). M05-2X performs poorly for kinetics prediction; the closest agreement with experiments achieved is with the cc-pVTZ basis set, which has a MAPEk value of 76.84%. Once more, we remind the reader that M05-2X/cc-pVTZ has been identified as the best option for capturing the kinetics in solvent acetonitrile (cf. Figure 4.18). M06-2X performs reasonably well with Dunning’s basis set cc- pVDZ, giving a respective a MAPEk value of 61.44%, which is the third lower value overall. On the other hand, using Pople’s basis sets leads to significant deterioration of the functional’s performance. The same is true for the wB97xD functional, which also results in very high MAPEk values when combined with 6-31+G(d) and 6-311++D(d,p) basis sets. In general, the use of Dunning’s basis sets leads to improved performance for DFT functionals, such as M05-2X and M06-2X, with exceptions such as B3LYP, where the performance is only slightly worsened by their use. No remarks are made for the wB97xD functional, as it is not assessed in combination with this group of basis sets. The MAPE values for the two most accurate levels of theory identified for selectivity and kinetics predictions for the Williamson reaction studied in solvents acetonitrile and methanol at 298 K are summarized in Table 5.13. M05-2X/6-31G(d) is identified as the best option for predicting the Gibbs free-energy difference between the two transition states and the selectivity ratio of the two pathways, and B3LYP/6-31+G(d) for predicting the absolute rate constants of the two pathways of the reaction. Given the approximations made for describing the solvent in the implicit solvation model SMD, the diverse nature of the solvents comprising the test set and the presence of species – some of them even in ionic forms – not available in the training set of the SMD model, the overall agreement with experimental data for the specific metrics studied is considered satisfactory. The findings of Table 5.13 highlight another important aspect of computational research: deciding upon a level of theory is intrinsically linked to the property or metrics one wants to predict. The good performance of one level of theory in a certain property or metric is not necessarily guaranteed when a different property or metric is investigated. In our study, M05-2X/6-31G(d) is an exemplar of a level of theory whose good performance in predicting selectivity ratio is severely compromised by the poor performance in predicting absolute rate constants. Fortuitous cancellation of errors as well as inherent functional specifications are among the decisive factors of a method’s overall performance. Of course, there are levels of theory that perform reasonably well for a series of properties and a range of metrics. In our case, B3LYP/6-31+G(d) falls more into this category of computational methods. Chapter 5. Kinetic investigation of a Williamson reaction in methanol 189

Table 5.13: Mean absolute percentage error (MAPE) values reported for the two levels of theory identiﬁed as the most accurate for QM predictions for selectivity (M05-2X/6-31G(d)) and kinetics (B3LYP/6-31+G(d)) for solvents acetonitrile and methanol at 298 K.

M05-2X/6-31G(d) B3LYP/6-31+G(d)

MAPE ◦ (%) MAPE (%) MAPE (%) ∆GC-O kO/kC k 14.17 16.84 44.05

5.6 Summary

In this chapter, we extended the work started in Chapter 4 by investigating the Williamson reaction between sodium β-naphthoxide and benzyl bromide in a diﬀerent solvent: methanol-d4.

Being capable of hydrogen-bonding, methanol-d4 significantly alters the final product distribution (compared to acetonitrile) and results in a substantial shift to the direction of the carbon alkylation reaction. We started by monitoring the kinetics of the reaction in methanol-d4 at temperatures 298, 313 and 328 K using in situ 1H NMR spectrometry. In the preliminary investigation of the reaction, we identified a side-reaction between one of the reactants (benzyl bromide) and the solvent (methanol-d4), which was not mentioned in previous experimental studies for this reaction. Moreover, further reaction of the C-alkylated product (in its deprotonated form) with benzyl bromide results in the formation of the double C-alkylated product, as mentioned in the study of Kornblum et al.14 The identified side-reaction as well as the extra alkylation reaction were further considered in our kinetic analysis. We developed and implemented in gPROMS a 4-reaction proton-exchange model, which accounts for the kinetics of the two main pathways (O- and C-alkylation) of the Williamson reaction as well as of the side-reaction between benzyl bromide and methanol-d4, the extra alkylation reaction for the formation of the double C-alkylated product and the equilibrium between protonated and deprotonated forms of sodium β-naphthoxide and the C-alkylated product. Based on this model, we obtained a new set of reaction rate constants for the two main pathways at Expt. Expt. 298, 313 and 328 K. The percentage rate constant ratio of kO :kC at 298 K has a value of 74:26; this value is not in accordance with previous experimental studies14,23 due to inconsistencies identified in the experimental procedures of those studies. Because of the way the 4-reaction proton-exchange model was developed, significant amount of uncertainty is involved in its parameters, as may be appreciated by the large estimated values of the 95% confidence intervals. However, the model constitutes a complete mathematical representation of the kinetic and thermodynamic effects of the reaction system investigated in methanol and provides Chapter 5. Kinetic investigation of a Williamson reaction in methanol 190 physical insight for a number of chemical phenomena occurring simultaneously with the main reaction. For these reasons, its use, despite the weakness of specifying with high precision the model parameters, is completely justified in this case. As a next step, we analysed the reaction rate constants obtained from the 4-reaction proton- exchange model at 298, 313 and 328 K following the Arrhenius and Eyring equations. Large values were obtained for the activation energy parameter (Eã) of the Arrhenius equation for the two pathways (74.94 kJ mol−1 for the O-alkylation and 66.29 kJ mol−1 for the C-alkylation). These values were larger than the ones specified in acetonitrile, which is consistent with the literature observation that the rate of the reaction is slower in protic solvents than in aprotic ones.14,23,271 From the parameters obtained from fitting the Eyring equation, we were particu- ◦,Expt. larly interested on the value of the quantity ∆GC−O at 298 K, which determines the selectivity preference of the reaction at this temperature. By fitting to our experimental rate-constant data −1 ◦,Expt. in methanol-d4, a value of 2.40 kJ mol was obtained for ∆GC−O at 298 K in favour of the O-alkylation pathway. This value is lower than the one obtained in acetonitrile, indicating a substantial increase in the amount of C-alkylated product yielded when the reaction takes place in methanol-d4. In the second half of this chapter, we performed quantum-mechanical calculations assessing several DFT functionals and basis sets (levels of theory) for the prediction of reaction selectivity and kinetics in methanol. For illustrative purposes, we reported our structural calculations at the B3LYP/6-31+G(d) level of theory. First, we determined the minimum energy conformation of sodium β-naphthoxide, which is located in a position where the sodium cation forms an ion-pair with the oxygen atom (similar to the study in vacuum and in acetonitrile). We then optimized the reactants and transition-state structures for the two possible pathways (identified in Chapter 4), both in the gas and liquid phase for each level of theory tested. We used the CTST expression in combination with the solvation model SMD to compute selectivity metrics ◦,QM QM QM QM (∆GC−O and kO /kC ) and kinetic metrics (k ) of the reaction of interest. Compared to experiments at 298 K, the selectivity preference towards the O-alkylation of the reaction in methanol was captured successfully by the B3LYP functional combined with any basis set tested as well as by the M05-2X/6-31G(d) level of theory; for these levels of theory, positive values ◦,QM were obtained for ∆GC−O . The other levels of theory assessed resulted in negative values for ◦,QM ∆GC−O predicting an overturn in the route of the reaction. The same pattern was observed for the selectivity ratio predictions. M05-2X/6-31G(d) achieved the best performance with experimental data when selectivity criteria were considered, predicting at 298 K a value of 1.93 kJ Chapter 5. Kinetic investigation of a Williamson reaction in methanol 191

−1 ◦,QM QM QM mol for ∆G (APE ◦ =19.59%) and a value of 2.27 for k /k (APE =19.08%). C−O ∆GC-O O C kO/kC B3LYP/6-311G(2d,d,p) achieved the best performance with experimental data when kinetics criteria were considered, capturing the two alkylation rate constants kQM at 298 K with an

APEk value of 37.99%. We extended our QM calculations to higher temperatures (313 and 328 K) using the B3LYP/6- 311G(2d,d,p) level of theory. Although SMD has been parametrized at 298 K, relatively small discrepancies occurred for the calculations of rate constants at temperatures other than 298 K. The temperature-dependent QM-calculated rate-constant values at the three temperatures studied were in reasonable agreement with the corresponding experimental values, with a MAPEk value of 54.46%. We further analysed the temperature-dependent reaction rate constants in terms of the Arrhenius and Eyring equations and we compared the corresponding parameter values obtained when fitting to experimental data. At 298 K, excellent agreement (with a maximum difference of 2.90 kJ mol−1) was achieved for the individual values of ∆‡G◦,QM for the two alkylation pathways when fitted to experimental and QM-calculated values. In the last section of this chapter, we identified levels of theory that accurately capture the selectivity and kinetics of the Williamson reaction in both solvents acetontrile and methanol.

M05-2X/6-31G(d) was proved the best option to predict selectivity metrics, with a MAPE ◦ ∆GC-O value of 14.17% and a MAPEkO/kC value of 16.84%. B3LYP/6-31+G(d) was proved the best QM option to predict the rate constants k , with a MAPEk value of 44.05%. Identifying levels of theory, which successfully predict reaction metrics in a small, but rather disparate, set of solvents is particularly useful. Such levels of theory may be proved extremely valuable when we extend the investigation of the reaction to more solvents; this will be the focus of the next chapter. Chapter 6

Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations

The promising findings of Chapters4 and5 encourage the investigation of the Williamson reaction between sodium β-naphthoxide and benzyl bromide in more solvents. In this chapter, an exclusive computational investigation of the reaction of interest in a diverse set of solvents is presented. At first, we perform DFT calculations in a set of eight solvents of varying polarity, for which we optimize anew the transition-state structures of the O- and C-alkylation pathways of the reaction. Subsequently, using the CTST expression with the SMD solvation model (as presented in Chapter4) we compute selectivity and kinetic metrics. We discuss the performance of the various density functional theory methods and basis sets tested for computing these metrics. Furthermore, we categorize all the available experimental data for the reaction and discuss the level of confidence placed on the data of each category. Subsequently, we compare our QM-calculations at a specific level of theory with experimental data in terms of solvent ranking considering selectivity criteria.

192 Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 193 6.1 Computational methodology

We follow the computational methodology described in Section 4.5. The same set of levels of theory is assessed: the functionals B3LYP, M05-2X, M06-2X and wB97xD with the basis sets 6-31G(d), 6-31+G(d), 6-31+G(d,p), 6-311+G(d,p), 6-311++G(d,p), 6-311G(2d,d,p), cc- pVDZ and cc-pVTZ in 18 diﬀerent combinations. The structures of the reactants and the two transition states are optimized for each level of theory and for each solvent investigated. The reaction rate constants for the two alkylation pathways are computed using the expression of CTST with the SMD solvation model (Equation (4.20)) and the selectivity ratio is computed using Equation (4.56). All the rate-constant calculations are performed at 298 K. In descending order of dielectric constant (ε), the following solvents are investigated: acetonitrile (ε=35.94), methanol (ε=32.66), ethanol (ε=24.55), acetone (ε=20.56), tetrahydrofuran (THF) (ε=7.58), 1,2-dimethoxyethane (1,2-DME) (ε=7.20), ethyl acetate (ε=6.02) and 1,4- dioxane (ε=2.21). The solvents acetonitrile and methanol have been already analysed in depth in Chapters4 and5. We remind the reader that the SMD model requires seven macroscopic descriptors (the dielectric constant (ε), the Abraham’s hydrogen bond acidity (α) and basicity

(β), the refractive index (nD), the bulk surface tension at 298 K (γ), the aromaticity (φ) and the electronegative halogenicity (ψ)) in order to describe the solvent medium. The default list of solvents for the SMD model does not contain the solvent 1,2-DME. Therefore, for this particular solvent we need to specify manually the descriptors: ε is obtained from experiments,303 whereas all the other properties are estimated using the group contribution methods developed by Shel- don et al.304 and Foli´cet al.42 The values used for the bulk solvent descriptors for 1,2-DME are listed in Table 6.1. For the rest of solvents investigated, the default values speciﬁed in Gaussian 09 are used.208

Solvation data used in the training set of the SMD model

For the set of solvents we investigate here, the following solvation data are available in the SMD model training set:24 for acetonitrile, solvation free energies for seven neutral and 69 ionic solutes; for methanol, solvation free energies for 80 ionic solutes; for ethanol, solvation free energies for eight neutral solutes; for THF, solvation free energies for seven neutral solutes; for ethyl acetate, solvation free energies for 21 neutral solutes and four solvent-water transfer free energies. Transfer free energies are included in the training set in the absence of experimental absolute solvation free-energy data for some solutes.24 For acetone, 1,2-DME and 1,4-dioxane Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 194

Table 6.1: Macroscopic descriptors for the solvent 1,2-dimethoxyethane (1,2-DME) determined either from experiments (Expt.) or group-contribution (GC) techniques. The various symbols are deﬁned in the text.

Property Value Technique Ref. ε 7.225 Expt. [303] α 0.000 GC [42,304] β 0.525 GC [42,304] nD 1.166 GC [42,304] γ 32.71 GC [42,304] φ 0.000 GC [42,304] ψ 0.000 GC [42,304] The surface tension (γ) is given in units of −2 cal mol−1 A˚ . The other properties are di- mensionless. no solvation data are available in the training set of the SMD model. Of the reactants and products that are involved in the Williamson reaction studied here, none of them is included in the list of 318 solutes for which solvation data have been used in the training set of the SMD model.24 Naturally, one would expect that the SMD model would result in more accurate solvation free-energy predictions when the reactive species investigated are included, or closely resemble species that are included, in the model’s training set. The mean signed error (MSE) and mean unsigned error (MUE) for these solvents, reported by Marenich et al.,24 are shown in Table 6.2. When neutral solutes are considered, the MUE values for aprotic solvents are fairly low, ranging from 1.63 kJ mol−1 to 2.97 kJ mol−1. However, the small number of data points (< 10) used in the training set (apart from ethyl acetate) should not be overlooked. None of the aprotic solvents has a MUE as large as that of the protic solvent ethanol (MUE=6.99 kJ mol−1). When ionic solutes are considered, the reported MUE values are more pronounced than those of neutral solutes. In particular, the MUE for acetonitrile, averaged over 69 data points, is 24.70 kJ mol−1 and the MUE for methanol, averaged over 80 data points, is 8.79 kJ mol−1. Therefore, it is expected that solvation energy calculations involving ionic (charged) solutes and protic solvents (such as methanol and ethanol) will be susceptible to larger errors than calculations involving neutral solutes and non-protic solvents. As it has been highlighted in Chapters 4 and 5, the number of available solvation data in the training set of the solvation SMD model, as well as the accuracy that the model captures these data, has a signiﬁcant impact on the accuracy of the model’s predictions. Therefore, it is useful to be aware of the level of accuracy expected from the SMD model. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 195

Table 6.2: Mean signed error (MSE) and mean unsigned error (MUE) in SMD calculations for solvation free energies and transfer free energies for neutral and ionic solutes per solvent, reported by Marenich et al.24 The SMD calculations have been performed at the M05-2X/6-31G(d) level of theory.

Solvent Data points Solutes MSE/kJ mol−1 MUE/kJ mol−1 Acetonitrile 7 Neutral −2.59 2.64 69 Ionic 12.56 24.70 Methanol 80 Ionic 2.51 8.79 Ethanol 8 Neutral −6.91 6.99 THF 7 Neutral 0.71 2.05 Ethyl Acetate 21 Neutral 2.09 2.97 Ethyl Acetate/water 4 Neutral −1.63 1.63

6.2 Computational results

6.2.1 Transition-state structures for the O- and C-alkylation

The transition-state structures of the O- and C-alkylation pathways are optimized at each level of theory considered and for all the different solvents investigated. Using the iterative closest point (ICP) algorithm (as discussed in Section 4.6.2), we are able to provide a quantitative measurement of the structural differences between the transition-state conformations obtained in the liquid phase and in vacuum for both alkylation pathways. In order to be consistent with the results presented in Chapter 4, the structures optimized at the B3LYP/6-31+G(d) level of theory are used for comparisons. We implement the ICP algorithm using as target point set the optimized Cartesian coordinates of the transition-state structure in vacuum and as source point set the optimized Cartesian coordinates of the transition-state structure in each one of the different solvents investigated for both alkylation pathways. For the transition-state structure of the O-alkylation pathway, only the TS 2 conformation (as identified in Section 4.6.2) is considered for comparisons. The root-mean-square error (RMSE) values obtained for the different solvents are presented in Table 6.3 (and plotted in Figure B.4 of AppendixB) in descending order of dielectric constant of solvent. As it may be seen in Table 6.3, the more the dielectric constants of the various solvents di- verge from the dielectric constant of vacuum (which is equal to 1), the more the transition-state structures in the liquid phase differ from the transition-state structure in vacuum (indicated by larger RMSE values). There is a distinct pattern of increasing RMSE values with increasing dielectric constant which is noticeable in case of both the O- and the C-alkylation pathways. The RMSE values cover for the O-alkylation transition-state structures a range from 0.9571 A˚ to Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 196

0.1251 A˚ and for the C-alkylation transition-state structures a narrower range from 0.1144 A˚ to 0.0611 A.˚ The transition-state structure leading to the O-alkylation pathway involves the charge transfer from the oxygen atom to the bromide ion against the attractive force applied by the sodium cation; these electrostatic interactions are greatly influenced by solvent’s dielectric effect. As a result, significant structural variations are expected for solvents with varying dielectric constant. Conversely, the transition-state structure leading to the C-alkylation pathway is comparatively less sensitive to the dielectric constant of the solvent medium, as it involves relatively little removal of the charge from the vicinity of the sodium cation. Therefore, even though some deviations from the vacuum transition-state structure of the C-alkylation pathway are noticeable as the dielectric constant of the solvent medium varies, these deviations are relatively smaller than the deviations observed for the O-alkylation pathway. Overall, these deviations suggest that the effect of solvent on the geometries of the transition-state structures from the gas phase to the liquid phase may be significant and they should be generally taken into account by re- optimizing the corresponding gas-phase structures within the field generated by the dielectric of the solvent.

Table 6.3: Root-mean-square error (RMSE) values for the structural changes between the transition-state structures obtained in diﬀerent solvents and in vacuum for the O- and C- alkylation pathways optimized at the B3LYP/6-31+G(d) level of theory obtained using the iterative closest point (ICP) algorithm. The solvents are listed in descending order of dielectric constant ε. For the O-alkylation pathway, only the transition-state structure TS 2 is considered.

RMSE (A)˚ Solvent ε For O-alkylation For C-alkylation Acetonitrile 35.94 0.9571 0.1128 Methanol 32.66 0.2664 0.1144 Ethanol 24.55 0.2691 0.1114 Acetone 20.56 0.2803 0.1109 THF 7.58 0.1550 0.1013 1,2-DME 7.20 0.1583 0.0999 Ethyl acetate 6.02 0.1485 0.0961 1,4-Dioxane 2.21 0.1251 0.0611 Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 197

6.2.2 Selectivity and reaction kinetics at 298 K

◦,QM QM QM QM Our calculated values at 298 K for ∆GC-O , kO /kC and k for all the solvents and all the levels of theory tested are summarized in Figures 6.1-6.4. The solvents are listed along the x-axis in descending order of dielectric constant. The different symbols and colours correspond to the 18 different combinations of functionals and basis sets tested. All the calculated values are reported in Tables 6.4-6.9. ◦,QM As illustrated in Figure 6.1, the range of QM-calculated values for ∆GC-O is fairly large. ◦,QM For the different solvents and levels of theory, the QM-calculated ∆GC-O values cover a range of more than 15.00 kJ mol−1, including both positive and negative values. For each solvent, ◦,QM the calculated ∆GC-O values with the various levels of theory are quite scattered. For solvents with high dielectric constant (e.g., ε> 20), namely acetonitrile, methanol, ethanol and acetone, ◦,QM −1 corresponding ∆GC-O ranges of approximately 11.0 kJ mol are obtained, whereas for solvents with low dielectric constant (e.g., ε ≤ 20), namely THF, 1,2-DME, ethyl acetate and 1,4-dioxane, ◦,QM −1 corresponding ∆GC-O ranges of 9.70 kJ mol are obtained. The level of accuracy achieved by the SMD model for the calculation of solvation free energies ◦,QM is crucial on determining accurately the value of ∆GC-O across the various solvents. For protic solvents, namely methanol and ethanol, quite a few levels of theory predict negative values ◦,QM for ∆GC-O . In the literature, it has been established that, in these solvents, the formation of the C-alkylated product is mainly because of the effect of hydrogen-bonding interactions.14 Such interactions are insufficiently captured by the continuum SMD model. This factor, in combination with appreciable errors associated with the SMD parametrisation of the solvation free energies for ionic solutes in methanol and the lack of solvation data for ionic solutes in ◦,QM ethanol (cf. Table 6.2), unavoidably leads to large discrepancies in the predicted ∆GC-O values. For 1,4-dioxane, according to experimental data from the literature,23,269 the dominant product of the reaction is the C-alkylated product (cf. Tables 4.1 and 4.2). A few levels of theory capture this trend, but the majority gives the opposite result. This may be attributed to the lack of solvation free-energy data for 1,4-dioxane in the SMD training set. Among the different functionals and basis sets used, some general trends are apparent. For ◦,QM example, calculations with the B3LYP functional yield solely positive values for ∆GC-O , which are always higher than the values obtained from any other functional evaluated. Hence, the ◦,QM B3LYP functional family predicts the largest positive values for ∆GC-O ; this is achieved for high-dielectric constant solvents using B3LYP/cc-pVDZ and for low-dielectric constant solvents using B3LYP/cc-pVTZ, with the exemption of 1,4-dioxane for which the highest value is obtained Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 198 using B3LYP/6-31+G(d). The M05-2X and M06-2X functionals predict positive values for ◦,QM ∆GC-O in solvents acetonitrile, acetone, THF and ethyl acetate and both positive and negative ◦,QM values for ∆GC-O in solvents methanol, ethanol, 1,2-DME and 1,4-dioxane – although, for 1,2-DME the only negative value, predicted with M06-2X/6-311++G(d,p), is marginally below ◦,QM zero. The wB97xD functional predicts positive values for ∆GC-O in acetone, THF, 1,2-DME ◦,QM and ethyl acetate; negative values for ∆GC-O in methanol and 1,4-dioxane; and both positive ◦,QM and negative values for ∆GC-O in acetonitrile – although marginally when combined with the 6-311++G(d,p) basis set – and ethanol. ◦,QM Interestingly, the range of calculated ∆GC-O values per solvent using a single functional, e.g., B3LYP, is significantly less dispersed than such range over all the functionals. In order to ◦,QM make consistent comparisons between the different functionals, calculations for ∆GC-O with the two basis sets that are common to all functionals (6-31+G(d) and 6-311++G(d,p)) are shown in Figure 6.2. Across the various solvents and functionals, the results with the large basis set (6-311++G(d,p)) are always lower than those with the small basis set (6-31+G(d)), on average by 1.20 kJ mol−1, 1.96 kJ mol−1, 1.55 kJ mol−1 and 2.05 kJ mol−1 for functionals B3LYP, M05- 2X, M06-2X and wB97xD, respectively. Using the 6-311++G(d,p) and the 6-31+G(d) basis ◦,QM sets with the wB97xD functional, the decrease in the range of ∆GC-O has the smallest value at 0.78 kJ mol−1 in the solvent methanol and the largest value at 4.17 kJ mol−1 in the solvent ethanol. On the other hand, the energy differences observed between the various functionals with the same basis set, for example between B3LYP/6-31+G(d) and M05-2X/6-31+G(d), are ◦,QM less homogeneous. For B3LYP, the predicted ∆GC-O values are distinctly different from the other functionals and these differences are well above the discrepancies discussed before between the same functional with different basis sets. The largest differences observed in the case of B3LYP with other functionals, using the same basis set, are the following: a value of 8.23 kJ mol−1 in solvent 1,4-dioxane between B3LYP/6-31+G(d) and wB97xD/6-31+G(d) and a value of 8.61 kJ mol−1 in solvent methanol between B3LYP/6-311++G(d,p) and M06-2X/6- 311++G(d,p). Smaller energy differences are observed among functionals M05-2X, M06-2X and wB97xD considering either of the two basis sets. The energy differences range from 0.13 to 3.04 kJ mol−1 for the 6-31+G(d) basis set and from 0.12 to 1.63 kJ mol−1 for the 6-311++G(d,p) basis set. The preceding analysis allows us to deduce that although the basis set plays an important role, it is actually the functional that determines the output of the energy calculation. As discussed in Chapter2, the form of the functional selected for the exchange-correlation energy constitutes the key difference among the different DFT methods.58 As Jensen pointed Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 199 out,58 the quality of each functional has to be settled eventually by comparing the performance with experiments or high-level wave mechanics. Conversely, highly accurate experimental values play a pivotal role in selecting the level of theory that captures best the experimental behaviour. However, it should be emphasized that any conclusions drawn with respect to the performance of a certain level of theory are only valid for the specific system and property investigated. Fortuitous cancellations of systematic errors for a level of theory might also benefit or damage its performance depending on whether an absolute or relative quantity is predicted. Different levels of theory will portray different trends for different systems and properties studied. No guarantee is provided regarding the performance of a level of theory identified as best for a particular system and property, when studying a different system and property. As scientific research progresses on the area of DFT, newly developed and more robust functionals are likely to appear. As discussed in Chapter2, the form of the functional selected for the exchange-correlation energy constitutes the key difference among the different DFT methods.58 As Jensen pointed out,58 the quality of each functional has to be settled eventually by comparing the performance with experiments or high-level wave mechanics. Conversely, highly accurate experimental values play a pivotal role in selecting the level of theory that captures best the experimental behaviour. However, it should be emphasized that any conclusions drawn with respect to the performance of a certain level of theory are only valid for the specific system and property investigated. Fortuitous cancellations of systematic errors for a level of theory might also benefit or damage its performance depending on whether an absolute or relative quantity is predicted. Different levels of theory will portray different trends for different systems and properties studied. No guarantee is provided regarding the performance of a level of theory identified as best for a particular system and property, when studying a different system and property. As scientific research progresses on the area of DFT, newly developed and more robust functionals are likely to appear. QM QM ◦,QM As has been shown in Equation (4.56), the ratio kO /kC depends exponentially on ∆GC-O , and so a small change on this value may result in a notably different output. Due to the wide range of values for the Gibbs free-energy difference calculated per solvent, large variations are QM QM expected in the predicted selectivity ratio values. The calculated selectivity ratios kO /kC per solvent are summarized in Figure 6.3. The black line at a ratio value of 1.0 assists the reader to distinguish the selectivity preference predicted by each level of theory towards the O-alkylated product (above 1.0) or the C-alkylated product (below 1.0) of the reaction. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 200

12.0 B3LYP/6-31G(d) 10.0 B3LYP/6-31+G(d) B3LYP/6-311+G(d,p) 8.0 B3LYP/6-311++G(d,p) B3LYP/6-311G(2d,d,p)

-1 6.0 B3LYP/cc-pVDZ B3LYP/cc-pVTZ 4.0 M05-2X/6-31G(d) M05-2X/6-31+G(d)

/kJ mol M05-2X/6-31+G(d,p) 2.0 M05-2X/6-311++G(d,p)

º,QM C-O M05-2X/cc-pVTZ

G | 0.0 M06-2X/6-31+G(d) ∆ M06-2X/6-311++G(d,p) -2.0 M06-2X/cc-pVDZ M06-2X/cc-pVTZ -4.0 wB97xD/6-31+G(d) wB97xD/6-311++G(d,p)

-6.0 Acetonitrile Methanol Ethanol Acetone THF 1,2-DME Ethyl Acetate1,4-dioxane

Solvents

Figure 6.1: QM-calculated Gibbs free-energy difference between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in different solvents at 298 K obtained for various levels of theory, indicated by the different symbols and colours, and the SMD solvation model implemented in Gaussian 09. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 201

12.0

10.0

8.0

-1 6.0 B3LYP/6-31+G(d) B3LYP/6-311++G(d,p) 4.0 M05-2X/6-31+G(d) M05-2X/6-311++G(d,p)

/kJ mol M06-2X/6-31+G(d) 2.0 M06-2X/6-311++G(d,p)

º,QM C-O wB97xD/6-31+G(d)

G | 0.0 wB97xD/6-311++G(d,p) ∆ -2.0

-4.0

-6.0 Acetonitrile Methanol Ethanol Acetone THF 1,2-DME Ethyl Acetate1,4-dioxane

Solvents

Figure 6.2: Gibbs free-energy difference between the C- and the O-alkylation transition-state structures of the reaction between sodium β-naphthoxide and benzyl bromide in different solvents at 298 K obtained for selected levels of theory, indicated by the different symbols and colours, and the SMD solvation model implemented in Gaussian 09. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 202

100.0 B3LYP/6-31G(d) B3LYP/6-31+G(d) B3LYP/6-311+G(d,p) B3LYP/6-311++G(d,p) B3LYP/6-311G(2d,d,p) 10.0 B3LYP/cc-pVDZ B3LYP/cc-pVTZ M05-2X/6-31G(d) QM C

k M05-2X/6-31+G(d) / M05-2X/6-31+G(d,p)

QM O M05-2X/6-311++G(d,p) k M05-2X/cc-pVTZ 1.0 M06-2X/6-31+G(d) M06-2X/6-311++G(d,p) M06-2X/cc-pVDZ M06-2X/cc-pVTZ wB97xD/6-31+G(d) wB97xD/6-311++G(d,p)

0.1 Acetonitrile Methanol Ethanol Acetone THF 1,2-DME Ethyl Acetate1,4-dioxane

Solvents

Figure 6.3: Selectivity ratio of the reaction between sodium β-naphthoxide and benzyl bromide in diﬀerent solvents at 298 K calculated using Equation (4.56) for various levels of theory, indicated by the diﬀerent symbols and colours, and the SMD solvation model implemented in Gaussian 09. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 203

103 102 101 -1

s 100 -1 -1 10 O-alkylation mol

3 C-alkylation 10-2 /dm 10-3 QM k 10-4 10-5 -6 10 Acetonitrile Methanol Ethanol Acetone THF 1,2-DME Ethyl Acetate1,4-dioxane

Solvents

Figure 6.4: Rate constants for the O- and the C-alkylation pathways of the reaction between sodium β-naphthoxide and benzyl bromide in diﬀerent solvents at 298 K calculated using Equa- tion (4.20) for various levels of theory, indicated by the diﬀerent symbols and colours, and the SMD solvation model implemented in Gaussian 09. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 204

As may be appreciated by reviewing Figure 6.3, the variation is indeed remarkable. For solvents acetonitrile, acetone, THF, 1,2-DME and ethyl acetate, the predictions of all levels of theory for the selectivity preference of the reaction towards the O-alkylated product are in line with experimental observations reported in the literature (cf. Tables 4.1 and 4.2). For protic solvents, namely methanol and ethanol, a number of levels of theory predict a reverse selectivity ratio. For 1,4-dioxane, the selectivity preference of the reaction according to experimental data23,269 is in favour of the C-alkylated product (cf. Tables 4.1 and 4.2). This preference is captured by a few levels of theory, even though the majority gives the opposite result. For the sake of clarity, the ranges, instead of the explicit values, of the calculated reaction rate constants for the O- and C-alkylation pathways are graphically displayed in Figure 6.4. It is QM easily noticeable that across the different solvents the range of kC values is consistently wider QM than the range of kO values. This result may be explained by the inherently slower rate of the C-alkylation reaction and hence the smaller rate-constant value,14,23,269–271 which is more QM difficult to be captured. As a consequence, the ranges obtained for the calculated values of kC given by the various levels of theory extend towards higher values, suggesting an overestimation QM QM of the kC values. The serendipitous result of obtaining the widest range of kC values in 1,4- dioxane, in which the rate constant for the C-alkylation reaction has the smallest value amongst all the solvents studied, corroborates this explanation. While the predicted results appear quite dispersed for all solvents, a reasonable justification seems to be possible for the low-dielectric constant solvents as well as the protic solvents. In these solvents, electrostatic and strong hydrogen-bonding interactions, particularly in the first solvation-shell, govern the mechanism of the C-alkylation reaction. Under the continuum solvation approximation, these interactions are more difficult to capture; therefore, a more detailed level of description of the solvent medium would be crucial to improve the solvation model’s performance. We make a few additional remarks with respect to solvent modelling. It has been shown by Ando and Morokuma,305 who studied an alkylation reaction between lithium enolate and methyl chloride in THF, that a consideration of explicit THF molecules in the QM calculations gave a completely different picture of the occurring solvation phenomena. In their study, they used the IEF-PCM model to account for the bulk solvent effects, as well as a number of explicit THF molecules (from 0 to 6) to account for site-specific interactions. These interactions are crucial to identify the lowest energy pathway and may not be captured sufficiently by the sole use of the IEF-PCM model. Their findings suggest that microsolvation plays an important role Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 205 in the determination of the transition-state structures and that the energies of the reactants and the various transition states vary depending on the number of explicit THF molecules included. The lowest energy pathway was found to be the one where three THF molecules were included, and the stabilization was due to the involvement of two of them in the lithium cation-chlorine anion interactions. The study of Ando and Morokuma305 points towards the need of an explicit treatment of THF molecules, which may be beneficial in our calculations and improve our insight for the solvated system. A similar treatment would seem advantageous also for 1,4-dioxane, due to its structural similarities with THF (only one additional oxygen atom). In the study of Kelly et al.,294 it was shown that when studying systems with ionic species, in particular monatomic anions, the inclusion of a single explicit solvent molecule (in their case, a water molecule) auxiliary to the implicit solvation SM6 model was sufficient to account for strong short-range effects and to improve significantly the accuracy of predictions. This approach has the extra advantages that is straightforward and avoids the problems of including many solvent molecules explicitly in the calculations (e.g., increased computational cost, conformational sampling). Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 206 1 − s 1 − 3 1 4 5 4 5 5 5 6 6 2 4 2 3 5 1 1 2 mol − − − − − − − − − − − − − − − − − − 3 ) and the rate constants of the 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 QM C × × × × × × × × × × × × × × × × × × /dm /k QM C QM 8.913 1.042 2.054 4.602 k 1.360 1.135 2.895 8.813 1.349 1.416 6.328 8.589 3.470 3.436 2.348 6.474 9.315 5.162 O k 1 − s 1 − 3 1 5 4 4 4 4 5 5 1 2 4 3 4 1 2 1 3 mol − − − − − − − − − − − − − − − − − − 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × /dm QM O k ), the selectivity ratio ( O QM − , QM C ◦ C 28 2.455 43 1.488 66 3.046 32 2.707 7651 2.357 88 2.173 33 8.703 10 3.283 38 2.279 2.140 4424 1.515 8.263 45 1.243 1902 3.097 1.498 73 4.727 38 1.950 31 2.928 ...... G /k QM O k 1 − 25 0 92 1 11 0 4593628882 20 91 7 9 24 16 3 60 1 1264 0 0 3690 13 11 8881 0 1 9752 0 0 ...... /kJ mol 3 1 2 3 0 2 2 O − − − − − − − QM − , ◦ C G ) of the reaction in ethanol at 298 K at all the levels of theory tested. QM k wB97xD/6-311++G(d,p) M06-2X/cc-pVTZ wB97xD/6-31+G(d) 0 B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p)B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 7 M05-2X/6-31G(d) 4 M05-2X/6-31+G(d) 5 7 M06-2X/6-31+G(d) 6 M06-2X/6-311++G(d,p) 2 M06-2X/cc-pVDZ 0 Level of theoryB3LYP/6-31G(d)B3LYP/6-31+G(d) ∆ 6 5 M05-2X/6-31+G(d,p) M05-2X/6-311++G(d,p) M05-2X/cc-pVTZ 0 two alkylation pathways ( Table 6.4: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 207 1 − s 1 − 2 1 5 4 3 2 3 2 1 1 5 4 5 5 4 6 6 mol − − − − − − − − − − 0 − − − − − − − 3 ) and the rate constants of the 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 QM C × × × × × × × × × × × × × × × × × × /dm /k QM C QM 1.575 2.774 7.172 3.513 4.449 5.747 1.289 6.382 1.168 9.821 1.946 k 3.926 1.909 3.493 3.429 1.124 2.533 2.062 O k 1 − s 1 − 3 2 1 4 3 3 1 3 1 3 3 3 3 4 4 mol − − − − − − − − − 0 0 − − − − − − 0 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × /dm QM O k ), the selectivity ratio ( O QM − , QM C ◦ C 58 2.495 50 9.717 59 1.857 90 1.369 13 5.040 18 1.252 38 8.222 95 1.242 2887 4.199 3.635 5014 9.450 50 1.158 15 1.046 60 3.388 79 1.662 42 1.191 1.569 53 1.630 ...... G /k QM O k 1 − 09 1 16 3 27 2 29 3 20 1 86 2 49 6 504554 4 1 1 6262413933 49 99 33 33 30 30 65 57 13 21 41 ...... /kJ mol O QM − , ◦ C G ) of the reaction in acetone at 298 K at all the levels of theory tested. QM k wB97xD/6-311++G(d,p) 1 wB97xD/6-31+G(d) 3 M06-2X/cc-pVTZ 2 M06-2X/cc-pVDZ 3 M06-2X/6-311++G(d,p) 0 M06-2X/6-31+G(d) 1 M05-2X/cc-pVTZ 4 M05-2X/6-31+G(d)M05-2X/6-31+G(d,p)M05-2X/6-311++G(d,p) 3 1 1 B3LYP/6-31+G(d)B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p)B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 8 9 M05-2X/6-31G(d) 8 8 10 9 6 Level of theoryB3LYP/6-31G(d) ∆ 9 two alkylation pathways ( Table 6.5: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 208 1 − s 1 − 4 2 1 4 4 3 1 3 1 5 5 5 4 6 6 1 mol − − − − − − − − − − − − − − 0 0 − − 3 ) and the rate constants of the 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × QM C × × × × × × × × × × × × × × × × × /dm /k QM C QM 1.889 3.971 1.401 8.072 8.238 1.154 2.747 k 2.368 1.300 7.916 3.622 3.565 1.423 4.204 2.408 3.667 2.218 4.536 O k 1 − s 1 − 2 4 3 2 1 2 1 3 3 3 3 4 4 2 mol − − 0 − − − − − − − − − − − − 0 +1 0 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × /dm QM O k ), the selectivity ratio ( O QM − , QM C ◦ C 27 2.407 72 1.477 06 4.292 94 3.183 36 1.119 34 2.700 33 2.013 29 2.972 6391 3.058 53 1.395 92 1.432 22 1.281 65 4.728 45 2.339 68 1.479 71 4.649 05 1.045 9.316 ...... G /k QM O k 1 − 53 1 33 3 69 3 32 3 65 1 05 2 83 7 0205058163 38 93 58 15 39 19 35 74 33 68 55 93 61 12 4 2 2 ...... /kJ mol O QM − , ◦ C G ) of the reaction in THF at 298 K at all the levels of theory tested. QM k wB97xD/6-311++G(d,p) 0 wB97xD/6-31+G(d) 3 M06-2X/cc-pVTZ 2 M06-2X/cc-pVDZ 3 M06-2X/6-311++G(d,p) 0 M06-2X/6-31+G(d) 2 M05-2X/cc-pVTZ 4 B3LYP/6-311+G(d,p) 9 Level of theoryB3LYP/6-31G(d)B3LYP/6-31+G(d)B3LYP/6-311++G(d,p) ∆ B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 10 9 M05-2X/6-31G(d) 8 M05-2X/6-31+G(d) 8 M05-2X/6-31+G(d,p)M05-2X/6-311++G(d,p) 9 10 6 3 1 1 two alkylation pathways ( Table 6.6: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 209 1 − s 1 − 2 1 4 3 1 2 3 1 4 4 5 5 4 6 6 1 mol − − − − − − − − − − − − − − 0 0 − − 3 ) and the rate constants of the 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 QM C × × × × × × × × × × × × × × × × × × /dm /k QM C QM 2.751 5.914 2.107 1.223 1.776 1.223 4.151 k 1.955 1.151 3.370 5.033 4.954 1.994 5.990 3.366 5.653 3.405 6.953 O k 1 − s 1 − 4 3 2 1 2 1 3 2 3 3 3 4 4 2 mol − 0 − − − − − − − − − − − − − 0 +1 +1 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × /dm QM O k ), the selectivity ratio ( O QM − , QM C ◦ C 36 3.729 99 1.767 33 4.901 95 3.607 01 1.241 92 3.414 54 2.298 70 3.316 2484 3.597 74 1.612 77 1.598 58 1.426 94 5.498 27 2.752 64 1.726 82 5.448 67 1.302 1.159 ...... G /k QM O k 1 − 73 1 79 2 01 2 61 2 09 1 56 1 14 5 5054512617 31 46 47 70 31 51 28 22 27 16 45 20 51 9 3 1 1 ...... /kJ mol 0 O − QM − , ◦ C G ) of the reaction in 1,2-DME at 298 K at all the levels of theory tested. QM k wB97xD/6-311++G(d,p) 0 wB97xD/6-31+G(d) 2 M06-2X/cc-pVTZ 2 M06-2X/6-311++G(d,p) M06-2X/cc-pVDZ 2 M06-2X/6-31+G(d) 1 M05-2X/cc-pVTZ 4 Level of theoryB3LYP/6-31G(d)B3LYP/6-31+G(d)B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p) ∆ B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 8 8 9 M05-2X/6-31G(d) 8 M05-2X/6-31+G(d) 8 M05-2X/6-31+G(d,p)M05-2X/6-311++G(d,p) 9 9 5 3 1 1 two alkylation pathways ( Table 6.7: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 210 1 − s 1 − 2 1 4 4 3 1 3 1 5 4 5 5 4 6 6 1 mol − − − − − − − − − − − − − − − − 0 0 3 ) and the rate constants of the 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 QM C × × × × × × × × × × × × × × × × × × /dm /k QM C QM 1.535 3.371 1.310 7.925 7.640 1.094 2.650 k 1.247 7.457 1.928 2.832 2.791 1.556 3.746 1.940 3.945 2.169 4.432 O k 1 − s 1 − 2 1 4 3 3 1 2 1 3 2 3 3 3 4 4 mol − − − − − − − − − − − − − − − 0 +1 0 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × /dm QM O k ), the selectivity ratio ( O QM − , QM C ◦ C 37 2.096 29 7.723 04 3.978 05 3.211 31 9.997 51 2.749 25 1.920 42 3.025 6710 2.884 28 1.197 45 1.198 65 1.073 14 5.234 91 2.066 16 1.259 02 4.798 19 1.089 9.702 ...... G /k QM O k 1 − 71 1 09 2 67 3 39 4 56 1 22 2 81 7 0318229866 38 91 62 28 42 09 38 90 33 84 55 08 64 12 5 2 2 ...... /kJ mol O QM − , ◦ C G ) of the reaction in ethyl acetate at 298 K at all the levels of theory tested. QM k wB97xD/6-311++G(d,p) 0 wB97xD/6-31+G(d) 2 M06-2X/cc-pVTZ 2 M06-2X/cc-pVDZ 3 M06-2X/6-311++G(d,p) 0 M06-2X/6-31+G(d) 2 M05-2X/cc-pVTZ 4 Level of theoryB3LYP/6-31G(d)B3LYP/6-31+G(d)B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p) ∆ B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 10 9 9 M05-2X/6-31G(d) 8 M05-2X/6-31+G(d) 8 M05-2X/6-31+G(d,p)M05-2X/6-311++G(d,p) 9 10 6 3 2 1 two alkylation pathways ( Table 6.8: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 211 1 − s 1 − 1 3 2 1 1 3 3 4 4 3 5 5 mol − − 0 − − − − − − − − − +1 +2 +2 0 − 0 3 ) and the rate constants of the 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 QM C × × × × × × × × × × × × × × × × × × /dm /k QM C QM 1.799 4.664 6.043 3.845 4.318 2.551 k 1.209 2.580 2.476 2.304 2.302 1.332 7.812 2.016 3.763 1.211 2.337 5.471 O k 1 − s 1 − 3 2 1 1 2 2 3 3 2 4 4 1 mol 0 − − − − 0 − − − − − − − − +2 +2 0 +2 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × /dm QM O k ), the selectivity ratio ( O QM − , QM C ◦ C 8660 5.208 1.082 16 5.429 17 4.492 54 1.378 98 4.237 63 3.182 9575 2.050 49 6.126 11 3.568 14 3.248 63 1.350 41 9.871 25 4.721 98 1.223 2.397 76 4.159 86 2.016 ...... G /k QM O k 1 − 3023 0 0 31 1 33 1 61 0 09 0 1293755271 7 28 24 77 15 83 14 60 10 12 23 31 3 1 2 4679 0 0 ...... /kJ mol 0 1 1 0 0 0 O − − − − − − QM − , ◦ C G ) of the reaction in 1,4-dioxane at 298 K at all the levels of theory tested. QM k wB97xD/6-31+G(d) wB97xD/6-311++G(d,p) M06-2X/cc-pVTZ 0 M06-2X/6-311++G(d,p) M06-2X/cc-pVDZ 0 M06-2X/6-31+G(d) Level of theoryB3LYP/6-31G(d)B3LYP/6-31+G(d)B3LYP/6-311+G(d,p)B3LYP/6-311++G(d,p) ∆ B3LYP/6-311G(2d,d,p)B3LYP/cc-pVDZB3LYP/cc-pVTZ 5 6 7 M05-2X/6-31G(d) 6 M05-2X/6-31+G(d) 5 M05-2X/6-31+G(d,p) M05-2X/6-311++G(d,p) M05-2X/cc-pVTZ 6 7 2 1 2 two alkylation pathways ( Table 6.9: QM-calculated values for the Gibbs free-energy difference (∆ Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 212 6.3 Selection of available experimental data

In order to know which level of theory describes best the experimental data, we will have to compare the results with either highly accurate experimental data or benchmarked high-level theoretical calculations in the limit of a complete basis set.47 For the set of solvents in which the Williamson reaction is computationally studied here, experimental data may be found in the literature,14,23,269–271 as well as in this work. These data are summarized in Table 6.11. Based on the experimental technique used, we organize them in two groups: in one group the results of Ganase23 and this work, where in situ 1H NMR spectrometry is used to monitor the progress of the reaction; and in another group the results of other researchers,14,269–271 where work-up techniques are used to isolate and purify the final products. As has been discussed already in Section 4.2, the results of previous experimental studies that use work-up techniques are prone to errors due to material losses, which may be significant, and inconsistencies due to different experimental reaction conditions (e.g., different temperatures) and work-up procedures. In addition, as they do not monitor the reaction in situ, concentration changes, for example due to side-reactions, are not tracked. On the other hand, NMR spectrometry, which is by no means a flawless analysis technique, is a viable alternative of monitoring the reaction in situ. With respect to the work of Ganase,23 while scrutinizing the kinetics of the reaction in acetonitrile and methanol, a number of oversights has been identified (cf. Chapters4 and5). Furthermore, some discrepancies and abnormal product ratio profiles have been observed in the kinetic experiments for the solvents ethyl acetate and 1,4-dioxane, suggesting unresolved issues during the experimental procedure; in the case of ethyl acetate, small-scale loss of the starting material benzyl bromide was also alluded in the original study. For the solvents investigated in the original study of Ganase,23 which are selected here for comparison with the QM-calculated values, we graphically extract the range of variation of the product ratio O:C based on the experimentally measured raw concentration data across the three different experiments performed in each solvent. The results may be seen in Table 6.10. In some solvents, the variation is larger than others. For example, the O:C ratio in the solvent acetone-d6 is very similar in all three experiments. On the contrary, in the solvent 1,4-dioxane the O:C ratio varies for the three experiments approximately from a value of 0.39 up to 0.70 (corresponding to percentage product O:C ratio values between 28:72% and 41:59%). These ratio values were also constantly increasing at the whole time-length that the three experiments Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 213 were monitored. This profile is observed most likely because of the consumption of the C- alkylated product in the additional alkylation reaction resulting in the formation of the double C-alkylated product. Large ranges of values are also seen for other solvents (e.g., ethanol), and even an inversion of the dominant product is suggested in the case of the solvent ethyl acetate.

Table 6.10: Graphically extracted range of variation for the product ratio of the O- over the C- alkylated product (O:C) – indicated as P1/P2 in the original study – based on the experimentally measured raw concentration data in six solvents tested in the study of Ganase.23 Only the solvents selected from that study for comparison with the QM-calculated values of this work are listed. The percentage product ratio O:C is also reported.

Solvent O:C O:C (%)

Acetone-d6 19.2-20.0 95:5 - 95:5 1,2-DME 5.00-5.88 83:17 - 85:15 Ethanol-d6 2.25-4.17 69:31 - 81:19 THF-d8 1.88-2.43 65:35 - 71:29 Ethyl acetate 0.80-1.60 44:56 - 61:39 1,4-dioxane-d8 0.39-0.69 28:72 - 41:59

With respect to this work, our conﬁdence for the quality and accuracy of the experimental kinetic data is supported by the following reasons:

• The high-standard quality of the experimental equipment used, (500 MHz Bruker Cry- oProbe Prodigy spectrometer) which allows for excellent temperature control and enhanced signal-to-noise-ratio.277

• The high resolution of the obtained spectra as a result of the previous reason, which enables a seamless integration of peak heights of the reaction species followed.

• The careful analysis of the chemical reactions occurring and their consideration in the kinetic models developed.

• The reliability of the gPROMS models developed, which is reflected in their good fitting and well-defined parameters. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 214

In summary, experimental data for the various solvents based on work-up techniques would be deemed unsuitable insofar as quantitative comparisons with the QM-calculated values would be needed; therefore, they are not henceforth considered for this task. On the other hand, the experimental data obtained with in situ 1H NMR spectroscopy are considered more reliable, and therefore are compared with the QM-calculated data in terms of the percentage rate constant ratios. The experimental data for solvents acetonitrile and methanol from the study of Ganase23 are discarded (for reasons discussed in Chapters4 and5) and instead, the results obtained in this thesis are used. In the absence of other available experimental data determined using in situ techniques for solvents ethanol, ethyl acetate and 1,4-dioxane, the results are not discarded; however, we appreciate that large uncertainty exists in the estimated ratio values for these solvents. Therefore, when comparisons are made with the QM-predicted values for these solvents, conclusions are drawn with due consideration. Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 215 [ 270 ] (%) Ref. − H NMR spectroscopy was used, Expt. C 1 k : 64:3674:2671:2995:5 This [ 23 ] 69:31 work 83:17 [ 23 ] 51:4934:66 [ 23 ] [ 23 ] [ 23 ] [ 23 ] [ 23 ] 95:5 This work 82:18 [ 23 ] O:C (%) Ref. Expt. O k 1 1 − − s s 1 1 4 4 4 4 4 5 5 4 5 4 − − − − − − − − − − − − 10 10 10 10 10 10 10 10 10 10 mol mol × × × × × × × × × × -naphthoxide and benzyl bromide at 298 K for the set 3 3 β /dm /dm 2.917 2.702 3.867 4.817 7.194 6.333 8.907 1.565 9.667 6.117 Expt. Expt. C C k k 1 1 − − s s 1 1 4 4 4 3 2 4 4 4 5 3 − − − − − − − − − − − − 10 10 10 10 10 10 10 10 10 10 mol mol × × × × × × × × × × 3 3 /dm /dm 7.581 1.298 ) are reported. Expt. Expt. O O At T=377-379 K. c Expt. C k : 6655 5.233 56 9.533 8.573 582002 1.400 21 4.440 1.635 5.000 94 n/a94 2.757 n/a 94:6 [ 271 ] 66 n/a n/a 63:37 [ 14 ] 55 n/a n/a 65:35 [ 14 ] 56 n/a n/a 100: 58 n/a n/a 63:37 [ 14 ] 20 n/a n/a 76:24 [ 14 ] 21 n/a n/a 47:53 [ 269 ] ...... ε k ε k 7 2 2 32 24 20 35 35 20 Expt. O k 8 3 a 4 6 6 c b 8 At T=329-331 K. b Methanol-d THF-d 1,2-DMEEthyl acetate1,4-Dioxane-d 6 7 Acetonitrile Acetonitrile-d Methanol 32 Ethanol 24 THF 7 1,2-DME 7 Dioxane At T=297 K. H NMR Ethanol-d 1 spectrometry Acetone-d Technique Solvent Technique Solvent a Work-up Acetone Table 6.11: Available experimental dataof of solvents the investigated Williamson in this reaction work.the between sodium percentage The data product are ratio classiﬁed valuesthe based (O:C) percentage on – rate the based constant experimental ratio on technique values the used. ( isolated When product work-up techniques amounts were – used, are reported. When Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 216 6.4 Solvent ranking

A somewhat diﬀerent – albeit equally useful and interesting – way of comparing the QM- calculated and experimental values may be on the context of solvent ranking. Solvent ranking may be used to facilitate solvent selection when the order of solvents, rather than the exact value of a property in each one of them, is of interest in a practical application. An example of such application is the design of optimal solvents using CAMD for rate maximization on a single-pathway reaction.45,306 In our work, in order to rank the eight solvents investigated computationally and experimentally, we consider as a criterion the selectivity preference of the reaction towards the O-alkylated or the C-alkylated product, according to the ratio value of the rate constants of the two alkylation pathways. First, we rank the solvents according to the experimental percentage rate Expt. Expt. Expt. Expt. Expt. constant ratio values kO over kC (kO :kC ), in descending order of kO , as shown in Table 6.12. We also rank the solvents after dividing them according to their dielectric constant (ε) into two groups: solvents with ε> 20 and solvents with ε ≤ 20, as shown in Table 6.13. This division serves to discern patterns related to the liquid-phase electrostatic interactions, a measure of which is the dielectric constant of a solvent.30

Table 6.12: Solvent ranking according to the experimental percentage rate constant ratio of Expt. Expt. Expt. kO :kC in descending order of kO . The experimental values from this work and the study of Ganase23 are used.

Expt. Expt. Solvent Rank kO :kC (%) Ref.

Acetonitrile-d3 1 95:5 This work Acetone-d6 2 95:5 [23] 1,2-DME 3 83:17 [23] Methanol-d4 4 74:26 This work Ethanol-d6 5 71:29 [23] THF-d8 6 69:31 [23] Ethyl acetate 7 51:49 [23] 1,4-dioxane-d8 8 34:66 [23] Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 217

Table 6.13: Ranking of high- and low-dielectric constant (ε) solvents according to the experimen- Expt. Expt. Expt. tal percentage rate constant ratio of kO :kC in descending order of kO . The experimental values from this work and the study of Ganase23 are used.

Expt. Expt. ε Solvent Rank kO :kC (%) Ref.

Acetonitrile-d3 1 95:5 This work Acetone-d 2 95:5 [23] ε> 20 6 Methanol-d4 3 74:26 This work Ethanol-d6 4 71:29 [23] 1,2-DME 1 83:17 [23] THF-d 2 69:31 [23] ε ≤ 20 8 Ethyl acetate 3 51:49 [23] 1,4-dioxane-d8 4 34:66 [23]

Based on our experience from studying the Williamson reaction in acetonitrile and methanol, we pick the best level of theory (M05-2X/6-31G(d)) identiﬁed for selectivity predictions in these two solvents and report the results we obtain for the entire set of solvents studied in this chapter. QM QM We list our QM-calculated values for the percentage rate constant ratio of kO :kC at the M05- 2X/6-31G(d) level of theory for all solvents in Table 6.14 and for high- and low-dielectric constant QM solvents in Table 6.15. In both tables, we rank the solvents in descending order of kO and compare with the ranking based on the experimental values.

Table 6.14: Solvent ranking according to the QM-calculated percentage rate constant ratio of QM QM QM kO :kC at the M05-2X/6-31G(d) level of theory in descending order of kO . The solvents are also ranked according to the corresponding experimental values for comparison. The experiments for solvents acetonitrile, acetone, methanol, ethanol, THF and 1,4-dioxane have been performed using deuterated solvents.

QM Expt. QM QM Expt. Expt. Solvent Rank kO :kC (%) Rank kO :kC (%) Ref. Acetonitrile 1 94:6 1 95:5 This work Acetone 2 93:7 2 95:5 [23] THF 3 93:7 6 69:31 [23] Ethyl acetate 4 92:8 7 51:49 [23] 1,2-DME 5 91:9 3 83:17 [23] Ethanol 6 77:23 5 71:29 [23] 1,4-dioxane 7 76:24 8 34:66 [23] Methanol 8 69:31 4 74:26 This work Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 218

Table 6.15: Ranking of high- and low-dielectric constant (ε) solvents according to the QM- QM QM calculated percentage rate constant ratio of kO :kC at the M05-2X/6-31G(d) level of theory QM in descending order of kO . The solvents are also ranked according to the corresponding experimental values for comparison. The experiments for solvents acetonitrile, acetone, methanol, ethanol, THF and 1,4-dioxane have been performed using deuterated solvents.

QM Expt. QM QM Expt. Expt. ε Solvent Rank kO :kC (%) Rank kO :kC (%) Ref. Acetonitrile 1 94:6 1 95:5 This work Acetone 2 93:7 2 95:5 [23] ε> 20 Ethanol 3 77:23 4 71:29 [23] Methanol 4 69:31 3 74:26 This work THF 1 93:7 2 69:31 [23] Ethyl acetate 2 92:8 3 51:49 [23] ε ≤ 20 1,2-DME 3 91:9 1 83:17 [23] 1,4-dioxane 4 76:24 4 34:66 [23]

By inspecting Table 6.14, we see that QM predictions and experiments agree in the order of the first two solvents, acetonitrile and acetone. Even more, the corresponding ratio values are in excellent quantitative agreement. In these two solvents, the reaction through the O-alkylation QM pathway is the dominant reaction, with the highest percentage value of kO compared with any other solvent tested. The fact that QM predictions and experiments are in consonance, both qualitatively and quantitatively, is encouraging. For the rest of solvents, the absolute ranking order is not strictly followed. A more insightful analysis is made by looking the solvents when categorized according to their dielectric constant. When inspecting Table 6.15, it is easily noticeable that for high-dielectric constant solvents the ranking order is nearly followed, with the QM-calculated ratio values for ethanol and methanol being in reverse order. Yet, the corresponding QM-predicted and experimental values for these two solvents are in good quantitative agreement: for ethanol, the predicted value of 77:23 is overestimated compared with the experimental value of 71:29, although the predicted value is still included within the experimental range of variation of the experimental value (as shown in Table 6.10); for methanol, the predicted value of 69:31 is underestimated compared with the experimental value of 74:26. Possible sources of discrepancy arise most likely from the poor treatment of hydrogen-bonding interactions in the SMD solvation model. Hydrogen-bonding is the driving force in the formation of the C-alkylated product of the reaction in protic solvents. For low-dielectric constant solvents, the QM predictions and the experiments match only in the order of the last solvent 1,4-dioxane. Yet for solvents THF, ethyl acetate and 1,4-dioxane, QM Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 219 calculations and experiments follow the same relative order. Compared with the experiments, the QM-predicted values for this group of solvents appear to be significantly overestimated towards the O-alkylated product. Despite the shortcomings with regard to accuracy, the overall performance of the M05-2X/6- 31G(d) level of theory for solvent ranking when compared with experimental values is promising. The reader may bear in mind the diversity of polarity in the set of solvents investigated, the large errors for solvation free-energy calculations of the SMD model when ionic species are included and the lack of any solvation free-energy data for certain solvents (e.g., 1,4-dioxane) in the SMD training set. In addition, we appreciate that the experimental values may include a considerable amount of uncertainty, as discussed in Section 6.3. As mentioned earlier, an explicit consideration of a number of solvent molecules might result in a more realistic, and thus more accurate, representation of the specific solute-solvent interactions. We expect that this treatment would be particularly advantageous in the case of low-dielectric constant solvents.

6.5 Summary

In this chapter, we performed a systematic computational analysis of the selectivity and kinetics of the Williamson reaction between sodium β-naphthoxide and benzyl bromide in a set of eight solvents. The dielectric constant (ε) of the solvents in this set varies from 2.21 to 35.94. Following the computational methodology described in Chapter 4, we undertook quantum-mechanical calculations using several DFT functionals and basis sets (levels of theory). The SMD model was used to account for the different solvent media. For the set of solvents investigated, we briefly reviewed the solvation free-energy data included in the training set of the SMD model and the level of accuracy in which the model captures them. We optimized the structures of the reactants and the transition states of the O- and C-alkylation pathways at 298 K both in the gas and liquid phase for every level of theory and every solvent tested. In particular, for the transition- state structures of both alkylation pathways we calculated the structural differences between the liquid and the vacuum conformations due to the presence of the solvent medium, using the iterative closest point algorithm implemented in MATLAB. For both alkylation pathways, our findings followed a pattern of increasing root-mean-square-error with increasing value of the dielectric constant of the solvent, suggesting that larger structural differences occur in the liquid phase as we deviate from the dielectric constant of vacuum. ◦,QM As a next step, we calculated the value of ∆GC−O across the various levels of theory and Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 220

◦,QM solvents tested. We found that the QM-calculated ∆GC−O values were rather scattered among the different levels of theory, varying within ranges of approximately 10 kJ mol−1 for the different solvents studied. For the solvents acetonitrile, acetone, THF, 1,2-DME and ethyl acetate, only ◦,QM positive ∆GC−O values were predicted. This result pointed towards the O-alkylated product being the dominant reaction product, which is consistent with literature evidence. For the ◦,QM solvents methanol and ethanol both positive and negative ∆GC−O values were obtained. The insufficient number of solvation free-energy data in the training set of the SMD model for these solvents as well as the inherent difficulty of the model to capture accurately strong specific short- range interactions24 (such as hydrogen-bonding interactions that are present in protic solvents) were suggested as the main reasons for these discrepancies. For the solvent 1,4-dioxane, in which the C-alkylated product dominates in the reaction according to experimental evidence, the majority of methods predicted the reverse result; this is most likely due to the absence of solvation free-energy data in the training set of the SMD model for this solvent. Through a ◦,QM systematic comparison of the QM-calculated values for ∆GC−O for selected levels of theory, we showed that the energy differences obtained between different functionals with the same basis sets were larger than the energy differences obtained between the same functional with different basis sets. Through this analysis, we identified the key role of functional in the determination ◦,QM of the ∆GC−O value. ◦,QM As expected from the wide ranges obtained for the ∆GC-O values, the ranges of the QM- QM QM calculated selectivity ratios kO /kC across the various solvents were also quite wide. The ◦,QM selectivity preference followed the trend found for the ∆GC-O values. For the QM-calculated QM QM absolute k values across the different solvents, we found that the ranges of kC values were QM consistently wider than the ranges of kO values. We speculate that this is due to the inherently smaller rate-constant value of the C-alkylation pathway reaction, which is susceptible to larger computational errors. We appreciate that the inclusion of specific solvent molecules would improve the precision of our QM calculations, as this strategy would ultimately provide a more complete and accurate representation of the solvated system at the molecular level. With the aim of comparing with the corresponding QM-calculated values, we collected and evaluated the ensemble of available liquid-phase experimental data for the Williamson reaction investigated in this thesis. Based on our analysis, we concluded that experimental data obtained in this thesis and in a previous study23 using in-situ NMR spectroscopy were more suitable for this task. As a next step, we compared the QM-calculated and experimental percentage rate Expt. Expt. constant ratio values of kO :kC of the reaction with the aim of ranking the solvents. Since Chapter 6. Systematic investigation of the kinetics of a Williamson reaction in various solvents using QM calculations 221 the M05-2X/6-31G(d) level of theory has been previously identified as the best option for selectivity predictions for solvents acetonitrile and methanol, we selectively presented the ranking order according to the results from this level of theory. For the list of eight solvents, qualitative agreement with the experimental values was achieved for the first two solvents, namely Expt. acetonitrile and acetone, in which the percentage rate constant ratio value of kO was the highest. The corresponding QM-calculated and experimental values for these two solvents were also in excellent quantitative agreement. We further divided the solvents in two groups: those with high dielectric constant (ε> 20) and those with low dielectric constant (ε ≤ 20). For high- dielectric constant solvents, we found that the predicted percentage rate constant ratio values nearly followed the ranking of the experimental values and they were also in good quantitative agreement, with the larger deviations obtained for the protic solvents (methanol and ethanol). These deviations are because hydrogen-bonding interactions that are dominant in these solvents are not sufficiently modelled in the continuum solvation SMD model. For low-dielectric constant solvents, the relative order of three solvents (THF, ethyl acetate and 1,4-dioxane) was followed, although the predicted percentage rate constant ratio values were significantly overestimated Expt. towards kO compared with the experimental values. This may be attributed to the existence of significant short-range electrostatic interactions that again were not adequately captured by the SMD model. Overall, the performance of the M05-2X/6-31G(d) level of theory for solvent ranking was promising. Important aspects such as the diversity of polarity of the solvents investigated, the treatment of specific solute-solvent interactions, the large solvation free-energy errors obtained with the SMD model when ionic species were considered and the lack of solvation free-energy data for some solvents need to be taken into consideration. Building on the promising performance of this method, future avenues may be explored which would further improve the accuracy of the method and predictive capability. Some of these suggestions for future work are discussed in Chapter 7. Chapter 7

Conclusions and future work

7.1 Summary

The temperature and the solvent medium are widely recognized as two of the most influential factors on chemical reaction rates, and as such, they have been the subject of significant scientific research for decades. The importance of predicting temperature and solvent effects on reaction rates is immense for a plethora of engineering processes that involve chemical reactions. The prediction of temperature and solvent effects is a crucial step towards the design of an optimum reaction environment – a discipline in which newly emerged techniques, such as computer-aided molecular design techniques, thrive. This notion essentially constitutes the core element of this thesis, in which we have developed a systematic methodology to predict temperature and solvent effects on the kinetics and selectivity of chemical reactions in the gas and liquid phase. First, we introduced in Chapter 2 the fundamental kinetic concepts, among others the most widely used kinetic theory, conventional transition state theory (CTST), which we used for the prediction of the rate constant of a reaction. In the calculation of the reaction rate constant in the liquid phase, the solvation free energy difference between the reactants and the transition state of a reaction is a key quantity. Several approaches have been presented in the literature for modelling the solvent molecules and calculating the solvation free energy term. We briefly reviewed these approaches; the implicit solvation models, where the solvent is described as a continuum medium characterized by a number of bulk properties, appear to provide the best compromise of computational cost versus accuracy. From this class of models, we selected the SMD solvation model24 to use in our liquid-phase calculations. Our review also revealed that modelling temperature effects on solvation free energy has been severely neglected and only a

222 Chapter 7. Conclusions and future work 223 few computational tools are available for solvation free energy predictions (mostly in aqueous systems) at temperatures other than 298 K. Recognizing the importance of developing accurate modelling tools for the prediction of reaction kinetics, we considered the task of predicting temperature effects on the rate constant of a gas-phase hydrogen abstraction reaction between ethane and the hydroxyl radical (Chapter 3). For this task, we conducted a thorough investigation of the suitability of quantum-mechanical (QM) calculations. We computed the activation energy barrier of the reaction using various computational methods, such as B3LYP, M05-2X, M06-2X, MP2 and PMP2, CBS-QB3 and W1BD, with a selection of basis sets. A broad range of values was obtained, including negative barriers for all of the calculations with B3LYP. We also obtained the rate constants for each method, using CTST, and we compared our findings with available experimental values at 298 K. The best agreement was achieved with the M05-2X functional with cc-pV5Z basis set. Rate constants calculated at this level of theory were also found to be in good agreement with experimental values at different temperatures, resulting in a mean absolute error of the logarithm of the calculated values of 0.213 over a temperature range of 200-1250 K. We also developed hybrid models that combine a limited number of QM calculations and experimental data in order to increase the model reliability. We proposed a series of hybrid models and found that they provide good correlated rate-constant values and are comparable in terms of accuracy with conventional kinetic models, namely the Arrhenius and the three- parameter Arrhenius models. This combination of QM-calculated and experimental data sources proved to be particularly beneficial when fitting to scarce experimental data. A test case based on four data points was set, for which the model built on the hybrid strategy had parameters with significantly reduced uncertainty (reflected in the much narrower 95% confidence intervals) compared with the conventional kinetic models. In summary, the hybrid approach provides a useful and novel strategy for kinetic model development. In addition to temperature effects, we investigated the effect of different solvents on the reaction rate constant, by extending our methodology to the liquid phase (Chapter 4). By combining CTST with the formalism of the SMD model, we have derived an expression for the liquid-phase reaction rate constant, which we have used for kinetic and selectivity predictions. For the study of a reaction taking place in the liquid phase, we targeted a Williamson reaction between sodium β-naphthoxide and benzyl bromide, for which experimental findings in the lit- erature14 suggest significant solvent effects on the regioselectivity of the reaction leading to an ether (O-alkylated product) and an alcohol (C-alkylated product) at various product ratios. We Chapter 7. Conclusions and future work 224 performed detailed kinetic experiments, using in situ 1H NMR spectroscopy, for the reaction in acetonitrile at temperatures 298, 313 and 323 K. Our preliminary investigation shed some light on the underlying chemical phenomena involved in this reaction, revealing an equilibrium established between protonated and deprotonated reaction species. This equilibrium was identified and properly described for the first time for this reaction. We developed a detailed kinetic model using the solvers provided in gPROMS and reported values of the reaction rate constants for the two alkylation pathways and selectivity ratios. Furthermore, by analysing the kinetic temperature-dependent data through the Arrhenius and Eyring equations, we provided relevant ◦,Expt. thermodynamic parameters, such as ∆GC−O at 298 K. Additionally, we performed QM calculations assessing several DFT functionals in combination with a number of basis sets. The transition-state structures for the two alkylation pathways of the reaction were identified for the first time for this reaction. All the levels of theory assessed resulted in selectivity ratios at 298 K favouring the O-alkylation product, consistent with the experimental data. Among them, M05-2X/6-31G(d) proved to be the most accurate for selectivity predictions, while M05-2X/cc-pVTZ proved to be the most accurate for kinetic predictions. The fact that both of these levels of theory are among the levels of theory used in the parametrization of the SMD model seems to play a pivotal role for the good agreement achieved with experiments. When fitting the Eyring equation to QM-calculated rate constant data, the obtained values for the thermodynamic parameter ∆‡G◦,QM at 298 K for each alkylation pathway were in excellent agreement with the corresponding values obtained when fitting to experimental values; the impact of extrapolating the use of the SMD model at temperatures slightly higher (+ 25 K) than 298 K was minor for the calculation of thermodynamic quantities, and thus for the kinetic rate constant data. Building on the successful application of our methodology on the Williamson reaction in acetonitrile, we further investigated the reaction in the protic solvent methanol (Chapter 5). Important findings of our preliminary experimental investigation in this solvent were first, the identification of a side reaction occurring between one of the reactants and the solvent which has not been observed in previous studies; and second, a further alkylation reaction of the C- alkylated product with one of the reactants, which leads to the formation of a double C-alkylated product. In order to account for the kinetics and thermodynamics of the reaction system in this solvent, we extended our kinetic modelling in gPROMS, from which we obtained values of the rate constants for the two alkylation pathways and the selectivity ratios, as well as rate constants for the other two reactions and the equilibrium constant. By fitting the Arrhenius and Eyring Chapter 7. Conclusions and future work 225 equations to our temperature-dependent experimental kinetic data, we provided the value of the ◦,Expt. thermodynamic parameter ∆GC−O at 298 K in methanol. We subsequently performed QM calculations using the same levels of theory as in the previous chapter and the SMD solvation model. Besides having only ionic solvation data available in methanol for training the SMD model, the short-range solute-solvent interactions, which are present and rather important when the reaction takes place in protic solvents, were not explicitly accounted under the continuum solvent approximation, making the task of capturing the experimental outcome of the reaction in methanol challenging. Several levels of theory assessed gave consistent predictions with experiments for the O-alkylation product being the dominant product of the reaction in methanol at 298 K. However, the majority of methods predicted the opposite result. This is likely to be because the presence of strong hydrogen-bonding interactions were not sufficiently modelled. Nevertheless, the fact that M05-2X/6-31G(d) proved again to be the most accurate level of theory for selectivity predictions is promising. B3LYP/6-311G(2d,d,p) proved to be the most accurate for kinetic predictions. The good performance of this level of theory may be attributed most likely to the extensive basis set used, which accurately captured the bulk electrostatic interactions of the system, counterbalancing the parametrization shortcomings of the empirical non-electrostatic term of the SMD model. Moreover, the QM-calculated values for ∆‡G◦,QM at 298 K for each alkylation pathway agreed well (with a difference less than 3.00 kJ mol−1) with the respective experimental values. Considering the reaction in both solvents (acetonitrile and methanol), we found that the best agreement with experimental values is achieved by M05-2X/6-31G(d) for selectivity predictions (with a MAPE ◦ value of ∆GC-O

14.17% and a MAPEkO/kC value of 16.84%) and by B3LYP/6-31+G(d) for kinetic predictions

(with a MAPEk value of 44.05%). Overall, the use of levels of theory that have been used in the parametrization of the SMD model, or similar to these, appeared as a critical factor of achieving good accuracy with the experimental values. We extended the investigation of the reaction to a set of a further eight solvents with varying polarity (Chapter 6). We performed DFT calculations for each solvent at every level of theory that had been previously considered and we calculated the selectivity and kinetic metrics of interest. We discussed the impact on the calculations of the components of a level of theory, namely the functional and the basis set, identifying the functional as the component with the largest impact. Other important factors presumably influencing the final outcome of the calculations were the following: the number of solvation data included in the training set of the SMD solvation model for each solvent and the accuracy in which these data were captured by the Chapter 7. Conclusions and future work 226 model; the negligence of specific solute-solvent interactions under the continuum solvent regime which, depending on a solvent’s nature, might be important for the course of the reaction; and the propitious cancellation of systematic errors for relative calculated quantities, such as selectivity ratio. For comparisons with experimental values, the acquisition of reliable experimental values was recognized as a prerequisite. We compared the QM-calculated and experimental data by ranking the solvents according to their selectivity preference for the O- and the C-alkylation pathway. We selected as more reliable the data calculated with M05-2X/6-31G(d), as this level of theory performed best for acetonitrile and methanol, and the experimental data obtained with in situ NMR spectroscopy from this work and from a previous study.23 Excellent qualitative and quantitative agreement of the QM-calculated and experimental results was achieved for the first two solvents, acetonitrile and acetone, in which a clear preference towards the O-alkylated product was shown. When a distinction was made between high-dielectric constant solvents (ε> 20) and low-dielectric constant solvents (ε ≤ 20), we were able to identify critical factors that determined the overall promising performance of M05-2X/6-31G(d), and more importantly, future directions to further improve the predictive capabilities of our methodology.

7.2 Main contributions

• A novel methodology has been developed for the prediction of temperature eﬀects in the rate constant of a gas-phase reaction using various QM methods. The methodology has been successfully applied to a hydrogen abstraction reaction over an extended temperature range. Further combination of the calculated values with experimental data has resulted in the development of a novel hybrid strategy for deriving parameters for correlative models; the hybrid strategy brings new insight as to value and contribution of the data obtained via QM calculations and via measurement and is particularly useful in reactions with scarce experimental data.

• High-quality kinetic experiments have been performed involving in situ 1H NMR data acquisition for a regioselective reaction between sodium β-naphthoxide and benzyl bromide (Williamson reaction) with two main products. Thorough experimental investigation has revealed the existence of an underlying equilibrium taking place between two of the reaction species, as well as an additional reaction occurring among reaction species and leading to the formation of a third reaction product; both these phenomena are facilitated or suppressed by the choice of the solvent medium used. An additional side-reaction has been Chapter 7. Conclusions and future work 227

established at the presence of a particular solvent medium (methanol). These chemical and thermodynamic phenomena have been properly described by detailed kinetic modelling built in the gPROMS platform. New sets of rate constants and selectivity ratios have been obtained for the reaction in acetonitrile and methanol at various temperatures, as well as pertinent thermodynamic parameters based on temperature-dependent data.

• The Williamson reaction between sodium β-naphthoxide and benzyl bromide has been investigated using QM methods for the ﬁrst time. Transition-state structures for the O- alkylation pathway (two structures) and for the C-alkylation pathway (one structure) have been identiﬁed and fully characterized in vacuum and in eight solvents.

• In order to account for solvent effects, the proposed methodology developed for gas-phase rate-constant predictions has been extended to the liquid phase using the continuum SMD solvation model. The methodology has been applied on the Williamson reaction of interest for selectivity and kinetic predictions in a diverse set of eight solvents. Density functional theory calculations have been performed, using 18 different combinations of functionals and basis sets (levels of theory), to calculate the rate constants and ratios of the reaction in these solvents. The M05-2X/6-31G(d) level of theory has consistently given the closest agreement with experiments for selectivity predictions in the set of solvents studied experimentally in this work, while its performance for solvent ranking according to final rate-constant ratios for the complete set of solvents has been promising. The B3LYP/6- 31+G(d) level of theory has been identified as a good option for kinetic predictions.

7.3 Future work

Alternative solvation schemes

In this work, only the continuum solvation SMD model was considered. It would be useful to investigate the predictive capabilities of other continuum solvation models, such as the popular COSMO-RS model112,113 and the temperature-dependent SM8T model.153 Moreover, it would be interesting to explore alternative solvation schemes considering a number of explicit solvent molecules or other powerful approaches, such as QM/MM; albeit, with an anticipated increase in the computational cost and complexity.94 Thanks to the more detailed description of the solvated system with these approaches, we would expect an improvement in the accuracy of Chapter 7. Conclusions and future work 228 the predictions, especially for solvents in which the speciﬁc solute-solvent interactions play a fundamental role (e.g., protic solvents).

Hybrid strategy in the liquid phase

The hybrid strategy proposed and implemented in the gas phase for developing correlative models for rate-constant predictions at various temperatures could be also applied in the liquid phase. For example, implementation on the Williamson reaction would require at least one additional kinetic experiment at a diﬀerent temperature to be performed and the corresponding reaction-rate constant to be speciﬁed (for example, in acetonitrile and methanol that three temperature-dependent experiments are readily available), as the methodology for building a correlative model requires a minimum of four temperature-dependent points.

Experimental work on the Williamson reaction

This thesis illustrates a good example of how preliminary experimental work on the kinetics and selectivity of a reaction may be used to validate an extensive computational study, which may then be used to suggest future directions for more experimentation, and so on. On this spirit, additional validated experimental data would be useful to conﬁrm the reliability of the existing experimental data and to compare with the computational results performed for the extended set of solvents. In addition, more detailed analysis could be done for secondary chemical phenomena potentially occurring during the reaction, for example the formation of the double O,C-alkylated product (benzyl 1-benzyl-2-naphthyl ether), and determine their impact on the kinetics and selectivity of the two main reaction pathways in various solvents.

Incorporating solvent eﬀects on selectivity in CAMD methodology

Acknowledging the great influence that solvents may have on the selectivity of a reaction, in this work we have confirmed the suitability and reliability of advanced QM methods in predicting these effects, paving the way towards the integration of the proposed methodology within the CAMD framework. In a recent CAMD application for solvent design,45 incorporating solvent effects on selectivity was identified as a major challenging application. The formulation of a solvent design problem may be extended by postulating selectivity as one of the key performance indicators for reaction optimization, setting different targets for solving the problem and con- structing a reaction performance map. As as result, novel or unexplored solvent designs might be identified which will greatly facilitate targeted experimentation. Bibliography

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Electronic structure calculations for the gas-phase reaction

257 Appendix A 258

Table A.1: QM-calculated ideal-gas electronic energy (Eel,IG), zero-point vibrational energy 0,IG (ZPVE) and ideal-gas molecular partition function at 298 K (q (298)) of ethane (C2H6) at all the levels of theory tested.

Level of theory Eel,IG/a.u. Particle−1 ZPVE/a.u. Particle−1 q0,IG(298) B3LYP/3-21G -79.400644 0.072930 1.2684E-23 B3LYP/6-21G -79.749320 0.072776 1.4988E-23 B3LYP/6-31G -79.812739 0.072715 1.5896E-23 B3LYP/6-31G(d) -79.830421 0.072338 2.3367E-23 B3LYP/6-31+G(d,p) -79.841645 0.071733 4.4824E-23 B3LYP/6-311++G(d) -79.856575 0.071443 6.0814E-23 B3LYP/6-311G(2d,d,p) -79.856263 0.071506 5.6849E-23 M052X/6-31G(d) -79.801774 0.073647 5.7718E-24 M052X/6-31+G(d,p) -79.811766 0.075891 5.2530E-25 M052X/6-311++G(d,p) -79.826572 0.075715 6.3037E-25 M052X/6-311+G(2d,p) -79.828910 0.075694 6.4516E-25 M052X/CBSB7 -79.826413 0.075751 6.0681E-25 M052X/aug-cc-pVTZ -79.838335 0.075744 6.1179E-25 M052X/cc-pVQZ -79.843329 0.075752 6.0526E-25 M052X/aug-cc-pVQZ -79.843658 0.075764 5.9783E-25 M052X/cc-pV5Z -79.845768 0.075735 6.1935E-25 M062X/6-31G(d) -79.771836 0.075953 4.9008E-25 M062X/6-31+G(d,p) -79.780314 0.075286 1.0019E-24 M062X/6-311++G(d,p) -79.797126 0.075121 1.1893E-24 M062X/CBSB7 -79.796880 0.075178 1.1185E-24 M06-2X/aug-cc-pVTZ -79.806133 0.075057 1.2739E-24 MP2/6-31G(d) -79.494742 0.072756 1.4664E-23 MP2/6-31+G(d,p) -79.545792 0.072583 1.7511E-23 MP2/6-311++G(d,p) -79.571672 0.071424 6.0966E-23 MP2/6-311+G(2d,p) -79.586312 0.075722 6.2322E-25 MP2/cc-pVTZ -79.629908 0.071339 6.6924E-23 MP2/adj2-cc-pVTZ -79.629908 0.071339 6.6924E-23 MP2/cc-pVQZ -79.657462 0.071353 6.6083E-23 PMP2/6-31G(d) -79.494742 0.072756 1.4664E-23 PMP2/6-31+G(d,p) -79.545792 0.072583 1.7511E-23 PMP2/6-311++G(d,p) -79.571672 0.071424 6.0966E-23 PMP2/6-311+G(2d,p) -79.586312 0.075722 6.2322E-25 PMP2/cc-pVTZ -79.629908 0.071339 6.6924E-23 PMP2/adj2-cc-pVTZ -79.629908 0.071339 6.6924E-23 PMP2/cc-pVQZ -79.657462 0.071353 6.6083E-23 CBS-QB3 -79.630568 0.074376 2.6449E-24 W1BD -79.842445 0.074381 2.6249E-24 Appendix A 259

Table A.2: QM-calculated ideal-gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of the hydroxyl radical (·OH) at all the levels of theory tested.

Level of theory Eel,IG/a.u. Particle−1 ZPVE/a.u. Particle−1 q0,IG(298) B3LYP/3-21G -75.303139 0.007123 3.5212E+04 B3LYP/6-21G -75.611457 0.007042 3.8612E+04 B3LYP/6-31G -75.708548 0.007592 2.0552E+04 B3LYP/6-31G(d) -75.723454 0.007980 1.3256E+04 B3LYP/6-31+G(d,p) -75.739014 0.008107 1.1511E+04 B3LYP/6-311++G(d) -75.762412 0.008119 1.1270E+04 B3LYP/6-311G(2d,d,p) -75.754527 0.008110 1.1360E+04 M052X/6-31G(d) -75.708924 0.008215 1.0193E+04 M052X/6-31+G(d,p) -75.722217 0.008650 6.3945E+03 M052X/6-311++G(d,p) -75.745969 0.008662 6.2654E+03 M052X/6-311+G(2d,p) -75.746865 0.008645 6.3993E+03 M052X/CBSB7 -75.740246 0.008660 6.2744E+03 M052X/aug-cc-pVTZ -75.753581 0.008605 6.6599E+03 M052X/cc-pVQZ -75.755665 0.008622 6.5225E+03 M052X/aug-cc-pVQZ -75.756679 0.008616 6.5635E+03 M052X/cc-pV5Z -75.759252 0.008636 6.4171E+03 M062X/6-31G(d) -75.688663 0.008487 7.6796E+03 M062X/6-31+G(d,p) -75.701750 0.008615 6.6640E+03 M062X/6-311++G(d,p) -75.726596 0.008629 6.5208E+03 M062X/CBSB7 -75.720614 0.008633 6.4828E+03 M06-2X/aug-cc-pVTZ -75.733810 0.008593 6.7672E+03 MP2/6-31G(d) -75.521033 0.008030 1.2476E+04 MP2/6-31+G(d,p) -75.541045 0.008214 1.0148E+04 MP2/6-311++G(d,p) -75.579908 0.008240 9.7691E+03 MP2/6-311+G(2d,p) -75.594489 0.008639 6.4423E+03 MP2/cc-pVTZ -75.618907 0.008203 1.0141E+04 MP2/adj2-cc-pVTZ -75.612025 0.008168 1.0532E+04 MP2/cc-pVQZ -75.643566 0.008210 1.0030E+04 PMP2/6-31G(d) -75.522572 0.008030 1.2476E+04 PMP2/6-31+G(d,p) -75.542692 0.008214 1.0148E+04 PMP2/6-311++G(d,p) -75.581557 0.008240 9.7691E+03 PMP2/6-311+G(2d,p) -75.596298 0.008639 6.4423E+03 PMP2/cc-pVTZ -75.620714 0.008203 1.0141E+04 PMP2/adj2-cc-pVTZ -75.613825 0.008168 1.0532E+04 PMP2/cc-pVQZ -75.645490 0.008210 1.0030E+04 CBS-QB3 -75.649769 0.008431 8.0909E+03 W1BD -75.782140 0.008415 8.2234E+03 Appendix A 260

Table A.3: QM-calculated imaginary frequency of the transition-state structure (ν‡), ideal- gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of the transition-state structure of the reaction between ethane (C2H6) and the hydroxyl radical (·OH) at all the levels of theory tested.

0 Level of theory ν‡/cm−1 Eel,IG/a.u. Particle−1 ZPVE/a.u. Particle−1 q ,IG(298) B3LYP/3-21G −1175.23 −154.704423 0.076946 1.0583E-22 B3LYP/6-21G −1209.86 −155.358870 0.076702 1.4089E-22 B3LYP/6-31G −1196.02 −155.519060 0.077301 5.3853E-23 B3LYP/6-31G(d) −1061.26 −155.551637 0.077313 6.3246E-23 B3LYP/6-31+G(d,p) −193.23 −155.581421 0.078946 2.7038E-23 B3LYP/6-311++G(d) −219.84 −155.619550 0.078598 1.7683E-23 B3LYP/6-311G(2d,d,p) −450.88 −155.611426 0.077954 3.2472E-23 M052X/6-31G(d) −1436.71 −155.502140 0.079117 7.8560E-24 M052X/6-31+G(d,p) −893.30 −155.528231 0.082227 2.2776E-25 M052X/6-311++G(d,p) −890.93 −155.566744 0.082094 2.4529E-25 M052X/6-311+G(2d,p) −908.13 −155.570165 0.082039 2.7441E-25 M052X/CBSB7 −1098.57 −155.561207 0.082305 1.4329E-25 M052X/aug-cc-pVTZ −860.75 −155.587049 0.082079 3.0620E-25 M052X/cc-pVQZ −873.08 −155.594084 0.082174 2.6364E-25 M052X/aug-cc-pVQZ −827.58 −155.595448 0.082147 3.0119E-25 M052X/cc-pV5Z −847.14 −155.600168 0.082054 4.0927E-25 M062X/6-31G(d) −1189.36 −155.452629 0.081626 7.9031E-25 M062X/6-31+G(d,p) −723.73 −155.476616 0.081885 3.6143E-25 M062X/6-311++G(d,p) −726.71 −155.518137 0.081714 4.5504E-25 M062X/CBSB7 −895.36 −155.512359 0.081835 2.8938E-25 M06-2X/aug-cc-pVTZ −691.60 −155.535029 0.081749 5.0528E-25 MP2/6-31G(d) −2045.37 −154.997873 0.077914 3.3193E-23 MP2/6-31+G(d,p) −1569.37 −155.073551 0.078077 2.5348E-23 MP2/6-311++G(d,p) −1422.14 −155.139601 0.077222 5.5204E-23 MP2/6-311+G(2d,p) −1460.06 −155.169126 0.081652 5.2270E-25 MP2/cc-pVTZ −1441.90 −155.238608 0.077144 8.5246E-23 MP2/adj2-cc-pVTZ −1456.24 −155.231874 0.077058 1.1733E-22 MP2/cc-pVQZ −1372.74 −155.291311 0.077120 8.9836E-23 PMP2/6-31G(d) −2045.37 −155.003170 0.077914 3.3193E-23 PMP2/6-31+G(d,p) −1569.37 −155.078179 0.078077 2.5348E-23 PMP2/6-311++G(d,p) −1422.14 −155.144287 0.077222 5.5204E-23 PMP2/6-311+G(2d,p) −1460.06 −155.173855 0.081652 5.2270E-25 PMP2/cc-pVTZ −1441.90 −155.243348 0.077144 8.5246E-23 PMP2/adj2-cc-pVTZ −1456.24 −155.236599 0.077058 1.1733E-22 PMP2/cc-pVQZ −1372.74 −155.296009 0.077120 8.9836E-23 CBS-QB3 −450.78 −155.276737 0.081083 1.0587E-24 W1BD −345.74 −155.618706 0.081293 9.6101E-25 Appendix A 261

Table A.4: QM-calculated ideal-gas molecular partition functions (q0,IG) at diﬀerent temperatures for the components of the reaction between ethane (C2H6) and the hydroxyl radical (·OH) at the M05-2X/cc-pV5Z level of theory. TS=Transition state.

q0,IG(T )

T/K For C2H6 For ·OH For TS 200 8.3147E-43 1.7820E+01 8.5741E-45 210 3.0518E-40 4.0464E+01 5.5859E-42 213 1.6142E-39 5.1059E+01 3.4775E-41 220 6.6195E-38 8.5930E+01 2.0546E-39 225 8.1804E-37 1.2244E+02 3.2564E-38 230 9.0852E-36 1.7210E+02 4.5950E-37 240 8.3527E-34 3.2737E+02 6.6465E-35 250 5.3963E-32 5.9496E+02 6.5495E-33 270 9.1364E-29 1.7474E+03 2.3821E-29 273 2.5436E-28 2.0296E+03 7.3777E-29 280 2.5531E-27 2.8467E+03 9.4268E-28 298 6.1935E-25 6.4171E+03 4.0927E-25 299 7.8841E-25 6.6517E+03 5.3464E-25 300 1.0456E-24 6.9377E+03 7.3079E-25 310 1.5972E-23 1.0433E+04 1.4976E-23 325 7.0429E-22 1.8474E+04 9.9826E-22 327 1.1378E-21 1.9869E+04 1.7001E-21 355 5.4439E-19 5.1138E+04 1.6210E-18 396 1.0144E-15 1.6608E+05 7.2130E-15 400 1.9553E-15 1.8429E+05 1.5026E-14 499 9.3801E-10 1.5456E+06 3.6355E-08 500 1.0445E-09 1.5735E+06 4.1052E-08 595 6.3084E-06 6.9110E+06 8.0088E-04 600 9.3102E-06 7.3935E+06 1.2483E-03 705 1.0624E-02 2.5593E+07 3.9692E+00 800 1.5389E+00 6.3104E+07 1.2437E+03 974 1.6498E+03 2.3164E+08 4.2385E+06 1000 3.9516E+03 2.7336E+08 1.1811E+07 1225 2.2359E+06 9.2464E+08 2.0940E+10 1250 4.0640E+06 1.0387E+09 4.2557E+10 Appendix B

Electronic-structure calculations for the Williamson reaction

B.1 Optimized geometries

Vacuum

H(30)

C(29) H(28) Na(20) H(32) C(27) C(31) H(16) H(18) C(25) C(33) O(19) H(14) C H(26) (15) C(1) C(17) H(34) C(24)

C(2) C(13) C(3) Br H(23) (21) C(11) C(4) H(22)

C(8) C(9) C(6) H(5) H(12)

H(10) H(7)

Figure B.1: A transition-state structure (TS 1) for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The optimized values for the bond lengths, bond angles and dihedral angles are reported in Table B.1.

262 Appendix B 263

Table B.1: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3919112 3 C 2 2.8041261 1 155.0597498 4 C 2 1.4294120 1 118.3099253 3 -179.9974010 5 H 4 1.0844401 2 120.0241580 1 -179.9978025 6 C 4 1.3771181 2 120.6606263 5 179.9980653 7 H 6 1.0881035 4 119.3481522 2 -179.9997075 8 C 6 1.4202779 4 121.8195585 2 0.0000000 9 C 8 1.4209886 6 122.6750889 4 180.0000000 10 H 9 1.0882004 8 118.6755876 6 -0.0003237 11 C 9 1.3789488 8 120.9531268 6 179.9995695 12 H 11 1.0867756 9 120.3165963 8 180.0000000 13 C 11 1.4186228 9 119.8531872 8 0.0000000 14 H 13 1.0873936 11 119.5545023 9 -180.0000000 15 C 13 1.3786611 11 120.5783542 9 0.0000000 16 H 15 1.0885244 13 120.1789389 11 -180.0000000 17 C 15 1.4246904 13 120.9613782 11 0.0000000 18 H 1 1.0905111 2 119.3598443 4 179.9996242 19 O 2 1.3485949 1 119.4767799 17 179.9997352 20 Na 19 2.1668064 2 137.0364750 1 -0.0041178 21 Br 19 4.4694823 2 118.5275655 1 -179.9988661 22 H 3 1.0771025 19 95.5711050 2 -59.7111134 23 H 3 1.0771025 19 95.5717490 2 59.7021782 24 C 3 1.4719024 19 5.3128574 2 179.9954623 25 C 24 1.4091966 3 20.2355092 19 -84.4390017 26 H 25 1.0866947 24 19.1251296 3 -11.9889916 27 C 25 1.3978789 24 20.6675265 3 168.5994838 28 H 27 1.0871672 25 19.6630137 24 178.9568530 29 C 27 1.4032979 25 20.3624604 24 -0.2506517 30 H 29 1.0865032 27 20.3305098 25 179.4388620 31 C 29 1.4032981 27 19.3213442 25 0.9578473 32 H 31 1.0871665 29 19.9696893 27 178.2471632 33 C 31 1.3978791 29 120.3624602 27 -0.9577451 34 H 33 1.0866951 31 120.2046817 29 179.6554902 Appendix B 264

H(30)

H(32) C(29) Na(20) C(31) H(28) C(33) C(27) O H(34) (19) C(25) C C (24) (2) H(26) H(5) C(4) C(3)

C(17) H(18) Br(21) C(6) H(22) C(8) C(1) H(23) H(7) H(16) C(9) C(15) H (10) C(13)

H(14) H(12) C(11)

Figure B.2: A transition-state structure (TS 2) for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The optimized values for the bond lengths, bond angles and dihedral angles are reported in Table B.2. Appendix B 265

Table B.2: Internal coordinates (Z-matrix) of the transition-state structure TS 2 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3867285 3 C 2 2.7122776 1 119.9934426 4 C 2 1.4277180 1 119.2041693 3 135.1140135 5 H 4 1.0871891 2 118.5635966 1 -179.7079005 6 C 4 1.3757901 2 120.6465003 5 -179.3785902 7 H 6 1.0881689 4 120.0018313 2 179.7617867 8 C 6 1.4231519 4 121.2120855 2 -0.2467762 9 C 8 1.4208294 6 122.4178738 4 -179.9590674 10 H 9 1.0881511 8 118.7318122 6 -0.0753447 11 C 9 1.3792964 8 120.9315648 6 179.8415835 12 H 11 1.0868703 9 120.2047895 8 -179.9176264 13 C 11 1.4179447 9 120.0216095 8 -0.1349361 14 H 13 1.0871967 11 119.6044698 9 -179.7872054 15 C 13 1.3789642 11 120.4386191 9 0.0341126 16 H 15 1.0881088 13 120.2768118 11 -179.7174583 17 C 15 1.4234583 13 120.9326915 11 0.0195397 18 H 1 1.0882919 2 119.2027373 4 179.0543634 19 O 2 1.3555944 1 121.1094302 17 178.4653772 20 Na 19 2.1464896 2 143.0010806 1 -75.6351528 21 Br 19 4.4921722 2 113.1922307 1 100.4346619 22 H 3 1.0782637 19 96.4347805 2 61.9500182 23 H 3 1.0788936 19 95.7521488 2 -56.5337286 24 C 3 1.4707169 19 96.6572925 2 -177.2859305 25 C 24 1.4103247 3 120.1810485 19 83.3756715 26 H 25 1.0866764 24 119.0289506 3 13.6681249 27 C 25 1.3975455 24 120.7619358 3 -167.1904167 28 H 27 1.0872665 25 119.6403491 24 -178.8370651 29 C 27 1.4034866 25 120.4128109 24 0.3979701 30 H 29 1.0864959 27 120.3696616 25 -179.4952207 31 C 29 1.4032416 27 119.2386830 25 -1.0240064 32 H 31 1.0872602 29 119.9556563 27 -178.2051968 33 C 31 1.3977982 29 120.3979788 27 0.9530106 34 H 33 1.0866964 31 120.1681531 29 -179.4096836 Appendix B 266

Na H(30) (20) C(29) H(32) C O(19) (31) H(28)

C(33) C(27) H (5) H(34) C(2) C(25)

H(18) C(4) H(26) C(24)

C(3) C(6)

H(7) Br(21) C(17) H(22) H(23) C(8) H(16)

C(9) C(15)

H (10) C(13)

C(11)

H(12) H(14)

Figure B.3: A transition-state structure for the C-alkylation pathway optimized at the B3LYP/6- 31+G(d) level of theory in vacuum. The optimized values for the bond lengths, bond angles and dihedral angles are reported in Table B.3. Appendix B 267

Table B.3: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in vacuum. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4416769 3 C 2 2.8502086 1 48.0616595 4 C 2 1.4511836 1 116.8793163 3 91.0506127 5 H 4 1.0866673 2 117.3245749 1 -173.0780333 6 C 4 1.3646970 2 121.1549917 5 -177.0228568 7 H 6 1.0889378 4 119.7371446 2 178.4108872 8 C 6 1.4398436 4 122.2885719 2 0.2954098 9 C 8 1.4117717 6 121.9360897 4 175.6030173 10 H 9 1.0880984 8 118.9558474 6 1.9772553 11 C 9 1.3868267 8 120.7904048 6 -177.7774664 12 H 11 1.0864638 9 120.2782633 8 -179.9378714 13 C 11 1.4075504 9 119.6167661 8 0.0402577 14 H 13 1.0870223 11 119.8085445 9 179.7384100 15 C 13 1.3887162 11 120.4626048 9 -0.4637753 16 H 15 1.0884469 13 119.8157671 11 -179.7975018 17 C 15 1.4108797 13 120.9308695 11 0.2257012 18 H 1 1.0890101 2 115.8161714 4 -166.8295753 19 O 2 1.2756710 1 122.5475164 17 164.7499936 20 Na 19 2.1018148 2 134.7395568 1 40.3082881 21 Br 19 5.9349912 2 59.6890533 1 62.1643513 22 H 3 1.0783710 19 92.7438496 2 52.3624494 23 H 3 1.0774315 19 125.1196082 2 -74.5395508 24 C 3 1.4655427 19 69.7576345 2 173.1997621 25 C 24 1.4108360 3 120.9231529 19 103.5471723 26 H 25 1.0869639 24 119.2416733 3 2.4671096 27 C 25 1.3984220 24 121.0388925 3 179.9326884 28 H 27 1.0873268 25 119.6658224 24 -177.6664261 29 C 27 1.4051896 25 120.3037785 24 0.2725979 30 H 29 1.0868505 27 120.3931495 25 -177.4345972 31 C 29 1.4057901 27 119.1347343 25 0.5501858 32 H 31 1.0871800 29 119.9715522 27 -178.7780047 33 C 31 1.3965275 29 120.4069796 27 -0.0575287 34 H 33 1.0857936 31 119.8535811 29 -178.7241123 Appendix B 268

Acetonitrile

Table B.4: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetonitrile. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3953836 3 C 2 2.8507099 1 126.9074237 4 C 2 1.4358900 1 118.2254153 3 -130.1878021 5 H 4 1.0876090 2 118.3083413 1 179.8587484 6 C 4 1.3760120 2 121.2296797 5 179.4606934 7 H 6 1.0886672 4 120.0515585 2 -179.7130808 8 C 6 1.4251923 4 121.2229164 2 -0.0398753 9 C 8 1.4217558 6 122.4310089 4 -179.9710062 10 H 9 1.0885682 8 118.6738398 6 0.1897951 11 C 9 1.3816025 8 120.9230686 6 -179.8328042 12 H 11 1.0872529 9 120.2785917 8 -179.9855636 13 C 11 1.4195740 9 119.9170332 8 0.0252948 14 H 13 1.0876720 11 119.5688198 9 -179.9599986 15 C 13 1.3807984 11 120.5300722 9 0.0445225 16 H 15 1.0883779 13 120.2906093 11 179.9467167 17 C 15 1.4264509 13 120.9781146 11 0.0562042 18 H 1 1.0887846 2 119.0718063 4 -179.4632667 19 O 2 1.3339834 1 121.7213302 17 -178.6219468 20 Na 19 2.1653373 2 141.8084819 1 48.4939097 21 Br 19 4.6188256 2 107.8389103 1 -112.4519885 22 H 3 1.0765245 19 89.9004133 2 -70.7626541 23 H 3 1.0776392 19 89.4127843 2 48.3304400 24 C 3 1.4613242 19 94.5001509 2 168.6609558 25 C 24 1.4086402 3 120.1635055 19 -86.8028089 26 H 25 1.0871050 24 119.4447374 3 -6.2016110 27 C 25 1.3970892 24 120.3775220 3 172.8942017 28 H 27 1.0867476 25 119.8011556 24 -179.6521615 29 C 27 1.4025529 25 120.1382447 24 0.1130743 30 H 29 1.0866409 27 120.0910463 25 -179.5902726 31 C 29 1.4018692 27 119.8041336 25 0.2125367 32 H 31 1.0867763 29 120.0538440 27 -179.9522467 33 C 31 1.3978134 29 120.1645560 27 -0.1948010 34 H 33 1.0870746 31 120.1365776 29 -179.1997541 Appendix B 269

Table B.5: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetonitrile. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4320595 3 C 2 2.9254022 1 52.9464326 4 C 2 1.4537565 1 116.7399712 3 91.9818680 5 H 4 1.0877151 2 117.4981175 1 -174.6342753 6 C 4 1.3662183 2 121.5377147 5 -177.8660884 7 H 6 1.0889624 4 119.9808522 2 178.6377077 8 C 6 1.4385424 4 121.9172720 2 -0.0044167 9 C 8 1.4142405 6 121.9816262 4 176.6948981 10 H 9 1.0883685 8 118.8224315 6 1.3296750 11 C 9 1.3876070 8 120.8009337 6 -178.3593153 12 H 11 1.0870570 9 120.2897959 8 179.8680702 13 C 11 1.4116930 9 119.6521347 8 0.0548517 14 H 13 1.0876375 11 119.7022686 9 179.5708436 15 C 13 1.3880507 11 120.5361378 9 -0.3936047 16 H 15 1.0882788 13 120.0669515 11 -179.9026130 17 C 15 1.4170749 13 120.9138720 11 0.2186015 18 H 1 1.0877168 2 117.6307877 4 -171.8673509 19 O 2 1.2803858 1 122.8723359 17 168.3818662 20 Na 19 2.1458781 2 136.1151809 1 26.7201239 21 Br 19 6.0265338 2 58.9589076 1 66.7596573 22 H 3 1.0778827 19 86.7973659 2 53.1418620 23 H 3 1.0771221 19 120.7022425 2 -68.8481425 24 C 3 1.4503847 19 68.2917435 2 178.2250249 25 C 24 1.4122383 3 120.4331236 19 108.5323415 26 H 25 1.0870079 24 119.3966720 3 0.0677749 27 C 25 1.3960420 24 120.6219779 3 -179.8697897 28 H 27 1.0867348 25 119.8842496 24 -179.8884167 29 C 27 1.4062008 25 120.1404256 24 -0.5285928 30 H 29 1.0867809 27 120.1145569 25 -179.2876519 31 C 29 1.4034162 27 119.6889803 25 0.8706417 32 H 31 1.0867347 29 120.0184594 27 179.6343280 33 C 31 1.3975863 29 120.1494251 27 -0.2062842 34 H 33 1.0863207 31 119.8927212 29 179.7777906 Appendix B 270

Methanol

Table B.6: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in methanol. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3937507 3 C 2 2.8388448 1 124.3560486 4 C 2 1.4343640 1 118.4260526 3 130.5504888 5 H 4 1.0878192 2 118.3833455 1 179.9595188 6 C 4 1.3759443 2 121.1438711 5 -179.1642994 7 H 6 1.0886855 4 120.0182597 2 179.6239175 8 C 6 1.4248928 4 121.1963617 2 -0.0808315 9 C 8 1.4217183 6 122.4202841 4 179.9367642 10 H 9 1.0886239 8 118.7241810 6 -0.0885062 11 C 9 1.3813087 8 120.8991490 6 179.9480615 12 H 11 1.0873872 9 120.2598666 8 179.9757999 13 C 11 1.4193074 9 119.9526020 8 -0.0347389 14 H 13 1.0877633 11 119.5776307 9 179.9595056 15 C 13 1.3805602 11 120.5235213 9 -0.0462172 16 H 15 1.0884512 13 120.2703076 11 -179.9470922 17 C 15 1.4260012 13 120.9509934 11 -0.0285512 18 H 1 1.0888471 2 119.1806467 4 179.4634030 19 O 2 1.3389537 1 121.6910229 17 178.6825316 20 Na 19 2.1762890 2 141.2293399 1 -47.7295776 21 Br 19 4.6224105 2 107.5626673 1 108.7879830 22 H 3 1.0769285 19 90.1655322 2 71.4468007 23 H 3 1.0780742 19 90.0834058 2 -47.5085939 24 C 3 1.4611887 19 94.8886208 2 -167.8557454 25 C 24 1.4088030 3 120.1044305 19 88.0331897 26 H 25 1.0872251 24 119.4815837 3 6.4815024 27 C 25 1.3968899 24 120.3750842 3 -172.7931126 28 H 27 1.0868710 25 119.8083878 24 179.7294314 29 C 27 1.4024788 25 120.1325914 24 -0.0982448 30 H 29 1.0867565 27 120.0964682 25 179.6053988 31 C 29 1.4016497 27 119.8142379 25 -0.1861974 32 H 31 1.0869035 29 120.0543154 27 179.8979760 33 C 31 1.3977188 29 120.1708978 27 0.1790437 34 H 33 1.0871797 31 120.1231179 29 179.1581307 Appendix B 271

Table B.7: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in methanol. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4298897 3 C 2 2.9037743 1 52.4786386 4 C 2 1.4499960 1 117.0897101 3 92.1330051 5 H 4 1.0878347 2 117.6500139 1 -174.6126569 6 C 4 1.3665694 2 121.3862758 5 -177.7587447 7 H 6 1.0888538 4 119.9009377 2 178.5888436 8 C 6 1.4376908 4 121.9183871 2 0.0315575 9 C 8 1.4143742 6 121.9482694 4 176.5763170 10 H 9 1.0883473 8 118.8686209 6 1.3587648 11 C 9 1.3870863 8 120.7442169 6 -178.3269406 12 H 11 1.0871435 9 120.2660857 8 179.8894670 13 C 11 1.4113281 9 119.6927626 8 0.0576926 14 H 13 1.0876765 11 119.7095393 9 179.5664947 15 C 13 1.3880246 11 120.5377736 9 -0.3660444 16 H 15 1.0882736 13 120.0358647 11 -179.9426758 17 C 15 1.4160261 13 120.8577622 11 0.1839036 18 H 1 1.0876735 2 117.6445429 4 -171.1452592 19 O 2 1.2873442 1 122.6724352 17 168.1374437 20 Na 19 2.1599451 2 135.1434195 1 25.5922591 21 Br 19 6.0290158 2 58.9110334 1 66.4290525 22 H 3 1.0781684 19 87.2619032 2 53.8936851 23 H 3 1.0773708 19 121.8450060 2 -68.6751174 24 C 3 1.4510934 19 68.7091638 2 178.5514623 25 C 24 1.4120473 3 120.3862529 19 108.2442682 26 H 25 1.0870860 24 119.4352934 3 0.3929888 27 C 25 1.3960710 24 120.6237827 3 -179.6103397 28 H 27 1.0868503 25 119.8857369 24 -179.8360755 29 C 27 1.4060768 25 120.1297425 24 -0.4825969 30 H 29 1.0868975 27 120.1167236 25 -179.3126089 31 C 29 1.4033148 27 119.6887694 25 0.8696727 32 H 31 1.0868160 29 120.0122773 27 179.6160001 33 C 31 1.3975337 29 120.1646095 27 -0.2324260 34 H 33 1.0864937 31 119.8649719 29 179.8270162 Appendix B 272

Ethanol

Table B.8: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethanol. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3936348 3 C 2 2.8379612 1 121.5324807 4 C 2 1.4345600 1 118.4347116 3 131.3022632 5 H 4 1.0878432 2 118.3477680 1 -179.8060226 6 C 4 1.3760776 2 121.1457704 5 -179.4892690 7 H 6 1.0887035 4 120.0447125 2 179.6688627 8 C 6 1.4249182 4 121.1783359 2 -0.1124371 9 C 8 1.4217559 6 122.4216345 4 179.9326645 10 H 9 1.0886282 8 118.7238387 6 -0.0859134 11 C 9 1.3812791 8 120.9037996 6 179.9351146 12 H 11 1.0873819 9 120.2556840 8 179.9897100 13 C 11 1.4192923 9 119.9574722 8 -0.0613917 14 H 13 1.0877508 11 119.5795277 9 -179.9798847 15 C 13 1.3805055 11 120.5139594 9 0.0170458 16 H 15 1.0884365 13 120.2709843 11 -179.9648424 17 C 15 1.4259919 13 120.9600513 11 -0.0405274 18 H 1 1.0887034 2 119.1818602 4 179.6049971 19 O 2 1.3383139 1 121.6663466 17 178.8595999 20 Na 19 2.1663081 2 144.2796823 1 -76.7729698 21 Br 19 4.6190147 2 107.5385262 1 102.9617900 22 H 3 1.0772992 19 90.4704365 2 59.6581791 23 H 3 1.0776965 19 89.5181371 2 -59.2592372 24 C 3 1.4615152 19 94.8778950 2 -179.7541910 25 C 24 1.4083517 3 120.2572448 19 86.1249820 26 H 25 1.0871420 24 119.4908479 3 6.4889865 27 C 25 1.3973376 24 120.3780649 3 -172.8167942 28 H 27 1.0869022 25 119.7874422 24 179.7689827 29 C 27 1.4020460 25 120.1607106 24 -0.0299106 30 H 29 1.0867652 27 120.1122687 25 179.6170339 31 C 29 1.4019359 27 119.7893656 25 -0.2016191 32 H 31 1.0868638 29 120.0554830 27 179.9551526 33 C 31 1.3973684 29 120.1593144 27 0.1921848 34 H 33 1.0871888 31 120.1008040 29 179.2777795 Appendix B 273

Table B.9: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethanol. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4305493 3 C 2 2.9078152 1 52.4725246 4 C 2 1.4507861 1 117.0075163 3 91.9913620 5 H 4 1.0878216 2 117.6214933 1 -174.5520728 6 C 4 1.3664285 2 121.4106042 5 -177.7458703 7 H 6 1.0888930 4 119.9051670 2 178.5857043 8 C 6 1.4378853 4 121.9281239 2 0.0295127 9 C 8 1.4142507 6 121.9542825 4 176.5512662 10 H 9 1.0883619 8 118.8643902 6 1.3708835 11 C 9 1.3871511 8 120.7555226 6 -178.3134478 12 H 11 1.0871333 9 120.2696505 8 179.8930443 13 C 11 1.4112522 9 119.6835684 8 0.0546684 14 H 13 1.0876783 11 119.7085916 9 179.5659836 15 C 13 1.3880276 11 120.5375427 9 -0.3729174 16 H 15 1.0883027 13 120.0237010 11 -179.9507319 17 C 15 1.4160999 13 120.8691952 11 0.1877163 18 H 1 1.0877244 2 117.5984940 4 -171.2220870 19 O 2 1.2857463 1 122.7096856 17 168.0105602 20 Na 19 2.1557505 2 135.3415738 1 26.5601533 21 Br 19 6.0294774 2 58.8521311 1 66.4539408 22 H 3 1.0781038 19 87.2670252 2 53.6401812 23 H 3 1.0773247 19 121.6520291 2 -68.8574692 24 C 3 1.4512685 19 68.6580300 2 178.3183388 25 C 24 1.4120329 3 120.4163500 19 108.2483708 26 H 25 1.0870903 24 119.4365447 3 0.3612662 27 C 25 1.3961050 24 120.6283722 3 -179.6729873 28 H 27 1.0868590 25 119.8767806 24 -179.8134750 29 C 27 1.4060359 25 120.1405569 24 -0.5377502 30 H 29 1.0869026 27 120.1233049 25 -179.2346694 31 C 29 1.4033406 27 119.6750443 25 0.9033104 32 H 31 1.0868252 29 120.0191968 27 179.6539132 33 C 31 1.3975423 29 120.1644708 27 -0.2203751 34 H 33 1.0864603 31 119.8552547 29 179.8306703 Appendix B 274

Acetone

Table B.10: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetone. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3955773 3 C 2 2.8613394 1 133.1428509 4 C 2 1.4359480 1 118.1699046 3 130.7711384 5 H 4 1.0872933 2 118.5417110 1 -179.4973930 6 C 4 1.3759080 2 121.2028972 5 -179.7584242 7 H 6 1.0886938 4 119.9663661 2 179.8367628 8 C 6 1.4248974 4 121.2995789 2 0.2065431 9 C 8 1.4215943 6 122.4653145 4 179.8240669 10 H 9 1.0886152 8 118.6858666 6 -0.2283480 11 C 9 1.3815373 8 120.9260381 6 179.7782726 12 H 11 1.0872899 9 120.2878505 8 179.9853769 13 C 11 1.4194592 9 119.9027354 8 -0.0365106 14 H 13 1.0877353 11 119.5685530 9 179.9799405 15 C 13 1.3807285 11 120.5386533 9 -0.0241230 16 H 15 1.0884499 13 120.2698125 11 -179.9468378 17 C 15 1.4263759 13 120.9814861 11 -0.0508255 18 H 1 1.0890762 2 119.0874739 4 179.2518351 19 O 2 1.3340984 1 121.5871900 17 178.0532986 20 Na 19 2.1710062 2 139.7271729 1 -33.9022515 21 Br 19 4.6148160 2 108.8835372 1 122.3988352 22 H 3 1.0758958 19 90.0607363 2 74.0101048 23 H 3 1.0775391 19 89.7524642 2 -45.1086843 24 C 3 1.4619928 19 94.3819042 2 -165.5910205 25 C 24 1.4085517 3 120.3314174 19 87.6369390 26 H 25 1.0871287 24 119.4836232 3 6.4477782 27 C 25 1.3970790 24 120.3709239 3 -172.8028090 28 H 27 1.0868236 25 119.7889659 24 179.7529021 29 C 27 1.4026467 25 120.1628150 24 -0.1087033 30 H 29 1.0866966 27 120.0911240 25 179.6604400 31 C 29 1.4017661 27 119.7865619 25 -0.3056584 32 H 31 1.0868315 29 120.0675613 27 -179.9331191 33 C 31 1.3979506 29 120.1511454 27 0.2363247 34 H 33 1.0871306 31 120.1397986 29 179.2765074 Appendix B 275

Table B.11: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in acetone. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4324805 3 C 2 2.9234754 1 52.8190259 4 C 2 1.4537832 1 116.7166073 3 91.8567867 5 H 4 1.0876971 2 117.4986229 1 -174.5455413 6 C 4 1.3660839 2 121.5275312 5 -177.8153516 7 H 6 1.0890081 4 119.9584968 2 178.6172537 8 C 6 1.4385976 4 121.9389104 2 -0.0032509 9 C 8 1.4140542 6 121.9814623 4 176.6391119 10 H 9 1.0883971 8 118.8330202 6 1.3614299 11 C 9 1.3875390 8 120.8027531 6 -178.3315172 12 H 11 1.0870838 9 120.2879560 8 179.8882198 13 C 11 1.4114289 9 119.6498138 8 0.0582938 14 H 13 1.0876679 11 119.7041621 9 179.5708224 15 C 13 1.3880450 11 120.5347576 9 -0.3997066 16 H 15 1.0883456 13 120.0316717 11 -179.9274718 17 C 15 1.4168322 13 120.9159042 11 0.2101552 18 H 1 1.0878270 2 117.5517146 4 -171.7029740 19 O 2 1.2799030 1 122.8710237 17 168.1572205 20 Na 19 2.1443824 2 136.0840147 1 27.2359632 21 Br 19 6.0237842 2 58.9561380 1 66.7071507 22 H 3 1.0778762 19 86.9362658 2 53.0774210 23 H 3 1.0771311 19 120.8541923 2 -68.9792769 24 C 3 1.4509476 19 68.2798552 2 178.0984816 25 C 24 1.4121515 3 120.4620935 19 108.5505033 26 H 25 1.0870628 24 119.4134648 3 0.1226257 27 C 25 1.3961020 24 120.6341062 3 -179.9893716 28 H 27 1.0868168 25 119.8688154 24 -179.7535390 29 C 27 1.4060951 25 120.1536232 24 -0.4912374 30 H 29 1.0868556 27 120.1268302 25 -179.1888789 31 C 29 1.4033780 27 119.6632995 25 0.8602150 32 H 31 1.0868025 29 120.0263028 27 179.7186606 33 C 31 1.3975615 29 120.1600929 27 -0.2094933 34 H 33 1.0863062 31 119.8621873 29 179.8597927 Appendix B 276

THF

Table B.12: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in THF. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3934383 3 C 2 2.8259264 1 123.7470328 4 C 2 1.4342220 1 118.4035997 3 131.2505151 5 H 4 1.0876288 2 118.3702051 1 -179.7425245 6 C 4 1.3758726 2 121.1242681 5 -179.5086218 7 H 6 1.0886028 4 120.0399873 2 179.7466625 8 C 6 1.4246834 4 121.2146285 2 -0.0418880 9 C 8 1.4214668 6 122.4365497 4 179.9838366 10 H 9 1.0885270 8 118.7028717 6 -0.1692217 11 C 9 1.3810349 8 120.9293073 6 179.8134614 12 H 11 1.0872307 9 120.2603868 8 -179.9793480 13 C 11 1.4190154 9 119.9436043 8 -0.0737134 14 H 13 1.0876306 11 119.5797766 9 -179.9483066 15 C 13 1.3803186 11 120.5009437 9 -0.0088139 16 H 15 1.0883936 13 120.2704578 11 -179.8797246 17 C 15 1.4256753 13 120.9770027 11 -0.0366961 18 H 1 1.0886400 2 119.1071713 4 179.3086382 19 O 2 1.3381099 1 121.6434786 17 178.7057027 20 Na 19 2.1564125 2 142.9313922 1 -62.1394819 21 Br 19 4.5971760 2 108.4315267 1 107.0448705 22 H 3 1.0766466 19 90.8677975 2 65.3639040 23 H 3 1.0776524 19 90.3836598 2 -53.4179933 24 C 3 1.4632698 19 94.6693282 2 -173.9355721 25 C 24 1.4082634 3 120.2202142 19 86.7482587 26 H 25 1.0870426 24 119.4546395 3 7.1553110 27 C 25 1.3973139 24 120.4384856 3 -172.4599953 28 H 27 1.0868982 25 119.7531265 24 -179.9089413 29 C 27 1.4022175 25 120.1829389 24 0.0075782 30 H 29 1.0866791 27 120.1328004 25 179.9030168 31 C 29 1.4018062 27 119.7160007 25 -0.3054596 32 H 31 1.0868902 29 120.0691788 27 -179.6161141 33 C 31 1.3978598 29 120.1868278 27 0.2570831 34 H 33 1.0871074 31 120.0858950 29 179.6049414 Appendix B 277

Table B.13: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in THF. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4338212 3 C 2 2.9111731 1 52.2024248 4 C 2 1.4534105 1 116.7141583 3 91.5652613 5 H 4 1.0874882 2 117.4798121 1 -174.2535565 6 C 4 1.3656966 2 121.4575487 5 -177.6651737 7 H 6 1.0889724 4 119.9117389 2 178.5812634 8 C 6 1.4388604 4 122.0110006 2 0.0541918 9 C 8 1.4134683 6 121.9745423 4 176.4324864 10 H 9 1.0883229 8 118.8550256 6 1.4944875 11 C 9 1.3873475 8 120.8010602 6 -178.2173907 12 H 11 1.0869662 9 120.2837320 8 179.9433734 13 C 11 1.4105286 9 119.6475970 8 0.0549878 14 H 13 1.0875510 11 119.7189299 9 179.5971753 15 C 13 1.3880854 11 120.5233705 9 -0.4235896 16 H 15 1.0883967 13 119.9612626 11 -179.9415783 17 C 15 1.4157529 13 120.9156742 11 0.2055706 18 H 1 1.0880700 2 117.2945699 4 -171.0084067 19 O 2 1.2788188 1 122.8144516 17 167.4723613 20 Na 19 2.1368831 2 136.0301969 1 29.0140960 21 Br 19 6.0123756 2 58.9597715 1 66.2278086 22 H 3 1.0777925 19 87.6162735 2 52.9875161 23 H 3 1.0770692 19 121.5753396 2 -69.5496331 24 C 3 1.4533042 19 68.4113228 2 177.5877332 25 C 24 1.4117522 3 120.5123863 19 108.2229475 26 H 25 1.0870847 24 119.4083508 3 0.5165204 27 C 25 1.3963840 24 120.7090106 3 -179.9395489 28 H 27 1.0868860 25 119.8374853 24 -179.4244853 29 C 27 1.4058798 25 120.1644718 24 -0.5822516 30 H 29 1.0868871 27 120.1688080 25 -178.7769334 31 C 29 1.4034741 27 119.5838962 25 0.8967726 32 H 31 1.0868331 29 120.0388112 27 -179.9810943 33 C 31 1.3975704 29 120.2015105 27 -0.1532288 34 H 33 1.0861593 31 119.8223625 29 -179.9996708 Appendix B 278

1,2-DME

Table B.14: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,2-DME. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3931501 3 C 2 2.8264267 1 122.3914063 4 C 2 1.4341600 1 118.4492938 3 131.9452409 5 H 4 1.0876658 2 118.3549647 1 -179.5609148 6 C 4 1.3759835 2 121.1085910 5 -179.6849478 7 H 6 1.0886291 4 120.0581281 2 179.7595746 8 C 6 1.4247002 4 121.1959541 2 -0.0545573 9 C 8 1.4215003 6 122.4267546 4 179.9354102 10 H 9 1.0885394 8 118.7050133 6 -0.1387277 11 C 9 1.3810550 8 120.9280163 6 179.8478315 12 H 11 1.0872452 9 120.2556118 8 -179.9794716 13 C 11 1.4190853 9 119.9477236 8 -0.0915544 14 H 13 1.0876316 11 119.5836540 9 -179.9243771 15 C 13 1.3803288 11 120.4967926 9 0.0169299 16 H 15 1.0883949 13 120.2730332 11 -179.8846143 17 C 15 1.4256587 13 120.9773695 11 -0.0375150 18 H 1 1.0885540 2 119.0952548 4 179.4340722 19 O 2 1.3384656 1 121.6457477 17 178.5450479 20 Na 19 2.1554164 2 143.2244651 1 -71.2211092 21 Br 19 4.5933471 2 108.7852815 1 104.4366456 22 H 3 1.0769290 19 91.0596456 2 61.1819039 23 H 3 1.0776504 19 90.3081018 2 -57.5948182 24 C 3 1.4636729 19 94.5922629 2 -178.1764957 25 C 24 1.4081604 3 120.2992616 19 86.1955259 26 H 25 1.0870261 24 119.4842272 3 7.2871403 27 C 25 1.3976165 24 120.4364249 3 -172.3439393 28 H 27 1.0868784 25 119.7533433 24 -179.8962705 29 C 27 1.4021238 25 120.1937623 24 0.0000000 30 H 29 1.0867181 27 120.1428099 25 179.8866895 31 C 29 1.4018835 27 119.7031992 25 -0.3185061 32 H 31 1.0868877 29 120.0634808 27 -179.6362013 33 C 31 1.3975783 29 120.1799296 27 0.2852685 34 H 33 1.0870607 31 120.0981759 29 179.5805967 Appendix B 279

Table B.15: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,2-DME. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4338993 3 C 2 2.9104113 1 52.1221600 4 C 2 1.4534319 1 116.7147776 3 91.5706681 5 H 4 1.0874970 2 117.4798005 1 -174.2498362 6 C 4 1.3657266 2 121.4543040 5 -177.6420180 7 H 6 1.0889827 4 119.9095967 2 178.5636447 8 C 6 1.4388843 4 122.0121938 2 0.0277732 9 C 8 1.4134788 6 121.9713285 4 176.4432740 10 H 9 1.0883395 8 118.8570128 6 1.4872891 11 C 9 1.3873910 8 120.8004653 6 -178.2314804 12 H 11 1.0869762 9 120.2836537 8 179.9521538 13 C 11 1.4105601 9 119.6469410 8 0.0500406 14 H 13 1.0875583 11 119.7242335 9 179.6078865 15 C 13 1.3881307 11 120.5191686 9 -0.4065387 16 H 15 1.0884139 13 119.9576121 11 -179.9495833 17 C 15 1.4157011 13 120.9171693 11 0.1886140 18 H 1 1.0880723 2 117.2817360 4 -170.9555009 19 O 2 1.2788095 1 122.8138860 17 167.4568509 20 Na 19 2.1365697 2 136.0739078 1 29.0432981 21 Br 19 6.0100854 2 58.9780683 1 66.1469051 22 H 3 1.0778262 19 87.6461810 2 53.0303931 23 H 3 1.0770775 19 121.6222127 2 -69.5442914 24 C 3 1.4534978 19 68.3982661 2 177.6152797 25 C 24 1.4116543 3 120.5116741 19 108.1225998 26 H 25 1.0871086 24 119.4056770 3 0.5322220 27 C 25 1.3964340 24 120.7169422 3 -179.9448193 28 H 27 1.0869059 25 119.8404083 24 -179.4026607 29 C 27 1.4059092 25 120.1610887 24 -0.5734991 30 H 29 1.0869028 27 120.1733166 25 -178.7582636 31 C 29 1.4035344 27 119.5766417 25 0.9017863 32 H 31 1.0868537 29 120.0343330 27 -179.9736035 33 C 31 1.3975928 29 120.2052674 27 -0.1615253 34 H 33 1.0861820 31 119.8254359 29 -179.9930836 Appendix B 280

Ethyl acetate

Table B.16: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethyl acetate. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3927298 3 C 2 2.8211116 1 122.3355821 4 C 2 1.4337510 1 118.4941846 3 132.0312317 5 H 4 1.0876393 2 118.3735583 1 -179.5841414 6 C 4 1.3759423 2 121.0787477 5 -179.6691382 7 H 6 1.0886030 4 120.0523809 2 179.7736375 8 C 6 1.4246136 4 121.1995215 2 -0.0640077 9 C 8 1.4214631 6 122.4313515 4 179.9611482 10 H 9 1.0885281 8 118.7095115 6 -0.1427509 11 C 9 1.3809028 8 120.9304188 6 179.8316818 12 H 11 1.0872343 9 120.2542259 8 -179.9687573 13 C 11 1.4189314 9 119.9540719 8 -0.0941484 14 H 13 1.0876228 11 119.5816893 9 -179.9092133 15 C 13 1.3802053 11 120.4916781 9 0.0218976 16 H 15 1.0883841 13 120.2704958 11 -179.8694078 17 C 15 1.4255121 13 120.9762859 11 -0.0397141 18 H 1 1.0885356 2 119.0987807 4 179.3860513 19 O 2 1.3394561 1 121.6181332 17 178.5996136 20 Na 19 2.1545921 2 143.2430088 1 -71.6349938 21 Br 19 4.5901093 2 108.8821443 1 104.3651716 22 H 3 1.0768955 19 91.2961877 2 61.0282882 23 H 3 1.0776680 19 90.5547282 2 -57.6879696 24 C 3 1.4641129 19 94.6700982 2 -178.3022466 25 C 24 1.4081750 3 120.3046432 19 86.1083494 26 H 25 1.0870043 24 119.4689133 3 7.5182816 27 C 25 1.3976201 24 120.4518291 3 -172.1834769 28 H 27 1.0869000 25 119.7488329 24 -179.8096673 29 C 27 1.4020803 25 120.2003880 24 0.0097794 30 H 29 1.0867266 27 120.1495772 25 179.9521963 31 C 29 1.4018512 27 119.6876952 25 -0.3351475 32 H 31 1.0869047 29 120.0581659 27 -179.5429175 33 C 31 1.3975812 29 120.1866520 27 0.3048027 34 H 33 1.0870400 31 120.0922026 29 179.6436571 Appendix B 281

Table B.17: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in ethyl acetate. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4305494 3 C 2 2.9078204 1 52.4718223 4 C 2 1.4507854 1 117.0075093 3 91.9907657 5 H 4 1.0878221 2 117.6214436 1 -174.5516418 6 C 4 1.3664274 2 121.4106365 5 -177.7456016 7 H 6 1.0888924 4 119.9052759 2 178.5856050 8 C 6 1.4378862 4 121.9280724 2 0.0296441 9 C 8 1.4142513 6 121.9542611 4 176.5507932 10 H 9 1.0883620 8 118.8644211 6 1.3710243 11 C 9 1.3871509 8 120.7555003 6 -178.3129151 12 H 11 1.0871334 9 120.2698460 8 179.8927462 13 C 11 1.4112520 9 119.6835384 8 0.0543033 14 H 13 1.0876782 11 119.7085351 9 179.5660521 15 C 13 1.3880284 11 120.5375744 9 -0.3728160 16 H 15 1.0883028 13 120.0237098 11 -179.9507028 17 C 15 1.4160999 13 120.8691779 11 0.1878678 18 H 1 1.0877249 2 117.5983973 4 -171.2218903 19 O 2 1.2857469 1 122.7097976 17 168.0091853 20 Na 19 2.1557580 2 135.3389022 1 26.5585295 21 Br 19 6.0294877 2 58.8514699 1 66.4536388 22 H 3 1.0781029 19 87.2669596 2 53.6405593 23 H 3 1.0773250 19 121.6520095 2 -68.8571490 24 C 3 1.4512705 19 68.6583966 2 178.3186919 25 C 24 1.4120321 3 120.4163322 19 108.2490215 26 H 25 1.0870903 24 119.4365240 3 0.3612331 27 C 25 1.3961050 24 120.6283928 3 -179.6732625 28 H 27 1.0868590 25 119.8767802 24 -179.8131710 29 C 27 1.4060368 25 120.1405378 24 -0.5374199 30 H 29 1.0869025 27 120.1232814 25 -179.2349134 31 C 29 1.4033411 27 119.6750321 25 0.9030284 32 H 31 1.0868238 29 120.0191825 27 179.6539989 33 C 31 1.3975423 29 120.1644654 27 -0.2202828 34 H 33 1.0864608 31 119.8552592 29 179.8306581 Appendix B 282

1,4-dioxane

Table B.18: Internal coordinates (Z-matrix) of the transition-state structure TS 1 for the O- alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,4-dioxane. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.3899636 3 C 2 2.7776192 1 122.1246625 4 C 2 1.4305140 1 118.7503342 3 133.6115989 5 H 4 1.0873712 2 118.5003780 1 -179.4487624 6 C 4 1.3754497 2 120.9058663 5 -179.6285940 7 H 6 1.0883911 4 119.9724490 2 179.8055380 8 C 6 1.4236164 4 121.2342788 2 -0.1340823 9 C 8 1.4208486 6 122.4491461 4 -179.9718763 10 H 9 1.0883613 8 118.7496916 6 -0.1837766 11 C 9 1.3798561 8 120.9323646 6 179.7727733 12 H 11 1.0870863 9 120.2301817 8 -179.9535384 13 C 11 1.4179587 9 119.9874043 8 -0.1458982 14 H 13 1.0874595 11 119.5942398 9 -179.8469759 15 C 13 1.3793396 11 120.4700589 9 0.0036035 16 H 15 1.0883191 13 120.2350432 11 -179.7574034 17 C 15 1.4242281 13 120.9561474 11 0.0022833 18 H 1 1.0885851 2 119.1459398 4 179.0478005 19 O 2 1.3466201 1 121.3235018 17 178.2849490 20 Na 19 2.1522451 2 141.9633450 1 -64.7417681 21 Br 19 4.5498180 2 111.1041867 1 104.4687694 22 H 3 1.0769884 19 93.5553213 2 66.5671690 23 H 3 1.0781189 19 93.0132813 2 -51.9720567 24 C 3 1.4673162 19 95.0212040 2 -172.6693736 25 C 24 1.4082294 3 120.2270399 19 84.8899063 26 H 25 1.0868956 24 119.2943864 3 9.9016205 27 C 25 1.3974289 24 120.5791118 3 -170.1663012 28 H 27 1.0870842 25 119.7004930 24 -179.3577271 29 C 27 1.4023560 25 120.2731361 24 0.1446903 30 H 29 1.0866413 27 120.2370470 25 -179.6921595 31 C 29 1.4019425 27 119.5061162 25 -0.7067734 32 H 31 1.0870687 29 120.0304737 27 -178.7619772 33 C 31 1.3979127 29 120.2676866 27 0.6311618 34 H 33 1.0869271 31 120.0953140 29 -179.8175390 Appendix B 283

Table B.19: Internal coordinates (Z-matrix) of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in 1,4-dioxane. The internal coordinates for each atom are deﬁned by the bond lengths, bond angles and dihedral angles with respect to previously deﬁned atoms listed in columns NA (for the bond length), NB (for the bond angle) and NC (for the dihedral angle).

Atom Symbol NA Bond length/A˚ NB Bond angle/◦ NC Dihedral angle/◦ 1 C 2 C 1 1.4376401 3 C 2 2.8792258 1 50.1975026 4 C 2 1.4519356 1 116.7539991 3 90.7871541 5 H 4 1.0870312 2 117.4634646 1 -173.4642656 6 C 4 1.3648509 2 121.2858975 5 -177.2354706 7 H 6 1.0889483 4 119.7676596 2 178.4601922 8 C 6 1.4392007 4 122.1814561 2 0.1848263 9 C 8 1.4122087 6 121.9621462 4 175.8497980 10 H 9 1.0881710 8 118.9351627 6 1.7451578 11 C 9 1.3867647 8 120.7857781 6 -177.9834298 12 H 11 1.0867290 9 120.2703693 8 -179.9476541 13 C 11 1.4083817 9 119.6422311 8 0.0516701 14 H 13 1.0872967 11 119.7638555 9 179.6620720 15 C 13 1.3882036 11 120.5010441 9 -0.4500039 16 H 15 1.0884729 13 119.8106521 11 -179.9372823 17 C 15 1.4129058 13 120.9127342 11 0.1956252 18 H 1 1.0886314 2 116.5504899 4 -169.0514609 19 O 2 1.2772223 1 122.6798455 17 165.5691507 20 Na 19 2.1216381 2 134.8077268 1 35.0850960 21 Br 19 5.9865041 2 58.8106113 1 64.5572805 22 H 3 1.0778952 19 89.8767560 2 52.5255173 23 H 3 1.0770796 19 123.4643015 2 -71.7950105 24 C 3 1.4598055 19 69.1399188 2 175.4476463 25 C 24 1.4108626 3 120.6956184 19 106.5302420 26 H 25 1.0870650 24 119.3645285 3 1.4550754 27 C 25 1.3972540 24 120.9269275 3 -179.8146849 28 H 27 1.0871098 25 119.7546836 24 -178.6618605 29 C 27 1.4053251 25 120.1903984 24 -0.6216924 30 H 29 1.0869361 27 120.2857018 25 -177.8812827 31 C 29 1.4041104 27 119.3640890 25 0.8845212 32 H 31 1.0870066 29 120.0232103 27 -179.2763248 33 C 31 1.3973175 29 120.3240374 27 -0.0068216 34 H 33 1.0860044 31 119.7900049 29 -179.5521745 Appendix B 284

B.2 Optimized energies

Table B.20: QM-calculated imaginary frequency of the transition-state structure (ν‡), ideal- gas electronic energy (Eel,IG), zero-point vibrational energy (ZPVE) and ideal-gas molecular partition function at 298 K (q0,IG(298)) of the transition-state structures for both the alkylation pathways optimized at the B3LYP/6-31+G(d) level of theory in vacuum. For the O-alkylation pathway, only the transition-state structure TS 2 is considered.

Transition State ν‡/cm−1 Eel,IG/a.u. Particle−1 ZPVE/a.u. Particle−1 q0,IG(298) O-alkylation -391.82 -3465.548177 0.250160 2.8725E-92 C-alkylation -404.20 -3465.548515 0.250206 2.4206E-93

Table B.21: QM-calculated imaginary frequency of the transition-state structure (ν‡), liquid- phase electronic energy (Eel,L), zero-point vibrational energy (ZPVE) and the non-electrostatic term of the solvation free energy (GCDS,L) at 298 K of the transition-state structure TS 2 for the O-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in diﬀerent solvents.

Solvent ν‡/cm−1 Eel,L/a.u. Particle−1 ZPVE/a.u. Particle−1 GCDS,L/kcal mol−1 Acetonitrile -396.59 -3465.599236 0.249584 -5.08 Methanol -399.63 -3465.597292 0.249766 -3.24 Ethanol -402.03 -3465.597326 0.249715 -3.98 Acetone -392.82 -3465.598592 0.249733 -5.72 THF -401.61 -3465.592576 0.249828 -5.68 1,2-DME -402.74 -3465.589934 0.249762 -4.20 Ethyl acetate -403.01 -3465.591491 0.249871 -6.30 1,4-Dioxane -405.05 -3465.573392 0.250236 -4.15

Table B.22: QM-calculated imaginary frequency of the transition-state structure (ν‡), liquid- phase electronic energy (Eel,L), zero-point vibrational energy (ZPVE) and the non-electrostatic term of the solvation free energy (GCDS,L) at 298 K of the transition-state structure for the C-alkylation pathway optimized at the B3LYP/6-31+G(d) level of theory in diﬀerent solvents.

Solvent ν‡/cm−1 Eel,L/a.u. Particle−1 ZPVE/a.u. Particle−1 GCDS,L/kcal mol−1 Acetonitrile -360.75 -3465.597841 0.249588 -4.98 Methanol -371.93 -3465.597689 0.249688 -2.89 Ethanol -370.55 -3465.597414 0.249647 -3.61 Acetone -361.89 -3465.597261 0.249645 -5.75 THF -369.27 -3465.591081 0.249812 -5.75 1,2-DME -370.76 -3465.588636 0.249781 -4.36 Ethyl acetate -371.72 -3465.589945 0.249848 -6.30 1,4-Dioxane -390.63 -3465.572705 0.250235 -4.37 Appendix B 285

B.3 Root-Mean-Square Errors

1.0

0.8

Å 0.6

0.4 RMSE/

0.2

0.0 Acetonitrile Methanol Ethanol Acetone THF 1,2−DME Ethyl Acetate 1,4−dioxane

Solvents

O−alkylation C−alkylation

Figure B.4: Root-mean-square error (RMSE) values for the structural changes between the transition-state structures obtained in diﬀerent solvents and in vacuum for the O- and C- alkylation pathways optimized at the B3LYP/6-31+G(d) level of theory obtained using the iterative closest point (ICP) algorithm. The solvents are presented in descending order of dielectric constant ε. For the O-alkylation pathway, only the transition-state structure TS 2 is considered. Appendix C

Overall scheme of the Williamson reaction in methanol-d4

O NaBr

O Na Br

OH NaBr

O Na O Br NaBr

CD3 Br O D3C OH HBr

O Na OH OH O Na

Figure C.1: Overall reaction scheme of the Williamson reaction between sodium β-naphthoxide and benzyl bromide in methanol-d4.

286 Appendix D

Methanol-d4: 2-reaction model results

The experimental concentration proﬁles for the components of the reaction between sodium

β-naphthoxide and benzyl bromide in methanol-d4 at temperatures 298, 313 and 328 K are shown in Figures D.1, D.2 and D.3, respectively. The components monitored in this reaction are the following: the reactants, sodium β-naphthoxide and benzyl bromide, and the products, the O-alkylated product and the C-alkylated product. The concentration of each reaction component is represented by a different symbol and colour (the blue circles correspond to sodium β-naphthoxide, the red triangles to benzyl bromide, the green diamonds to the O-alkylated product and the pink squares to the C-alkylated product). The curves of the corresponding colours represent the calculated concentration values for each reaction component obtained by implementing the 2-reaction model in gPROMS. For each experiment at a different tempera- Expt. Expt. ture, the estimated reaction rate constants kO and kC , together with their 95% confidence intervals, are presented in Table D.2. The 95% confidence intervals are estimated on the basis of a constant variance model assuming an error of 0.01 dm3 mol−1 s−1 for the concentration values. Superscript “Expt.” refers to the use of experimental concentration data to fit those parameters. Additionally, the estimated initial concentration values for the reactants at time zero, [1]0 and [2]0, and their 95% confidence intervals, are presented in Table D.1.

287 Appendix D 288

0.1 Sodium β-naphthoxide Benzyl bromide O-alkylated product 0.08 C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 2000 4000 6000 8000 10000 12000 14000 16000 t/sec

Figure D.1: Experimental concentration data as a function of time in methanol-d4 at 298 K measured using in situ 1H NMR. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS. Appendix D 289

0.1 Sodium β-naphthoxide Benzyl bromide O-alkylated product 0.08 C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 t/sec

Figure D.2: Experimental concentration data as a function of time in methanol-d4 at 313 K measured using in situ 1H NMR. The diﬀerent symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS. Appendix D 290

0.1 Sodium β-naphthoxide Benzyl bromide O-alkylated product 0.08 C-alkylated product -3 0.06

0.04 /mol dm c

0.02

0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 t/sec

Figure D.3: Experimental concentration data as a function of time in methanol-d4 at 328 K measured using in situ 1H NMR. The different symbols, as labelled in the legend, indicate the concentration values of the components measured during the reaction. The corresponding lines indicate the calculated concentration values for the reaction components obtained by implementing Equations (4.31)-(4.35) for the 2-reaction model in gPROMS. Appendix D 291 (%) Expt. C k : 69:31 67:33 68:32 Expt. O 95% CI k 3 3 4 − − − 3 10 10 10 − × × × 95% CI /mol dm -naphthoxide and benzyl bromide in methanol- e 0 β [2] [3.381-5.477] [0.917-1.442] [1.811-4.597] 3 − is also reported. 1 − s 1 Expt. C 3 3 4 − k − − − : /mol dm 10 10 10 mol w 0 × × × 3 Expt. O k -naphthoxide ([1]) and benzyl bromide ([2]), for the reaction studied β /dm 4.429 1.179 3.204 Expt. C k , with their 95% confidence intervals (CI), obtained by fitting the 2-reaction 3 3 4 95% CI [2] − − − Expt. C 10 10 10 k × × × 3 − and 95% CI Expt. O k -naphthoxide Benzyl bromide /mol dm β e 0 [7.953-11.58] [2.072-2.792] [5.361-8.351] [1] 1 − 3 s − Sodium 1 3 3 4 − − − − 10 10 10 mol × × × 3 /mol dm w 0 /dm Expt. O k 298313328 0.090 0.091 0.091 0.088 0.101 0.120 [0.082-0.095] [0.094-0.108] [0.111-0.129] 0.090 0.089 0.090 0.090 0.095 0.106 [0.084-0.096] [0.089-0.102] [0.099-0.114] T/K [1] 328 9.766 313 2.432 298 6.856 T/K model to experimental concentration data ford4 the at components of three the different reaction temperatures. between sodium The percentage rate constant ratio of Table D.1: Initial concentrations at time zero of the reactants, sodium Table D.2: Estimated reaction rate constants in methanol-d4 at temperaturesby 298, the superscript 313 “w”, andimplementing and 328 the the 2-reaction K. estimated model The in values, reported gPROMS. indicated by concentration the values superscript correspond “e”, to with the their weighted 95% amounts, confidence indicated intervals (CI), obtained by