Benchmark DFT Approach for Dissociation Energies of Chemically Important Bonds

By Naveen Kosar

CIIT/FA15-R66-002/ATD

PhD Thesis

Chemistry

COMSATS University Islamabad Abbottabad Campus - Pakistan

Spring, 2019

COMSATS University Islamabad

Benchmark DFT Approach for Dissociation Energies of Chemically Important Bonds

A Thesis Presented to

COMSATS University Islamabad, Abbottabad Campus

In partial fulfillment

of the requirement for the degree of

PhD (Chemistry)

Naveen Kosar

CIIT/FA15-R66-002/ATD

Spring, 2019

Benchmark DFT Approach for Dissociation Energies of Chemically Important Bonds

A Post Graduate Thesis submitted to the Department of Chemistry as partial fulfillment of the requirement for the award of Degree of Ph.D in Chemistry.

Name Registration Number

Naveen Kosar CIIT/FA15-R66-002/ATD

Supervisor

Dr. Tariq Mahmood Associate Professor Department of Chemistry COMSATS University Islamabad, Abbottabad Campus

Co-Supervisor

Dr. Khurshid Ayub Associate Professor Department of Chemistry COMSATS University Islamabad, Abbottabad Campus

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DEDICATION

I would like to dedicate my thesis to my beloved Parents, supportive Family and encouraging Teachers

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ACKNOWLEDGEMENTS

“All the praises and thanks be to Allah, Who has guided us to this, and never could we have found guidance, were it not that Allah had guided us!” (Aayah No. 43, Surah Al-A’raf, Chapter No. 7, Holy Qur’an). My thousands Drood and Slam for Holy Prophet Mohammad (SAW) who emphasized the Umma to seek knowledge from the cradle to grave. First, I am thankful to my supervisor Dr. Tariq Mahmood and co-supervisor Dr. Khurshid Ayub, for their excellent supervision and guidance throughout my PhD research. They always pushed me forward and encouraged to have firm belief in myself. I extend my gratitude to Dr. Umar Rashid, Department of Chemistry, for his motivation and moral support throughout my research work. I am very thankful to Dr. Syeda Samina Ejaz, Department of Biochemistry, Islamia University Bahawalpur, for her valuable guidance and suggestion regarding my research work. I am very thankful to Dr. Mazhar A. Gilani, Department of Chemistry, COMSATS University Islamabad, Lahore Campus, for his valuable critics regarding the writing of publications were worthwhile. I acknowledge the Higher Education Commision, Pakistan and COMSATS University Islamabad, Abbottabad Campus for providing funding against my research project which really helped to carry out this research work. Second, my deepest gratitude goes to my entire family for their continuous encouragement, constant support, and unconditional love that gave me strength during PhD. Special thanks to my cousin, Farukh Shahzad Khan, for his help, guidance and suggestion to achieve this success. Bundle of thanks to my parents, Zahir Ahmad Khan and Razia Sultana, who raised me with a love of science and supported me in all my pursuits. Thanks, are extended to my grandmothers and aunts, for their prayers and wishes. I would also like to thank to my dear sister (Ayesha Zaheer) and brothers (Saud Ahmad Khan and Masroor Ahmed Khan) for encouragement and support. I would like to pay my gratitude to my Uncles (Sajjad Ahmad Khan, Sabir Ahmad Khan, Shakir Ahmad Khan, Abdur Rashid Ahmad Khan, Riaz Ali Khan, Hadayth Ullah Khan, Azhar Iqbal Khan, Anwar Iqbal Khan and Ishtiaq Khan) and their families for their support, wishes and prayers. Finally, I am very thankful to all my friends (Abida, Sidra, Saira, Humira, Amna, Aqsa, Romana and Muneeba) for their encouragement and support throughout this journey. Next, Lab colleagues (Riffat, Maria, Saira, Saima, Sehrish, Tanzeela, Rabia, Zainab, Fatima, Maria, Sajida, Saba, Saima, Ridda, Iram, Arenda, Muneeba, Hira, Sana, Kiran, Annum, Maryam, Nadia, Sidra, Mamoona, Saba, Yasir, Tabish, Hasnain, Faizan, Asghar, Akhtar, Sajjad, Arsalan, Zulqarnain, Asad and Bilal) at the department, COMSATS University Islamabad, Abbottabad Campus, for their valuable feedback, discussion, comments and support throughout my PhD research.

Naveen Kosar CIIT/FA15-R66-002/ATD

ABSTRACT

Benchmark DFT Approach for Dissociation Energies of Chemically Important Bonds

A very fascinating aspect in quantum chemical research is to acquire accurate and cost- effective method for the calculation of electronic as well as structural properties. The benchmarking is a famous approach where theoretical results from low-level calculations are compared with accurate high-level quantum chemical methods or experimental results. The current study focuses on performance evaluation of density functional theory methods for accurate measurement of bond dissociation energies. Bond dissociation energy (BDE) measurement has got noteworthy attention due to its importance in all areas of chemistry. Keeping in view the importance of bond dissociation energy (BDE), the current study is focused on the exploration of accurate and low cost DFT method for bond dissociation energy calculations of different chemically important bonds. The selected bonds include C−X (X = Cl and Br), C−Sn,

C−CN, C−Mg and M−O2 bonds. Various functionals of DFT classes with a variety of basis sets are implemented for the calculation of BDE. The accuracy of a method is examined through different statistical tests including root mean square deviation (RMSD), standard deviation (SD), Pearson’s correlation (R) and mean absolute error (MAE). Theoretical results are compared with the already reported experimental BDE values of respective bonds. The method which has less deviation and error with a reasonable Pearson’s correlation is considered as the method of interest.

For the BDE measurement of carbon halogen (C−X where X = Cl and Br) bond, 33 different density functionals (DFs) with four basis sets are used for sixteen halogen containing compounds. Two basis sets (6-31G(d) and 6-311G(d)) are selected from Pople basis sets and other two are selected from Dunning basis sets (aug-cc-pVDZ and aug-cc-pVTZ). Among all selected DFs, ꞷB97X-D shows the best performance with least deviations (RMSD, SD), error (MAE) and a signiﬁcant Pearson's correlation (R) when compared with experimental data. Secondly, nineteen DFs from eight different DFT classes with four basis sets are selected for BDE calculation of the C−Sn bond of ten organotin compounds. Two basis sets containing pseudopotential basis sets (LANL2DZ and SDD) and other are selected from Karlsruhe basis sets (def2-SVP and

def2-TZVP). In this benchmark approach, BLYP-D3 functional of dispersion corrected GGA class with SDD basis set is observed as the best method for homolytic BDE calculation of C−Sn bond. Thirdly, for twelve organo-nitrile compounds thirty-one DFs with eight basis sets including Pople, Dunning and Karlsruhe basis sets are used. Thus, 6-31G(d), 6-31G(d,p), 6-311G(d,p), 6-31+G(d) and 6-311++G(d,p) basis sets are selected from Pople basis sets, aug-cc-pVDZ and aug-cc-pVTZ basis sets are selected from Dunning basis sets and def2-SVP basis set is selected from Karlsruhe basis sets. Overall, CAM-B3LYP functional of range separated hybrid GGA class with Pople’s 6- 311G(d,p) basis set provides the most accurate results for the BDE measurement of C−CN bond of nitrile compounds. Fourthly, twenty-nine DFs from thirteen DFT classes with four basis sets (Pople’s 6-31G(d) and 6-311G(d), Dunning’s aug-ccpVDZ and Karlsruhe’s def2-SVP basis sets) are implemented for BDE measurement of C−Mg bond of fifteen Grignard reagents. TPSS of meta-GGA class with 6-31G(d) basis set gave the accurate results. Finally, for BDE measurements of M−O2 bond in five metal complexes with dioxygen, fourteen DFs are chosen from seven DFT classes with two series of mixed basis sets. A combination of pseudopotential and Pople basis sets (LANL2DZ & 6-31G(d) and SDD & 6-31+G(d)) are used as a series of mixed basis sets. M06 functional with SDD & 6-31+G(d) gave outstanding results due to low deviations, error and the best R between experimental and theoretical data.

PUBLICATIONS ASSOCIATED WITH THESIS

1. Naveen Kosar, Tariq Mahmood, Khurshid Ayub. (2017). Role of dispersion corrected hybrid GGA class in accurately calculating the bond dissociation energy of carbon halogen bond: A benchmark study. Journal of Molecular Structure, 1150, 447-458. (IF = 2.120)

2. Naveen Kosar, Khurshid Ayub, Tariq Mahmood. (2018). Accurate theoretical method for homolytic cleavage of C−Sn bond: A benchmark approach. Computational and Theoretical Chemistry, 1140, 134-144. (IF = 1.344)

3. Naveen Kosar, Khurshid Ayub, Mazhar Amjad Gilani, Tariq Mahmood. (2019). Benchmark DFT studies on C−CN homolytic cleavage and screening the substitution effect on bond dissociation energy. Journal of Molecular Modeling, 25, 47-60. (IF = 1.335)

4. Naveen Kosar, Khurshid Ayub, Mazhar A Gailani, Faheem Shah, Tariq Mahmood. Benchmark approach in search of cost-effective and accurate density functional for homolytic cleavage of C-Mg bond of Grignard reagent. International Journal of Quantum Chemistry, 2019, e26106 (IF = 2.263)

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TABLE OF CONTENTS 1 Introduction ...... 1 1.1 Theoretical Chemistry ...... 2 1.1.1 Applications of Computational Chemistry ...... 3 1.1.2 Limitations ...... 3 1.2 Computational versus Experimental Chemistry ...... 4 1.3 Benchmark Study ...... 4 1.3.1 Applications of Benchmark Study ...... 5 1.3.1.1 Benchmark Approach useful for Different Properties Studies .... 5 1.3.1.2 Benchmark Study of Magnetic Properties ...... 6 1.3.1.3 Benchmark Study of Crystal Structures ...... 7

1.3.1.4 Benchmark Study of Different SN2 Type Reactions ...... 7 1.3.1.5 Benchmark Study of Metal and Non-Metal Clusters ...... 7 1.3.1.6 Benchmark Study for Simple and Complex Dimers ...... 8 1.3.1.7 Benchmarking of Non-Covalent Interactions ...... 8 1.3.1.8 Benchmark Study of Bond Activations of Transition-Metal Based Catalysts ...... 9 1.3.1.9 Benchmark Study for Late-Transition Metals Reactions ...... 9 1.3.1.10 Benchmark Study on Different Properties of Lanthanides Series …………………………………………………………………..9 1.3.1.11 Benchmark Study for Geometric and Thermodynamic Properties of Noble Gas Containing Molecules ...... 10 1.4 Bond Dissociation Energy ...... 10 1.4.1 Homolytic versus Heterolytic Bond Dissociation ...... 11 1.4.2 Techniques used for Determining BDE ...... 12 1.4.2.1 Electron Capture Detector (ECD) Method ...... 12 1.4.2.2 Collision Induced Dissociation (CID) Method ...... 12 1.4.2.3 Threshold Photoelectron Photoion Coincidence (TPEPICO) Spectroscopic Method ...... 12 1.4.2.4 Infrared Chemiluminescence Method ...... 12

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1.4.2.5 Photoionization Mass Spectrometry ...... 13 1.4.2.6 Combustion Calorimetric Method ...... 13 1.4.2.7 Reaction-Solution Calorimetric Method ...... 13 1.4.2.8 Photo Acoustic Calorimetric Method ...... 14 1.4.2.9 Heat of Formation ...... 14 1.4.2.10 Kinetic Studies ...... 15 1.4.3 Role of Computational Studies towards BDE Measurements ...... 15 1.4.4 Benchmark Study of BDE of Different Chemical Bonds ...... 18 1.5 Statement of Problem ...... 20 1.6 Motivations ...... 20 1.7 Aims and Objectives of the Present Work ...... 20 2 Computational Methodology ...... 21 2.1 Computational Methodology ...... 22 2.2 Molecular Mechanics ...... 22 2.2.1 Bond Stretching Energies ...... 22 2.2.2 Bending Energies ...... 23 2.2.3 Torsion Energy ...... 23 2.2.4 Non-Bonded Interaction Energies ...... 23 2.3 Semi-Empirical ...... 24 2.4 Ab Initio ...... 25 2.4.1 Quantum Mechanics ...... 25 2.4.2 Hamiltonian Operator ...... 26 2.4.3 The Born-Oppenheimer Approximation ...... 27 2.4.4 Wave Function ...... 28 2.4.5 Slater Determinant ...... 29 2.4.6 Variational Principle ...... 31 2.4.7 Single Reference Methods ...... 32 2.4.8 Electron Correlation ...... 32 2.4.9 Hartree-Fock Approximation ...... 33 2.4.9.1 Minimization of Energy and Fock Operator...... 34

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2.4.9.2 Hartree-Fock-Roothaan Method ...... 35 2.4.9.3 Restricted, Unrestricted and Restricted Open HF ...... 35 2.4.10 Post Hatree-Fock Methods ...... 36 2.4.10.1 Moller-Plesset Perturbation (MP) Methods ...... 36 2.4.10.2 Configuration Interactions (CI) Methods ...... 37 2.4.10.3 Coupled Cluster Methods ...... 39 2.4.10.4 Quadratic Configuration Interaction (QCI) Methods ...... 40 2.4.11 Multi-Reference Methods ...... 41 2.4.11.1 Complete Active Space Self-Consistent-Feld (CASSCF) Method …………………………………………………………………41 2.4.11.2 Restricted Active Space Self Consistent Field (RASSCF) method………………………………………………………………….42 2.4.11.3 Multi-Reference Perturbation Theory (MR-PT) Method ...... 42 2.4.11.4 Multi-Reference Configuration Interaction (MRCI) Method .... 42 2.4.11.5 Multi-Reference Couple Cluster (MR-CC) Method ...... 42 2.5 Monte Carlo and Quantum Monte Carlo ...... 43 2.6 Molecular Dynamics ...... 45 2.7 Density Functional Theory (DFT) ...... 46 2.7.1 Kohn-Sham Density Functional Theory (KS-DFT) ...... 47 2.7.2 Classification of DFT ...... 48 2.7.2.1 Local Density Approximation (LDA) ...... 48 2.7.2.2 Generalized Gradient Approximation (GGA) ...... 49 2.7.2.3 Meta-Generalized Gradient Approximation (meta-GGA) ...... 50 2.7.2.4 Hybrid GGA (H-GGA) ...... 50 2.7.2.5 Double Hybrid GGA (DH-GGA) ...... 51 2.7.2.6 Global Hybrid meta-GGA (GH meta-GGA) ...... 51 2.7.2.7 Range-Separated Hybrid GGA (RS H-GGA) ...... 52 2.7.2.8 Hybrid GGA with Dispersion Correction (H-GGA-D) ...... 52 2.7.2.9 Range-Separated Hybrid GGA with Dispersion Correction ..... 53 2.8 Basis Sets ...... 53

2.8.1 Gaussian Functions ...... 54 2.8.2 Types of Basis Sets ...... 56 2.8.2.1 Minimal Basis Sets ...... 56 2.8.2.2 Split Valence Basis Sets ...... 57 2.8.2.3 Correlation-Consistent (cc) Basis Sets ...... 58 2.8.2.4 Effective Core Potentials Basis Sets ...... 58 2.8.2.5 Karlsruhe Basis Sets ...... 59 2.8.2.6 Plane Wave Basis Sets ...... 60 2.9 Computational Methodology Adopted in the Current Work ...... 60 2.10 Statistical Tools ...... 62 2.10.1 Root Mean Square Deviation (RMSD) ...... 62 2.10.2 Standard Deviation (SD) ...... 62 2.10.3 Pearson's Correlation (R) ...... 63 2.10.4 Mean Absolute Error (MAE) ...... 63 3 Results and Discussion ...... 65 3.1 Benchmark Study for BDE of C−X (X = Cl and Br) Bond in Halogens Containing Compounds ...... 67 3.1.1 Importance of Halogen Containing Molecules ...... 67 3.1.2 Experimental and Theoretical Studies of BDE of C−X Bond ...... 67 3.1.3 Efficiency of Pople Basis Sets ...... 69 3.1.3.1 Evaluation of DFs with 6-31G(d) Basis Set ...... 69 3.1.3.2 Evaluation of DFs with 6-311G(d) Basis Set ...... 75 3.1.4 Efficiency of Dunning Basis Sets ...... 79 3.1.4.1 Evaluation of DFs with Aug-cc-pVDZ Basis Set ...... 79 3.1.4.2 Evaluation of DFs with Aug-cc-pVTZ Basis Set ...... 83 3.2 Benchmark Study for BDE of C−Sn Bond in Organotin Compounds 88 3.2.1 Transition Metal Catalyzed Cross Coupling Reactions ...... 88 3.2.1.1 Applications ...... 88 3.2.1.2 Stille Cross Coupling Reaction ...... 88 3.2.1.3 Importance of Organotin Compounds ...... 89

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3.2.1.4 Experimental Studies of C−Sn Bond ...... 89 3.2.1.5 Theoretical Studies of C−Sn Bond ...... 89 3.2.1.6 Homolytic Versus Heterolytic Cleavage ...... 91 3.2.2 Efficiency of Effective Core Potential (ECP) Basis Sets ...... 92 3.2.2.1 Evaluation of DFs with LANL2DZ Basis Set ...... 92 3.2.2.2 Evaluation of DFs with Stuttgart–Dresden (SDD) Basis Set .... 96 3.2.3 Efficiency of Karlsruhe Basis Sets...... 101 3.2.3.1 Evaluation of DFs with Def2-SVP Basis Set ...... 101 3.2.3.2 Evaluation of DFs with Def2-TZVP Basis Set ...... 104 3.3 Benchmark Study for BDE of C−CN Bond in Nitrile Compounds .. 108 3.3.1 Nitriles as Hazards ...... 109 3.3.2 Experimental Study ...... 110 3.3.3 Theoretical Study ...... 110 3.3.4 Efficiency of Pople Basis Sets ...... 111 3.3.4.1 Evaluation of DFs with 6-311G(d,p) Basis Set ...... 111 3.3.4.2 Evaluation of DFs with 6-31G(d) Basis Set ...... 117 3.3.4.3 Evaluation of DFs with 6-31G(d,p) Basis Set ...... 119 3.3.4.4 Evaluation of DFs with 6-31+G(d) Basis Set ...... 122 3.3.4.5 Evaluation of DFs with 6-311++G(d) Basis Set ...... 124 3.3.5 Efficiency of Dunning Basis Sets ...... 127 3.3.5.1 Evaluation of DFs with Aug-cc-pVDZ Basis Set ...... 127 3.3.5.2 Evaluation of DFs with Aug-cc-pVTZ Basis Set ...... 129 3.3.6 Efficiency of Karlsruhe Basis Sets...... 133 3.3.6.1 Evaluation of DFs with Def2-SVP Basis Set ...... 134 3.4 Benchmark Study for BDE of C−Mg Bond of Grignard Reagents ... 137 3.4.1 Importance of Grignard Reagents ...... 137 3.4.2 Cross Coupling Reaction using Grignard Reagents ...... 138 3.4.3 Ionic versus Radical Pathways ...... 138 3.4.4 Experimental Study on BDE of C−Mg Bond of Grignard Reagents ………………………………………………………………….139

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3.4.5 Benchmark DFT Study on BDE of C−Mg Bond of Grignard Reagents ...... 139 3.4.6 Efficiency of Pople Basis Sets ...... 140 3.4.6.1 Evaluation of DFs with 6-31G(d) Basis Set ...... 140 3.4.6.2 Evaluation of DFs with 6-311G(d) Basis Set ...... 144 3.4.7 Efficiency of Karlsruhe Basis Sets...... 148 3.4.7.1 Evaluation of DFs with Def2-SVP basis set ...... 148 3.4.8 Efficiency of Dunning Basis Sets ...... 153 3.4.8.1 Evaluation of DFs with Aug-cc-pVDZ Basis Set ...... 153

3.5 Benchmark Study for BDE of M−O2 Bond for Water Splitting ...... 157 3.5.1 Introduction ...... 157 3.5.2 Literature Review ...... 157 3.5.3 Experimental Study ...... 158 3.5.4 Theoretical Study ...... 159 3.5.5 Efficiency of Effective Core Potential and Pople Basis Sets ...... 160 3.5.5.1 Evaluation of DFs with SDD & 6-31+G(d) Basis Sets ...... 161 3.5.5.2 Evaluation of DFs with LANL2DZ & 6-31G(d) Basis Sets ... 165 4 Conclusions ...... 169 4.1 Conclusions ...... 170 5 References ...... 173

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LIST OF FIGURES

Fig. 3.1 The Structures of Halogen-Containing Compounds with Known Experimental BDEs of C−X (X = Cl and Br) Bond ...... 69

Fig. 3.2 Root Mean Square Deviation (RMSD) of Different Density Functionals with Four Basis Sets for C−X (X = Cl and Br) BDEs ...... 72

Fig. 3.3 Standard Deviation (SD) of Different Density Functionals with Four Basis Sets for C−X (X = Cl and Br) BDEs ...... 73

Fig. 3.4 Mean Absolute Error (MAE) of Different Density Functionals with Four Basis Sets for C−X (X = Cl and Br) BDEs ...... 76

Fig. 3.5 Pearson’s Correlation (R) of ωB97X-D with 6-311G(d) Basis Set for BDE Calculations of C−X (X = Cl and Br) Bond ...... 86

Fig. 3.6 The Structures of Organotin Compounds with Known Experimental BDEs of C−Sn Bond ...... 91

Fig. 3.7 Root Mean Square Deviation (RMSD) of Different Density Functionals with Four Basis Sets for C−Sn BDEs ...... 94

Fig. 3.8 Standard Deviation (SD) of Different Density Functionals with Four Basis Sets for C−Sn BDEs ...... 96

Fig. 3.9 Mean Absolute Error (MAE) of Different Density Functionals with Four Basis Sets for C−Sn BDEs ...... 98

Fig. 3.10 Pearson’s Correlation (R) of BLYP-D3 with SDD Basis Set for BDE Calculations of C−Sn Bond ...... 99

Fig. 3.11 The Structures of Organo-Nitrile Compounds with Known Experimental BDEs of C−CN Bond ...... 111

Fig. 3.12 Root Mean Square Deviation (RMSD) of Different Density Functionals with Three Basis Sets for C−CN BDEs ...... 114

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Fig. 3.13 Standard Deviation (SD) of Different Density Functionals with Three Basis Sets for C−CN BDEs ...... 115

Fig. 3.14 Mean Absolute Error (MAE) of Different Density Functionals with Three Basis Sets for C−CN BDEs ...... 130

Fig. 3.15 Pearson’s Correlation (R) of CAM-B3LYP with 6-311G(d,p) Basis Set for BDE Calculations of C−CN bond ...... 133

Fig. 3.16 The Structures of Grignard Reagents with Known Experimental BDEs of C−Mg Bond ...... 140

Fig. 3.17 Root Mean Square Deviation (RMSD) of Different Density Functionals with Four Basis Sets for C−Mg BDEs ...... 143

Fig. 3.18 Standard Deviation (SD) of Different Density Functionals with Four Basis Sets for C−Mg BDEs ...... 147

Fig. 3.19 Mean Absolute Error (MAE) of Different Density Functionals with Four Basis Sets for C−Mg BDEs ...... 151

Fig. 3.20 Pearson's Correlation (R) of TPSS with 6-31G(d) Basis Set for BDE Calculations of C−Mg Bond ...... 155

Fig. 3.21 Catalytic Water Oxidation and Dioxygen Evolution Mechanism by Ru- Complexes...... 158

Fig. 3.22 The Structures of Transition Metal Complexes having Oxygen Molecule with

Known Experimental BDEs of M−O2 Bond ...... 160

Fig. 3.23 Mean Absolute Error (MAE) of Different Density Functionals with Two

Series of Basis Sets for M−O2 BDEs ...... 162

Fig. 3.24 Standard Deviation (SD) of Different Density Functionals with Two Series of

Basis Sets for M−O2 BDEs ...... 163

Fig. 3.25 Root Mean Square Deviation (RMSD) of Different Density Functionals with

Two Series of Basis Sets for M−O2 BDE ...... 166

LIST OF TABLES

Table 2.1 Primitive Gaussian Functions (pgf) and Contracted Gaussian Functions (cgf) of Different Basis Sets ...... 55

Table 3.1 RMSD, SD, R and MAE of C−X (X = Cl and Br) BDEs Calculated with Different DFs While Using 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)……………………………………..74

Table 3.2 RMSD, SD, R and MAE of C−X (X = Cl and Br) BDEs Calculated with Different DFs While Using 6-311G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 77

Table 3.3 RMSD, SD, R and MAE of C−X (X = Cl and Br) BDEs Calculated with Different DFs While Using Aug-cc-pVDZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 80

Table 3.4 RMSD, SD, R and MAE of C−X (X = Cl and Br) BDEs Calculated with Different DFs While Using Aug-cc-pVTZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 83

Table 3.5 Comparison of C−Sn Bond Dissociation Energy for Homolytic vs. Heterolytic Cleavage Calculated at BLYP-D3/SDD Method (All Values are Given in kcal/mol) ...... 92

Table 3.6 SD, RMSD, R and MAE of C−Sn BDEs Calculated with Different DFs While Using LANL2DZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0). For Differentiation of R Results, the Values are Given up to Three Decimal Points...... 92

Table 3.7 SD, RMSD, R and MAE of C−Sn BDEs Calculated with Different DFs While Using SDD Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0). For Differentiation of R Results, the Values are Given up to Three Decimal Points ...... 97

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Table 3.8 SD, RMSD, R and MAE of C−Sn BDEs Calculated with Different DFs While Using Def2-SVP Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0). For Differentiation of R Results, the Values are Given up to Three Decimal Points...... 101

Table 3.9 SD, RMSD, R and MAE of C−Sn BDEs Calculated with Different DFs While Using Def2-TZVP Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0). For Differentiation of R Results, the Values are Given up to Three Decimal Points...... 104

Table 3.10 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-311G(d,p) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 112

Table 3.11 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 117

Table 3.12 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-31G(d,p) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 120

Table 3.13 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-31+G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 123

Table 3.14 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-311G++(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 125

Table 3.15 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using Aug-cc-pVDZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 128

Table 3.16 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using Aug-cc-pVTZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 131

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Table 3.17 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using Def2-SVP Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 135

Table 3.18 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 141

Table 3.19 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using 6-311G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 145

Table 3.20 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using Def2-SVP Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 149

Table 3.21 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using Aug-cc-pVDZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 153

Table 3.22 RMSD, SD, R and MAE of M−O2 BDEs Calculated with Different DFs While Using SDD & 6-31+G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 164

Table 3.23 RMSD, SD, R and MAE of M−O2 BDEs Calculated with Different DFs While Using LANL2DZ & 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0) ...... 165

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LIST OF ABBREVIATIONS Ar Argon Br Bromine Cl Chlorine CN Nitrile CC Couple Cluster cc Correlation Consistent DFs Density Functionals DFT Density Functional Theory DH Double Hybrid Generalized Gradient Approximation E Energy GGA Generalized Gradient Approximation GH meta-GGA Global Hybrid meta-GGA H-GGA Hybrid GGA HF Hartree−Fock H Hydrogen LDA Local Density Approximation Li Lithium MAE Mean Absolute Error NMR Nuclear Magnetic Resonance MM Molecular Mechanics Mg Magnesium TM Transition Metal O Oxygen RMSD Root Mean Square Deviation R Pearson’s Correlation Rn Radon RS H-meta-GGA Range Separated Hybrid meta-GGA RS H-GGA-D Range Separated Hybrid meta-GGA with Dispersion Correction Sn Tin SD Standard Deviation vs. versus

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X Halogen 푥푐 Exchange Correlation

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Chapter 1

1 Introduction

1.1 Theoretical Chemistry

Theoretical chemistry uses the mathematical models based on fundamental laws of physics for studying different chemical processes. Molecules are composed of atoms, which are held together through charged particles [1]. The coulombic interaction between these charged particles is the only physical force in terms of chemical phenomena. Each molecule is different from the other on basis of number of electrons and protons, therefore interacting forces are also different. Various properties of the molecules can be calculated with the help of theoretical chemistry [2].

Walter Heitler and Fritz London were the pioneers of the theoretical chemistry and they performed first calculation in 1927. They reported theoretical explanation of the combination of two hydrogen atoms to form a molecule. Before the development of modern computers (earlier 1950s), limited systems were examined with high accuracy. During 1960-1970, the emergence and availability of modern computers throughout the world has increased their use tremendously in many branches of science. This revolution resulted in a new area of chemistry known as computational chemistry. In computational chemistry, the computer programs are used as an “experimental” tool for various studies of atoms and molecules. Development of new theoretical models using computers enable the scientists to study problems, and the results from simulation may improve the theory by overcoming the limitations [3].

In computational chemistry, mathematical models are used for solving chemical problems. The computers are used for generating information such as various properties of atoms or molecules and for justifying experimental results via simulations. Materials difficult to find or expensive to purchase are easy to investigate with the help of computational tools [4]. Computational chemistry also makes predictions and observations for the experimentalists before they start their work in wet lab. There are two sub types of theoretical chemistry approaches to solve chemical problems i.e. computational and non-computational quantum chemistry. The former is related to the numerical computation of molecular electronic structures via different computational methods. Whereas, latter is related with the formulation of analytical expressions for various properties of desired molecules. The importance of computational chemistry is obvious from the Noble Prizes of 1998 and 2013. The Nobel Prize of 1998 was given to Kohn and Pople for their efforts towards the development of DFT and basis sets,

2 respectively [5]. The Nobel Prize of 2013 was awarded to M. Karplus, M. Levitt and A. Warshel, for the development of powerful computer programs, used for the understanding and prediction of chemical problems [6].

1.1.1 Applications of Computational Chemistry

Computational chemistry is a set of mathematical tools used for the investigation of various chemical problems. Major applications of computational chemistry are:

(i) Molecular geometry; It describes the geometric parameters of molecules including bond lengths, bond angles and dihedral angles. (ii) Energetic analyses of molecules including reactants, products and their transition states; The measurements of energies give the idea about the feasibility of reactions and stability of the resultant products. (iii) Chemical reactivity; In which the charge distribution on electron rich as well as electron deficient sites is described in a specific molecule. (iv) Spectroscopic properties; Various spectroscopic properties such as vibrational, absorption and nuclear magnetic resonance are calculated computationally. (v) Quantitative structure-activity relationships; It is a strategy designed to find a relationship between chemical structure and activity of compounds. (vi) Physical properties; e.g. mechanical hardness and softness of a system is well described. These properties depend on the nature of a molecule individually and in bulk material. (vii) Molecular dynamics; It includes reaction rates and mechanisms. Specifically, protein folding phenomenon is theoretically studied in certain frame of time.

1.1.2 Limitations

The computational chemistry fails in solving rate equations and deconvoluting spectra. The applicability of computational chemistry depends on the precision of the level of theory for any application. The second task is the efficiency and execution of computer programs which depends on the CPU power for the desired task and computer resources.

1.2 Computational versus Experimental Chemistry

Since last decade, advancement take place in the field of computational chemistry due to the development of fast, efficient, safe and low-cost computer devices. The availability of advanced instruments has made possible accurate characterization of various physical as well as chemical properties of matter [7]. Besides these techniques, the simultaneous emergence of theoretical methods with the support of computers are also valuable tools for scientiﬁc community. Keeping these tools in hand, the researchers have the capability to predict, design and extract valuable information of ﬁnite and extended systems i.e. from simple to very complex. Computational chemistry is a helping hand towards experimental study. Computational chemistry explains and compliments experimental results. It goes to the point, where experiment can’t go such as: electronic distribution study, reaction energies, reaction barriers and molecular dynamics. Therefore, computational chemistry can make predictions and guide experiments. Overall, two main aspects of computational chemistry are to set a starting point for the assessment of experimental work and to find out the possibility of new molecules [8].

1.3 Benchmark Study

Benchmark approach is the performance evaluation of computational methods based on cost (cost in terms of time) and accuracy. Most of the times, theoretical methods selected for a given system are based on hit and trial and fail to reproduce the accurate results. Benchmark studies provide an effective accurate level of theory which can help in understanding mechanism of an important step especially rate-determining step and other important chemical bonds. The accurate method obtained via benchmark study is used for further theoretical studies of these bonds. Besides theoretical importance, this method can compliment and explain experimental work as well as make predictions and guide experiments. Therefore, benchmarking approach is helpful both for the experimentalist and theoretician for their future studies. During benchmark study, the theoretical results of cost-effective methods are compared either with experimental data or accurate theoretical results from higher-level calculations [9]. This approach has been successfully applied to different classes of compounds including inorganic

4 compounds, transition metals, metal complexes, metal clusters, organic compounds (hydrocarbons, organometallics and polymers) and bioorganic molecules [10].

The benchmark study has been applied successfully in order to explore the low-cost and accurate methods for various structural and electronic properties, spectroscopic properties, thermochemistry, kinetics, magnetic properties, dipole moment, polarizabil -ity, hyperpolarizability, nucleophilic substitution reactions, barrier heights, atom transfer reactions, unimolecular reactions, covalent and non-covalent interactions [11]. To access the accuracy of computational methods, different scientists have adopted benchmark approach to study the molecules. DFT methods are more economical for system with moderately large size, therefore these are the most common evaluated methods. Kohn–Sham density functional theory (KS-DFT) is widely used due to its low cost and accuracy. Various DFT methods have been developed in recent past, and assessed for their accuracy in calculating different properties of compounds [12, 13]. A model system is designed to study the characteristics of DFs by comparing to high- level methods i.e. CCSD(T) or other high-level methods. These ab initio methods are highly accurate methods and comparison with these methods is a valuable tool to understand the systematic errors of different DFs. Some ab initio methods are used as the gold standard in extrapolation schemes to give a clear picture (accurate answers) for a model system. These methods have systematic basis sets and are known as valuable sources in a computational chemist’s arsenal. Due to these reasons, comparison of DFT results to the highly accurate data obtained from ab initio methods allows to predict the errors observed for a density functional (DF) [14].

1.3.1 Applications of Benchmark Study

The literature search reveals that benchmarking approach is adopted to explore accurate and cost-effective methods for different properties of various molecules.

1.3.1.1 Benchmark Approach useful for Different Properties Studies

Benchmark studies are used for different properties measurements from small to large molecules. Curtiss et al., applied benchmark study for the measurement of atomization energy of 31 small molecules including CH4, LiH, NH2NH2 and CO2 etc. for accessing the accuracy of G1 and G2 computational methods. Later, this data set was expanded to 454 different molecules with new versions of G3 and G4 composites (accuracy was

5 much improved in case of G4 method) [15]. Later on, Zhao and Truhlar examined different DFT functionals and other computational methods for the study of thermodynamic properties, non-covalent interactions and activation barriers of a large number of compounds [16]. To remove any chemical biases, Grimme proposed a data set of artificial molecules, and further elaborated it with Georigk to generated GMTKN30 (main group elements) database covering kinetics, thermochemistry and non-covalent interactions [17]. Zhang et al., worked on the benchmarking of heat of formation of 70 transition metal complexes by implementing 42 different density functionals [18]. Weymuth et al., used benchmark approach using different density functionals for BDE measurements of ten transition metals complexes and results show that PBE0 and TPSSH have better results with less deviations (mean absolute deviations and largest absolute deviations) compared to experimental data. However, they appraised that benchmark study depends on the chosen molecule set and desired properties study [19]. Different theoretician applied various methodologies for the accurate measurement of BDE. Dunning examined BDE of diatomic molecules of 3d transition metals with the help of CCSD(T) method, and concluded that it shows the best agreement with experimental results having standard deviation of 9 kJ mol-1 [20].

1.3.1.2 Benchmark Study of Magnetic Properties

Benchmark studies for the magnetizability of various atoms and molecules are reported. The second-order response of the molecule to an external magnetic ﬁeld is known as magnetizability. For indirect spin–spin coupling constants measurements of different compounds, low cost DFT methods are the best choice. Lutnaes et al., applied B3LYP with a series of dunning basis sets (cc-pVXZ, cc-pVXZ, cc-pCVXZsu5 and HIIIsu0 etc.) and observed that B3LYP/HIIIsu0 level show the best performance for desired data set in comparison to experimental data [21]. In another study, they also examined the shielding effects and indirect spin-spin coupling constants of o-benzyne using DFT and CCSD methods. The study of such systems is very difficult because of the biradical nature of the molecules. Among different density functionals, the results of PBE with cc-pVTZ basis set are similar to the CCSD for spin-spin coupling. Spin-spin coupling of triple bonds in free o-benzyne is 205 Hz which is comparable to the 178 Hz obtained from experimental data [22].

1.3.1.3 Benchmark Study of Crystal Structures

Kronik et al., applied the benchmark DFT approach to find an accurate method for a description of crystal structure. Different DFs with and without minimally empirical pairwise Tkatchenko-Scheﬄer correction are used in that study. The best performance is achieved with Van der Waal PBE method (vdW-PBE). PBE-vdW gave equilibrium lattice parameters of crystal in very close agreement (∼0.05 Å), to the experimental data. Besides structural parameter, results about energies are also in good agreement to the experimental data [23].

1.3.1.4 Benchmark Study of Different SN2 Type Reactions

Martin et al., performed combined DFs and ab initio methods-based benchmark study

to determine the reaction energetics of halogenated compounds in gas phase via SN2 pathway. It is observed that among all selected DFs, mPW1K is the best functional for

calculations of activation barriers for SN2 reaction [24]. Hajdu and Czako explored the potential energy surfaces (PESs) of the SN2 reaction at high-level CCSD(T)-F12b − method for X + NH2Y systems, where X and Y are F, Cl, Br and I. For orbital description of these dynamic measurements, they used aug-cc-pVXZ (X = D, Q) for all atoms (whereas, effective core potential basis sets are used for Br and I atoms). Among all selected computational methods, CCSD(T)-F12b/aug-cc-pVQZ method gave the best results. They exclusively studied the minima and transition state for proton transfer. Besides these proton transfer minima, they located minima for halogen abstraction which plays a crucial role for accessing collision energies [25].

1.3.1.5 Benchmark Study of Metal and Non-Metal Clusters

Ayub and his coworkers carried out benchmark study of scandium clusters and proved BPV86 with LANL2MB basis set as the best method for structural and vibrational frequency analysis of scandium clusters, as an alternative to the gold standard (CCSD(T)) method [26]. A benchmark study of H-bonded small water clusters is executed by Santra at DFT and MP2 methods with complete basis set. The results showed high accuracy of hybrid X3LYP, PBE0, MPW3LYP and PBE1W functionals [27]. Katin reported quantum chemical investigation of three small boron nitride clusters (B2N2, B3N3 and B4N4). In this benchmark study, they compared the results of twenty-four DFs with the results obtained from high level couple cluster

7 methods. B3P86V5 and B97 functionals performed better for the geometry optimization whereas OP and VWN functionals were superior for electronic properties (these DFs gave the closest results to CC results). B3LYP and PBE0 functionals showed lower accuracy for both properties [28].

1.3.1.6 Benchmark Study for Simple and Complex Dimers

Sebastian and Martin applied ab initio and DFT methods with aVnZ (n = 3 and 4) basis sets for geometries and BDEs of halogen-halogen bond in different halogens dimers, and the results showed that M06-2X and ωB97X-D are better for the desired data [29]. David and Krik observed that CCSD(T) method showed good agreement with different basis sets for calculation of bond length and thermodynamic properties of carbon dimer and hydroperoxyl radicals. Their results indicated that computational results at CCSD(T) were in agreement with experimental data [30]. Wong and his co- workers provided a comparative analysis of different exchange and dispersion corrected functionals for benchmark data set of halogen-bonding interactions in a variety of complex dimers. Their results illustrate that non-covalent interactions between halogen dimers can be accurately studied through dispersion corrected methods in selected data set [31].

1.3.1.7 Benchmarking of Non-Covalent Interactions

The study of non-covalent interactions is important in many areas of biology, materials science and chemistry. Correlation eﬀects between electrons and nucleus of systems strongly influence these non-covalent interactions. Voorhis et al., executed a benchmark study of various DFT methods and observed high accuracy of LC-VV10 functional for geometries and interaction energies measurements of van der Waals complexes such as benzene-H2S, pyridine dimer and complexes of CO2 with small molecules but the computational cost was high due to the addition of long range correlation in DFs [32]. Sherrill and his co-workers examined non-covalent interactions of prototype bimolecular complexes of formic acid, formamide, and formamidine. A wide range of DFT functionals are compared with standard CCSD(T)/CBS. They evaluated that DFT-D functional with Dunning basis set (ωB97X-D/aug-cc-pVDZ) performed best for noncovalent interactions because of its dependence on electrostatic interactions [33]. Razec et al., determined binding energies using an accurate CCSD(

T)/CBS method for 66 molecular complexes. Along with CCSD(T) method, several other QM methods are used, and SCS-MI-CCSD method gave the best performance (where root-mean-square error (RMSE) was 0.08 kcal/mol) [34].

1.3.1.8 Benchmark Study of Bond Activations of Transition-Metal Based Catalysts

The activation barriers and reaction energies of covalently bounded Ni and Pd based catalysts are studied by Steinmetz and Grimme. For this purpose, they used 23 different density functionals (DFs) from various classes of DFT with variety of basis sets. Among all selected DFs, PBE0-D3 showed the best performance. Their results illustrated the worse performance of MP2 for PdCl2 and Ni [35]. G. S. Girolami and his co-workers implemented benchmark study for the coordination reactions of methane with transition metals. They applied ab initio and different DFs (LSDA, BP86, PBE, B3LYP, MPW1K, PBE0, TPSSH, BB1K, BMK and M05-2X) for the geometrics and thermodynamics of minima and transition states of these molecules, and compared the results of those with CCSD(T) and experimental data. Their focus was on the reversible reaction of C−H bond activation to form osmium methyl hydride complex. MPW1K and M05-2X provide better results for the measurement of bonding interactions in comparison to CCSD(T) method. For weak interactions study, they recommended DFT method with dispersion correction and double-zeta basis sets [36].

1.3.1.9 Benchmark Study for Late-Transition Metals Reactions

Martin and his co-workers used benchmark DFT approach to study reactions of late- transition metals. In their outcomes, they suggested B1B95, B97-1, B97-2, PBE0, PW6B95, TPSS25B95 and TPSS1KCIS as best performers. Among these DFs, B1B95 and PW6B95 are better for reaction barriers and weak molecular interactions of late- transition metals and main group elements [37].

1.3.1.10 Benchmark Study on Different Properties of Lanthanides Series

Fantin and his co-workers studied valence triple zeta polarization functions (TZP) for the lanthanides series. They examined the performance of B3LYP with TZP and ATZP- DKH basis sets for bond lengths, charges, valence orbital analyses and dissociation energies of lanthanides and their monoxides. They compared their results with

9 experimental and theoretical data (including B3LYP/DKH, B3LYP/6-311++G(df) & ECP and ZORA-PBE/TZ2P). They concluded that ATZP-DKH basis set can be used as a reliable basis sets for molecular property calculations. The reason for better performance of the ATZP-DKH basis set is that both the valence and core electrons are combinedly treated in this basis set [38].

1.3.1.11 Benchmark Study for Geometric and Thermodynamic Properties of Noble Gas Containing Molecules

Hu and his co-workers executed a benchmark DFT approach on the noble gases, for the prediction of their geometric and thermodynamic properties. They used a collection of 13 different DFs from pure and hybrid DFT methods with a series of various basis sets. For their study, they implemented these methodologies on a data set of thirty-one inert gas complexes. The results obtained from DFT are compared with the results obtained from the CCSD(T)/ aug-cc-pVTZ for reference structures and CCSD(T)/CBS for reference energies calculations. In comparison, hybrid class DFs showed better performance compared to DFs of pure DFT class. Among selected methods, MPW1B95/6311+G(2df,2pd), DSD-BLYP/aug-cc-pVTZ, B2GP-PLYP/aug-cc-pVTZ, and BMK/aug-cc-pVTZ are good for bond energies and MPW1B95/6-311+G(2df,2pd), B3P86/6311+G(2df,2pd), MPW1PW91/6-311+G(2d- f,2pd), DSD-BLYP/aug-cc-pVTD and SD-BLYP/6-311+G(2df,2pd) methods are better for bond distance. Overall, for noble gas containing complexes, DSD-BLYP/aug- cc-pVTZ and MPW1B95/6-311+G(2df,2pd) methods are best approaches for theoretical studies [39].

1.4 Bond Dissociation Energy

Measurement of strength of a chemical bond is known as bond dissociation energy (BDE), it is also known as standard enthalpy change of a system at 298 K. Bond dissociation energy is the amount of energy required for the dissociation of a single bond whereas bond energy is the average of the BDEs of all similar chemical bonds in a system. Therefore, except diatomic molecules, all molecules have different BDE compared to bond energy. Cleavage of a chemical bond usually results in a radical or charged species [40].

1.4.1 Homolytic versus Heterolytic Bond Dissociation

Symmetrical dissociation of a chemical bond is known as homolytic bond dissociation while asymmetric cleavage is called heterolytic bond dissociation. In gaseous state, homolytic cleavage of a bond is the usual pattern for dissociation and its energy is lower than the heterolytic one. The reason for high energy of heterolytic dissociation is the more energy requirement for the separation of resultant opposite charges [41].

Example

The homolytic BDE of H-H bond in hydrogen molecule results in the formation of two hydrogen radicals. The enthalpy of the reaction is 104 kcal/mol (as shown in Eq. 1.1). On the other side, the heterolytic cleavage of this bond results in the formation of positively and negatively charged hydrogen. The enthalpy of the reaction is 400.4 kcal/mol in the gas phase (as shown in Eq. 1.2). In solvent phase, the BDE of heterolytic cleavage becomes lower (as shown in Eq. 1.3).

• H2 → 2 H ΔH = 104 kcal/mol Eq. 1.1

+ − H2 → H + H ΔH = 400.4 kcal/mol (gas phase) Eq. 1.2

+ − H2 → H + H ΔH = 66 kcal/mol (in water) Eq. 1.3

Dissociation of a bond is related to heterolytic or homolytic breaking of a chemical bond. Homolytic breaking of a chemical bond results in radical formation. Radicals are short-lived species and their lifespan is one second to be identified by electron spin resonance. Moses Gomberg reported the first free triphenyl phosphine radical in 1900. These radicals are important species in several reactions. Since there is limited knowledge about radical species in literature, particularly about their energetics. For · example, cyclopentadienyl radicals ( C5H5) are used as ligands in organometallic chemistry. To understand the nature of metal-C5H5, it is necessary to know the o o energetics i.e. heat of formation (ΔH f) of C5H5 radical. ΔH f of these radicals is calculated from the dissociation of C5H5-H bond [42]. Besides thermodynamic data, BDE also gives an idea about the stability of these radicals.

1.4.2 Techniques used for Determining BDE

Different techniques are used for the measurements of BDE of chemical bonds, the details of which are as follows:

1.4.2.1 Electron Capture Detector (ECD) Method

ECD, invented in 1957 by James Lovelock, is used for detection of atoms, ions and radicals. It consists of an ionization chamber having radioactive source of particles with two electrodes (polarized). It is possible to collect the thermal electrons by applying a potential difference to the ECD electrodes. For this purpose, ECD uses a metal foil (known as radioactive emitter) as a source of primary ionizing particles [43].

1.4.2.2 Collision Induced Dissociation (CID) Method

Collision induced dissociation is the most common activation technique being used in spectrometric instruments. CID mass spectrometric technique involves the fragmentation of ionic compounds and their molecules in gas phase. Therefore, the CID is very sensitive towards the masses of resultant species. Overall, two-step mechanism occurs during CID process. First step is excitation (precursors) and second step is their fragmentations. Both steps are separated based on time interval [44].

1.4.2.3 Threshold Photoelectron Photoion Coincidence (TPEPICO) Spectroscopic Method

A combination of threshold photoelectron spectroscopy and photoionization mass spectrometry is known as TPEPICO method. This technique is used for the measurement of internal energy with threshold of emitted electrons from selected ions. Threshold means detection of only electrons with zero kinetic energy and as a result, the internal energy imparted to the selected ions. Decay dynamics information of individual vibrionic states of positively charged molecular ions and radicals are obtained via this technique [45].

1.4.2.4 Infrared Chemiluminescence Method

John Polanyi developed infrared chemiluminescence technique (IRCT) based on the observation of excitation of a molecule, and then emission of infrared light. The

12 changes in emitted light during a chemical reaction are studied via spectroscopic analysis which are helpful to understand the exchange of chemical bonds. The basic principle of this technique is to study chemical reaction mechanisms by the measurement and analysis of weak infrared emissions from product molecules (which are formed during a chemical reaction). Formation of products with excess energy appears as excited vibrational states of the molecules which decay with emission of IR radiation. Information about the states in which the desired product molecules are formed, is obtained from spectroscopic investigation of this IR radiation [46].

1.4.2.5 Photoionization Mass Spectrometry

It is a spectrometric technique used for investigating the nature of low-pressure flames and elementary reaction kinetics. This method gives details about the mass, physical dimension and photon energy of the system. In this technique, a high-energy photon (X-ray or UV) is used to dissociate stable gaseous molecules in a carrier gas (He or Ar). This technique has two advantages; (a) multiple species are detected simultaneously with less time consumption and unexpected chemical intermediates are automatically detected, (b) the easily tunable photon energy allows to determine isomeric composition through the photoionization spectrum. The later point is more valuable for the researcher. For quantitative analysis, the data are compared to reference spectra. Where the % mass of the isomers is determined in two low-pressure flames [47].

1.4.2.6 Combustion Calorimetric Method

A type of calorimetry used to find out the heat of combustion (enthalpy change) of a chemical bond at constant-volume. Combustion calorimetry stands with high pressure during chemical analysis. The reference system is burn first, and the resultant exothermic reaction allows to measure the heat content of the calorimeter. Measurement of the enthalpy change of the final reaction gives an idea about the nature of the chemical bonds in these compounds [48].

1.4.2.7 Reaction-Solution Calorimetric Method

This technique is used to measure the change in heat during a chemical process. If an exothermic reaction takes place in a solution, the heat is produced by the system. On the other side, during an endothermic reaction, the heat required is removed from the

13 solution to proceed the reaction (temperature of the solution decreases). For the measurement of temperature change, a thermometer is used [49].

1.4.2.8 Photo Acoustic Calorimetric Method

This technique is used for high precision thermochemistry studies of short lived (ions and radicals) species, developed by Alexander Graham Bell in 1880. Tam and Patel for the first time presented the theory of photoacoustic effect while Peters and Braslavsky groups established its applications in pulsed laser photoacoustic calorimetry, around 1983. According to the basic principle, sample containing dilute solution of a photosensitive specie is deposited in the cell. For the initiation of reaction, the laser light is passed through the cell sample. During this process, a part of the energy is absorbed by the photoreactive species of the sample for cleavage of chemical bond in solution. Whereas, remaining energy is deposited as heat in the solution because laser light energy is more than the required energy for the cleavage of that bond. The solution locally expands due to abrupt and localized heating. Thus, a sound wave is produced which travels through the fluid and is detected by microphone-scope arrangement. It is a fast phenomenon and takes microseconds to detect heat changes [50].

1.4.2.9 Heat of Formation

The amount of heat evolved or absorbed during formation of a compound from its constituent elements at 25 °C and 1 atm pressure is known as heat of formation. Heat is released when a compound is formed, and the reaction is known as exothermic reaction. The heat is absorbed when a compound is dissociated, and the reaction is known as an endothermic reaction [51].

Standard heat of formation is defined as the enthalpy change for formation of one mole of a substance in standard state. ΔH°f is the symbols used for the standard heat of formation, it is also known as the standard enthalpy of formation. The heat of reaction is indirectly calculated for any reaction from the standard heats of formation at standard conditions, an application of the Hess’s law. When reaction occurs at standard conditions, then it is called standard heat of formation (ΔH°f).

ΔH°f = ∑ n ΔHf (product) - ∑ n ΔH°f (reactant) Eq. 1.4

Whereas, n represents that the heat formation of reactant and product is multiplied with a coefficient. The standard heats of reaction are calculated from standard heats of formation.

1.4.2.10 Kinetic Studies

Kinetics studies are also used to find out the rates of chemical reactions. Based on experimental conditions, kinetics gives an idea about the feasibility of the reaction [52]. For the first time, Peter Waage and Cato Guldberg developed chemical kinetics phenomenon in 1864. This phenomenon is based on the law of mass action. According to this concept, the kinetics of a reaction is directly proportional to the reacting species. Kinetics is essential for the fundamental understanding of the system designs and optimization of the catalytic chemical processes. Chemical reaction rate, thermodynamic effects and effect of various process variables are the important parameters of the chemical kinetics. Rate law and rate constants are derived from chemical kinetics. Law of mass action is followed by elementary steps but for the step wise reactions, rate law is derived from their individual steps. In the latter case, kinetics is measured from the rate determining step. The concentrations of the reactants, physical state of the reactants and temperature are the main factors influencing kinetics of a chemical reaction (in the presence or absence of any catalysts in the reaction).

1.4.3 Role of Computational Studies towards BDE Measurements

The concept of a chemical bond in chemistry has been introduced more than two centuries ago. The concept is based on the configuration of attractions between atoms, which results in the formation of a chemical compound. The main participants of these actions are the atom’s electrons and nuclei which result in bond formation. Theoretical chemistry briefly describes the bond concept based on electronic theory of matter [53]. Computationally, electronic density distribution (related to an atom in a molecule theory) and empirically interatomic distance (which are compared to standard bond lengths of different structures) between two atoms in a molecule are structural identifiers associated with a chemical bond. These interrelation parameters are used to illustrate bond formation, bond stretching and bond cleavage in a chemical system [54]. Mostly “bond transformations” are explained with respect to covalent bonds. From last decade, theoretical developments and their implementation in large software packages

15 made it easy-to-use quantum theory in the study of different chemical systems and their reactivity. Quantum chemical methods provide more accurate predictions of geometric structures, thermochemical data of chemical compounds and their reactions [55]. The basic purpose of optimization is to calculate minimum energy for a given chemical system. This is obtained at 0 K from a numerical solution of non-relativistic time- independent Schrӧdinger wave equation. The energy is a minimum for a set of atomic coordinates in correspondence to the optimized structure. Also, to obtain estimates of the enthalpy at the one desired temperature, corrections may be added to the calculated energy. These data can then be used to calculate thermochemical properties such as reaction energies or enthalpies etc. Among various energies parameters, bond dissociation energy is of interest to understand the reactivity of a chemical system [56].

Despite its importance, the already reported BDE values are limited and some of these required re-evaluations. The reason is the experimental difficulties for calculating BDE of resultant short-lived species. Computational chemistry is used as an efficient and reliable method for the BDE measurements from last few years. Computational chemists designed models of different chemical bonds which allowed an accurate computation of BDEs of these bonds. They implemented various computational tools for BDE measurements of these bonds. Most common methods are ab initio and DFT methods. Among all these methods, ab initio methods provide accurate predictions. But these methods are size consistent and highly expensive. DFT is particularly useful for studying thermochemical, electronic and structural properties from small to large molecules and its accuracy is also reliable [57].

Nunes et al., used experimental and ab initio methods for the measurement of BDE of carbon-hydrogen bond of 1,3-cyclopentadiene. Among theoretical methods, the results obtained from CCSD(T) are in good agreement with the experimental TR-PAC results [58]. Computational studies of BDE measurements play a vital role in understanding the mechanism of many reactions which is impossible experimentally. For example, DiLabio and his co-workers designed a procedure for the calculations of the BDE and ionization potential (IP) of phenolic antioxidants, based on DFT with locally dense basis sets methods. Several classes of phenolic antioxidants including food additives, sterically hindered phenols, flavonoids in tea, stilbenes related to resveratrol, aminophenols, and compounds related to Vitamin E are examined theoretically. After

16 the investigation of homolytic BDE of RO-H bond of phenol, their rate of reactions with free radicals (usually formed in body) are analyzed. Two mechanisms are expected for this reaction; (a) H-atom-transfer related to BDE and (b) a single-electron-transfer related to ionization potential (IP). They observed that in most cases the H-atom transfer via BDE is dominated. These types of studies are important in designing antioxidant for any required biological role [59]. Lu et al., measured two series of BDE of the S- NO bond in S-nitrosothiols (RSNOs) through experimental as well as theoretical approaches. Experimentally, the reaction is carried out in calorimetry while theoretically evaluated at B3LYP/6-31+G(d). They observed that data obtained from latter methods are in good agreement with the experimental results [60]. Francisco and his co-workers examined bond dissociation energies in haloalkanes at MP2 and CCSD(T) with cc-pVXZ (X = 3 and 5). Corrections for the spin-orbit (of atomic fragment) and zero-point energies yielded accurate bond dissociation energies in comparison to experimental results. They applied similar correction for multihalogenated compounds (CH2X2 and CHX3 (X = F, Cl, and Br), and again accurately reproduced experimental results [61]. Klein and Lukes studied O-H BDE of substituted phenols in correlation with the C−O bond length at B3LYP. Again, the data of B3LYP is in good agreement with the experimental results [62]. Klopper et al., determined the geometry and BDEs of water dimers at CCSD(T) and MP2 methods with all basis sets limit. Results of CCSD(T) are comparable to the experimental outcomes [63]. Rablen and Hartwig implemented G-2 and CBS-4 methods to explore the sequential homolytic BDEs of a series of cyclic and acyclic boranes. Specifically, the dissociation energies of B-H, B-C, and B-F bonds are explored in these compounds [64]. Senosiain et al., used two DFT functionals with CBS-Q method (when required), for BDE measurements of hydrocarbons. Among these selected, B3LYP and KMLYP functionals, the latter is better for the practical applicability [65].

Xun and his co-workers performed quantum chemical calculations to investigate bond length and BDEs of C–COOH bond of 15 acids. They used experimental data as a benchmark for comparison of different hybrid DFs such as B3LYP, B3PW91, B3P86 and PBE0. Among all selected hybrid DFs, B3P86 with 6311G(d,p) basis set gave the best results [66]. Yao et al., theoretically studied the homolytic BDE of main group elements (C−H, N−H, O−H, S−H, X−H, C−C, C−N, C−O, C−S, and C−halogen). They applied B3LYP, MPW1PW91, B3PW91, B3P86, and MPW1P86, to evaluate

17 comprehensively the BDEs of selected bonds and compared the results with available experimental BDE values. The results indicated that the MPW1P86 provides the best agreement with already reported results, whereas B3LYP delivered the largest deviations [67]. Therefore, this is observed that several DFs can be applied to get highly precise BDE values for a system. After these reports, a series of benchmark studies are executed in search of an accurate computational method for dissociation energies of different bonds.

1.4.4 Benchmark Study of BDE of Different Chemical Bonds

The literature search reveals that benchmarking approach is adopted for exploring accurate and cost-effective method for BDEs of various chemical bonds in different molecules. DiLabio and his co-workers demonstrated that thermodynamic properties such as non-covalent binding energies, covalent BDEs and activation barrier calculations of pericyclic reactions are improved by incorporation of dispersion correction to simple DFT methods [68]. Younker et al., measured the BDEs of C−O and C−C bonds in substituted β-O-4 Lignin model by adopting benchmark approach and confirmed that M06-2X works better than the gold standard CCSD(T) method [69].

Similarly, Dang and his co-workers studied BDEs of C−SO2R bond by using DFT methods and observed that M06-2X/6-31G(d) method shows the best performance [70]. Poliak and Vagánek evaluated the accuracy of four density functionals with two basis sets for BDE and ionization potential (IP) of C−H bonds and their benchmark study results showed that B3LYP with 6-31G(d) basis set performed best for both properties [71].

Amin and Truhlar performed benchmark study of Zn co-ordination compounds with O- 2 -2 - - , S , NH3, H2O, OH and (SCH3) ligands. BDE, bond distance and dipole moment are measured at semi-empirical, ab initio and DFT methods. PM3 and PM6 of semi- empirical and M05-2X, B97-2, and mPW1PW91 of DFT performed best for all these properties with small mean unsigned error (MUE < 1 kJ/mol) [72]. First time, Kepp executed benchmarking of experimental bond enthalpies of fifty-one AuX (X = atom or molecule) compounds and seven additional polyatomic and cationic gold clusters. The performance of twelve DFs from meta-GGA, hybrid, double-hybrid, dispersion- corrected hybrid GGA and non-hybrid GGA classes is evaluated. Dispersion corrected PBE or TPSS functionals performed well in comparison to other functionals due to the

18 implicit approximations in these functionals. Besides these DFs, effective core potentials (ECP) basis sets are used to model the relativistic bond contraction. These results illustrate that DFT with ECP basis sets is the best level of theory for understanding of gold (Au) chemistry [73].

Wang et al., theoretically studied homolytic BDE of C−B bond in organoborane compounds including ab initio methods and thirty-four density functionals. They observed that M06-HF performed the best (RMSE = 4 kJ/mol) [74]. Li et al., observed that M08-HX is a good functional (MUD = 1.0 kcal/mol) for BDE measurement of C−X (X = C, H and O) bonds in biodiesel molecules such as unsaturated methyl ester and methyl-linolenate. They used CCSD(T)/CBS as a reference for BDE calculations. All results are first compared to experimental BDE values and then, with the most expensive multi-reference averaged coupled pair functional (MRACPF2) method. The mean unsigned deviations (MUDs) of M08-HX from CCSD(T)/CBS and MRACPF2 are 1.0 kcal/mol and 1.1 kcal/mol, respectively [75]. Zhao et al., investigated the BDE of ylidic bond at different quantum mechanical methods. They observed performance of various DFs on the basis of lower mean unsigned error. The better DFs with their DFT classes include VSXC from LDA class, LC-BLYP, ωB97, M11, and ωB97X from GGA class, M06-L from meta-GGA, M05-2X, M06-2X, MPWB1K and M08-HX from GH meta-GGA class and SOGGA11 from GH meta-NGA class [76]. Retzer et al., 6 theoretically studied the geometries, stabilities and BDE of M−O2 and M-S bonds in d (pentacarbonyl) metal complexes at LSDA-PW91/TZ level. They observed that BDE depends on π back donation and softness/hardness of metal [77].

The bond lengths, vibrational frequencies and BDEs of P−P bonds in organophosphorus compounds are studied experimentally and theoretically by Chitnis et al., and they evaluated that homolytic cleavage is dominated over heterolytic dissociation [78]. Hemelsoet et al., used high-level ab initio and DFT methods for the thermodynamic properties measurement of phosphorus-containing compounds. Bond dissociation energies calculated at these methods are compared with G3(MP2)-RAD results. Results indicate that BMK, M05-2X, and SCS-ROMP2 methods gave reliable results. Analyses also indicated the BDEs of tricoordinate phosphorus compounds are lower in comparison to pentacoordinate phosphorus compounds and the order of increasing

BDE of selected compounds is (P−OPh) < (P−CH3) < (P−Ph) < (P−OCH3) [79].

1.5 Statement of Problem

Literature does not reveal any significant data about the BDEs of important chemical bonds (C−X (Cl and Br), C−Sn, C−CN, C−Mg and M−O2) involved in the rate determination step of well-known named reactions. Previously, some of these bonds are studied but the computational methodology used is expensive. Moreover, computational methodology, statistical analyses methods are also limited. For better results, scientist need to use a variety of statistical methods. An accurate and cost- effective density functional with suitable basis set is needed for the theoretical study of BDEs of chemically important bonds.

1.6 Motivations

Benchmarking plays a pivotal role in the accurate measurement of BDEs of different molecules. But the methodologies used in literature are expensive and deliver results with moderate accuracy. Our motivation is to search out less time consuming and accurate method for the measurement of BDEs of different bonds, particularly which play an important role in many key reactions.

1.7 Aims and Objectives of the Present Work

1. Quantum chemical investigations for bond dissociation energies of; • Unusual carbon-halogen bond (C−X, X = Cl and Br) and carbon nitrile bond (C−CN) • Carbon stannous bond (C−Sn) and carbon magnesium bond (C−Mg)

• Metal-oxygen bond for water splitting reaction (M−O2)

2. To apply benchmark approach by implementing a variety of density functionals and different basis sets with a wide range of statistical analyses.

3. Computationally determined bond dissociation energy values will be compared with experimentally known BDEs values of the respective bonds available in the literature, to explore cost-effective and accurate methods for respective bonds.

Chapter 2

2 Computational Methodology

2.1 Computational Methodology

Computational chemistry describes the structures, stability and energy of different states of molecular systems as well as bulk materials with the help of different computer programs [80]. It gives information about reaction processes within molecular systems. Different computational methods are developed by scientist over the past few decades which include:

(i) Molecular mechanics (ii) Semi-empirical (iii) Ab-initio (iv) Monte Carlo and quantum Monte Carlo (v) Molecular dynamics (vi) Density functional theory

2.2 Molecular Mechanics

Molecular mechanics (MM) describes molecular structures and properties on the basis of classical mechanics. The basic principle on which molecular mechanics depends is that atoms act as particles. Particles are spherical in shape having zero charge. Interactions between atoms are based on classical mechanics (relationship between the motion of an object and the forces acting on it is described by classical mechanics). These interactions explain the spatial distribution of particles and their associated energies [81]. The main objective of MM theory is to find out the total energy of a molecule. The energy equation of the MM can be described as:

E = Es + Eb + Et + Enbi Eq. 2.1

E represents total energy of a system. Whereas, Es, Eb, Et, Enbi are bond stretching, bending, torsion and non-bonded interaction energy terms, respectively between atoms of a system.

2.2.1 Bond Stretching Energies

This is based on the Hooke's law and the mathematical expression of stretching energy is given as:

2 퐸푠 = ∑푏표푛푑푠 푘푏 (푟 − 푟표) Eq. 2.2

The kb describes the strength of the respective bond, r represents the bond length while ro denotes its equilibrium bond length. This equation estimates the energy associated with vibration of bonds at equilibrium [82].

2.2.2 Bending Energies

This is also based on the Hooke's law and the mathematical expression of bending energy is given as in the following equation:

2 퐸푏 = ∑푎푛𝑔푙푒푠 푘ɵ (ɵ − ɵ표) Eq. 2.3

The kɵ describes the strength of the respective bond angle. ɵ represents bond angle between atoms and ɵ표 represents its value at equilibrium. This equation measures the energy of respective bond angle at equilibrium [83].

2.2.3 Torsion Energy

Torsion energy is used to capture steric and non-bonded interactions between two atoms (A and D) connected by B and C atoms. Mathematically torsion energy is expressed as:

퐸푡 = ∑푡표푟푠푖표푛푠 퐴 [1 + cos (푛휏 − Փ)] Eq. 2.4

A represents dihedral angle force constant, 푛 is the periodicity, 휏 represents torsion angle, and Փ is the representation of dihedral angle. Periodicity (푛) may vary, based on different contribution (dipole-dipole, hyperconjugation and hydrogen bonding etc.) toward the energy [84].

2.2.4 Non-Bonded Interaction Energies

Non-bonded interaction energy term is the combination of van der Waals and columbic (electrostatic energy term) forces (see in Eq. 2.5).

−퐴푖푗 퐵푖푗 푞푖푞푗 퐸푛푏푖 = ∑푖 ∑푗 6 + 12 + ∑푖 ∑푗 Eq. 2.5 푟푖푗 푟푖푗 ∈푟푖푗

van der Waals terms electrostatic term

Where the first term (van der Waals) is based on the Lennard-Jones or different potentials, −퐴푖푗 represents the repulsive term and 퐵푖푗 represents attractive term between atoms i and j, respectively. Electrostatic energy term in Eq. 2.5 is based on

Coulomb’s Law, where 푞푖푞푗 represents partial charges on the atoms (i and j) and ∈ represents effective dielectric constant. Similarly, 푟푖푗 defines interatomic distance between these atoms (i and j) and ∑ is a Greek letter means summation. For explicit treatment of hydrogen bonding interactions, additional non-bonded energy terms are also included. Popular methods of molecular mechanics include assisted model building with energy refinement (AMBER), chemistry at Harvard macromolecular mechanics (CHARMM), carbohydrate hydroxyls represented by external atoms (CHEAT), all-purpose organic or bio-organic molecule force field (DREIDING), empirical force field (EFF), merck molecular force field (MMFF), general-purpose organic force fields or molecular mechanics (MM1, MM2, MM3, MM4), universal force field (UFF) and interactive molecular mechanics (YETI) [85] etc.

2.3 Semi-Empirical

This is based on the Hartree-Fock method but additional approximations and empirical parameters are introduced in these methods to simplify calculations for computationally demanding terms. Approximation are carried out in semi-empirical methods by omitting some of the two-electron integral or by approximating them. Remaining integrals are parameterized for the correction of resultant error from these approximations. These parameters are fitted to the experimental data or high-level ab initio methods [86]. The complexity of calculations is reduced in these methods by explicit treatment of only valence electrons. The core electrons are treated via scaling down the nuclear charge or the coulomb effects of core electrons and nuclei are simultaneously treated by introducing functions. Basis sets used to treat the valence electrons and the number of functions representing the valence orbitals are limited to minimal basis sets. Due to which, in most of the semi-empirical methods, s- and p- orbitals are used and basis functions implemented are Slater type orbitals. For the first-

24 row elements, s- and p-type functions are used whereas s- and p- and d-type functions are used for transition metals.

Modern semi-empirical models are based on the neglect of diatomic differential overlap (NDDO) method. NDDO is a type of central approximation in which atomic orbitals residing on different atomic centers are insisted not to overlap, to reduce the overall computation [87]. So, the differential overlap of basis sets is set to zero as shown in Eq. 2.6.

휇퐴(1)휇퐵(1) = 0 A ≠ B Eq. 2.6

In Eq. 2.6, 휇퐴 and 휇퐵 represent atomic basis functions of atoms A and B, respectively.

In NDDO, the electron-electron interaction is reduced from N4 to N2 (N is the total number of basis functions in the Roothaan-Hall equation [88]. The development of semi-empirical method is based on the extended Huckel theory and used for electronic energy and molecular orbitals measurements. It ignores all electron-electron repulsions. This method is used for calculation of equilibrium geometries, transition states and excited states. However, semi-empirical methods are not suitable for thermochemical calculations and conformational analysis. Computationally, semi-empirical methods are less demanding than ab initio methods. Different types of semi-empirical methods are as follows: CNDO, SINDO, MINDO, ZINDO, NDDO, Huckel, PPP, MNDO, AM1, PM3 and PDDG/PM3 [89].

2.4 Ab Initio

Ab initio is a quantum mechanical method which is derived directly from Schrӧdinger wave equation without inclusion of experimental results [90]. The name “ab initio” is derived from Latin word meaning “from the beginning”.

2.4.1 Quantum Mechanics

Quantum mechanics (QM) describes the nature of matter and its interactions with energy on the atomic and sub atomic scales. QM explains the dual nature (wave as well as particle) of matter. In QM, the state of a system is entirely described by a wave function (Ψ). This Ψ depends on the 3D coordinates of all particles in a system.

휓푛 (푥1, 푥2, … … 푥푁, 푅1, 푅2, … … 푅푀) Eq. 2.7

Where, 푥 represents the space spin coordinates of N electrons and 푥 (푥 = si, ri) itself is equal to si and ri. si illustrate spin coordinates and ri represents cartesian coordinates of the electrons, whereas n (principal quantum number) is different for different systems. R represents cartesian coordinates of the M nuclei. One of the important conditions for a wave function is to always obey the antisymmetric property of a wave function [91].

휓푛 (푥1, 푥2, … … 푥푁) = −휓푛 (푥1, 푥2, … … 푥푁) Eq. 2.8

Most accurate ab initio methods are directly derived from wave function without any empirical parameterizations. Ab initio calculations are useful for transition and excite- d states. It is mathematically rigorous method due to which these are computationally expensive. The 휓 is a complete solution of Schrӧdinger wave equation.

Ĥ 휓n = 퐸푛휓n Eq. 2.9

Symbol Ĥ represents the quantum mechanical operator (known as Hamiltonian operator) for the energy, its eigenvalues 퐸푛 defines the energy of the system and the eigenfunctions 휓푛 is the corresponding molecular wave functions. Erwin Schrӧdinger reported this famous wave equation in 1926 and he shared the Nobel prize of 1933 with

Dirac for this discovery. If the wave function (휓푛) is known, then any time independent observable property can easily be computed by the measurement of required value [92].

∗ ∫ 휓n (휏)Ω̂휓n(휏)푑휏 Eq. 2.10

In Eq. 2.10, ʃ term is used for integration over all the spatial coordinates and over the full space 휏. Whereas, Ω̂ represents Hermitian operator (associated with the observable property) and the wave function (휓푛) of n state is assumed to be normalized. This equation is usually used for the measurement of the ground state energy [93].

2.4.2 Hamiltonian Operator

Hamiltonian operator of any system consisting of M nuclei and N electrons in the absence of external potentials is deﬁned as

Ĥ = 퐸̂푒 + 퐸̂푛 + 푉̂푛푒 + 푉̂푒푒 + 푉̂푛푛 Eq. 2.11

Where, Êe and Ên represent kinetic energies of the electrons and nuclei, respectively.

푉̂ ne, 푉̂ ee and 푉̂ nn represent nuclear-electron attraction, electron-electron repulsion and repulsion between two nuclei, respectively. These are further elaborated into different terms such as:

1 푁 2 Êe = - ∑ ∇ Eq. 2.12 2 푖=1 푖

푀 1 2 Ên = - ∑퐴=1 ∇퐴 Eq. 2.13 MA

̂ 푁 푀 푍퐴 푉푛푒 = - ∑푖=1 ∑퐴=1 Eq. 2.14 푟푖퐴

̂ 푁 푁 1 푉푒푒 = ∑푖=1 ∑푗>1 Eq. 2.15 푟푖푗

푀 푀 푍퐴푍퐵 ̂푉푛푛= ∑퐴=1 ∑퐵>퐴 Eq. 2.16 푅퐴퐵

By addition of above Eq. 2.12-Eq. 2.16 into Eq. 2.11, overall Hamiltonian operator equation in atomic unit can be written as Eq. 2. 17:

1 푁 2 푀 1 2 푁 푀 푍퐴 푁 푁 1 Ĥ = - ∑ ∇푖 - ∑퐴=1 ∇퐴 - ∑푖=1 ∑퐴=1 + ∑푖=1 ∑푗>1 2 푖=1 MA 푟푖퐴 푟푖푗

푀 푀 푍퐴푍퐵 + ∑퐴=1 ∑퐵>퐴 Eq. 2.17 푅퐴퐵

In Eq. 2.17, Z represents the nuclear charge, MA represents mass of nucleus A. The 2 2 ∇푖 푎푛푑 ∇퐴 define Laplacian operators of electron (i) and nucleus (A), respectively. RAB denotes the bond distance between two nuclei (A and B), rij defines the distance between electrons i and j, and riA represents distance between electron i and nucleus A [94].

2.4.3 The Born-Oppenheimer Approximation

Schrӧdinger wave equation is simplified by an approximation developed by two scientists Julius Robert Oppenheimer and Max Born. They treated the motion of electrons and nuclei separately. The separation becomes possible because the nuclei is much heavier in comparison to electrons i.e. mproton ~ 2000 melectron. Electrons show instantaneous response to the displacement of nuclei. So, one can assume the charge distribution due to the slow movement of the nucleus. Thus, it is a convenient approach

27 to ﬁx the nuclear positions and solve the Schrӧdinger wave equation for the electrons of a ﬁxed molecular system. This is known as Born-Oppenheimer approximation and according to this, the kinetic energies of nuclei are neglected, and columbic nuclear- nuclear repulsions are kept constant [95]. Hence, total energy of the system becomes as:

푀 푀 푍퐴푍퐵 퐸푡표푡 = 퐸푒푙푒푐 + ∑퐴=1 ∑퐵>퐴 Eq. 2.18 푅퐴퐵

However, electronic Hamiltonian itself is equal to the

Ĥ푒푙푒푐 = 퐸̂푒 + 푉̂푛푒 + 푉̂푒푒 Eq. 2.19

Where, 퐸푒푙푒푐 is derived solving electronic Hamiltonian as:

Ĥ푒푙푒푐 훹푒푙푒푐 = 퐸푒푙푒푐훹푒푙푒푐 Eq. 2.20

General body of electronic Hamiltonian can be further extended as:

1 푁 2 푁 푀 푍퐴 푁 푁 1 Ĥ푒푙푒푐= - ∑ ∇푖 - ∑푖=1 ∑퐴=1 +∑푖=1 ∑푗>1 Eq. 2.19 2 푖=1 푟푖퐴 푟푖푗

Thus, eigenvalue problem is simplified. The 훹푒푙푒푐 explicitly is dependent on the 3D coordinates of the electrons and parametrically on the nuclear coordinates (푅퐼).

휓 = 휓푒푙푒푐 ({푟푖}; {푅퐼}) Eq. 2.20

The total energy (퐸푡표푡) also depends parametrically on the 푅퐼, and the relationship between 퐸푡표푡 and 푅푖 gives the potential energy surface (PES). In accordance with the Dirac notation, Eq. 2.20 is rewritten as:

Ĥ |휓〉 = 퐸 |휓〉 Eq. 2.21

Where, | 휓 〉 = 휓 (푥1, 푥2 … … . . , 푥푁) is the wave function for the N electrons in the system.

2.4.4 Wave Function

Wave function (휓) is a continuous, single-valued and square integrable. The Schrӧdinger wave equation does not depend on the spin of the electron. But, according

28 to the Pauli principle, spin is an important observable property of a system and has the direct influence on the behavior of an electronic structure. Spin is included as ad hoc quantum effect in the 휓 using the spin functions 훼 (휔) and 훽 (휔) without spin-orbit coupling [96]. These spin functions represent the two possible spin states (1/2 (clockwise) and −1/2 (anti-clockwise)) for an electron and ω is used for the spin variable. These spin functions are orthonormal and are given as:

∫ 훼∗(휔)훼(휔)푑휔 = ∫ 훽∗(휔)훽(휔)푑휔 = 1 Eq. 2.22

∫ 훽∗(휔)훼(휔)푑휔 = ∫ 훼∗(휔)훽(휔)푑휔 = 0 Eq. 2.23

For many electron systems, the 휓 is dependent on the position and spin of electrons. This behavior can be mathematically expressed as:

휓 (푥1, 푥2 … … . . , 푥푁) Eq. 2.24

Where, x = {r (electronic coordinates) and ω (spin coordinate)}.

2.4.5 Slater Determinant

Total energy of a non-interacting electronic system (N) is equal to the sum of one 푐표푟푒 particle energy (ɛ푖 ). Hence, Hamiltonian operator Eq. 2.19 is reduced to one-particle operator (ĥ푖).

푁 푁 (∑푖=1 ĥ푖) |훹〉 = (∑푖=1 ɛ푖) |훹〉 Eq. 2.25

1 푁 2 푁 푀 푍퐴 (ɛ푖) = (- ∑ ∇푖 - ∑푖=1 ∑퐴=1 ) Eq. 2.26 2 푖=1 푟푖퐴

So, putting Eq. 2.26 in Eq. 2.25, the resultant Eq. 2.27 is as follows:

푁 푁 1 푁 2 푁 푀 푍퐴 (∑푖=1 ĥ푖) |휓〉 = (∑푖=1[- ∑ ∇푖 - ∑푖=1 ∑퐴=1 ]) |휓〉 Eq. 2.27 2 푖=1 푟푖퐴

The total 휓 of ĥ is rewritten as a product of eigen functions, due to the decoupling of quantum states for electrons such as:

|휓〉 = 휓퐻푎푟푡푟푒푒 (1, 2, … . , 푁) = 풳푖 (1)풳푗 (2) … … . 풳푘 (푁) Eq. 2.28

In Eq. 2.28, electron variables (1, 2) are replaced by electron index such as 푥1 and 푥2.

If 푥1 is replaced by 푥2, the sign of wave function would remain the same. This phenomenon is against the Pauli exclusion principle, according to which the 휓 for electrons must be antisymmetric with respect to the exchange of electrons [97]. Thus, Eq. 2.28 is modified to account for this antisymmetric behavior for the measurement of expected value for the energy. For example, a system of two non-interacting electrons the wave function equation is 휓Hartree (1,2) = 풳푖 (1) 풳푖 (2). According to Pauli exclusion principle this equation can be written as 풳푖 (1) 풳푖 (2) = −풳푖 (2) 풳푖 (1). It means that the two interacting electrons are indistinguishable, so an electron cannot ascribe to any given orbital. Overall, the wave function can be written as:

1 − 휓(1,2) = 2 2[풳푖 (1)풳푗 (2) − 풳푖 (2)풳푗 (1)] Eq. 2.29

1 − Where, 2 2 is used to normalize the equation so that

∗ 〈휓 (1, 2)| 휓(1, 2)〉 = ∫ ∫ 휓 (1, 2) 휓 (1, 2)푑푥1푑푥2 = 1 Eq. 2.30

휓∗ represents conjugated wave function in Eq. 2.30. If the coordinates of the electrons (1 and 2) are exchanged, their signs in 2.29 are also changed. Thus, for a system of N electrons there are 2N undistinguishable pathways for the distribution of electrons. The signals can be given by the parity of the electrons permutation and the equation is written as:

1 − 푁! 휓(1,2, … , 푁) = (푁!)2 2 ∑푛=1 푠푔푛(푛) 푃̂(푛){풳푖(1)풳푗(2) … 풳푘(푁) } Eq. 2.31

Permutation operator 푃̂ performed 푛th permutation of electrons with respect to the spin orbitals. These 푛th permutations are represented by a sign of 푠푔푛(푛) in Eq. 2.31. American physicist John Slater discovered this approach and called as Slater determinant. According to this approach, each individual electron is confined to wave functions termed as molecular orbitals in practice. Each of this wave function is calculated by assuming that the electrons are moving within an average field generated by its surrounding electrons. Thus, total 휓 is written in the form of a single determinant called as Slater determinant and can be written as:

풳푖(푥1) 풳푗(푥1) ⋯ 풳푘(푥1) 1 − 풳푖(푥2) 풳푗(푥2) ⋯ 풳푘(푥2) 휓(1,2, … , 푁) = |푖푗 … . . 푘〉 = (푁!) 2 | | Eq. 2.32 ⋮ ⋮ ⋮ 풳푖(푥푁) 풳푗(푥푁) 풳푘(푥푁)

One-electron wave function is represented by 풳i (푥) spin orbital or molecular orbitals (MO). This MO is equal to the product of spatial coordinate and spin coordinate i.e.

(풳i (푥) = 휓푖 (푟)푠(휔)), where (spin) is either α or β. The rows in a Slater determinant defines the electron coordinates, and the single electron wave functions, spin orbitals are along the columns [98].

2.4.6 Variational Principle

Variational principle is applied to derive approximate solutions of the electronic Schrӧdinger wave equation, and this principle is developed by Wang [99]. According to this, energy measured from any approximate ψ is always greater than the real ground state energy of 휓.

퐸푡푟푖푎푙 > 퐸표 Eq. 2.33

Thus, energy (퐸푡푟푖푎l) of the 휓푡푟푖푎푙 is derived as the expectation value of Hamiltonian operator and computed as:

⟨휓푡푟푖푎푙|Ĥ|휓푡푟푖푎푙⟩ 퐸 = Eq. 2.34 푡푟푖푎푙 ⟨휓|휓⟩

Different operators are used for studying the various properties of a system. But for energy measurement, Hamiltonian operator (Ĥ) is used because it is based on a variational principle.

The energy calculation depends on the interaction encountered by the Ĥ. Each interaction part in the Hamiltonian results in the decrease of energy. Hamiltonian will always be exact and minimum upon the inclusion of entire interactions. But in practice, the calculated Hamiltonian is higher than the real one because of neglecting some nonimportant interaction. The identiﬁcation of the energy value close to the real one, involves a minimization process of calculating energy by using a set of basis functions

31 and this is called the variational method. For a normalized wave function the denominator value of Eq. 2.34 become equal to zero, so this equation is written as:

퐸푡푟푖푎푙 = ⟨휓푡푟푖푎푙|Ĥ|휓푡푟푖푎푙⟩ Eq. 2. 35

Any trial wave function implemented to get minimized energy is given as:

∞ 휓trial = ∑푖=0 퐶푖휓푖 Eq. 2. 36

Where, 휓푡푟푖푎푙 is the trial wave function and Ci represent expansion coefficient of 휓푡푟푖푎푙 in the basis formed by the eigenvalue (휓푖). A 휓푡푟푖푎푙 composed of a single Slater determinant can be used to construct Hartree-Fock (HF) equation according to a variational principle [97].

2.4.7 Single Reference Methods

Ab initio methods are divided in to two main types of wave function-based approaches i.e. single reference and multi-reference methods. The method in which 휓 based on single Slater determinant is known as single reference method e.g. HF. It is described by single configuration function. Sigle reference method is used for simple closed shell molecules with paired electrons. They are zeroth-order methods described by a set of occupied MOs from which a Slater determinant is constructed. Single reference procedure starts with HF-SCF and then correlation (dynamic correlation) energy is captured by post-HF methods. Some of the post-HF methods are composed of multideterminant wave function but still they have same coefficient as in HF and classified as single reference [100].

2.4.8 Electron Correlation

HF methods based on single Slater determinant and electron-electron interaction is treated in the average sense. It neglects the instantaneous electron-electron correlation due to which coulomb energy is high. The only correlation HF accounts is partially by exchange interactions, which correlate positions of electrons with the same spin. The correlation energy is the amount of energy missing in HF calculation.

퐸푐표푟푟 = 퐸0 − 퐸퐻퐹 Eq. 2. 37

퐸푐표푟푟 is the correlation energy, 퐸퐻퐹 is the HF energy and 퐸0 real ground state energy of the system.

Dynamic Correlation

Electrons are charged particles moving in space. When electrons cross each other during movement (because their motion is correlated), they avoid each other. This correlation is known as dynamic correlation. This is a type of instantaneous electron- electron interactions missing in HF due to consideration of static electronic distribution. HF based on the single Slater determinant where a part of correlation is account from the exchange interactions. However, some of the post-HF methods capture dynamic correlation energy [101].

Static Correlation

In systems with multiple resonance states, electrons can avoid each other by occupying different ‘resonance states’. Electrons are allowed to spread out among more than one state which decreases the electron-electron repulsion. This type of correlation is known as static correlation or non-dynamic correlation and is not related to the correlation motion of electrons. Static correlation is better described by multideterminant wave function where each configuration has its own Slater determinant. A single Slater determinant only represents 1 resonant state, due to this reason HF cannot describe static correlation.

2.4.9 Hartree-Fock Approximation

For multi-electron system, the complexity of Schrӧdinger wave equation is minimized by adopting orbital approximation approach which is called Hartree-Fock (HF) approximation or central field approximation. Each electron in HF approximation is described separately by its own wave function with an additional Hartree potential. The HF approximation was originally introduced by D.R. Hartree in 1928 (before computers were available). Then, it is modiﬁed by V. Fock in 1930, where Pauli exclusion principle is considered [102].

The energy measured by the HF is always greater than the ground state energy. The reason for higher energy is neglecting the electron-electron correlation and relativistic effects of the core electrons.

2.4.9.1 Minimization of Energy and Fock Operator

The energy of a system is minimized by solving Schrӧdinger wave equation based on the variational principle. The spin orbital of Eq. 2.32 (used to build a single Slater determinant) is used to find the optimum set of spin-orbitals. The Ĥelec (in Eq. 2.19) includes one electron and two electrons operators. When Ĥelec is built based on Slater determinant, it is assumed as sum of one electron and two electrons terms as given below:

푁 푁 푁 퐸퐻퐹 = ⟨푖푗 … . 푘|Ĥ|푖푗 … . 푘⟩ = ∑푖=1⟨푖|ĥ|푖⟩ + ∑푖=1 ∑푗>푖⟨푖푗|| 푖푗⟩ Eq. 2. 38

Where

∗ 1 2 푀 푍퐴 ⟨푖|ĥ|푖⟩ = ℎ푖 = ∫ 풳푖 (1) (− ∇1 − ∑퐴=1 ) 푥푖 (1)d푥1 Eq. 2. 39 2 푟1퐴

⟨푖푗|| 푖푗⟩ = 퐽푖푗 − 퐾푖푗 Eq. 2. 40

=⟨푖푗|푖푗⟩ − ⟨푖푗|푖푗⟩ Eq. 2. 41

Then Eq. 2.41 is further expanded in terms of molecular orbitals (MOs), see as below

∗ ∗ −1 =∫ ∫ 풳푖 (1)풳푗 (2)푟12 풳푖 (1)풳푗 (2)푑푥1 푑푥2 − ∗ ∗ −1 ∫ ∫ 풳푖 (1)풳푗 (2)푟12 풳푗 (1)풳푖 (2)푑푥1 푑푥2 Eq. 2. 42

In Eq. 2.42, the 1st part represents Coulomb integral (which illustrates classical repulsion between electrons occupying spin orbitals) and 2nd part represents exchange integral [103]. Right-hand side shows the linear combination of all spin orbitals. ⟨푖푗|| 푖푗⟩ 1 means that i = j because electron does not interact with itself. is used to avoid double 2 counting of the terms. So, Eq. 2.38 is rewritten as:

푁 ∗ 1 2 푀 푍퐴 퐸퐻퐹 = ∑푖=1 ∫ 풳푖 (1) (− ∇1 − ∑퐴=1 ) 푥푖 (1)d푥1 + 2 푟1퐴 1 ∑푁 ∑푁 [∫ ∫ 풳∗ (1)풳∗(2)푟−1 풳 (1)풳 (2)푑푥 푑푥 − 2 푖=1 푗>1 푖 푗 12 푖 푗 1 2 ∗ ∗ −1 ∫ ∫ 풳푖 (1)풳푗 (2)푟12 풳푗 (1)풳푖 (2)푑푥1 푑푥2] Eq. 2. 43

In simplified notation Eq. 2.43 is rewritten as:

̂ 푁 ̂ 푓(1)풳푖(1) = [ĥ (1) + ∑푗=1 푗푖̂ (1) − 퐾푖 (1)] Eq. 2. 44

푓̂, ĥ, 푗̂, and 퐾̂ are Fock, kinetic energy, Coulomb and exchange operators, respectively. Fock operator is used to get minimized energy of the system. A series of transformations are implemented to get a set of spin orbitals for which matrix is orthogonal and written as:

̂ 푓(1)풳푖(1) = 휖푖풳푖 Eq. 2. 45

In Eq. 2.45, eigen function (풳푖) represents HF spin orbitals, and eigenvalue (휖푖) represents orbital energies [104].

2.4.9.2 Hartree-Fock-Roothaan Method

Clemens Roothaan and George Hall developed a method to define spin orbital (푥i) for a molecule in 1951. 푥i is a product of a spatial orbital by a spin function. Each such spatial orbital, 휓 is a linear combination of atomic orbitals (LCAO), Փ and mathematically represented as:

퐾 휓푖 (1) = ∑푎=1 퐶푎푖Փ푎 (1) Eq. 2. 46

K represents the size of basis set and C represents the coefficient. If the AO basis set is larger, the expansion will be more accurate [105].

2.4.9.3 Restricted, Unrestricted and Restricted Open HF

HF is divided into three types in accordance to Pauli exclusion principle. When a pair of electrons occupied same spatial orbital and having opposite spin. In such a situation, the HF is known as restricted (R) HF and its ψ is said to be R wave function. In closed- shell electronic conﬁgurations where the spatial orbitals are partly ﬁlled without any distinction between α and β electrons, the HF is known as restricted open (RO) HF and

35 its ψ is said to be RO wave function. Alternatively, if the spatial orbitals of α and β electrons are different, the 휓 in this case is unrestricted (U) and HF becomes UHF. The orthogonality of spin functions is used for simplification of formula. This type of process for a restricted 휓 and expanding the orbitals leads to the Roothaan-Hall equation:

FC = εSC Eq. 2. 47

Where, F is the Fock matrix, C represents basis set coefficients, ɛ is a diagonal eigenvalue matrix and S denotes overlap matrix. The spatial orbitals are different for α and β electrons and couple Fock equations have to be solved by matrix diagonalization. Latest computer software makes the implementation of the Hartree-Fock-Roothaan method [106]. HF is a type of variational calculation, where the energies are calculated in unit of Hartree and 1 Hartree is equal to 27.211eV.

2.4.10 Post Hatree-Fock Methods

After HF method, several methods are developed to study/account for many-electron systems based on a wave function. These methods are known as post-HF methods [106].

2.4.10.1 Moller-Plesset Perturbation (MP) Methods

MP perturbation theory is developed by Moller and Plesset in 1934. While, the development of this theory into a computational method was done by Binkley and Pople in 1975. This size-consistent MP methods are not variational [107]. This theory is developed by the addition of correlation term as a perturbation to the HF wave function. Simplest single reference MP perturbation method is based on Rayleigh–Schrӧdinger perturbation theory. HF becomes the first-order perturbation in using HF wave function for the development of MP formulation. A small perturbation (휆푉) is added to the unperturbed HF Hamiltonian, and the one electron Fock operator expression can be written as:

퐻̂푒푙푒푐 = 퐻̂0 + 휆푉̂ Eq. 2. 38

퐻̂0 is HF Hamiltonian and 휆푉̂ is a small perturbation.

For small perturbation the energy and 휓 can be derived as:

2 푛 퐸 = 퐸0 + 휆퐸1 + 휆 퐸2 … + 휆 퐸 푛 Eq. 2. 39

2 푛 휓 = 휓0 + 휓퐸1 + 휓 퐸2 … + 휓 퐸 푛 Eq. 2. 40

HF wave function and energy are represented by 퐸0 and 휓0, which are extended in a Tayler series in 휆. Putting the values of Eq. 2.48-Eq. 2.50 in Eq. 2.21, the nth order Moller-Plesset equation is written as:

2 n 2 n 퐻̂0 + 휆푉̂(휓0 + 휓퐸1 + 휓 퐸2 … + 휓 퐸n) = (퐸0 + 휆퐸1 + 휆 퐸2 … + 휆 퐸n)

2 n (휓0 + 휓퐸1 + 휓 퐸2 … + 휓 퐸n) Eq. 2. 41

These perturbed methods are differentiated based on correlation captured by each method and nth value. Different types of Moller-Plesset perturbation (MP) methods are developed such as MP2, MP3, MP4 and MP5, MP6 and MP7, respectively, where integers 2, 3, 4, 5, 6 and 7 represent the order of perturbation. MP2 method can capture up to 90% of correlation and its time consumption is N5. The MP2 energy is given by equation as below:

2 [⟨Փ Փ |Փ Փ ⟩−⟨Փ Փ |Փ Փ ⟩] (2) 표푐푐 푣푖푟 푖 푗 푎 푏 푖 푗 푏 푎 퐸푖 = ∑푖<푗 ∑푎<푏 Eq. 2. 42 휖푖+휖푗−휖푎−휖푏

Փi and Փj represent the occupied orbitals whereas Փa and Փb represent unoccupied orbitals, respectively. The 휖푖, 휖푗, 휖푎 and 휖푏 are the corresponding orbital energies. Increasing the nth value increases the complexity of the integrals, however in case of MP3 (94% correlation) computational cost is increased significantly but the accuracy is slightly increased compared to MP2. However, when n = 4 the integrals pertaining to 4th order correction (MP4). Although using MP4 method, the results are more accurate, but calculations are more time consuming. The accuracy of MP4 (capture 97% correlation) calculation is almost comparable to the accuracy of CISD calculation.

2.4.10.2 Configuration Interactions (CI) Methods

These are variational methods based on multiple-determinant wave function. These single reference CI methods are designed starting with the HF wave function and new

37 determinants are constructed by promoting electrons from the occupied to virtual orbitals. An approximation to the ψ, the linear combination of determinant in CI are mathematically expressed as:

퐶퐼 푟 푟 푟푠 푟푠 |휓⟩ = 푐0|휓0⟩ + ∑ 푐푎 |휓푎⟩ + ∑ 푐푎푏 |휓푎푏⟩ + … ar a

= |∑푖=0 푐푖휓푖⟩ Eq. 2. 43

휓0 and prefactor 푐 represent the HF determinant and coefficients respectively, 푟 휓푎 expresses the determinant for the singly excited state (in which an electron in spin 푟푠 orbital a is excited to orbital r) and 휓푎푏 is the determinant for the doubly excited state (in which electrons in a and b orbitals are excited to r and s orbitals, respectively) [108].

Whereas, ψiand 푐푖 represent infinite wave function and coefficient

CI calculations are the most accurate one among ab initio methods, but the cost in term of CPU time is very high. CI methods are classified on the basis of number of excitations used to construct each determinant and details of these subtypes are as follows:

CIS is CI with a single excitation

CISD is CI with a single and double excitations

CISDT is CI with a single, double and triple excitations

CISDTQ is CI with a single, double, triple and quartet excitations. All these classes are known as truncated CI methods

Full CI is configuration interaction with all possible types of excitations with a given basis sets. It is the most accurate method and gives an exact solution to the Schrӧdinger wave (time independent) equation. Full CI is mathematically expressed as:

퐶퐼 푟 r 푟푠 푟푠 |휓⟩ = 푐0|휓0⟩ + ∑ 푐푎 |휓a⟩ + ∑ 푐푎푏 |휓푎푏⟩ + ⋯ + ar a

표푐푐 푣푖푟푡 푟푠 (퐾−푁) = |∑푖<푗..<푁 푟<푠<⋯<(2퐾−푁) ∑2퐾−푁 푎푖푗…푁 휓푖푗…푁 ⟩ Eq. 2. 44

Full CI includes the entire set of excitations from N occupied orbitals into all possible combinations of 2K – N virtual orbitals.

2.4.10.3 Coupled Cluster Methods

The phenomena of a linear combination of many determinants in a wave function results in a new type of method known as coupled cluster method similar to configuration interaction methods. The explicit coupled-cluster equations for electrons in the language of quantum chemistry is presented by Cizek, in 1966. Later, Paldus applied the CC equations to solve chemical problems, in 1972 [109]. In CC method, cluster operator is similar to the correlation found in perturbation theory. This method provides the more accurate solution to Schrӧdinger wave equation because it offers the best treatment of correlation and exchange effects. The CC wave function is written as:

퐶퐶 푇̂ |휓 ⟩ = 푒 |휓퐻퐹⟩ Eq. 2. 45

푇̂ 휓푐푐 is a couple cluster wave function, 푒 exponential operator and 휓퐻퐹 represents HF Slater determinant. 푒푇̂ is further elaborated in terms of Taylor series and mathematically expressed as:

1 1 1 푒푇̂ = 1 + 푇̂ + 푇̂ 2 + 푇̂ 3 + ⋯ = ∑∞ 푇̂ 푘 Eq. 2. 46 2 3 푘=0 푘!

Where, cluster operator is given as:

푇̂ = 푇̂1 + 푇̂2 + 푇̂3 … … … . +푇̂푁 Eq. 2. 47

th All the i excited Slater determinants are generated by the 푇푖푡ℎ operator acting on reference wave function as:

̂ 표푐푐 푣푖푟 푎 푇1|휓퐻퐹⟩ = ∑푖 ∑푎 푎푖 |휓푖 ⟩ Eq. 2. 48

Similar to CI, there are different orders of CC expansion, known as CCSD, CCSD(T), CCSDT, and so on. Eq. 2.56 and Eq. 2.57 can be combined to get an idea about the excitation in CC theory as:

1 1 1 1 푒푇̂ = 1 + 푇̂ + 푇̂ 2 + 푇̂ 3 + ⋯ 푇̂ + 푇̂ 2 + 푇̂ 3 1 2! 1 3! 1 2 2! 2 3! 2

1 1 + ⋯ + 푇̂ + 푇̂ 2 + 푇̂ 3 Eq. 2. 49 푁 2! 푁 3! 푁

Hence, the complete CC wave function can be written as:

푇̂ 1 2 1 3 푒 |휓퐻퐹⟩ = 1 + 푇̂1 + 푇̂1 + 푇̂1 + ⋯ 퐶퐶 2! 3! |휓 ⟩ = [ 1 1 1 1 ] Eq. 2. 50 푇̂ + 푇̂ 2 + 푇̂ 3 + ⋯ + 푇̂ + 푇̂ 2 + 푇̂ 3 … |휓퐻퐹⟩ 2 2! 2 3! 2 푁 2! 푁 3! 푁

CCSD(T) is a type of calculations where triple excitations are included as perturbation rather than exact triple excitation. Coupled cluster calculations give variational energies. The accuracy of CCD and CID methods is almost similar. The advantage of doing coupled cluster calculations is its size extensive nature. A full CC calculation including all possible configurations is equivalent to a full CI calculation. This size- consistent CC method is not variational [110].

2.4.10.4 Quadratic Configuration Interaction (QCI) Methods

QCI method is an extension of CI method. QCI is used to correct the size consistency of CI method by the addition of quadratic terms but it loses the variational property of CI. Single- and double-excitation methods (QCISD) are the most popular types of QCI methods for which the mathematical expression are as follows:

⟨휓0|퐻̂|푇2휓0⟩ = 퐸푐표푟푟 Eq. 2. 51

̂ 푎 ⟨휓0|퐻|(푇1+푇2 + 푇1푇2)휓0⟩ = 푎푖 퐸푐표푟푟 Eq. 2. 52

1 ⟨휓 |퐻̂|(1 + 푇 + 푇 + 푇2)휓 ⟩ = 푎푎푏퐸 Eq. 2. 53 0 1 2 2 2 0 푖푗 푐표푟푟

1 푇 푇 and 푇2 are added as additional quadratic terms in Eq. 2.62 and Eq. 2.63. These 1 2 2 2 types of calculations are popular because they often capture optimum amount of correlation for high-accuracy calculations on organic molecules. On the other side, QCI calculations use less CPU time compared to CC calculations. To improve the efficiency of QCISD, triple excitations are also included which results in QCISD(T) method. The T in parentheses indicates that the triple excitations are included as perturbation instead of exact excitations as is observed for CCSD(T) and CISD(T). QCI method is

40 developed by Pople et al., in 1987. The accuracy of QCISD(T) is similar to the CCSD(T) method [111].

The order of single reference methods on the basis of accuracy and time is as follows:

HF < MP2 < MP3 = CISD = CCSD = QCISD < MP4 = CCSD(T) = QCISD(T) < MP5 = CISDT= CCSDT < MP6 < MP7 = CISDTQ= CCSDTQ < Full CI

2.4.11 Multi-Reference Methods

These multiple electronic configuration methods include multi-Slater determinants in the reference wave function to describe the static correlation [112]. These methods are used for non-trivial electronic structure, such as radicals, transition metal compounds and electronically excited molecules having unpaired electrons. Static correlation properly describes the unpaired electrons in a system. This correlation means that a single set of occupied orbitals is not a good zeroth-order description of a system [113].

2.4.11.1 Complete Active Space Self-Consistent-Feld (CASSCF) Method

CASSCF method is used as a standard method for the measurement of singlet diradicals and excited state species. It is a limited version of CI method in which electrons are carefully promoted from and to selected sets of MOs. This method gives a limited CI wave function which provides different properties (geometry and energies etc.) of the system. The basic principle of this method is the division of occupied orbital space into a set of active, inactive and secondary orbitals. All these chosen orbitals are highlighted, and the active (valence) electrons are completely distributed in all possible ways in accordance to the excitation in full CI. If the active spaces are properly selected, then MOs involved in the chemical process are completely examined [114].

The occupied active space MOs (ψi) in CASSCF are derived as;

푆퐶퐹 2 표푐푐 푛표. 푀푂푖 = ∑푛 (표푐푐 푛표. )푖, 푛 푎푛 Eq. 2. 54 n represents MOi containing conﬁguration state functions (CSFs), which is the sum of the product of occupation number and fractional contribution (α2) of the CSF to the total wave function.

2.4.11.2 Restricted Active Space Self Consistent Field (RASSCF) method

This method is restricted to the selected subspaces of active orbitals. For instance, RASSCF could restrict the number of electrons to at most 2 in some subset. For example, analyses of a double bond isomerization require an active space including all π-electrons and π-orbitals of the conjugated system (for the incorporation of the reactive double bond). It allows all possible variations in the overlap between the set of p- orbitals that are involved for the formation of reacting π-system along the reaction coordinate. However, the selection of the active space electrons and their orbitals is not so easy because the generation of flexible configuration space to describe the physical process is a difficult task [115].

2.4.11.3 Multi-Reference Perturbation Theory (MR-PT) Method

Complete treatment of dynamic and static correlation with single point perturbational measurements based on the CASSCF wave function is called CAS perturbational theory or CASPT2. It is a cost-effective multi-reference method compared to other multi- reference methods, this method often gives accurate vertical excitation energies in computational photochemistry and photobiology (with error up to 3 kcal/mol). The reason for higher accuracy is the balanced cancellation of errors [116].

2.4.11.4 Multi-Reference Configuration Interaction (MRCI) Method

MRCI starts with a CASSCF calculation, it includes nearly degenerate electron configurations in the ψ and describes the static electron correlation. Since, the dynamic electron correlation is included by substitutions of occupied orbitals by unoccupied (virtual) orbitals in the individual configuration state functions (CSFs). Because of the increase in the number of CSFs and simultaneous computational effort, the expansion space of this method is truncated to single and double substitutions. Hence, MRCI method is also known as truncated CI but it suffers from the lack of size extensivity [117].

2.4.11.5 Multi-Reference Couple Cluster (MR-CC) Method

Standard generalization of single reference couple cluster is known as MR-CC method. It is a cost-effective technique with inclusion of local static correlation within the

42 framework of CASSCF. MR-CC method is almost size-extensive, but it does not give a clear representation of active spaces in comparison to MRCI, as is observed in single reference methods [118].

All these multi-reference methods require an expert capable of properly selecting variables such as the active space and the number of all required states for calculations.

2.5 Monte Carlo and Quantum Monte Carlo

Monte Carlo is a random search stochastic technique used for the evaluation of large dimensional integrals. During Monte Carlo calculations, some points are randomly selected based on the probability distribution of a function and then the average value is numerically computed. Integral is evaluated as:

퐼 = ∫ 푑푅푔(푅) Eq. 2. 55

An essential function (which illustrate the probability density and its value is greater than zero) is added to Eq. 2.65 and mathematically represented as:

퐼 = ∫ 푑푅푓(푅)푃(푅) Eq. 2. 56

In MC calculations, the ﬁnite sample of points are averaged over 푃(푅) and then, the integral is estimated from it. The mathematical equation is represented below:

1 퐼 ≈ ∑푀 푓(푅 ) Eq. 2. 57 푀 푀 푚=1 푚

In MC simulation, at every step the conformation is randomly modified to get the new one. Conformations are either accepted or rejected based on ﬁlters. According to the Boltzmann distribution law, a large number of states are constructed where the energy of each state is measured using MM method. Metropolis MC is the most popular type of MC technique. The main advantage of Monte Carlo simulation is that the errors in the results are independent of dimension. These methods are developed by Moskowitz and Schmidt in 1984 [119] and further popularized by Pople. One of the most famous type of MC method is grand-canonical Monte-Carlo (GCMC) technique. Density fluctuations are explicitly treated through GCMC method at constant temperature and volume. The basic principle is based on the insertion and removal of molecules. This method is mostly used for the study of bulk materials properties. Monte Carlo method

43 is the most popular method for magnetic properties studies. While for spin quantization, quantum Monte Carlo (QMC) is the best approach.

The technique used for the solution of multi-electrons Schrӧdinger wave equation in a broad range of physical systems for accurate measurements are known as quantum Monte Carlo (QMC) technique. This method is based on an explicitly correlated wave function and using a Monte Carlo integration to evaluate integrals numerically. Highly parallel algorithms allow it to take full advantage of high-performance computing systems. The computational scaling of QMC methods is also favorable such as M3 or M4 in comparison to CCSD(T) and due to this reason, this method is best for large periodic system. QMC is further classified in to variational, diﬀusion, path integral, reptation and Gaussian quantum Monte Carlo methods, among which the most basic and simplest is VMC. In the VMC method, the actual ground state |휓T⟩ is approximated by a trial wave function |휓푇⟩ [120]. |휓푇⟩ depends on a set of parameters, and it is optimized to get the ground state energy of a system.

∗ ∫ 휓푇(푅)퐻̂휓푇(푅)푑푅 퐸휓푇 = ∗ Eq. 2. 58 ∫ 휓푇 (푅)휓푇(푅)푑푅

R represents 3D coordinates of all N number of atoms in a given system (푅 =

푟1, 푟2 … . 푟푁). According to a variational principle, energy calculated from |휓T⟩ is higher than the actual one. VMC method stochastically solves the high-dimensional integral where the non-separable wave function is treated directly. The VMC energy can be written as:

2 −1 ∫|휓푇(푅)| [휓푇 (푅)퐻̂휓푇(푅)]푑푅 퐸푉푀퐶 = 2 Eq. 2. 59 ∫|휓푇(푅)| 푑푅

The quantity 퐸퐿(푅) is suggested as the local energy (evaluated at each of these points), and the average energy accumulated is given as:

1 퐸 = ∑푁 퐸 (푅 ) Eq. 2. 60 푉푀퐶 푁 푖 퐿 푖

The error scales of stochastic variational energy EVMC can be resolved by ground state projection methods. Projection QMC methods treat the wave function only implicitly, and sample it to calculate expectation values. These QMC methods get the ground-state

wave function |휓푇⟩ applying the imaginary time projection operator to the chosen reference on trial wave function |휓푇⟩

−휏퐻̂ |휓0⟩ ∝ lim 푒 |휓푇⟩ Eq. 2. 61 휏→∞

The different classes of QMC methods are differentiated in the way they carry out this projection. Second method, diffusion Monte Carlo (DMC) represents actual ground state wave fucntion |휓푇⟩ in real space. DMC is numerically exact and finds ground state energy within a given error for any desired quantum system. In DMC simulation, the algorithm scales as a polynomial for bosons and exponentially for fermions, respectively with the system size. Due to which DMC simulations for fermionic systems suffer from the sign problem, the problem is arising from the antisymmetric nature of fermion wave function [121]. This problem is solved via fixed-node approximation. Nodes of the reference wave function elucidate the accuracy of this approximation whereas the fixed-node DMC energy is variational. For atomization and chemical reaction energies, DMC typically performs extremely well but the error cancellation becomes essential in the energy differences. Both, VMC and DMC methods are based on single Slater determinant and successfully used for large systems. These methods reproduce approximately 95% of the correlation energy within the fixed-node approximation [122]. They play a vital role in the development of computer codes i.e. CASINO package etc. Third one is an auxiliary field quantum Monte Carlo (AFQMC) which is one of the most accurate QMC methods and mostly used for materials studies [123], and its accuracy is comparable to CCSD(T) method for system having moderate correlation. It is size consistent method and parallel supercomputers are used for its implementation.

2.6 Molecular Dynamics

MD is a computational method used to study the time dependent motions of atoms and molecules. The basic principle of MD simulations is that the changes in structures due to vibrations and movements of atoms are analyzed [124]. These simulations even provides chemical and physical information about the molecules as one can observe in the real experiments. This method follows Newton’s law of classical mechanics.

퐹푖 = 푚푖푎푖 Eq. 2. 62

푖th represents each individual atom in a molecule and m is the atomic mass, a denotes acceleration (where a푖 = 푑2r푖/푑푡2) and F represents the force acting on atoms. MD is an approach to obtain a set of configurations or states distributed according to some statistical distribution principles. MD simulations start with an initial guess set of positions and velocities of atoms in a molecule and simultaneously time evolution is determined. As a result, the atoms are allowed to pass through all possible states according to Ergodic hypothesis. Total average time is used to calculate the properties such as kinetic energy of molecules. These simulations are of fixed duration, due to which enough phase space sampling is necessary. For example, a system of N atoms, the phase space is the 3D for all types of positions and momenta. The state (position and velocity) of an atom at any given time is described by a specific point in the phase space. The subsequent changes of a point in the phase space gives information about time evolution of the molecule. However, the disadvantage of the MD simulations is that it is difficult to study the whole phase space due to a larger volume and configurations of different molecules. In case of a constant temperature system, different probabilities can be observed with in the different regions of the phase space which are most frequent in MD calculations. For thermodynamic properties measurements it uses a Maxwell-Boltzmann (MB) theory and produces configurations of the molecules based on their Boltzmann weight. According to MB law, molecules have higher probability to be in a minimum energy state. MD techniques have applications in molecular science, chemistry, medicine, biophysics and biochemistry [125]. Different levels of methods are implemented for MD simulations.

2.7 Density Functional Theory (DFT)

It is a method in which the electronic density instead of many-body wave function is used for electronic structure description of a system. DFT is used for properties measurements from small to large molecules. This approach is much popular among different computational methods due to its simplicity and low computational cost. This theory is based on Schrӧdinger wave equation where the energy calculations are carried out in terms of electron probability density. Hohenberg and Kohn developed the modern form of DFT in 1964 [126].

2.7.1 Kohn-Sham Density Functional Theory (KS-DFT)

An excellent method for implementation of DFT theory was proposed by Kohn and Sham in 1965 [127]. The importance of this method stems from the fact that most of the researchers when used term DFT nowadays is actually KS-DFT. The density of non-interacting system is represented by a single Slater determinant in KS-DFT, and it is mathematically derived as:

1 휙푆퐷 = √푁! det {푥1(푥⃗⃗⃗1 )푥2(⃗푥⃗⃗⃗2 ) … . 푥푛(⃗푥⃗⃗⃗푛 )} Eq. 2. 63

The 휙푆퐷 defines the Slater determinant which approximates the N electron wave th function. 푥푖 denotes the i spin-orbital, and 푥1 denotes spin coordinates. The total energy of the system based on density is mathematically represented as:

퐸[휌] = 퐸푁푒[휌] + 푇[휌] + 퐽[휌] + 퐸푥푐[휌] Eq. 2. 64

퐸푁푒, T and 퐽 represent nucleus-electron attraction, kinetic energy and electron-electron repulsion, respectively. The 퐸푥푐 defines exchange-correlation (푥푐) energy, and this term is composed of all remaining portion of energy.

The KS equation for variational minimum of the total energy with respect to the electronic density (ρ) is mathematically written as:

퐾푆 ƒ̂ 휓푖 = 퐸푖 휓푖 Eq. 2. 65

퐾푆 Where, ƒ̂ represents Fock operator, and the equation for this operator is derived as:

퐾푆 푀 푍퐴 1 2 휌푟2 ƒ̂ 푟1 = − ∑퐴 − ∇푖 + ∫ 푑푟2 + 푉푥푐 (푟1) Eq. 2. 66 푟푖퐴 2 푟12

Four terms in Eq. 2.76 are the elaborated forms of the terms in Eq. 2.74. The 푥푐 potential is equal to the derivatives of the 푥푐 energy over electronic density (Eq. 2.77):

훿퐸 푉 = 푥푐 Eq. 2. 67 푥푐 훿휌

The Eq. 2.74 and Eq. 2.76 are solved by the self-consistent field (SCF) procedure, starting with an initial guess structure of the ρ and iteratively update all remaining parts

47 until the convergence is achieved. Then, total energy is calculated by integrating Eq. 2.74:

The KS equations are in principle exact, and if the exact Exc and Vxc forms are known, the exact energy of a system can be obtained. Unfortunately, scientists are unable to solve these terms (Exc and Vxc) correctly. So, computational chemist carried out different types of approximation in the Kohn-Sham equations to solve these terms. The approximation only enters when the explicit treatment of Exc and Vxc terms is required. Scientist made developments in the field of modern density functional, and they are keenly interested to get the good 푥푐 energy forms, from which the 푥푐 potential is obtained. Since 1965, a number of DFs are developed and the detail of their DFT classes are as follows:

2.7.2 Classification of DFT

DFT is divided into different classes based on the electronic density treatment such as:

(i) Local density approximation (ii) Generalized gradient approximation (iii) meta-GGA (iv) Hybrid GGA (v) Double hybrid GGA (vi) Global hybrid meta-GGA (vii) Range separated hybrid GGA (viii) Dispersion corrected GGA (ix) Dispersion corrected range-separated hybrid meta-GGA

2.7.2.1 Local Density Approximation (LDA)

Kohn and Sham introduced the first class of DFT known as LDA, as a result of approximation for the Exc in 1965. In KS approximation, the Exc per particle at each point in space for a homogeneous electronic gas is described by its value. This is also known as uniform electron gas (UEG) model, and exchange-correlation energy is represented by 퐸푥푐. A number of electrons (N) having volume (V) in a cube is known as UEG. To make the system neutral there is a uniform distribution of charges. Also,

48 the UEG is described as the limit where N and V are equal to infinity, then the density is also remaining infinite.

LSDA is the most famous DF of LDA class where approximations to the 퐸푥푐 only depends on spin densities (ρα and ρβ) [19].

퐿푆퐷퐴 퐸푥푐 [휌훼, 휌훽] = [∫ 푑푟 휌(푟 )ɛ푥푐(휌훼, 휌훽] Eq. 2. 68

퐸푥푐 stands for the exchange-correlation energy and ρα, ρβ are the spin densities of α and β spins per electron at a given point in space, respectively. This DF is simple and frequently implemented in solid-state physics, and gives good results for energy minima geometries and frequencies analyses. Other DFs of this class are VWN [128] (it was modified to SVWN [129] when spin is included in VWN), PZ81 [130] and PW92 [131] etc.

2.7.2.2 Generalized Gradient Approximation (GGA)

This class depends on the gradient of the density. GGA class was developed in 1980s, as a result of another type of approximation for the 퐸푥푐. (See in Eq. 2.79).

퐿푆퐷퐴 퐸푥푐 [휌(푟 )] = [∫ 푑푟 [휌(푟 ), ∇휌(푟 )] Eq. 2. 69

Basically, GGA approximation is based on the gradient expansion of the exchange hole. Originally the phenomenon was related to the real-space cutoffs selected for the confirmation that the hole is positively charged as a whole and represents one electron deficiency. In comparison to LSDA, GGAs shows improvement in measurement of bond length, energies and atomization energies etc. However, it overestimates the lattice constants in an ionic crystal and overcorrect LDA results. Thus, GGAs results are marginally better as compared to LDA results. The DFs of this class are B88 [132], BLYP, G96LYP, HCTH, MPWLYP1W, PBE, PW91 [133] and PBELYP1W etc. In similarity to GGA, the NGA class was developed where the functionals are parameterized with same spin densities and their reduced gradients instead of using separable form for exchange. In this class, the exchange and correlation terms are combinedly treated. Most popular DF of non-separable gradient approximation (NGA) class are N12 [134] and GAM [20] etc.

2.7.2.3 Meta-Generalized Gradient Approximation (meta-GGA)

When the kinetic energy density is included in GGA approximation, the class of meta- generalized gradient approximation is originated. Approximation are derived by the addition of this local ingredient (kinetic energy):

1 휏 = ∑푁 |∇휑 (푟)|2 Eq. 2. 70 푖 2 푖

KE allows the 푥푐 functional to recognize the regions having only a single electron, where self-interaction error is significant [20]. Some of the popular meta-GGAs are M06-L, τ-HCTH, TPSS, TPSSLYP1W and VSX etc. Similar to meta-GGA, addition of kinetic energy to NGA results in a new class called meta-NGA [135]. Two meta- NGA DFs, MN12-L and MN15-L are recently developed, those having the same ingredients as meta-GGA DFs. These DFs are based on the combined parameterizing of exchange and correlation terms.

2.7.2.4 Hybrid GGA (H-GGA)

Combined treatment of the exchange and correlation functionals through adiabatic connection formula results in the class of hybrid GGA. Becke investigated the first DF of this class, inspired by reexamination of the adiabatic formula [136]. Approximation for the Exc is done by mixing exact % HF exchange energy and the LDA Exc, as shows below in Eq. 2.81:

푒푥푎푐푡 퐿푆퐷퐴 퐸푥푐 = 0.5 퐸푥 + 0.5 퐸푥푐 [휌] Eq. 2. 71

This approach of developing exchange-correlation functional is also known as Becke half-and-half approach. To improve performance, % exact HF exchange (푥) is usually empirically optimized (taken in the range of 5–60%), and thus GGA is the best choice of use rather than LSDA. Functionals that incorporate a certain percentage of HF exchange are known as hybrid functionals. Some of hybrid DFs are B3LYP, B3PW91, B98, B97-1, B97-2, B98, MPW3LYP, MPW1K, MPW1PW91, PBE0 and X3LYP etc.

A three-parameter functional reported by Becke in 1993 is known as B3PW91 (20% HF) [137]. Its parameters are fitted against main-group atomization energy values. Stephens et al., replaced PW91 (GGA correlation functional) by LYP (GGA correlation

50 functional) [138] and obtained a new B3LYP functional. The exchange correlation energy of B3LYP is mathematically expressed as:

퐵3퐿푌푃 퐿푆퐷퐴 퐻퐹 퐿푆퐷퐴 퐿푆퐷퐴 퐸푥푐 = 퐸푥푐 + 훼0(퐸푥 − 퐸푥 ) + 훼푥∆퐸푥 +

퐵88 푃푊 훼푥∆퐸푥 + 훼푐∆퐸푐 Eq. 2. 72

B3LYP is the most popular DF in computational chemistry during recent years, due its widespread applicability in different properties measurements with least errors. For the first time, Truhlar and his co-workers developed a hybrid meta-NGA named MN15 [139]. MN15 has 44% HF exchange, and it gives good results for various chemical properties measurements.

2.7.2.5 Double Hybrid GGA (DH-GGA)

To improve the accuracy of hybrid GGA, second-order perturbative correlation (MP2) treatment of hybrid GGA results in the double hybrid GGA class of DFT. In this method, the semi-local exchange mix with exact exchange, and the semi-local correlation with MP2. This semi-empirical double hybrid method is developed by Grimme and has widespread applications in organic chemistry [140]. The mathematical form of exchange correlation energy of double hybrid GGA functionals are given as:

퐷퐻−퐺퐺퐴 퐻퐹 퐷퐹퐴 퐷퐹퐴 퐸푥푐 = 푎푥퐸푥 + (1 − 푎푥)퐸푥 + (1 − 푎푐)퐸푐

푀푃2 + 푎푐퐸푐 Eq. 2. 73

퐷퐹퐴 퐷퐹퐴 Where, 퐸푥 and 퐸푐 represent semi-local exchange (푥) and correlation (푐) energies terms. Several double hybrid DFs are developed among those some are follows: B2- PLYP [141], mPW2-PLYP [142], PBE0-DH [143], PBE-QIDH [144], TPSS-QIDH [145] and XYG3 [146] etc.

2.7.2.6 Global Hybrid meta-GGA (GH meta-GGA)

When % HF exchange is included in the meta-GGA approximation, it results in the development of a class known as GH meta-GGA. The DFs of GH meta-GGA depend on the localized spin density, its gradient and spin kinetic energy density. In this class, % 푥 of exact HF exchange is taken as a constant. When 푥 is treated as a constant in a

51 functional, the resultant functional is named as a global hybrid meta-GGA functional.

The Exc of global hybrid meta-GGA is mathematically expressed as:

푋 퐸𝑔ℎ 푚−퐺퐺퐴 = 퐸퐻퐹 + (1 − ) (퐸퐷퐹푇 − 퐸퐻퐹) + 퐸퐷퐹푇 Eq. 2. 74 푥푐 푥 100 푥 푥 푐

퐻퐹 Where, 퐸푥 represents the nonlocal Hartree-Fock exchange energy, 푥 represents the % 퐷퐹푇 퐷퐹푇 HF exchange, 퐸푥 denotes the local DFT exchange energy, and 퐸푐 is the local DFT correlation energy [147]. Some of the DFs of this class are BB95, BMK, B1B95, M05, M05-2X, M06, M06-2X, M06-HF, MPW1KCIS, PBE1KCIS and TPSS1KCIS etc. Another class is developed, known as global hybrid meta-NGA class where the exchange and correlation terms are combinedly treated. One of the DF from global hybrid meta-NGA class is MN15.

2.7.2.7 Range-Separated Hybrid GGA (RS H-GGA)

Range separated interactions at short range (SR) as well as long range (LR) play key role in describing the nature of chemical systems. To explore such types of interactions, computational chemists developed the class of range separated hybrid GGA. The exchange energy of long-range corrected DF [148] is defined as:

퐿퐶−퐷퐹 푆푅−퐷퐹 퐿푅−퐷퐹 퐷퐹 퐸푥푐 = 퐸푥 (휔) + 퐸푥 (휔) + 퐸푐 Eq. 2. 75

Some of the DFs of this class are CAM-B3LYP, LC-ωPBE, LC-BLYP, ωB97X and ωB97X-2 etc.

2.7.2.8 Hybrid GGA with Dispersion Correction (H-GGA-D)

The addition of dispersion energy term to KS-DFT energy gives the overall dispersion corrected energy of the system [149]. The general form dispersion energy is given by the equation as:

퐴퐵 퐷퐹푇−퐷 1 퐶푛 퐸푑푖푠푝 = − ∑퐴≠퐵 ∑푛=6,8,10,… 푠푛 푛 ƒ푑푎푚푝 (푅퐴퐵) Eq. 2. 76 2 푅퐴퐵

Here, the sum represents the overall atom pairs of given system, CAB defines the th averaged 푛 order dispersion coefﬁcient for AB atoms, and 푅퐴퐵 denotes distance between AB nuclei. sn are global scaling factors that are used for the correction of repulsive behavior of 푥푐-DF. Damping function ƒ푑푎푚푝 is a key ingredient in all DFT-

D methods and it determines the short-range behavior of DC (dispersion correction). Some of the DFs of this class are B97-D, B97-D3, B3LYP-D3BJ, BMK-D3BJ, M05- D3, M05-2X-D, M06-D3, M06-2X-D3, PBE-D, PBE-D3, BEEF-vdw [150], ωB97X- V [151] and B97M-V [152] etc.

2.7.2.9 Range-Separated Hybrid GGA with Dispersion Correction

The combined treatment of the dispersion correction and long-range correction to Kohn-Sham orbitals results in a new class known as range-separated hybrid GGA with dispersion correction (RS H-GGA-D). It also covers the space of non-locality of functionals.

푖푗 푁푎푡−1 푁푎푡 퐶6 퐸푑푖푠푝 = − ∑푖=1 ∑푗=푖+1 6 ƒ푑푎푚푝 (푅푖푗) Eq. 2. 77 퐽푖푗

Some of the DFs of this class are ωB97X-D, LC-ωPBE-D3, LC-BLYP-D3, LC-ωPBE- D3 and LC-ωPBE-D3BJ etc.

Most popular ωB97X-D is composed of three terms i.e. ωB97, X and D. Where ωB97 is LC hybrid functional with identical numbers of parameters in their GGA. In 푥c term, 푥 is added as short-range exact exchange. However, the non-locality of the correlation hole error (due to the lack of dispersion forces) is removed by addition of “D” term which represents dispersion correction [153]. ωB97X-D functional of this class gives more improved results for thermochemistry (reaction energies and atomization energies) and non-covalent interactions.

2.8 Basis Sets

It is a set of basis functions that are centered on a specific atom. These basis sets usually include at least 1 basis function for each type of occupied orbital on the atom. These basis functions are not actually the real atomic orbitals but are set of mathematical functions (basis functions) used for manipulation and the linear combinations of these basis functions yield molecular orbitals. These molecular orbital approximations are also known as linear combination of atomic orbitals (LCAO). Electronic distribution around atoms are described by basis functions and their combinations give an idea about the electronic distribution in a molecule. First time basis sets are used in simple and extended Huckel methods. Simple Huckel method includes s and p atomic orbitals and

53 extended Huckel method includes a set of atomic valence orbitals of each atom in a molecule [154].

2.8.1 Gaussian Functions

Different types of functions are used for the representation of the electronic distribution around the nucleus. For example, hydrogen functionals (based on the solution of Schrӧdinger equation), polynomial functions (with adjustable parameters), Slater functions and Gaussian functions (Slater type functions). Among all these types of functions, Slater functions and Gaussian functions are simplest known functionals [118]. Semiempirical calculations such as extended Huckel methods used Slater functions while, on the other side, ab initio methods used Gaussian functions. In Gaussian basis functions, the product of two Gaussian functions on two centers is equal to the Gaussian function on third center. For a real function, consider s-type Gaussian centered on two nuclei A and B.

−훼퐴 2 −훼퐵 2 푔퐴 = 훼퐴푒 |푟 − 푟퐴| , 푔퐵 = 훼퐵푒 |푟 − 푟퐵| Eq. 2. 78

2 2 2 2 |푟 − 푟퐴| =(푥 − 푥퐴) + (푦 − 푦퐴) + (푧 − 푧퐴) Eq. 2. 79

2 2 2 2 |푟 − 푟퐴| =(푥 − 푥퐴) + (푦 − 푦퐴) + (푧 − 푧퐴) Eq. 2. 80

These equations represent electronic and nuclear positions in cartesian coordinates. The product of two nuclei function (푔퐴푔퐵) is equal to the third function (푔푐) centered on C atom nuclei (푟푐), which is expressed mathematically as:

2 −훼푐|푟−푟퐶| 푔퐴푔퐵 = 푒 = 푔푐 Eq. 2. 81

However, it is difficult for a single Gaussian function to provide the exact description of wave function as provided by Slater function because of its poor approximation. Gaussian function (i.e. STO-1G) is rounded near 푟 = 0 while the Slater function has a cusp. Gaussian functions decays more readily at large r as compared to Slater function. To overcome this limitation, several Gaussian functions are used to approximate Slater function. Based on this approximation, the two types of Gaussian functions are STO- 1G and STO-3G. Approximation of single Slater-type orbital by one Gaussian function is called STO-1G whereas Slater-type orbital approximated by three Gaussian function

54 is called STO-3G. STO-1G is observed better for an illustration of some of the reported HF calculations and the STO-3G is the smallest function used for the high-level ab initio calculations. STO-3G is a better choice in comparison to STO-2G and STO-4G because of compromisation between better speed and accuracy [155].

Slater functions with exp(−ξ푟) local coordinate system for molecular calculations is almost completely displaced by the development of Gaussian, with exp(−훼푟2) Cartesian coordinates. STO-3G consist of three primitive Gaussian functions which are contracted to form one contracted Gaussian function. One basis function for each core and valence orbitals are implemented in standard ab initio calculation. For example, atomic orbitals of oxygen (1s, 2s, 2p) in water molecule are represented by Փ1, Փ2 and

Փ3 while atomic orbital of hydrogen (1s) is represented by Փ1. Thus, the number of different heavy atoms in a molecule is identified the number of atomic orbitals used for getting information via basis sets [156].

Table 2.1 Primitive Gaussian Functions (pgf) and Contracted Gaussian Functions (cgf) of Different Basis Sets

Hydrogen Main group elements Basis sets (pgf)/[cgf] (pgf)/[cgf]

STO-6G (6s)/[1s] (12s6p)/[2s1p]

6-31G (4s)/[2s] (10s4p)/[3s2p]

6-31G(d) (4s)/[2s] (10s4p1d)/[3s2p1d]

6-31G(d,p) (4s1p)/[2s1p] (10s4p1d)/[3s2p1d]

6-311G(d,p) (5s1p) [3s1p] (11s5p1d)/[4s3p1d]

6-31+G(d) (5s1p) [3s1p] (11s5p1d) /[4s3p1d]

6-311++G(d) (6s1p)/[4s1p] (12s6p1d)/[5s4p1d]

aug-cc-pVDZ (5s2p)/[3s2p] (10s5p2d)/[4s3p2d]

def2-SVP (4s)/[2s1p] (7s4p)/[3s2p1d]

2.8.2 Types of Basis Sets

A single Gaussian function poorly describes a Slater function, this limitation is minimized by approximation via use of linear combination of Gaussians functions approach [157]. Based on these approximations, basis sets are divided into different types such are:

(i) Minimal basis sets (ii) Split valence basis sets (iii) Dunning basis sets (iv) Karlsruhe basis sets (v) Pseudopotential basis sets (vi) Plane wave basis sets

2.8.2.1 Minimal Basis Sets

Basis sets having minimum number of Gaussian functions are known as minimal basis sets. STO-1G is the simplest function used for the illustration of functions, but it is not useful for theoretical calculations. This function has one primitive Gaussian functions. However, few atoms have extra functions (orbitals more than they required) for the accommodation of their all electrons. Atoms of the periodic table from H−Ar (Ar represents argon) have one basis function in correspondence to the description of the atomic orbital. For example, hydrogen and helium atoms have 1s basis functions. Each atom of the ﬁrst-row (Li−Ne) has 1s, 2s, and 2px, 2py and 2pz functions, as a whole there are ﬁve basis functions for each individual atom. Since, lithium (Li) and beryllium (Be) atoms are often not using p orbitals, still the same basis functions are used for all the atoms of this row. Construction of contraction functions from primitive Gaussian functions stems from STO-3G. STO-3G basis set had been used explicitly for the analysis of the electronic structures of 3 and 4-membered rings and unusual pyrimidine molecules [158]. STO-3G is computationally inexpensive basis set (3 times faster than

3-21G) and it also easily dissects the molecular orbitals into atomic orbitals. But the accuracy is not good for geometries and thermodynamic measurements, therefore it is not acceptable for research work.

2.8.2.2 Split Valence Basis Sets

These are also known as Pople basis sets because they are developed by Pople and his coworkers. These basis sets are more general and use various types of contracted Gaussian functions to simulate Slater functions (orbitals). In Pople basis sets, the core and valence orbitals are separately treated. The simplest basis set of this series is 3-21G, where 3 represents a function used for the core orbital while 2 and 1 functions are combinedly used for valence orbitals. Function 3 means that each core orbital is represented by 3 contracted Gaussian function. The splitting of the valence shell into different functions gives more ﬂexible SCF algorithm for adjusting the contributions of the basis functions to the MOs. Thus, a more realistic electronic distribution simulation is achieved. Later, 4–21G basis set was also used in 1988s, but now it is out of date [159]. Other examples of Pople basis sets are 6-31G(d) [160], 6-311G, 6–31G(d), 6– 31G(d,p) [161], 6–311G(df,p) and 6–311G(3df,3pd) [162] etc.

Polarization Functions

Function included in a basis functions that describe higher angular momentum than the electrons in the atom are known as polarization functions. To improve the accuracy of basis sets, polarization functions are used. The polarization functions allow the electronic distribution to be polarized and displaced more along a specific direction. The polarization functions * or d or other higher functions are used for heavy atoms (other than hydrogen and helium). For hydrogen and helium, polarization functions are denoted by ** or p functions. Some of the Pople basis sets with polarization function are 6-31G(d), 6-31G(d,p) and 6-311G(d), 6-311G(d,p) etc. Addition of polarization functions improves the performance of basis set for measurement of geometric and thermodynamic properties [163].

Diffuse Functions

Core as well as bonding electrons of a molecular system are strongly bound to the nucleus. Whereas, non-bonding or virtual orbital electrons are loosely held to the

57 nucleus. Diffuse functions are used to simulate the behavior of non-bonding electrons and ionic species. Diffuse Gaussian functions have small α values which causes e-αr2 to fall off slowly at large distance (r) from nucleus. Typically, a basis set with diffuse function has single Gaussian function for each valence atomic orbital. For heavy atoms other than H and He, diffuse functions are denoted by + sign e.g. 6-31+G(d). Instead of + sign, the diffuse functions are also denoted by aug in Dunning basis sets which stands for augmented. Besides heavy atoms, diffuse functions are also used for hydrogen and helium and denoted by ++ or aug. Some of the Pople basis sets with diffuse function are 6–31+G(d) [160], 6–31+G(d,p) [164] and 6–31++G(d,p) [164] etc.

2.8.2.3 Correlation-Consistent (cc) Basis Sets

A class of basis sets where core and valence orbitals are combinedly treated, these basis sets are introduced by Dunning research group [132]. Dunning cc basis sets are optimized using correlated methods i.e. CIS, CISD, etc. and converge smoothly towards the complete basis set limit. These basis sets are computationally more demanding compared to split valence sets. The cc term is used for these basis sets to distinguish it from couple cluster, designated as cc.

The cc basis sets are denoted as cc-pVXZ [165], where p defines polarization function, V for valence orbital, X stands for a number of valence shells, and Z stands for zeta. When the diffuse functions are added to cc basis sets, the term aug is used their designation e.g. aug-cc-pVXZ basis set. Different types of cc basis sets are cc-pVDZ, cc-pVTZ [166], cc-pVQZ, cc-pV5Z, aug-cc-pVDZ and aug-cc-pVTZ [167] etc. cc basis sets are specially designed for post-HF calculations, in which electron correlation are treated excellently as compared to HF level. These basis sets are intended to give post-HF calculations with better results in each step. The results are further improved with increasing size of basis set (X = 2, 3, 4, 5, 6…) and with correlation treatment, due to which these basis sets are named as correlation-consistent (cc) basis sets.

2.8.2.4 Effective Core Potentials Basis Sets

Hans Hellmann developed pseudopotential or effective core potential in 1934, which is an approximation used to treat complex systems with a simple description [168]. Pseudopotential explicitly treats only the chemically active valence electrons while the core electrons are 'frozen', being considered together with the nuclei as rigid non-

58 polarizable ion cores. This cut-off results by the reduction of basis set size, reduces the number of electrons and decreases the computational cost by focusing on valence electrons only. Moreover, once derived from relativistic calculations, eﬀective core potentials can include relativistic eﬀects, which are important in heavy elements. Some of the well-known pseudopotential basis sets are LANL2DZ [169], SDD [170] and SDDALL [171] etc.

Sometimes distinction is made between an ECP and a pseudopotential, the latter is used for the valence electrons only, whereas former is used to express a function with fewer orbital nodes than the correct functions. ECPs is useful for molecules with transition metals for speeding up calculations and indeed, ECPs are generally relativistic in nature. In simple molecules, there is no advantage of using pseudopotential calculations because the Pople basis sets are actually better for geometry optimization. But, the use of these ECP basis sets is more necessary for calculations on molecules containing very heavy atom; particularly transition metal. These molecules rely on ab initio or DFT calculations with pseudopotentials basis sets [172].

2.8.2.5 Karlsruhe Basis Sets

These are second-generation default or ‘‘def2’’ basis sets developed by Ahlrichs and coworkers at Karlsruhe [173]. def2 basis sets are very fascinating because these constitute graded quality from partially polarized DZ to heavily polarized QZ for all elements (H−Rn). The most popular Karlsruhe basis sets are def2-SV(P) [174], def2- SVP [175], def2-TZVP, def2-TZVP, def2-QZVP [176]and def2-QZVPP [177] etc.

SV represents split valence term, TZV corresponds to valence triple zeta, QZV defines valence quadruple zeta, (P) denotes partially polarized and PP denotes heavily polarized. Weigend and Ahlrichs recommended that partially polarized basis sets are suitable for DFT calculations and PP-type basis sets are suitable for correlated wave function calculations. They have two drawbacks; (a) those are designed without inclusion of diffuse functions and (b) scalar relativistic ECP is not allowed in the fourth period of the periodic table especially for the 3d and 4p block elements, where relativistic effects are beginning to become chemically important [174].

2.8.2.6 Plane Wave Basis Sets

The basis functions are implemented at modelling atomic orbitals i.e. STO’s and GTO’s and then, taking their linear combination to describe the orbitals (electronic distribution) of the whole system. But there are some of the basis functions which are directly implemented at the periodic system, these are known as plane wave basis sets. Infinite basis functions are required for modelling large system (of infinite range) e.g. a unit cell with periodic boundary conditions. Metals having free electrons in the outer most shell, the solution free electrons are used as a basis function. These plane wave basis sets are larger in size than all electrons Gaussian basis sets [178]. Their size depends on the size of the unit cell (not on the actual system described with in the cell). Besides periodic systems, these basis sets are useful for molecular system through supercell approaches. In this approach, the molecule is placed in large unit cell that it cannot interact with neighboring molecules. Many plane wave basis functions are used to minimize the self-interaction error [179] by placing the small molecule in a large cell. In such cases, localized Gaussian function are more beneficial to handle the situation. Plane wave (PW) basis sets are also useful for 3-diamentional periodic system as compared to atomic centered functions. Valence bonds in metals having delocalized electronic densities (slowly varying) are easily described by plane wave basis set. The core electrons are strongly localized on the nuclei while valence orbitals show rapid oscillation in the core region (to maintain orthogonality). Adequate description of core region requires a large number of oscillating functions (PW basis set has a large Kmix). The disadvantage of PW basis sets is that it is not applicable for nucleus-electron potential singularity. To overcome this limitation, PW basis sets are used with pseudopotential basis sets for the description of nuclear charge and modelling the effect of core electrons. Pseudopotential only gives an idea of potential near nucleus, although there are no core electrons in hydrogen [180].

2.9 Computational Methodology Adopted in the Current Work

Gaussian 09 software is used for calculation of bond dissociation energy [181]. For structural visualization GuessView 5.0 is used [182]. Selected DFs cover all classes of DFT including local density approximation (LDA) class (LSDA [183]), generalized gradient approximation (GGA) class (BLYP [148], G96LYP [184], HCTH [185], MPW3LYP1W [185], PBELYP1W [186], PBE1W [187] etc.), meta-GGA class [188]

(M06-L [189], τ-HCTH [190], TPSS [191], TPSSLYP1W [186] etc.), hybrid GGA (H-GGA) class (B3LYP [192], B3PW91 [193], B98 [194], B97-1 [185], B97-2 [195], MPW1K, MPW1PW91 [195], MPW3LYP [196], X3LYP [197] etc.), global hybrid meta-GGA (GH meta-GGA) class (BMK [198], B1B95 [198], BB95 [192], M05 [189], M05-2X [199], M06 [200], M06-2X [201], M06-HF [200], MPW1KCIS [202], PBE1KCIS [203], TPSS1KCIS [204] etc.), dispersion corrected GGA (GGA-D) class (B97-D [205], B97-D3 [153], BLYP-D3 [92] etc.) [153], range separate hybrid GGA (RS H-GGA) class (CAM-B3LYP, ωB97) [206], , generalized gradient approximation with dispersion correction (GGA-D) class (B97-D, B97-D3 and BLYP-D3 etc.), global hybrid meta-generalized gradient approximation with dispersion correction (GH meta-GGA-D) class (BMK-D3BJ [207], M05-D3, M06-D3, M05-2X-D3 and M06-2X-D3 etc.) [153], hybrid generalized gradient approximation with dispersion correction (H-GGA-D) class (B3LYP-D3 [148, 153, 193], B3LYP-D3BJ etc.) [207], double hybrid generalized gradient approximation with dispersion correction (DH-GGA-D) class (B2PLYP-D3) [17], non-separable gradient approximation (NGA) class (N12) [208], meta-non-separable gradient approximation (meta-NGA) class (M11-L [208], MN12-L [140]), range separated hybrid meta-NGA (RS H-NGA) class (M11 [147]) and range separated hybrid GGA with dispersion correction (RS H-GGA-D) [153] class (CAM-B3LYP-D3BJ, LC- ωPBE-D3, ωB97X-D [206]). For the proper description of occupied orbitals, different classes of basis sets are selected. These basis sets include: 6-31G(d), 6-31G(d,p), 6- 311G(d), 6-311G(d,p), 6-31+G(d) and 6-311++G(d) basis sets of Pople basis sets [163,

209] for C−X, M−O2 and C−CN bonds, aug-cc-pVDZ and aug-cc-pVTZ of Dunning basis sets [132, 165] for C−X and C−CN bonds, LANL2DZ and SDD of effective core potential (ECP) basis sets for C−Sn and M−O2 bonds and def2-SVP, def2-DZVP and def2-TZVP basis sets of Karlsruhe basis sets, [169, 172] for C−Sn and C−CN bonds. For C−X bond, 33 different density functionals with four basis sets are used. Nineteen DFs from eight different classes of DFT with four basis sets are selected for BDE calculation of C−Sn bond. All organo-nitrile compounds and their respective radicals are optimized with 31 different DFs with eight basis sets [174]. For the optimization of all structures of Grignard reagents, thirteen different classes of DFT are selected including twenty-nine DFs with four basis sets. Fourteen DFs are chosen from seven

DFT classes with two series of mixed basis sets for BDE of M−O2 bond.

For all selected molecules, optimization and frequency are calculated at the same level of theory to confirm all the structures as true minimum. The zero-point corrected energies are taken for BDE of all selected bonds at 298 K and 1 atm. and the results are compared with already reported experimental results. Spin polarization are analyzed for homolytic and heterolytic cleavage of some bonds (C−Sn). For M−O2 bond, all the complexes and their resultant radicals are studied up to four lowest spin states.

2.10 Statistical Tools

For validation of theoretical methods with experimental data, various statistical analysis tools are used.

2.10.1 Root Mean Square Deviation (RMSD)

The RMSD is used to compare differences between two data sets those may vary, neither of which is accepted as the “standard”. Mathematically, RMSD (root mean square deviation) is defined by the following expression:

1 2 ⁄2 (xi − yi) ⁄ RMSD = [∑ N] Eq. 2. 82

Where, N is the number of compounds, xi represents theoretical data for each specie and yi represents the experimental data accordingly. Experimental data is subtracted from theoretical data then the square root of the average of squared data gives RMSD.

2.10.2 Standard Deviation (SD)

Standard deviation is used to quantify the amount of variation [210]. A low standard deviation indicates that the theoretical values tend to be close to the mean value, while a high standard deviation indicates that the theoretical values are spread out over a wider range of values. Mathematically, SD (standard deviation) is defined by the following expression:

1 2 ⁄2 (xi − yi) ⁄ SD = [∑ N − 1] Eq. 2. 83

Where, N is the number of compounds used in study, i represents number of selected compounds, xi represents the calculated value for each species, x is the mean of the

62 calculated values, N-1 is variance. Square the difference between individual theoretical values and its mean then square root of the difference is divided by variance which is known as SD.

2.10.3 Pearson's Correlation (R)

R is a measure of the linear correlation between two variables x and y on a scatter plot. A statistical test to determine just how strong is the relationship between those two variables. A perfect positive linear relationship has value of one (R = 1). Mathematica -lly, R (Pearson's correlation) is defined by the following expression:

1 푅 = 푛(∑xy) − (∑ 푥) (∑ 푦)/[{푛 ∑ 푥2 − (∑ 푥)2][푛 ∑ 푦2 − (∑ 푦2)}]2 Eq. 2. 84

(Where, x and y represent the experimental value and theoretical values for each specie, respectively). The linear correlation (R) is confirmed from Shapiro-Wilks test (a type of normality test according to the null hypothesis of the test, the population is normally distributed when p-value is greater than alpha value then its correlation is significant).

2.10.4 Mean Absolute Error (MAE)

A quantity that is used to measure how much estimated values (theoretical) differ from actual values (experimental). Mathematically, MAE (mean absolute error) is defined by the following expression:

∑n |y −x | ∑n |e | MAE = i=1 i i = i=1 i Eq. 2. 85 n n

Where, n denotes the number of selected compounds, yi represents experimental value for each specie, xi represents calculated value for each specie. The error up to ±1 kcal/mol is significant, under binding MAE is positive and over binding MAE is negative.

Some studies recommend the use of MAE instead of RMSD because it possesses advantages of interpretability over RMSD. Moreover, MAE is the average absolute difference between two variables, and it is fundamentally easier to understand than the RMSD. In RMSD, square of difference is taken and depends on prominent errors not

63 on small errors. While MAE count the small errors. Beside this, each error contributing to MAE is proportional to the absolute value of the error, which is not the case for RMS [211]. MAE measures the average magnitude of the errors in data set of predictions, without considering their direction. R depends both on the strength and direction of relationship between two variables. In Pearson correlation, relationship is linear when one variable is changed, the other variable also changes at the same rate. Every statistical term has its own merits and demerits. In current study, all these four statistical tools are used and the density functional which give best RMSD, SD, R and MAE results compared to experimental data, is considered the desire functional for BDE measurement of selected chemical bonds.

Chapter 3

3 Results and Discussion

Results & Discussion

A benchmark DFT study has been executed for five chemically important (C−X (X =

Cl and Br), C−Sn, C−CN, C−Mg and M−O2 (M = Rh, Ir, Cu, Co)) bonds, to explore cost-effective and accurate density functionals. These chemical bonds have been selected due to their widespread applications in different fields of chemistry. Rate determining steps of famous named reactions involve the dissociation of these chemical bonds. Literature survey reveals that a lot of experimental work is done on the BDE of these bonds. But, literature is limited about the theoretical studies of BDE of these bonds. Theoretical studies of BDE measurements of these bonds are important for mechanistic understanding of respective chemical reactions. The cost-effective and accurate theoretical method for each bond is achieved through benchmark study. Results and discussion section is divided in to 5 subsections and short details are as follows:

3.1. Benchmark DFT study on BDE of C−X (X = Cl and Br) bond is studied in the first section. This study includes a number of DFs from different DFT classes with a variety of basis sets (Pople and Dunning basis sets). Results of all chosen DFs are compared with experimental data. For summarization and further validation of data, four statistical tools (RMSD, SD, R and MAE) are used [211].

3.2 This section corresponds to the benchmark DFT study on homolytic cleavage of C−Sn bond of organotin compound. Again, several DFs are selected from seven DFT class, and tested for their accuracy with pseudopotential and Karlsruhe basis sets [212].

3.3 Benchmark DFT study on BDE of C−CN bond of nitrile compounds includes in this section. Theoretical data collected from a number of DFs and basis sets are compared with experimental data [213].

3.4 This section comprises of benchmark DFT study on BDE on C−Mg bond of Grignard reagents. A number of Grignard reagents are tested via several DFs with different basis sets.

3.5 Benchmark DFT study on M−O2 bond for water splitting is briefly discussed in this section.

In all above sections, Theoretical data is compared with experimental results and then validated via statistical tests including RMSD, SD, MAE and R.

3.1 Benchmark Study for BDE of C−X (X = Cl and Br) Bond in Halogens Containing Compounds

3.1.1 Importance of Halogen Containing Molecules

Halogens containing molecules have been extensively studied for industrial applications, organic, bioorganic synthesis and for their involvement in environment as pollutants [214]. Daniel et al., observed that degradation and detoxification of halogen molecules require dehalogenation pathways [215]. The broad use of halogenated molecules as precursors in synthetic chemistry, has driven the experimentalist and theoretician to explore the BDE of C−X bonds. Carbon-halogen bond dissociation energy plays an important role in many vital chemical reactions, which reflects the importance of this bond.

3.1.2 Experimental and Theoretical Studies of BDE of C−X Bond

Experimentally, C−X bonds have been studied widely by ECD (Electron capture detector) technique via dissociative thermal electron attachment [43], collision induced dissociation (CID) method [44], threshold photoelectron photoion coincidence (TPEPICO) spectroscopy [45], infrared chemiluminescence technique [46] and photoionization mass spectrometry [47]. Since the development of DFT, different density functionals (DFs) have been used to measure C−X BDEs of various compounds and calibrated with experimental values for the validation of DFs. Ana et al., through combined theoretical and experimental studies estimated standard molar heat of o o formation (∆fH m) and heat of sublimation (∆subH m) of halo- derivatives of acetic acid. Their results were used to calculate bond dissociation energy of C−X bond in chloro, bromo- and iodo-acetic acid [216].

Martin et al., in 2001, implemented ab initio and DFT methods to determine the reaction energetics of halogenated SN2 reactions in gas phase, and the results showed that mPW1K functional is the best method for calculating activation barriers [216]. Sebastian and Martin performed combined DFs and ab initio methods-based benchmark study with aVTZ/aVQZ basis sets for accurate geometries and BDEs of

67 halogen-halogen bond in different halogens dimers. It is observed that among all selected DFs, M06-2X and ωB97X-D performed better [29]. Houk et al., applied B3LYP and G3B3 (more accurate) level of theories to measure reactivity order of C−X bond in five and six membered halogenated heterocycles in relation to BDE of C−X bond and realized that their BDEs are less than alkyl halides [217]. Wang et al., studied some non-conventional density functionals (B3P86, MPW1K, TPSS1KCIS, X3LYP, BMK) with B3LYP for C–Cl BDE of aryl chlorides and demonstrated that B3P86/6- 311++G(2df,2p) method gave most accurate result (mean absolute deviation is 6.58 kJ/mol) [218]. Feng et al., used composite ab initio methods (CBS-Q, G3) and DFT functionals (B3LYP, BH and HLYP, B3P86 and B3PW91) for calculation of BDEs of 200 compounds of different classes and compared the results with the validated experimental data. Their results indicate that among all DFs, B3LYP, B3P86, and B3PW91 show superior performance (SD 12.1-18.0 kJ/mol) however performance of BH and HLYP functional was very poor [219]. Radom and his coworkers reported that GH meta-GGA functionals give best performance for bond dissociation energy of R−X

(R= Me, Et, i-Pr and t-Bu and X = H, CH3, OCH3, OH and F) bond of hydrocarbons [220].

For the current benchmark study, sixteen structurally diverse organic compounds containing carbon halogen (C−X, X = Cl and Br) bond have been selected (Fig. 3.1 (1- 16)) [221–226]. The optimized geometries of all compounds are in singlet spin state and their resultant radicals are in doublet spin states. In the present study, density functionals with different split valence basis sets are implemented to test their accuracy for C−X bond dissociation energy. Analogous to the previous literature results, the accuracy is judged by comparing linear correlation factor (Pearson's correlation (R)), root mean square deviation (RMSD), standard deviation (SD), mean absolute error (MAE) and linear correlation (R) between experimental and theoretical values [211]. The detailed results of these statistical analyses are given in Tables 3.1-3.4 and graphically represented in Figs. 3.2-3.4.

Fig. 3.1 The Structures of Halogen-Containing Compounds with Known Experimental BDEs of C−X (X = Cl and Br) Bond

3.1.3 Efficiency of Pople Basis Sets

To explore the efficiency of all selected DFs with Pople basis sets, 6-31G(d) and 6- 311G(d) basis sets are analyzed.

3.1.3.1 Evaluation of DFs with 6-31G(d) Basis Set

First of all, 6-31G(d) basis set with different density functionals (DFs) is implemented and compared statistically. RMSD, SD and MAE of C−X bond dissociation energy with respect to different DFs are provided in Tables 3.1-3.4, and graphically shown in Figs. 3.2-3.4.

Range separated hybrid GGA with dispersion correction (RS H-GGA-D) class shows better performance with 6-31G(d) basis set because all the dispersion corrected functionals (LC-ωPBE-D3 and ωB97X-D) performed efficiently with 6-31G(d) basis set. RMSD and SD values are in the range of 3.77 to 5.54 kcal/mol where R is in order of 0.95-0.96, respectively. The correlation of ωB97X-D is the most favorable than the former functional and performed better for C−X bond dissociation energy calculations. Over bounded MAE is far away from -1 kcal/mol. All these results are comparable to the previously reported literature e.g. range separated hybrids have been parameterized by Grimme with cost-effective dispersion correction in order to improve their performance for thermochemical properties [153]. Sherrill and Tschumper showed that dispersion corrected functionals deviate 5-10% from gold standard CCSD(T) for calculation of bond dissociation energies, conformational energies and non-covalent interactions [227]. Similarly, these are superior in overall performance for thermal dissociation energy measurements e.g. ωB97X-D is well known functional for measurement of reaction energies, atomization energies and non-covalent interaction energies. On the other side, second functional of this class is LC-ωPBE-D3 which is parametrized with long-range correlation and dispersion correction of range separated hybrid functionals. Both of these functionals show improvement over conventional density functionals and give accurate results for molecular properties, such as thermochemistry and reactions energies [228]. The same results are obtained in the current study. Here, range separated hybrid GGA (RSH GGA) class and dispersion corrected GGA (GGA-D) class functionals also show good results with 6-31G(d) basis set for C−X bond dissociation energy. RMSD and SD of B97-D and B97-D3 are in the range of 4.41-4.99 kcal/mol. On the other hand, R is 0.94 and over bounded MAE ranges from 0.06 to 2.57 kcal/mol. Statistical analysis concluded that GGA-D class is less efficient than RSH GGA class. However, long range corrected class show slightly less performance compared to dispersion corrected long range class. RMSD, SD, R and MAE of CAM-B3LYP from RS H-GGA class are 4.10 kcal/mol, 3.94 kcal/mol, 0.95 and 1.51 kcal/mol, respectively (Table 3.1), which reflects moderate performance of this functional. Previous literature revealed that CAM-B3LYP has some improved long-range properties, therefore thermodynamic quantities can be calculated more accurately as compared to B3LYP [229]. Similarly, in our study, CAM-B3LYP performed good as compared to B3LYP. GGA class explicitly depends on gradient of the density, this class showed moderate performance with 6-31G(d) basis set. HCTH

70 has better performance for C−X BDE calculation. Its (HCTH) RMSD and SD are 3.43 kcal/mol and 3.59 kcal/mol, respectively. A good Pearson's correlation (0.96) and over bounded MAE of 1.36 kcal/mol is observed. Deviations and error are less than the former classes of DFs (dispersion corrected GGA, RSH GGA and RSH GGA with dispersion corrected classes). The second functional of this class is PBE1W for which SD is less i.e. 3.87 kcal/mol and R is 0.95 (similar to HCTH). In contrast, its RMSD and MAE are high which made functional less efficient for C−X BDE calculations. Martin et al., has reported that BLYP shows good performance for BDE calculations of organic halides in combination with higher basis set [24]. Similar results are obtained in present C−X BDE study, where performance of BLYP is good. RMSD and SD are up to 4 kcal/mol while R and MAE are 0.95 and 0.48 kcal/mol, respectively. These results are slightly less significant than HCTH. For G96LYP, RMSD and SD are 3.99 kcal/mol, 3.87 kcal/mol, respectively. The significant R for G96LYP is 0.95 and has under bound MAE of 1.36 kcal/mol. Deviations of MPW3LYP1W and PBELYP1W are more than 3 kcal/mol. The correlation of these functionals is 0.95 and MAE (over bound) is more than 1 kcal/mol (Table 3.1). These results appraised the above functionals for their reasonable performance against C−X bond dissociation energy calculation except HCTH which shows better performance compared to other functionals. Conclusively, this class is rational performer for C−X BDE calculations.

30 6-31G(d) 6-311G(d) Aug-cc-pvdz Aug-cc-pvtz

10 RMSD (kcal/mol)

B98

PBE

M05 M05

BB95

BMK

TPSS

B97-1

B97-2

BLYP

B97-D

B1B95

LSDA LSDA

M06-L

HCTH

B3LYP

X3LYP

B97-D3

M05-2X

t-HCTH

G96LYP

B3PW91

MPW1K

WB97X-D

PBE1KCIS

MPW3LYP

TPSS1KCIS

PBELYP1W

MPW1KCIS

MPW1PW91

CAM-B3LYP

TPSSLYP1W MPWLYP1W Density Functionals LC-WPBE-D3

Fig. 3.2 Root Mean Square Deviation (RMSD) of Different Density Functionals with Four Basis Sets for C−X (X = Cl and Br) BDEs

The performance of GH meta-GGA class functionals (BMK, B1B95, M05, M05-2X, MPW1KCIS, PBE1KCIS and TPSS1KCIS) is also tested with 6-31G(d). BMK and Truhlar functional, TPSS1KCIS with 6-31G(d) basis set have RMSD and SD more than 3 kcal/mol and MAE > 2 kcal/mol (over bound error from zero baseline). The Pearson's correlation (R) for all compounds (1-16) is 0.95 (nearer to one) from experimental bond dissociation energies and demonstrates the moderate significance of results. Minnesota functionals M05, M05-2X and remaining Truhlar functionals MPW1KCIS, PBE1KCIS and B1B95, have RMSD and SD in the range of 7-11 kcal/mol. R in the range of 0.86- 0.96 and over bound MAE. These results of GH meta-GGA class indicate large BDE values of C−X bond. Based on the current statistical analysis, these functionals are less significant for carbon-halogen bond dissociation energies as previously reported for BDEs of C–S bonds [70]. Although, the results of TPSS1KCIS signify good performance but the results of other functionals designate the whole class as moderate performer for C−X BDE calculation.

Most popular B3LYP functional with B3PW91, B98, B97-1, B97-2, MPW1K, MPW1PW91, MPW3LYP and X3LYP belong to hybrid GGA class, which highly depends on occupied molecular orbitals. B3LYP shows outstanding performance in calculating energies, structures and properties of organic compounds [230]. Unexpectedly, for C−X bond dissociation energy investigation even at 6-31G(d) basis set, deviations of B3LYP and all other selected hybrid functionals from experimental values are high. These functionals performed less significantly than other classes of DFT. Deviations and error are more for B3PW91, B98, B97-1, MPW1K, MPW1PW91, MPW3LYP and X3LYP functionals and R is less than 0.96 with over bound MAE in the range of 3.07 to 5.67 kcal/mol (more away from base line). Pearson's correlation is good in case of B97-2 (0.96) but the trend of RMSD, SD and MAE is similar to other functionals. Which indicates the less efficiency of the hybrid class for BDE of C−X bond.

6-31G(d) 6-311G(d) 25 Aug-cc-pvdz Aug-cc-pvtz

10 SD(kcal/mol)

B98

PBE

M05 M05

BB95

BMK TPSS

B97-1 B97-2

BLYP

B97-D

B1B95

LSDA LSDA M06-L HCTH

B3LYP

X3LYP B97-D3

M05-2X

t-HCTH

G96LYP B3PW91

MPW1K

WB97X-D

PBE1KCIS

MPW3LYP

TPSS1KCIS

PBELYP1W

MPW1KCIS

MPW1PW91

CAM-B3LYP TPSSLYP1W MPWLYP1W Density Functionals LC-WPBE-D3

Fig. 3.3 Standard Deviation (SD) of Different Density Functionals with Four Basis Sets for C−X (X = Cl and Br) BDEs

For performance assessment of C−X BDE calculation, four functionals M06-L, 휏- HCTH, TPSS and TPSSLYP1W are selected from meta-GGA class. Performance of TPSSLYP1W to calculate C−X BDE is good as compared to the other functionals. TPSSLYP1W has less RMSD (4.39 kcal/mol) and SD (4.54 kcal/mol), less significant R (0.94) and the under bound MAE of 0.21 kcal/mol. RMSD, SD and MAE of 휏-HCTH and TPSS functionals are more than 3 kcal/mol and R in the range of 0.88 to 0.93, compared to experimental results. M06-L has the poorest performance, deviations (RMSD and SD) and MAE are high as compared to former functionals and showed large BDEs of required C−X bond. Moreover, correlation (R) is less significant (Table 3.1). Based on this analysis, meta-GGA class is considered as a poor class for bond dissociation energy calculation of C−X bond.

LSDA, local density dependent, is also tested for calculation of BDE of C−X bond for selected compounds (Fig. 3.1). SD, and R of this functional are 3.83 kcal/mol and 0.95 respectively, whereas RMSD and MAE are above 26 kcal/mol. LSDA based on uniform electron gas and generally tends to over bound (hence, bond length are too short and bond energies are too high) and results in a very high MAE [183]. These results of LSDA functional indicate poorest performance of the LDA class.

Classes of DFT DFs RMSD SD R MAE LDA LSDA 26.62 3.83 0.95 26.36 BLYP 3.95 4.05 0.95 0.48 G96LYP 3.99 3.87 0.95 1.36 HCTH 3.59 3.43 0.96 1.36 GGA MPWLYP1W (MPWV5LYP) 4.53 3.95 0.95 2.43 PBELYP1W (PBEV5LYP) 5.42 4.08 0.95 3.72 PBE (PBEPBE) 8.36 3.87 0.95 7.47 M06-L 11.18 9.28 0.88 6.65 휏-HCTH (THCTHH) 11.03 8.51 0.88 7.33 meta-GGA TPSS 6.15 4.83 0.93 3.99 TPSSLYP1W (TPSSV5LYP) 4.39 4.54 0.94 0.21

B3LYP 9.05 8.79 0.87 3.07 B3PW91 7.27 5.89 0.92 4.51 B98 5.05 3.89 0.95 3.37 B97-1 5.95 4.02 0.95 4.50 Hybrid GGA B97-2 5.59 3.66 0.96 4.33 MPW1K (MPWPW91) 6.84 3.95 0.95 5.67 MPW1PW91 10.57 8.95 0.87 6.05 MPW3LYP (MPWLYP) 4.53 3.95 0.95 2.43 X3LYP 9.08 8.65 0.87 3.52 BMK 5.79 4.06 0.95 4.26 B1B95 10.18 7.77 0.86 6.85 BB95 29.36 18.47 0.26 23.29 M05 9.33 8.36 0.91 4.62 GH meta-GGA M05-2X 11.33 7.73 0.88 8.51 MPW1KCIS (MPWKCIS) 7.18 3.70 0.96 6.22 PBE1KCIS (PBEKCIS) 8.50 3.58 0.96 7.76 TPSS1KCIS (TPSSKCIS) 5.49 3.96 0.95 3.93 B97-D 4.53 4.67 0.94 0.06 GGA-D B97-D3 4.99 4.41 0.94 2.57 RS hybrid GGA CAM-B3LYP 4.10 3.94 0.95 1.51 LC-ωPBE-D3 5.32 4.04 0.95 3.60 RS hybrid GGA-D ωB97X-D 5.54 3.77 0.96 4.16

3.1.3.2 Evaluation of DFs with 6-311G(d) Basis Set

The performance of all DFs is also tested with higher basis set (6-311G(d)) for C−X bond dissociation energy calculation. All classes show improved performance with 6- 311G(d) basis set except LDA class. The statistical analyses data (RMSD, SD, R and MAE) are given in Table 3.2 and graphically represented in Figs. 3.2-3.5.

The dispersion corrected class shows best performance with 6-311G(d) basis set. The results are consistent with 6-31G(d) basis set but more accurate when compared with experimental data. Among selected functionals of dispersion corrected hybrid GGA class, ωB97X-D functional shows the best results (Fig. 3.5), which is consistent with the results of 6-31G(d) basis set. RMSD, SD, R and MAE of ωB97X-D are 3.14

75 kcal/mol 3.05 kcal/mol, 0.97 and 1.07 kcal/mol, respectively (interestingly less as compared to 6-31G(d) basis set). Surprisingly, LC-ωPBE-D3 also shows good performance with less deviations and error. The R is also significant (Table 3.2). Both functionals give most accurate results with 6-311G(d) basis set and overall this class is designated as best class for C−X BDE measurements.

6-31G(d) 6-311G(d) Aug-cc-PVDZ 5 Aug-cc-PVTZ

) -5

-10

kcal mol (

-15 MAE

-20

-25

-30

B98

PBE

M05 M05

BB95

BMK

TPSS

B97-1

B97-2

BLYP

B97-D

B1B95

LSDA LSDA

M06-L

HCTH

B3LYP

X3LYP

B97-D3

M05-2X

t-HCTH

G96LYP

B3PW91

MPW1K

WB97X-D

PBE1KCIS

MPW3LYP

TPSS1KCIS

PBELYP1W

MPW1KCIS

MPW1PW91

CAM-B3LYP

TPSSLYP1W MPWLYP1W

Density Functionals LC-WPBE-D3

Fig. 3.4 Mean Absolute Error (MAE) of Different Density Functionals with Four Basis Sets for C−X (X = Cl and Br) BDEs

In dispersion corrected GGA class, the performance of B97-D3 at higher 6-311G(d) basis set is better than B97-D functional, while in case of 6-31G(d) basis set, B97-D is superior in its performance against experimental C−X BDEs. RMSD (3.35 kcal/mol), SD (3.4 kcal/mol) are less with significant R (0.97), and lower MAE (0.63 kcal/mol). The performance of dispersion corrected GGA (GGA-D) class is better for C−X bond

76 dissociation energy but less as compared to range separated hybrid meta-GGA class with dispersion correction (RSH GGA-D class).

CAM-B3LYP (range separated hybrid GGA class) efficiency is decreased with 6- 311G(d) basis set and observed as inferior functional for C−X bond dissociation energy calculation, although it is slightly superior in case of 6-31G(d) basis set. Thus, range separated hybrid meta-GGA class is slightly less efficient with 6-311G(d) for desired data. PBELYP1W has low deviations and error with good R and suggest it as the best performer of this class. This observation is consistent with the results of 6-31G(d) basis set, however, the results are improved with 6-311G(d) basis set. Compared with 6- 31G(d) basis set, MPW3LYP1W and HCTH functionals have less error, less deviations and more significant correlation between experimental and theoretical values with 6- 311G(d) basis set (Table 3.2). RMSD, SD, R and MAE of BLYP functional are 4.55 kcal/mol, 3.09 kcal/mol, 0.97 and 3.43 kcal/mol, respectively. These results indicate the improved performance of BLYP with 6-311G(d) basis set than 6-31G(d) basis set. Whereas, G96LYP and PBE1W functionals show moderate performance and G96LYP performed inferior with 6-311G(d) basis set compared to 6-31G(d) basis set. In case of GH meta-GGA class, TPSS1KCIS has significant results at higher basis set. RMSD and SD of BMK, MPW1KCIS and TPSS1KCIS are in the range of 3-4 kcal/mol. R values of these DFs range from 0.96 to 0.97, and MAE are less (up to 2 kcal/mol). PBE1KCIS also shows less deviations with significant R of 0.97 but the MAE is more (Table 3.2).

Classes of DFT DFs RMSD SD R MAE LDA LSDA 26.34 20.22 0.59 17.62 BLYP 4.55 3.09 0.97 3.43 G96LYP 5.59 3.21 0.97 4.65 HCTH 3.73 3.09 0.97 2.22 GGA MPWLYP1W 3.60 3.24 0.97 1.77 PBELYP1W 3.07 3.16 0.97 0.31 PBE 5.17 3.12 0.97 4.20

M06-L 3.72 3.71 0.96 0.98 휏-HCTH 12.62 12.79 0.73 2.42 meta-GGA TPSS 4.69 4.79 0.93 0.66 TPSSLYP1W 4.95 3.33 0.97 3.75 B3LYP 8.52 8.79 0.89 0.44 B3PW91 9.08 8.96 0.88 2.68 B98 3.02 3.11 0.97 0.13 B97-1 3.24 3.05 0.97 1.32 Hybrid GGA B97-2 13.11 13.42 0.67 1.71 MPW1K 3.73 3.52 0.96 1.50 MPW1PW91 8.89 8.72 0.89 2.79 MPW3LYP 3.78 3.19 0.97 2.19 X3LYP 9.75 10.03 0.75 0.88 BMK 3.72 3.47 0.96 1.59 B1B95 9.74 8.85 0.89 4.64 BB95 21.36 14.95 0.49 15.71 M05 8.66 8.89 0.91 0.90 GH meta-GGA M05-2X 7.74 7.23 0.92 3.30 MPW1KCIS 3.89 3.06 0.97 2.52 PBE1KCIS 4.77 3.05 0.97 3.74 TPSS1KCIS 2.98 3.07 0.97 0.33 B97-D 4.32 3.26 0.97 2.94 GGA-D B97-D3 3.35 3.4 0.97 0.63 RS hybrid GGA CAM-B3LYP 4.23 4.16 0.94 1.28 LC-ωPBE-D3 3.32 3.4 0.96 0.40 RS hybrid GGA-D ωB97X-D 3.14 3.05 0.97 1.07

On the other side, BB95, B1B95, M05 and M05-2X functionals show poor performance in whole class quite similar to the 6-31G(d) basis set. Comparative statistical analysis depicts that TPSS1KCIS can act as good functional for the measurement of C−X dissociation energy and the trend observed is same as that for 6-31G(d) basis set. Hence, GH meta-GGA class show good performance for the C−X BDE but slightly improved than 6-31G(d) basis set. The performance of M06-L and TPSSLYP1W (meta-GGA class) for C−X BDE calculation is good, however, TPSS displays inferior performance

78 compared to former two functionals. The trend of significance performance of 6- 311G(d) basis set is similar to 6-31G(d) basis set. For 휏-HCTH functional, deviations and error are more as compared to experimental values and correlation is also less significant (Table 3.2). These results demonstrate that meta-GGA class acts as moderate performer for C−X BDE calculations.

Hybrid GGA class is rational toward BDE of carbon-halogen bond. Hybrid GGA functionals (B3LYP, B3PW91, B98, B97-1, B97-2, MPW1K, MPW3LYP, MPW1PW91 and X3LYP) with Pople basis set are examined where 6-311G(d) basis set have improved performance compared to 6-31G(d) basis set. The performance of B98, B97-1 and MPW1K functionals is better in comparison to all other functionals of hybrid class. RMSD and SD are 3.02-3.73 kcal/mol, R of 0.97 and MAE is up to 2.19 kcal/mol, respectively. These three functionals having less efficiency in 6-31G(d) basis set but show significant results with 6-311G(d) basis set. Whereas all other functionals have moderate performance with 6-311G(d) basis set, quite similar to 6-31G(d) basis set except B97-2, for which deviations (RMSD and SD) > 10 kcal/mol but MAE is 1.71 kcal/mol with a very poor R. B97-2 functional has superior performance in 6-31G(d) becomes inferior in higher basis set for C−X BDE measurement. Overall, the whole class represent moderate performance for C−X BDE calculations. LSDA (LDA class) has poor results for calculation of CX BDE with 6-311-G(d) basis set as previously observed in 6-31G(d) basis set, which indicates LDA as a poor class.

All these results of different classes of DFT show the best competence of 6-311G(d) basis set for C−X BDE.

3.1.4 Efficiency of Dunning Basis Sets

To explore the efficiency of all selected DFs with Dunning basis sets, aug-cc-pVDZ and aug-cc-pVDZ basis sets are analyzed.

3.1.4.1 Evaluation of DFs with Aug-cc-pVDZ Basis Set

Dunning et al., developed augmented correlation consistent polarized valence-only double zeta basis set which gives more reliable results in combination with DFT methods for calculating different properties of molecules [227]. Here, double zeta basis set is evaluated with different DFs to calculate the bond dissociation energy of C−X

79 bond in selected compounds (1-16). The statistical analysis data (RMSD, SD, R and MAE) are given in Table 3.3 and graphically represented in Figs. 3.2-3.4. All DFs classes show slightly less efficiency with aug-cc-pVDZ basis set but the trend of H- GGA-D class good performance is maintained here.

It is observed that dispersion corrected range separated hybrid GGA class shows improved performance among all classes of DFT. Whereas, ωB97X-D is observed as best performer in range separated hybrid GGA with dispersion corrected (RSH GGA- D) class. RMSD, SD, R and MAE of ωB97X-D/ aug-cc-pVDZ method are 3.4 kcal/mol, 2.98 kcal/mol, 0.97 and 1.79 kcal/mol, respectively. Efficiency of ωB97X-D with aug-cc-pVDZ is consistent with Pople (6-31G(d) and 6-311G(d) basis sets) basis sets but results are improved with aug-cc-pVDZ basis set. LC-ωPBE-D3 functionals with aug-cc-pVDZ basis set also shows better accuracy when compared with experimental data. But the deviations, error values are higher, and correlation is lower than ωB97X-D. These results are also same as observed in Pople basis sets (6-31G(d)).

Classes of DFT DFs RMSD SD R MAE LDA LSDA 22.31 5.31 0.94 21.71 BLYP 9.32 9.56 0.87 1.14 G96LYP 10.59 10.78 0.77 1.81 HCTH 3.34 3.01 0.97 1.63 GGA MPWLYP1W 13.73 14.03 0.72 1.97 PBELYP1W 11.61 11.18 0.61 4.21 PBE 7.02 6.62 0.86 2.84 M06-L 9.56 8.74 0.9 4.45 휏-HCTH 4.65 4.12 0.95 2.39 meta-GGA TPSS 7.29 7.31 0.86 1.79 TPSSLYP1W 4.75 3.15 0.97 3.64 B3LYP 5.73 5.60 0.93 1.84 Hybrid GGA B3PW91 6.36 6.29 0.93 1.84 B98 5.25 5.36 0.91 0.79

B97-1 4.36 3.85 0.95 2.27 B97-2 3.29 2.76 0.98 1.91 MPW1K 6.67 6.22 0.87 2.85 MPW1PW91 10.21 9.78 0.87 3.81 MPW3LYP 11.31 10.8 0.79 4.29 X3LYP 6.43 6.63 0.93 0.21 BMK 13.93 14.36 0.70 0.91 B1B95 7.86 7.86 0.87 1.97 BB95 17.72 11.75 0.65 13.58 M05 7.01 7.19 0.92 0.79 GH meta-GGA M05-2X 14.5 12.89 0.80 7.38 MPW1KCIS 17.77 14.62 0.68 10.75 PBE1KCIS 10.01 10.16 0.71 1.86 TPSS1KCIS 7.05 6.85 0.89 2.39 B97-D 4.57 3.86 0.96 2.63 GGA-D B97-D3 4.16 4.19 0.94 0.92 RS hybrid GGA CAM-B3LYP 5.95 6.07 0.88 0.89 LC-ωPBE-D3 3.44 3.42 0.96 0.95 RS hybrid GGA-D ωB97X-D 3.40 2.98 0.97 1.79

Results of dispersion corrected generalized gradient approximation (GGA-D) class indicate that between B97-D and B97-D3 functionals, B97-D functional shows reasonable performance. Deviations and error of B97-D are lower than 4 kcal/mol with better R (0.96). However, B97-D3 has under bound MAE of 0.92 kcal/mol but due to more deviation and less significant R it is less efficient than the former functionals of the same class. Previous literature revealed that dispersion corrected DFT functionals are reliable for thermo-chemical measurements and the same results are observed [183]. Here, with double zeta basis set, more pronounced results are observed for both dispersion corrected range separated hybrid GGA and dispersion corrected GGA classes. The only difference for C−X bond dissociation energy calculations is that performance of RS H-GGA-D is more remarkable. On the other side, RS H-GGA class has low performance with aug-cc-pVDZ. The results indicated lower efficiency of CAM-B3LYP functional with aug-cc-pVDZ basis set when compared with Pople basis

81 sets. Deviations of this functional are high with less significant correlation (Although error is low), which signify less efficiency of RS-H-GGA class for measurement of BDE of C−X bond (Table 3.3). Hybrid GGA class shows good performance for C−X BDE calculations. Herein, B97-2/aug-cc-pVDZ method shows better performance compared to B97-2 performance with Pople basis sets. RMSD value is 3.29 kcal/mol, SD is 2.76 kcal/mol with more significant R (0.98) and MAE is 1.91 kcal/mol from experimental BDE of C−X bond. Remaining hybrid functionals such as B3LYP, B3PW91, B97-1 and X3LYP show moderate results except some of the functionals i.e. B98, MPW1K, MPW1PW91 and MPW3LYP (Table 3.3). These results illustrate the moderate performance of hybrid class with aug-cc-pVDZ basis set compared to previous results.

Previously, 휏-HCTH (meta-GGA class) has poor performance with Pople (6-31G(d) and 6-311G(d)) basis sets. Whereas, with aug-cc-pVDZ basis set, 휏-HCTH has RMSD (4.65 kcal/mol) and SD (4.12 kcal/mol) with over bound MAE (2.39 kcal/mol) and R is 0.95, which indicates good performance and signify that 휏-HCTH can act as a slightly efficient functional for C−X bond dissociation energy at aug-cc-pVDZ basis set. On the other side, efficiency of all remaining functionals of this class is inferior in comparison to Pople basis sets. For M06-L, TPSS and TPSSLYP1W functionals, the deviations and error are more from experimental values with poor calculated correlation (< 0.9). So, none of them exhibits a precise result to predict the bond dissociation energy of unusual carbon halogens bonds. Although, deviations and error of TPSSLYP1W functional are lower than the other DFs of meta-GGA, still these values are higher compared to experimental values. The R value of TPSSLYP1W is 0.97. Thus, the class is designated as a poor executor for C−X BDE calculations. Global hybrid meta-GGA, density dependent gradient (GGA) and local density approximation (LDA) classes give poorer results with aug-cc-pVDZ basis set. The R value noticed is 0.97 which is a significant value. Therefore, in the presence of aug-cc-pVDZ basis set these are considered as less preferable classes for calculation of bond dissociation energy of C−X bond. Exceptional behavior is observed for HCTH of GGA class where deviations are up to 3 kcal/mol with lower error of 1.63 kcal/mol.

3.1.4.2 Evaluation of DFs with Aug-cc-pVTZ Basis Set

Statistical analyses of density functionals with aug-cc-pVTZ basis set are summarized in Table 3.4. The trend of results for aug-cc-pVTZ basis is almost similar to that observed for aug-cc-pVDZ basis set except improvement in dispersion corrected range separated hybrid meta-GGA class results.

RS H-GGA-D class has phenomenal performance with aug-cc-pVTZ basis set (improved performance) which is consistent with previous results of 6-31G(d), 6- 311G(d) and aug-cc-pVDZ basis sets. ωB97X-D functional shows less deviations; RMSD and SD are 3.13-3.56 kcal/mol. MAE is 1.87 kcal/mol with more significant R of 0.97. Whereas, LC-ωPBE-D3 has MAE of 1.48 kcal/mol with same R of 0.97 but RMSD and SD are 3.74 kcal/mol, 3.55 kcal/mol, respectively (more than ωB97X-D). Based on literature studies, LC-ωPBE-D3/aug-cc-pVTZ method is considered as the best for reaction barriers, kinetics and specially thermochemistry and long range charge transfer [231], and in current study the performance of LC-ωPBE-D3 is also better for carbon halogen BDE measurements. But again, the efficiency of LC-ωPBE-D3 is less compared to ωB97X-D, Hence, all the functionals performed very efficiently (Table 3.4) and signify the dispersion corrected range separated class as the best class (with aug-cc-pVTZ) among all the classes of DFT for C−X BDE measurement.

Dispersion corrected GGA class, B97-D3 performed best compared to B97-D functional of this class. RMSD and SD of B97-D3 and B97-D are in the range of 2.92- 3.8 kcal/mol. The calculated R is 0.97 and MAE for both functionals is 0.21 and 2.53 kcal/mol, respectively. Error in B97-D3 is less than 1 kcal/mol which confirms that B97-D3 functional with aug-cc-pVTZ basis set is the best for C−X BDE measurement. Similarly, B97-D performance is more improved in aug-cc-pVTZ basis set as compared to previous basis set results.

Classes of DFT DFs RMSD SD R MAE LDA LSDA 24.17 3.18 0.97 26.71

BLYP 6.17 6.21 0.92 1.38 G96LYP 6.62 6.4 0.92 2.31 HCTH 3.52 3.27 0.97 1.53 GGA MPWLYP1W 6.03 6.17 0.93 0.83 PBELYP1W 12.73 12.95 0.65 2.19 PBE 5.75 4.61 0.94 3.63 M06-L 10.82 10.19 0.89 4.44 휏-HCTH 13.94 12.81 0.85 6.35 meta-GGA TPSS 3.02 3.10 0.97 0.32 TPSSLYP1W 6.16 6.06 0.92 1.86 B3LYP 8.51 8.79 0.89 0.08 B3PW91 9.78 9.49 0.89 3.36 B98 6.88 5.89 0.93 3.85 B97-1 6.93 6.08 0.93 3.67 Hybrid GGA B97-2 6.40 6.08 0.93 2.51 MPW1K 6.84 7.04 0.92 0.61 MPW1PW91 13.98 12.95 0.71 6.17 MPW3LYP 7.32 6.07 0.93 4.37 X3LYP 12.75 12.99 0.85 2.1 BMK 7.6 6.43 0.92 4.36 B1B95 13.72 11.34 0.77 8.22 BB95 13.24 13.22 0.87 3.36 M05 12.05 10.07 0.87 7.09 GH meta-GGA M05-2X 7.47 6.14 0.93 4.52 MPW1KCIS 10.55 7.53 0.88 7.62 PBE1KCIS 6.02 5.79 0.93 2.18 TPSS1KCIS 13.72 11.34 0.77 8.22 B97-D 3.80 2.92 0.97 2.53 GGA-D B97-D3 2.98 3.07 0.97 0.21 RS hybrid GGA CAM-B3LYP 3.72 3.52 0.96 1.48 LC-ωPBE-D3 3.74 3.55 0.96 1.48 RS hybrid GGA-D ωB97X-D 3.56 3.13 0.97 1.87

BDEs of C−X data set are also simulated using CAM-B3LYP (range separated hybrid GGA class) with aug-cc-pVTZ. RMSD and SD are above 3 kcal/mol, R ~ 0.96 and has under bound error of 1.48 kcal/mol. On comparative analysis with Pople and Dunning (aug-cc-pVDZ) basis sets, here with Dunning basis set, the results are better and comfirm that CAM-B3LYP/aug-cc-pVTZ method can be suitable for BDE measurement of C−X bond. But less preferable compared to dispersion corrected range separated hybrid GGA class and range separated GGA class. From GGA class, only HCTH and PBE1W functionals give good results. Among these two functionals, HCTH functional has RMSD, SD, MAE and R values 3.52 kcal/mol, 3.27 kcal/mol, 0.97 and 1.53 kcal/mol, respectively which indicate the better performance of HCTH with aug- cc-pVTZ basis set similar to Pople basis sets and aug-cc-pVDZ basis set. On the other hand, PBE1W has moderate performance with aug-cc-pVTZ basis set analogues to Pople basis set (Table 3.4). More significant GGA performer (BLYP) with Pople basis sets, represents poor performance with aug-cc-pVTZ basis set. RMSD and SD ~ 6 kcal/mol, R of 0.92 and MAE is 1.38 kcal/mol, respectively. These results are consistent with aug-cc-pVDZ basis set but slightly improved functioning is observed in case of aug-cc-pVTZ basis set. All other functionals of this class show less efficient performance and based on their results, GGA class is considered as moderate performer for C−X BDE calculations.

110 y = a + b*x ) Equation Direct Weig Weight hting Residual 423.58006 100 Sum of Squares Pearson's r 0.97135 Adj. R-Squa 0.93948 Value Standard Err B Intercept -2.816 5.2402 90

B Slope 1.0225 0.06686 WB97X-D/6-311G(d) ( 80

60 Theoretical C-X BDE C-X Theoretical 50 60 70 80 90 100

Experimental C-X BDE (kcal/mol)

Fig. 3.5 Pearson’s Correlation (R) of ωB97X-D with 6-311G(d) Basis Set for BDE Calculations of C−X (X = Cl and Br) Bond

TPSS from meta-GGA class (moderate results with previous basis sets) becomes superior with aug-cc-pVTZ basis set. RMSD and SD of TPSS are ~ 3 kcal/mol, and correlation is 0.97 with less MAE (0.32 kcal/mol). The results confirmed that TPSS functional is more efficient for BDE calculations of C−X bond at aug-cc-pVTZ level. TPSSLYP1W performed better in Pople basis set but less significant with aug-cc-pVDZ basis set and same results are obtained with aug-cc-pVTZ basis set (with slight improvement). M06-L and τ-HCTH functionals have deviations (RMSD and SD) and MAE above 4 kcal/mol, whereas correlation is less than 0.89 which make them poor functionals for C−X bond dissociation energy calculations. In contrast to previous basis sets results, M06-L and 휏-HCTH functional performance is more discouraging here. Therefore, meta-GGA class is a slightly less significant class for BDE of C−X bond. Hybrid GGA, local density approximation and GH meta-GGA classes have poor

86 performance with aug-cc-pVTZ basis set and not suitable for BDE calculations of C−X bond.

Among all selected DFs with four basis sets, ωB97X-D/6-311G(d) method is observed as the best method for BDE measurement of C−X bond. The results of our study are superior, compared to the previous literature in a few ways. Most of the work in the literature is based on the hybrid GGA and GH-GGA classes i.e. Feng et al., used these DFs for BDE of C−X (X = halogens) bonds [219]. Whereas, dispersion corrected and long-range correction DFs were not considered in their study. The dispersion and range separated interactions are important in organohalides compounds. ωB97X-D which performed best in our study for these compounds which was not tested by Feng et al. Although, Sebastian and Martin describe the use of DFs i.e. M06-2X and ωB97X-D but on the other side they used expensive Dunning basis sets (aVTZ/aVQZ basis sets) and restricted their study to these basis sets. Pople 6-31G(d) basis set performed better in our study which has low cost as compared to Dunning basis sets Instead of using expensive basis sets, it is necessary to checkout the different basis sets and then, choose the one which is of low cost and more compatible with any DF for a particular system [29]. Moreover, a maximum of one or two statistical tools, either deviation, error or correlation are used in the literature to compare theoretical results with experimental data. For example, Wang et al., used SD for comparison of experimental and theoretical data [218]. Further, for the validation of theoretical results, a proper experimental data is necessary because scientists can never deny the natural truth of experimental studies. In previous literature, such as, Garcia et al., analyzed the BDE of C−X (X = halogens) bond in different heterocycles and compared the results of DFT functional (B3LYP) with high level G3B3 method [217]. They used theoretical data obtained from G3B3 as a benchmark due to the lack of their experimental data. In current thesis, we selected the molecules for which the experimental BDE values are already given in literature. In our study, a set of different C−X bonds are selected with a ladder of different DFs. For proper description of orbitals, Pople and Dunning basis sets are used with summarization of results using a series of statistical parameters. Thus, we provide a low cost and accurate methodology (ωB97X-D/6-311G(d)) for the dissociation energies measurement of C−X bond. These results fulfill all these deficiencies in literature with applicability for both usual and unusual C−X bonds. All the results are compared with already reported experimental data based on statistical analyses.

3.2 Benchmark Study for BDE of C−Sn Bond in Organotin Compounds

3.2.1 Transition Metal Catalyzed Cross Coupling Reactions

Discovery of transition metal catalyzed cross coupling reactions was a breakthrough in the field of organic chemistry. Over the years, several coupling reactions are developed [232]. The most famous examples of transition metal mediated cross coupling reactions are Suzuki [233], Sonogashira coupling [234], Heck [235], Stille [236], Hayama [237], Negishi [238] and Tsuji–Trost cross coupling reactions [239].

3.2.1.1 Applications

Metal-mediated cross-coupling reactions have applications in pharmaceutical industry [239], natural products synthesis, conjugated organic materials [150], solar cells, nanomaterials and ligand. Besides these practical applications, cross coupling reactions are in use for the synthesis of non-fluorescent fluorescein-derived sensors. The sensors are used for detection of Palladium/Platinum by monitoring fluorescence [240]. These coupling reactions were the subject of several experimental and theoretical mechanistic studies. In these reactions, C–M cleavage (transmetallation) is the key step.

3.2.1.2 Stille Cross Coupling Reaction

Stille coupling is more frequently used in organic synthesis due to ease of availability of reagents, stable towards air and moisture, functional groups tolerance and execution of the reaction under mild conditions. Stille coupling has been extensively used for the synthesis of substituted hydrocarbons, highly functionalized aromatic compounds, solid phase synthesis of biphenyls, carotenoid butanolides, triflones, alkaloids, pyrones and coumarins. Stille coupling, is the reaction of organotin compounds with a variety of electrophiles via palladium-catalyzed reaction. Rate determination step of Stille coupling is transmetallation, which involves the dissociation of C−Sn bond. For thermodynamic feasibility of this reaction, bond dissociation energy (BDE) of C−Sn bond of organotin compounds is much valuable phenomena [241].

3.2.1.3 Importance of Organotin Compounds

Apart from Stille coupling, organotin compounds are also used as catalysts for regioselective substitution in organic synthesis, stabilizers in poly vinyl chloride plastic, antifungal agents, neuroprotective and antitumor drugs. C−Sn bond plays a crucial role in thermolysis of organometallic compounds which proves the synthetic utility of these compounds. C−Sn bond is also involved in photoalkylation, photobenzylation and photoallylation of differently substituted organic compounds by using electron transfer photodissociation technique. C−Sn bonds of trimethyl tin (TMT) as well as halogenated organotin compounds have technological importance in the synthesis of SnO2 based thin films for gas sensing fabrication, solar cells, emissive glass and flat panel display [242].

3.2.1.4 Experimental Studies of C−Sn Bond

Combustion calorimetry, reaction-solution calorimetry, photo acoustic calorimetry, heat of formation, threshold photoelectron photoion coincidence (TPEPICO) spectrometry and kinetic studies are used to measure BDE of C−Sn bond. Xu et al., observed the high reactivity of Me3Sn of trimethylstannylated analogues over Me3Si of trimethylsilylated complexes in the formation of homogenous catalyst as metallocene complexes, where carbon-metal bond dissociation played an important role [243].

3.2.1.5 Theoretical Studies of C−Sn Bond

Beside experimental reports of BDE of C−Sn bond, theoretician also studied this bond with the help of different quantum chemical methods. Rosa et al., employed C−Sn bond of tetramethyltin (TMT) as photo precursor for the gas-phase detection of dimethylstannylene (heavy carbenes) through insertion reactions [244]. Matt et al., computationally evaluated the BDE of eighteen organotin compounds by using ab initio (CCSD(T), MP2) and DFT (B3LYP, M06-2X, ωB97X-D) methods with relativistic effect of tin core electrons and concluded that dissociation energy value depends on method and relativistic effect [245]. Grindley and his co-workers used DFT-B3LYP and MP2 methods with relativistic core potential basis sets (SDB-aug-cc-pVTZ and

LANL2DZdp) for the homolytic BDE calculation of trimethyltin species ((CH3)3Sn-X,

X = F, H, CH3, Cl, OH, CH2CH3, NH2 etc.). They compared their theoretical data with experimental results and observed that relativistic effective core potential (ECP) is a

89 convenient methodology for BDE calculations of C−Sn bond [246]. Accurate values of homolytic BDE are mandatory for a chemist who aims to understand and predict the products of free radical reactions as well as a standard for quantum chemical calculations.

Structurally diverse organotin compounds (17-26) are selected as a data set and are shown in Fig. 3.6. The optimized geometries of all compounds are in singlet spin state and their resultant radicals are in doublet spin states. Results and discussion portion include two sections, based on the basis sets used in this benchmark study. Inside each section, the influence of nineteen DFs from eight different classes of DFT is thoroughly discussed with two types of basis sets i.e. ECP and Karlsruhe basis sets [212]. Each section is further divided in to two subsections based on the selected basis set from these basis sets. LANL2DZ and SDD basis sets are selected from ECP basis sets and def2-SVP and def2-TZVP basis sets are selected from Karlsruhe basis sets. Theoretical BDE results are compared with already known experimental BDEs values (see in Fig. 3.6) [221, 244, 247]. The comparative graphical representation of RMSD, SD, R and MAE of four basis sets with nineteen DFs with experimental data is shown in Figs. 3.7- 3.10 and Tables 3.6-3.9.

Fig. 3.6 The Structures of Organotin Compounds with Known Experimental BDEs of C−Sn Bond

3.2.1.6 Homolytic Versus Heterolytic Cleavage

In the first step of the study, homolytic pathway is compared with the heterolytic pathway in order to realize the low energy dissociation pathway. The bond dissociation energies calculated at BLYP-D3/SDD (Table 3.5) clearly reflect that the homolytic path is much favorable over the heterolytic pathway. Moreover, the values calculated with radical pathway are in agreement with the experimental observations. Therefore, further benchmarking study is performed only with the radical pathway.

Table 3.5 Comparison of C−Sn Bond Dissociation Energy for Homolytic vs. Heterolytic Cleavage Calculated at BLYP-D3/SDD Method (All Values are Given in kcal/mol)

Compounds Radical Carbanion Carbocation Experimental value

(CH3)3SnCH3 63.58 230.23 267.19 64.00

(C2H5)3SnC2H5 57.51 221.33 220.16 56.93

(CH3)3SnBz 50.93 181.50 187.27 54.00

(CH3)3SnCH2CH3 58.54 227.46 223.43 60.00 i (CH3)3Sn Pr 54.69 228.27 199.24 55.40

(CH3)3SnPh 74.65 398.58 236.79 78.00 t (CH3)3Sn Bu 52.16 219.48 178.26 50.40

(Ph)3SnCCPh 96.87 165.02 252.94 102.10

(Ph)3SnCH=CH2 69.77 200.74 221.22 67.40

(Ph)3SnPh 75.10 192.74 214.99 65.50

3.2.2 Efficiency of Effective Core Potential (ECP) Basis Sets

Among ECP basis sets, two basis sets (LANL2DZ and SDD) are selected for this section.

3.2.2.1 Evaluation of DFs with LANL2DZ Basis Set

LANL2DZ is a combination of effective core potential (ECP) and a valence double- zeta basis set to describe the valence electrons. This is a shape-consistent basis set and derived from reference calculations on an isolated atom within relativistic Dirac-Fock theory which includes mass velocity and Darwin terms. The choice of the LANL2DZ pseudopotential basis set is governed by the limited availability of quality atomic orbital basis sets for tin [98]. First of all, performance of LANL2DZ basis set in combination with different DFs is examined for whole data set of organotin compounds. The numerical values of SD, RMSD, MAE and R are given in Table 3.6.

Table 3.6 SD, RMSD, R and MAE of C−Sn BDEs Calculated with Different DFs While Using LANL2DZ Basis Set (All Values are Given in kcal/mol, Except R Which is

Presented as Fraction of 1.0). For Differentiation of R Results, the Values are Given up to Three Decimal Points

Classes of DFT DFs SD RMSD R MAE B97-D 4.17 7.31 0.965 6.15 GGA-D B97-D3 4.20 7.19 0.961 5.99 BLYP-D3 4.05 5.59 0.964 4.06 meta-GGA M06-L 4.14 6.55 0.962 5.25 H-GGA-D B3LYP-D3BJ 4.20 9.13 0.962 8.21 M05 3.42 5.37 0.974 4.28 M05-2X 3.56 13.33 0.973 12.89 GH meta-GGA M06 3.53 10.77 0.973 10.23 M06-2X 3.28 15.38 0.976 15.06 M06-HF 3.84 14.22 0.969 13.75 BMK-D3BJ 4.24 17.11 0.960 16.63 M05-2X-D3 3.84 14.22 0.969 13.75 GH meta-GGA-D M06-2X-D3 3.51 15.86 0.973 15.51 M05-D3 3.71 6.94 0.970 5.98 M06-D3 3.75 11.87 0.969 11.32 DH-GGA-D B2PLYP-D3 3.47 8.63 0.978 7.98 RSH-GGA-D CAM-B3LYP-D3BJ 5.16 9.54 0.958 8.19 ωB97X-D 4.08 11.06 0.964 10.37 LC-GGA-D LC-ωPBE-D3BJ 4.29 11.09 0.966 10.32

Generalized gradient approximation (GGA) class of DFT depends on gradient of density. When Grimme dispersion is included in GGA, a new class generalized gradient approximation (GGA) with Grimme dispersion (GGA-D) is evolved. Three density functionals, B97-D, B97-D3 and BLYP-D3 are selected from GGA-D class. Among these three DFs, the performance of BLYP-D3 is good. For BLYP-D3, SD and RMSD values are 4.05 and 5.59 kcal/mol, respectively. The linear correlation (R) is 0.964 and error (MAE) is 4.06 kcal/mol. The other two functionals (B97-D and B97-D3) have almost average performance for BDE calculation of C−Sn bond. SD and RMSD values are in the range of 4.17-4.20 kcal/mol and 7.19-7.31 kcal/mol, respectively. R is in the range of 0.965-0.961, whereas MAE values are 6.15 kcal/mol and 5.99 kcal/mol,

93 respectively. The results show the overall good performance of GGA class with LANL2DZ basis set for C−Sn BDE calculation. M06-L functional of meta-GGA is also tested. Statistical analysis shows that the performance of M06-L is moderate. Deviations (SD and RMSD) and error (MAE) are in the range of 4-6 kcal/mol and R value is 0.962.

Def2-SVP LC-WPBE-D3BJ Def2-TZVP WB97X-D SDD CAM-B3LYP-D3BJ LANL2DZ B2PLYP-D3 M06-D3 M05-D3 M06-2X-D3 M05-2X-D3 BMK-D3BJ M06-HF M06-2X M06 M05-2X

Density Functionals M05 B3LYP-D3BJ M06-L BLYP-D3 B97-D3 B97-D

0 5 10 15 20 RMSD (kcal/mol)

Fig. 3.7 Root Mean Square Deviation (RMSD) of Different Density Functionals with Four Basis Sets for C−Sn BDEs

Double hybrid GGA combine exact HF exchange with an MP2-like correlation to a DFT calculation. Recently, this class is modified by inclusion of dispersion correction which is known as DH-GGA-D. One functional, B2PLYP-D3 is selected from this class. The SD (3.47 kcal/mol) and correlation (0.978) are good. However, RMSD (8.63 kcal/mol) and MAE (7.98 kcal/mol) are higher which illustrate the low performance of this functional towards BDE calculations of C−Sn bond. GH meta-GGA depends on meta-GGA with HF exchange. M05, M05-2X, M06, M06-2X and M06-HF functionals are chosen from this class for the current study. Minnesota functional M05 shows good

94 performance with less deviations and error with excellent correlation. The SD, RMSD, MAE and R are 3.42 kcal/mol, 5.37 kcal/mol, 4.28 kcal/mol and 0.974, respectively. Remaining functionals, M05-2X, M06, M06-2X and M06-HF have SD ⁓ 3 kcal/mol and R in the range of 0.973-0.976, respectively. Unfortunately, RMSD and MAE are very high i.e. 10 to 15 kcal/mol, respectively (Table 3.6). The statistical analyses reflect the poor performance of GH meta-GGA class for BDE calculations of C−Sn bond.

From GH meta-GGA-D class, M05-D3 functional of this class is found efficient for BDE calculation of C−Sn bond. SD, RMSD, R and MAE of M05-D3 are 3.71 kcal/mol, 6.94 kcal/mol, 0.97 and 5.98 kcal/mol, respectively. For the remaining functionals (M06- D3, M05-2X-D3, M06-2X-D3 and BMK-D3BJ), the SD and linear correlation are good but RMSD and MAE are high. For example, SD, RMSD, R and MAE of M06- D3 are 3.75 kcal/mol, 11.87 kcal/mol, 0.969 and 11.32 kcal/mol, respectively. The most popular Minnesota functional, M05-2X-D3 has SD, RMSD, R and MAE of 3.84 kcal/mol, 14.22 kcal/mol, 0.969 and 13.75 kcal/mol, respectively. Fourth selected functional (M06-2X-D3) has less standard deviation (3.51 kcal/mol) with good R (0.973). But root mean square deviation and error are high and their values are 15.86 kcal/mol (RMSD), and 15.51 kcal/mol (MAE), respectively. and BMK-D3BJ has SD, RMSD, R and MAE 4.24 kcal/mol, 17.11 kcal/mol, 0.96 and 16.63 kcal/mol, respectively. Overall, this class is less efficient for BDE calculations of C−Sn bond.

Hybrid GGA class with dispersion correction is modified to another class known as dispersion corrected hybrid GGA (H-GGA-D). SD, RMSD, R and MAE of selected B3LYP-DBJ functional of hybrid GGA-D class are 4.2 kcal/mol, 9.13 kcal/mol, 0.962 and 8.21 kcal/mol, respectively. Based on the statistical analyses of this functional, H- GGA-D class is poor performer for calculating BDE of C−Sn bond.

CAM-B3LYP-D3BJ functional is a range separated hybrid GGA with dispersion correction. It has very poor performance as compared to the functionals of other classes. This poor efficiency is evident from results where deviations and error are in the range of 5-10 kcal/mol and R is 0.958. Long range corrected GGA with dispersion correction class includes non-locality, dispersion and long-range correction. Two functionals ωB97X-D and LC-ωPBE-D3BJ are selected from LC-GGA-D class. Both of these functionals have almost same SD (up to 4 kcal/mol). The R values of ωB97X-D and LC-ωPBE-D3BJ are 0.964 and 0.966, respectively. However, RMSD and MAE are

10.32 kcal/mol to 11.09 kcal/mol. Hence, the results indicate the worse performance of LC-GGA-D class among all DFT classes.

LanL2DZ 10 SDD Def2-SVP Def2-TZVP

8 )

kcal mol (

4 SD

M05

M06

B97-D

M06-L

B97-D3

M05-2X

M06-2X

M05-D3

M06-D3

M06-HF

BLYP-D3

WB97X-D

M05-2X-D3

M06-2X-D3

BMK-D3BJ

B2PLYP-D3

B3LYP-D3BJ LC-WPBE-D3BJ

Density Functionals CAM-B3LYP-D3BJ

Fig. 3.8 Standard Deviation (SD) of Different Density Functionals with Four Basis Sets for C−Sn BDEs

Overall, results of different DFT classes with LANL2DZ basis set show that GGA-D class has good performance where particularly, BLYP-D3 functional is more efficient for BDE calculation of C−Sn bond.

3.2.2.2 Evaluation of DFs with Stuttgart–Dresden (SDD) Basis Set

Stuttgart-Dresden (SDD) basis set describes the inner core electrons with a pseudopot -ential and the valence electrons with triple zeta valence basis set [170]. This energy- consistent ECP basis set is constructed to reproduce experimental observables of a sin -gle atom like ionization potentials and excitation energies by relativistic Dirac-Fock

96 theory. The numerical values of SD, RMSD, MAE and R of selected DFs with SDD basis sets are given in Table 3.7.

Classes of DFT DFs SD RMSD R MAE B97-D 4.24 4.52 0.961 2.06 GGA-D B97-D3 4.47 4.73 0.956 2.08 BLYP-D3 4.11 3.90 0.963 0.01 meta-GGA M06-L 8.28 9.50 0.936 5.35 H-GGA-D B3LYP-D3BJ 4.82 7.18 0.950 5.54 M05 3.52 3.82 0.973 1.86 M05-2X 3.76 10.34 0.969 9.70 GH meta-GGA M06 3.60 7.49 0.973 6.67 M06-2X 4.13 13.01 0.962 12.41 M06-HF 4.53 20.02 0.955 19.56 BMK-D3BJ 10.01 14.48 0.792 10.93 M05-2X-D3 4.28 11.99 0.959 11.28 GH meta-GGA-D M06-2X-D3 4.34 13.58 0.958 12.94 M05-D3 4.44 5.92 0.957 4.16 M06-D3 5.57 10.14 0.930 8.66 DH GGA-D B2PLYP-D3 4.36 6.97 0.963 5.61 CAM-B3LYP- RS H-GGA-D 5.11 8.69 0.949 7.21 D3BJ ωB97X-D 5.40 8.22 0.936 6.43 LC-GGA-D LC-ωPBE-D3BJ 4.82 9.98 0.952 8.87

LANL2DZ SDD 5 Def2-SVP Def2-TZVP

0 )

-5

kcal mol (

-10 MAE

-15

-20

M05

M06

B97-D

M06-L

B97-D3

M05-2X

M06-2X

M05-D3

M06-D3

M06-HF

BLYP-D3

WB97X-D

M05-2X-D3

M06-2X-D3

BMK-D3BJ

B2PLYP-D3 B3LYP-D3BJ

Density Functionals LC-WPBE-D3BJ

CAM-B3LYP-D3BJ

Fig. 3.9 Mean Absolute Error (MAE) of Different Density Functionals with Four Basis Sets for C−Sn BDEs

Overall, the performance of GGA-D class is better with SDD basis set. The deviations and errors are less, and linear correlation between experimental and theoret i-cal BDE values of C−Sn bond is significant. Among three functionals (BLYP-D3, B 97-D and B97-D3) of GGA-D class, the efficiency of BLYP-D3 is excellent for C−Sn bond BDE calculation. The SD, RMSD, R and MAE of BLYP-D3 are 4.11 kcal/mol, 3.9 kcal/mol, 0.963 and 0.01 kcal/mol, respectively. The correlation of experimental a -nd theoretical data of BLYP D3 is shown in Fig. 3.10. In comparison to LANL2DZ b asis set, BLYP-D3 shows outstanding performanc-e with SDD basis set. The results of B97-D functional at SDD are comparable to the results of LANL2DZ basis set. B97- D3 has almost same SD and R as observed in case of LANL2DZ basis set but RMSD and MAE are significantly reduced. The SD, RMSD, R and MAE are 4.47 kcal/mol, 4.73 kcal/mol, 0.956 and 2.08 kcal/mol, respectively. Minimization of RMSD and

MAE indicates that GGA-D class with SDD basis set is a better choice for BDE calculations of C−Sn bond compared to LANL2DZ basis set.

120 Equation y = a + b*x Weight Instrumental Residual Sum 8.8923 of Squares 4 Pearson's r 0.9630 100 Adj. R-Square 0.9183 Value Standard Er B Interce -2.083 6.81081 B Slope 1.0317 0.10204

60 Theoretical C-Sn BDE (BLYP-D3/SDD) BDE C-Sn Theoretical

40 50 60 70 80 90 100

Experimental C-Sn BDE (kcal/mol)

Fig. 3.10 Pearson’s Correlation (R) of BLYP-D3 with SDD Basis Set for BDE Calculations of C−Sn Bond

The trend observed for BDE measurement of C−Sn bond with selected functionals of GH meta-GGA class is similar to that observed with LANL2DZ basis set. Among five selected functionals, M05 shows better performance, having less SD (3.52 kcal/mol) and a significant R (0.973). RMSD (3.82 kcal/mol) and MAE (1.86 kcal/mol) are further decreased compared to LANL2DZ basis set. M06 with SDD basis set has a moderate performance for BDE calculation of C−Sn bond. RMSD (7.49 kcal/mol) and MAE (6.67 kcal/mol) are high despite good SD (3.6 kcal/mol) and efficient R (0.973). RMSD and MAE are reduced ⁓ 3 kcal/mol for M05-2X and M06-2X (modified forms of M05 and M06) as compared to LANL2DZ basis set. In case of M06-2X functional, SD increases 1 kcal/mol and less significant correlation is observed with SDD basis set. On the other side M06-HF is the poorest functional for BDE calculation of C−Sn bond

99 in this class. SD and R of M06-HF are 4.53 kcal/mol and 0.955 respectively, whereas RMSD and MAE are up to 20 kcal/mol. Overall results reflect that the GH meta-GGA class has an average performance for BDE of C−Sn bond.

SD, RMSD, R and MAE of M05-D3 functional from GH meta-GGA-D class are 4.44 kcal/mol, 5.92 kcal/mol, 0.957 and 4.16 kcal/mol, respectively. This functional is moderately good for BDE calculations. However, the remaining functionals of this class have poor performance. SD is 5.57 kcal/mol and R is 0.93 for M06-D3 functional, which shows less accuracy compared to that with LANL2DZ. On the other hand, RMSD and MAE decrease to 10.14 kcal/mol and 8.66 kcal/mol, respectively. Results of M05-2X-D3 are almost similar to M06-2X-D3 (see Table 3.7). The efficiency of Becke functional with Grimme dispersion (BMK-D3BJ) is poor among all functionals of this class. Deviations and error are high with lower R and these analyses show the similar trend of this class as is observed in case of GH meta-GGA class.

Dispersion corrected double hybrid functional, B2PLYP-D3 shows a similar trend of results as observed for GH meta-GGA-D class. Compared to LANL2DZ basis set, in case of SDD basis set B2PLYP-D3 functional has RMSD and MAE reduces up to 2 kcal/mol but the SD increases up to 1 kcal/mol, respectively. Large deviations, error and poor correlation with experimental data signifies the moderate performance of this class for BDE calculations of C−Sn bond. Although on the basis of comparative analyses to the LANL2DZ basis set, still its deviations and error are reduced but the performance of double hybrid GGA is poor compared to the above DFT classes. Therefore, as a whole, it can be considered as a less efficient class for homolytic cleavage of C−Sn bond.

CAM-B3LYP-D3BJ functional has high SD (5.11 kcal/mol), RMSD (8.69 kcal/mol), MAE (7.21 kcal/mol) and less significant R (0.949). The statistical analyses reflect the poor performance of CAM-B3LYP-D3BJ for homolytic cleavage of C−Sn bond. On comparison to LANL2DZ basis set RMSD and MAE are decreased by 1 kcal/mol besides almost same SD and less significant R. With ECP basis sets (SDD and LANL2DZ), the performance of M06-L of meta-GGA class is very unsatisfactory against BDE calculation of C−Sn bond compared to the functionals of other classes. In this case, SD and RMSD are high, MAE further increases up to 0.10 kcal/mol and R decreases to 0.936. ωB97X-D and LC-ωPBE-D3BJ functionals (LC-GGA-D class)

100 have almost similar performance, just a difference of 1 kcal/mol in deviations and 2 kcal/mol in MAE is observed between both functionals. The R in both cases is less significant (see Table 3.7). In comparison to LANL2DZ basis set, RMSD and MAE are minimized but SD is high with less significant R. So, this class has the poorest performance for BDE calculation of C−Sn bond among all selected DFT classes with SDD basis set. GGA-D class with SDD basis set has more efficient performance among all selected DFT classes and specifically, BLYP-D3 functional shows outstanding performance for BDE calculation of C−Sn bond. BLYP-D3 functional performs better (already discussed in computational methodology section). In comparison to other DFs of all DFT classes, its performance is good due to its low deviations and error with significant R. Among different DFs, BLYP-D3 has a larger dispersion correction due to smaller scaling factor (Sr,6) value which strongly depends on the gradient- enhancement factor in low-density regions. May be due to this reason, BLYP-D3 has also outstanding results in the current study [153].

3.2.3 Efficiency of Karlsruhe Basis Sets

In search of an appropriate basis set for homolytic cleavage (BDE) of C−Sn bond, two basis sets (i) def2-SVP (second-generation default with split valence polarization) (ii) def2-TZVP (second generation default with valence triple-zeta polarization) are selected from Karlsruhe basis sets [168].

3.2.3.1 Evaluation of DFs with Def2-SVP Basis Set

The accuracy of selected nineteen DFs for BDE calculation of C−Sn bond in organotin compounds is also tested with def2-SVP basis set [174]. The numerical values of SD, RMSD, MAE and R are given in Table 3.8.

Classes of DFT DFs SD RMSD R MAE B97-D 3.96 4.25 0.980 1.98 GGA-D B97-D3 3.86 3.99 0.970 1.59

101

BLYP-D3 4.04 4.83 0.969 2.94 meta-GGA M06-L 5.42 6.13 0.935 3.34 H-GGA-D B3LYP-D3BJ 3.99 4.23 0.965 1.87 M05 3.06 3.96 0.981 2.70 M05-2X 3.47 6.69 0.974 5.82 GH meta-GGA M06 3.60 6.27 0.978 5.26 M06-2X 3.41 4.76 0.975 3.49 M06-HF 4.29 5.62 0.959 3.87 BMK-D3BJ 4.16 11.90 0.963 11.23 M05-2X-D3 2.24 5.96 0.990 5.57 GH meta-GGA- M06-2X-D3 3.34 5.03 0.975 3.91 D M05-D3 5.91 8.28 0.923 6.10 M06-D3 3.57 5.66 0.979 4.54 DH GGA-D B2PLYP-D3 7.62 9.28 0.971 5.82 RS H-GGA-D CAM-B3LYP-D3BJ 3.89 5.51 0.967 4.09 ωB97X-D 3.90 7.94 0.967 7.02 LC-GGA-D LC-ωPBE-D3BJ 3.89 8.13 0.968 7.25

Interestingly the selected functionals (B97-D3, B97-D and BLYP-D3) of GGA-D class with def2-SVP basis set have better performance having less deviations and error from the experimental data (significant correlation). SD and RMSD of B97-D3 are 3.86 kcal/mol and 3.99 kcal/mol, respectively. The significant R of this functional (B97-D3) whereas MAE is 1.59 kcal/mol. B97-D is at second position in its performance for BDE calculation of C−Sn bond where deviations and error are in the range of 1 kcal/mol to 4 kcal/mol with a more significant R of 0.98. The most efficient BLYP-D3 with SDD basis set is less efficient here (more error and deviations). However, the deviations and errors are reduced compared to LANL2DZ basis set (Table 3.8). B3LYP-D3BJ functional (H-GGA-D class) has good results with def2-SVP basis set which was less efficient with LANL2DZ and SDD basis sets. The SD is reduced up to 1 kcal/mol and RMSD is minimized to a factor of 2 kcal/mol with a lower MAE (1.87 kcal/mol). R is more significant compared to LANL2DZ and SDD basis sets. These results reflect the good efficiency of this class with def2-SVP basis set for BDE calculations for C−Sn bond.

102

M05 functional of GH meta-GGA class maintained its efficiency among all selected functionals of this class. The range of SD, RMSD and MAE of M05 with def2-SVP basis set is between 3 kcal/mol to 4 kcal/mol with a significant R value (0.981). The statistical analyses results, here are better compared to the LANL2DZ basis set. On the other hand, SD and R are better than SDD basis set but RMSD and MAE are high which make it less efficient compared to SDD basis set. The performance of remaining selected functionals (M05-2X, M06, M06-2X and M06-HF) with def2-SVP basis set is lower but in comparison to LANL2DZ and SDD basis sets their statistical (SD, RMSD, R, MAE) values are better. SD is in the range of 3.41-4.29 kcal/mol and RMSD is in the range of 4.76 to 6.69 kcal/mol, respectively. R values range from 0.959 to 0.978 and MAE is in the range of 3.49 to 5.82 kcal/mol, respectively. The overall data reflect the moderate performance of GH meta-GGA class for desired homolytic cleavage (BDE) of C−Sn bond. The M06-L functional of meta-GGA class shows average performance for desired data set with def2-SVP basis set. Compared to LANL2DZ and SDD basis sets, deviations and error are improved (except SD of LANL2DZ basis set) but R between experimental and theoretical data is less significant.

Unexpectedly, M05-2X-D3 functional with def2-SVP basis set appears better than the selected functionals of GMH-GGA-D class. In this case, SD is 2.24 kcal/mol with the most significant R (0.99). The RMSD and MAE values are high, but when compared to LANL2DZ and SDD basis sets, these values are quite small. M06-D3 and M06-2X- D3 functionals have similar results to each other. SD, RMSD, R and MAE are ~ 3 kcal/mol, ~ 5 kcal/mol, ~ 0.97, ~ 4 kcal/mol, respectively. These statistical values (deviations and errors) are further reduced with a better R compared to LANL2DZ and SDD basis sets which express their good efficiency for desired property measurement. Previously M05-D3 is described as a better performer with LANL2DZ and SDD basis sets but here, its performance is suppressed with def2-SVP basis set due to more deviations (SD = 5.91 kcal/mol, RMSD = 8.28 kcal/mol) and errors (MAE = 6.1 kcal/mol) with a very poor R (0.923). Again, BMK-D3BJ is inferior in its performance in comparison to functionals of its own class and also follows the same trend of results as was observed in previous basis sets. As a whole, all results indicate the less efficiency of this class for BDE calculation of C−Sn bond of organotin compounds. For CAM- B3LYP-D3BJ, the deviations are decreased and appear in the range of 2-4 kcal/mol. The error is reduced up to 4 kcal/mol and the calculated R value is 0.967 as compared

103 to LANL2DZ and SDD basis sets. But in competition with other DFs of different DFT classes at def2-SVP level, the efficiency of CAM-B3LYP-D3BJ is poor which demonstrates this class is less capable for desired property measurement. In comparison to LANL2DZ and SDD basis sets, performance of B2PLYP-D3 (DH-GGA-D class) for calculation of BDE of C−Sn bond is poor with def2-SVP basis set. Both long range corrected functionals (ωB97X-D and LC-ωPBE-D3BJ) from LRC-GGA-D class have less degree of deviations, error and significant R compared to LANL2DZ and SDD basis sets. But these functionals have more deviations (SD and RMSD), MAE and less significant correlation (R) compared to all selected DFs at def2-SVP basis set (Table 3.8). So, these two DFs are poorest among all DFs of different selected DFT classes with def2-SVP basis set and this class is designated as the poorest for measurement of C−Sn bond dissociation energies. Among all DFT classes, dispersion corrected GGA class performs best with def2-SVP basis set and B97-D3 functional performance is better among three selected functionals (B97-D, B97-D3 and BLYP-D3) of this class.

3.2.3.2 Evaluation of DFs with Def2-TZVP Basis Set def2-TZVP is a combination of second-generation default of basis set and TZVP denotes valence triple zeta polarization [174]. The numerical values of SD, RMSD, MAE and R of all DFs with def2-TZVP are given in Table 3.9.

Classes of DFT DFs SD RMSD R MAE B97-D 4.44 4.60 0.969 1.87 GGA-D B97-D3 4.27 4.52 0.966 2.01 BLYP-D3 4.09 5.46 0.969 3.84 meta-GGA M06-L 3.42 3.28 0.981 0.49 H-GGA-D B3LYP-D3BJ 4.04 4.03 0.965 1.26 M05 3.68 4.04 0.972 2.05 GH meta-GGA M05-2X 3.57 6.88 0.973 5.99 M06 3.75 5.51 0.976 4.21

104

M06-2X 3.87 5.16 0.967 3.63 M06-HF 4.12 5.05 0.963 3.20 BMK-D3BJ 3.98 11.86 0.965 11.24 M05-2X-D3 3.62 7.57 0.972 6.74 GH meta-GGA-D M06-2X-D3 3.26 4.62 0.977 3.43 M05-D3 3.82 5.30 0.970 3.86 M06-D3 4.08 11.98 0.920 11.53 DH GGA-D B2PLYP-D3 4.41 5.57 0.976 3.68 RS H-GGA-D CAM-B3LYP-D3BJ 3.80 4.94 0.968 3.38 ωB97X-D 4.25 7.00 0.965 5.73 LC GGA-D LC-ωPBE-D3BJ 3.72 7.63 0.970 6.76

Statistical analyses of meta-GGA functional M06-L with def2-TZVP basis set indicate that this functional has good performance and shows less deviations (SD is 3.42 kcal/mol and RMSD is 3.28 kcal/mol), error (MAE is 0.49 kcal/mol) and significant R (0.981). Compared to the other basis sets studied (LANL2DZ, SDD and def2-SVP), the performance of M06-L/def2-TZVP is better for the BDE calculation of C−Sn bond. Among three functionals of GGA-D class, B97-D here gives highest accuracy having least deviations and error with a significant correlation. The SD, RMSD, R and MAE are 4.44 kcal/mol, 4.6 kcal/mol, 0.969 and MAE 1.87 kcal/mol, respectively. Similarly, the results of B97-D3 and BLYP-D3 are also good. The performance of B97-D with def2-TZVP basis set is enhanced while in case of def2-SVP basis set, its efficiency is low. In comparison to pseudo potential basis sets, B97-D with def2-TZVP basis set has more SD but RMSD MAE are reduced with more significant R. B97-D3 and BLYP- D3 have SD in the range of 4.09-4.27 kcal/mol and RMSD is in the range of 4.52 to 5.46 kcal/mol, respectively. R values of both functionals are 0.966 and 0.969, respectively with MAE in the range of 2.01-3.84 kcal/mol, respectively. The efficiency of B97-D3 functional is lower in the presence of def2-TZVP basis set compared to the def2-SVP basis set. B97-D3 with def2-TZVP basis set has more SD but RMSD and MAE are reduced (except RMSD with SDD basis set) with less significant R compared to pseudo potential basis sets. Interestingly, the performance of BLYP-D3 with def2- TZVP basis set is consistent with the results of def2-SVP basis set. This functional has almost similar SD but the RMSD and MAE are lower with both pseudo potential basis

105 sets (def2-SVP and def2-TZVP) than LANL2DZ basis set. On the other side compared to SDD basis set, here RMSD and MAE are high with a significant R value. These results reflect the good performance of GGA-D class with def2-TZVP basis set for BDE calculation of C−Sn bond.

B3LYP-D3BJ functional of H-GGA-D class exhibits average performance, for homolytic cleavage of C−Sn bond. The deviations of this functional (B3LYP-D3BJ) are up to 4 kcal/mol. While, the R and MAE values are 0.965 and 1.26 kcal/mol, respectively. These results of B3LYP-D3BJ functional at def2-TZVP basis set are almost similar to the def2-SVP basis set except lower MAE. Despite the consistency of SD and R, RMSD and MAE are significantly reduced in this case, compared to LANL2DZ and SDD basis sets. In case of CAM-B3LYP-D3BJ functional of RS H- GGA-D class has deviations and error up to 4 kcal/mol with a good R (0.968). This illustrates the moderate capability of this class for the BDE calculation of C−Sn bond. Compared to the previous three basis sets (LANL2DZ, SDD and def2-SVP), the efficiency of this class with def2-TZVP basis set is enhanced which makes it moderate performer of BDE calculation of C−Sn bond.

B2PLYP-D3 functional has the same computational cost as for MP2 level of theory. The high computational cost is due to the inclusion of dispersion corrected MP2 with exact HF exchange. So, during the comparison of B2PLYP-D3 with this higher def2- SVP basis set, we calculated only single point energy for aryl stannous compounds (8, 9 and 10 see in Fig. 3.7). This functional also has moderate efficiency for homolytic cleavage of C−Sn bond. The SD, RMSD, MAE and R are 4.41 kcal/mol, 5.57 kcal/mol, 3.68 kcal/mol and 0.976, respectively. The deviations and error of B2PLYP-D3 with def2-TZVP basis set are lower, with more significant correlation compared to the earlier basis sets (LANL2DZ, SDD and def2-SVP). Hence, these results classify DH-GGA-D class as an ordinary performer for desired data. Five functionals of GH meta-GGA class are examined with def2-TZVP basis set. The overall performance of these functionals is proved less efficient for BDE calculation of C−Sn bond as compared to above DFT classes. Collectively, SD RMSD and MAE of all these functionals lies in between 2.05 to 6.88 kcal/mol. However, R value are in the range of 0.963-0.976. The results of def2-TZVP basis set are in consistent with results of def2-SVP basis sets. The performance of these functionals (M05-2X, M06, M06-2X and M06-HF) in case of

106

LANL2DZ and SDD basis sets was moderate, but here with def2-TZVP basis set is improved for BDE calculation of C−Sn bond. LC-GGA-D class functionals i.e. ωB97X-D and LC-ωPBE-D3BJ show poor performance for homolytic cleavage of desired C−Sn bond with def2-TZVP basis set. Although, SD and R are good but their RMSD and MAE are high (Table 3.9). However, in this class with def2-TZVP basis set, deviations (RMSD, SD and MAE) are further reduced and the R becomes efficient compared to LANL2DZ and SDD basis sets. On the other side, results of this class with def2-TZVP basis set are almost persistent to def2-SVP basis set, only MAE is further decreased with the former basis set. On the basis of all these results, this class is categorized as a poor performer for C−Sn BDE. Dispersion corrected Becke and Minnesota functionals of GH meta-GGA-D class show poorest performance for the BDE calculation of C−Sn bond. Their SD is in the range of 3.26 to 4.08 kcal/mol and RMSD in the range 4.62 to 11.98 kcal/mol. R is in the range of 0.920 to 0.977 with MAE up to 11.53 kcal/mol. M05-D3, M05-2X-D3 and M06-2X-D3 functionals with def2-TZVP basis set show enhancement in their results having lower deviations and error with a significant correlation in comparison to pseudopotential basis sets (LANL2DZ and SDD). A regular consistency is observed in the results of BMK-D3BJ functional in comparison to the results of this functional with def2-SVP basis set. M06- D3 show worse performance compared to previous basis set due to high deviations, error and lower R with experimental data. Remaining functionals of this class becomes less efficient at def2-TZVP basis set in comparison to their performance with def2-SVP basis set. Consequently, amongst all DFT classes, H-GGA-D and secondly GGA-D with def2-TZVP basis set show better performance for BDE calculation of C−Sn bond. Unfortunately, the results of def2-basis sets are not satisfactory. The reason maybe is that these do not allow for scalar relativistic effective core potentials in the fourth period of the periodic table (especially the 3d and 4p block elements) where relativistic effects are chemically important. These basis sets do include a relativistic effective core potential starting with Rb, at start of the ﬁfth row [173].

It is observed that BLYP-D3/SDD level of theory shows outstanding performance in reproducing BDE of C−Sn bond. If the results of C−Sn bond dissociation energies work are compared with the previous literature, it is observed that our study is vast and more attractive for organic and in-organic chemists. The main objective of our study included the C−Sn bond of different organotin compounds which was missing in literature. The

107 previous studies on C−Sn bonds are related to only tetramethyl tin bonds [244, 245].

Beside C−Sn (of CH4Sn) bond, Grindley and his co-workers examined Sn−X (X = Cl, Br, F, OH, S, O, and N etc.) dissociation energies are studied in other reported compounds [246]. Since the studied compounds belong to different classes therefore a direct comparison of the outcomes is not rationalized. The computational details of these reports also reflect that a limited number of DFs (up to three i.e. B3LYP, M06- 2X, ωB97X-D) are examined. We tried to study a number of DFs from different DFT classes for a variety of organotin compounds having C−Sn bond, including tetramethyl tin compounds. Pople basis sets failed for simulation of tin atom (atomic number of Sn is 50). Most of the literature reports are based on introducing different types of relativistic core potential basis sets for tin. For example, Matt et al., used MP2 and DFs i.e. B3LYP, M06-2X, ωB97X-D with various basis sets including DKH/TZP-DKH, def2-SVPD+6-311G(2d,p) (the former basis set is used for tin atoms while the latter basis set is selected to represent atoms other than Sn), SDB-aug-cc-pVTZ and ATZP methods for BDEs measurements of C−Sn bond. Expensive MP2 method with high level ECP basis sets are used in their report. Instead of these basis sets, we used low level ECP basis sets and obtained comparable results to the experimental data. Previous search is also based on models of tin compounds for which the experimental BDE is not known and CCSD(T) methods is used for calibration as reported by Russell and his co-workers [245]. The CCSD(T) method is although gold standard method but the drawback of this method is that, it is applied to a compound having 15-20 atoms. A best method is developed for dissociation energies measurement of C−Sn bond in current study which is important step in Stille reaction. These results will be more helpful for synthetic and material chemists working on C−Sn bond as compared to previous reports.

3.3 Benchmark Study for BDE of C−CN Bond in Nitrile Compounds

Nitrile is an important chemical entity having sp-hybridized nitrogen atom and has widespread applications in chemistry, biology and environmental science. Nitriles are important building blocks for various organic compounds such as pyridine, indophenol, tetrazoles, optically active glutarimides and C-nucleoside antibiotics (pyrazomycin, formycin, and showdomycin). In abiotic chemistry, nitriles have a vital role in the formation of biogenic amino acids and N-heterocycles, including purines (adenine,

108 guanine) and pyrimidines (cytosine and uracil). Specially, acetonitrile acts as an excellent precursor for the synthesis of nitrile derived amino acids. Nitrile containing compounds are also beneficial as bridge linkers in transition metal complexes and in native vitamin B12 [248].

Nitrile, as a substituent group, has a significant role in deciding the regioselectivity in many organic transformations. Adrian and his co-workers examined the removal of halogen group (for minimizing the environmental pollution) in 2-chloro-6- fluorobenzonitrile. They suggested that electron withdrawing nitrile-group had assisted in easy removal of chloro- group with high regioselectivity compared to the flouro- group [249]. Nitriles in combination with rare gases i.e. He, Ne, Ar are important species for the collision-induced electron energy transfer process in which production of cyano-radical facilitates the resonance between ground state and first two excited states [250]. Nitrile stretch of nitriles analogues are used as important IR probe because of the small size of nitrile compounds in comparison to typical fluorophores under investigation. The second reason is their little perturbation to the protein structure. Moreover, frequency shift of the nitrile stretch shows sensitivity towards local electrostatic environment and used as a potential candidate for antenna system or electric ﬁeld meter. [251]. Nitriles are also used as ligands in several bimetallic complexes having magnetic properties. Popular chromium based Prussian Blue

(CnAp[Cr(CN)6]q. xH2O) analogues are used as building blocks for the fabrication of three-dimensional molecular magnets (with high and tunable Curie temperature) [252]. Kom used nitriles for the detection of ammonia by using the Berthelot reaction [253]. Acetonitrile is an important intermediate in combustion processes and flame applications [249]. Acrylonitrile, an important raw material in chemical industry, is used for the production of plastics, rubbers, barrier resins etc. [250].

3.3.1 Nitriles as Hazards

Another aspect of nitriles is their poisonous nature in biology and environment. Nitrile compounds are volatile in nature and cause the formation of poisonous gases, which ultimately result in environmental pollution. The wastes containing nitrile compounds from different industries (gold mining, electroplating, and metallurgy) have disastrous consequences on the environment [254]. The nucleophilic cyano-group is harmful to human and environment. Although a number of strategies are reported and

109 implemented for minimizing their toxicity, still this drawback restricts its usage for practical applications. On the other hand, CN radical is non-toxic and generally produced by homolytic cleavage of C−CN bond [254]. Therefore, bond dissociation energy is a decisive factor to understand the reactivity of organic nitrile, C−CN. Despite the importance of this bond, experimental and theoretical reports on its dissociation are limited.

3.3.2 Experimental Study

Nibbering and co-workers used Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry for the determination of heat of formation of formyl and thioformyl nitriles from methoxyacetonitrile. During these chain reactions, CH3 and CN radicals are formed as intermediates, which are detected by using double-focusing quadrupole hybrid mass spectrometer [255]. Zou et al., studied photo induced reaction of nitriles with transition metal nitrates and observed that radical based cyanation could be a green method for preparation of metal cyanides [186].

3.3.3 Theoretical Study

DFT has been widely used for studying the homolytic bond dissociation of chemically important bonds. DFT methods provide an accurate description of BDE with high efficiency and less computational cost. Here, we intend to study the homolytic cleavage (BDE) of C−CN bond by adopting benchmark approach. To obtain an accurate density functional for C−CN bond cleavage, the results obtained from different functionals with an array of basis set are compared with the experimental values. The effect of electron- withdrawing groups (DWGs) and electron donating groups (EDGs) on BDE of C−CN bond has also been explored. NBO analysis and orbital properties have been calculated to support the results. This benchmark study will be helpful for experimentalists in understanding the reactivity of C−CN bond.

DFT methods are the best choice for BDE measurements of small to large molecules due to high accuracy and low computational cost, when compared with high-level ab initio methods [256]. In this benchmark study, we selected twelve structurally diverse nitrile compounds (27-38) whose experimental BDEs are reported in the literature [221, 222, 226, 257]. The optimized geometries of all compounds are in singlet spin state and

110 their resultant radicals are in doublet spin states. The representative structures of nitriles with experimental BDE values are given in Fig. 3.11.

Fig. 3.11 The Structures of Organo-Nitrile Compounds with Known Experimental BDEs of C−CN Bond

3.3.4 Efficiency of Pople Basis Sets

Five Pople basis sets i.e. 6-31G(d), 6-31G(d,p), 6-311G(d,p), 6-31+G(d) and 6- 311++G(d) are selected for BDE measurement of C−CN bond. All the Pople basis sets showed almost similar results. However, 6-311G(d,p) basis set provided the best results with all selected DFs in reproducing accurate results. The performance of DFs with 6- 311G(d,p) basis set is discussed in details as:

3.3.4.1 Evaluation of DFs with 6-311G(d,p) Basis Set

From the range separated hybrid GGA class, CAM-B3LYP functional with 6- 311G(d,p) basis set shows excellent role in the measurement of BDE of C−CN bond with high accuracy. The MAE value is the smallest (0.06 kcal/mol). The values of SD, RMSD and R are 2.79 kcal/mol, 2.67 kcal/mol and 0.96, respectively. The Pearson’s correlation between theoretical values calculated at CAM-B3LYP with experimental values of twelve nitriles is shown in Fig. 3.15 (graphical representation of SD, RMSD

111 and MAE is given in Figs. 3.12-3.14, respectively). Similar to the previous results in the literature [257], in the current benchmark study, CAM-B3LYP is proved as the best DF for BDE measurement of C−CN bond.

Table 3.10 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-311G(d,p) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 4.27 16.32 0.93 15.80 BP86 3.42 7.01 0.96 6.20 BLYP 2.70 9.52 0.98 9.16 G96LYP 3.38 12.08 0.95 11.64 GGA HCTH 6.06 9.10 0.87 7.01 MPWLYP1W 3.70 7.50 0.95 6.62 PBE 2.70 3.75 0.97 2.72 휏-HCTH 2.74 3.62 0.97 2.49 meta-GGA TPSS 3.94 8.05 0.94 7.11 TPSSLYP1W 3.60 9.64 0.95 9.00 B1LYP 2.81 6.01 0.97 5.37 BH and HLYP 2.86 3.32 0.98 1.87 B3LYP 3.12 5.10 0.97 4.14 B3PW91 2.48 3.60 0.97 2.70 B98 2.69 3.08 0.97 1.69 Hybrid GGA B97-1 2.71 2.69 0.97 0.72 B97-2 2.86 2.95 0.97 1.11 MPW1K 2.85 5.86 0.97 5.18 MPW1PW91 2.78 3.02 0.97 1.42 MPW3LYP 4.90 8.72 0.90 7.35 X3LYP 2.85 4.41 0.97 3.47 BMK 2.68 4.36 0.97 3.53 B1B95 3.45 3.45 0.96 1.00 BB95 3.50 5.41 0.96 4.25 GH meta-GGA MPW1B95 3.87 6.21 0.96 4.98 MPW1KCIS 3.97 3.87 0.92 0.71 PBE1KCIS 3.56 4.84 0.95 3.44

112

TPSS1KCIS 3.78 6.26 0.95 5.11 RS H-GGA CAM-B3LYP 2.79 2.67 0.96 0.06 RS H-GGA-D ꞷB97X-D 2.61 2.93 0.97 1.54 H-GGA-D B3LYP-D2 2.42 2.92 0.98 1.77

B3LYP-D2 functional of H-GGA with dispersion correction class is the next more accurate functional for calculating BDE of C−CN bond with 6-311G(d,p) basis set. Deviations and error are in the range of 1-3 kcal/mol with R of 0.98. D2-functionals include the missing intermolecular attractive dispersion effects, and these methods are good for inter and intramolecular non-covalent interactions, thermodynamic and kinetic properties of a system. The B3LYP-D2 functional shows better performance for homolytic cleavage of C−CN bond, but is less accurate compared to the CAM-B3LYP [229, 258].

ωB97X-D functional of range separated hybrid GGA with dispersion correction at 6- 311G(d,p) is also tested for BDE calculation of C−CN bond. Although, R (0.97) is more significant, yet SD (2.61 kcal/mol), RMSD (2.93 kcal/mol) and MAE (1.54 kcal/mol) are higher than those obtained from CAM-B3LYP. The high deviations and errors lower the accuracy of this class for the desired bond dissociation energy measurement. ωB97X-D suffers from self-interaction error at short range and free of SIE at long-range self-interaction. Conclusively, ωB97X-D briefly describes long range electronic interactions but underestimates the short-range correlation which is the reason for its lower performance than CAM-B3LYP.

113

20 6-311G(d,p) Aug-cc-pVTZ Def2-SVP

15 )

kcal/mol

( RMSD 5

B98

PBE

BP86

TPSS

BB95

BMK

B97-1

B97-2

LSDA

BLYP

B1B95

HCTH

B1LYP

B3LYP

X3LYP

t-HCTH

G96LYP

B3PW91

MPW1K

WB97X-D

MPW1B95

B3LYP-D2

PBE1KCIS

MPW3LYP

TPSS1KCIS

MPW1KCIS

MPW1PW91

TPSSLYP1W

CAM-B3LYP

MPWLYP1W BH and HLYP and BH Density Functionals

Fig. 3.12 Root Mean Square Deviation (RMSD) of Different Density Functionals with Three Basis Sets for C−CN BDEs

Seven functionals (BMK, B1B95, BB95, MPW1B95, MPW1KCIS, PBE1KCIS and TPSS1KCIS) from GH meta-GGA class are analyzed with 6-311G(d,p) basis set. The functionals of this class have combined characteristics of meta-GGA with % HF exchange. These exchange functionals have little effect on the respective bond dissociation [259]. These functionals have R in the range of 0.92-0.97, while SD is in the range of 2.68-3.97 kcal/mol. The RMSD values of selected DFs ranges from 3.45 to 6.26 kcal/mol, respectively and the MAEs of all functionals range from 3.53 to 5.11 kcal/mol, respectively. Though, the MAE of MPW1KCIS is 0.71 kcal/mol but R is less significant (0.92) due to more deviations and error. The performance of this class, as a whole, is moderate for BDE measurement of C−CN bond.

From H-GGA class, a number of functionals are selected including B1LYP, BH and HLYP, B3LYP, B3PW91, B98, B97-1, B97-2, MPW1K, MPW1PW91, MPW3LYP

114 and X3LYP. The BH and HLYP functionals show good efficiency for BDE calculations of C−CN bond in nitriles at 6-311G(d,p) basis set. Its SD, RMSD, R and MAE values are 2.86 kcal/mol, 3.32 kcal/mol, 0.98 and 1.87 kcal/mol, respectively. The well-known DF i.e. B3LYP shows lower efficiency. Although R (0.97) is good, but error and deviations are high. All the remaining functionals have SD and RMSD ranging from 2.69 to 8.72 kcal/mol. A similar value of R (0.97) is observed for all remaining DFs. However, MAEs range from 0.72-7.35 kcal/mol. Conclusively, H-GGA class shows moderate accuracy for measuring BDE of C−CN bond. Three functionals 휏-HCTH, TPSS and TPSSLYP1W of meta-GGA with 6-311G(d,p) basis set are tested for BDE of C−CN bond. This class shows lower performance for the required property measurement. SD and RMSD values are 2.74-9.64 kcal/mol, while R values range from 0.94-0.97. MAE of 휏-HCTH is less (2.49 kcal/mol). MAE values of the other two functionals (TPSS and TPSSLYP1W) are 7.11 and 9.00 kcal/mol, respectively. The accuracy of meta-GGA is high than the GGA in this study.

6-311G(d,p) Aug-cc-pVTZ Def2-SVP

6 )

kcal/mol

( SD SD

B98

PBE

BP86

TPSS

BB95

BMK

B97-1

B97-2

LSDA

BLYP

B1B95

HCTH

B1LYP

B3LYP

X3LYP

t-HCTH

G96LYP

B3PW91

MPW1K

WB97X-D

MPW1B95

B3LYP-D2

PBE1KCIS

MPW3LYP

TPSS1KCIS

MPW1KCIS

MPW1PW91

TPSSLYP1W

CAM-B3LYP

MPWLYP1W BH and HLYP and BH Density Functionals

Fig. 3.13 Standard Deviation (SD) of Different Density Functionals with Three Basis Sets for C−CN BDEs

115

BP86, BLYP, G96LYP, HCTH, MPWLYP1W and PBE functionals of GGA class are used with 6-311G(d,p) basis set. All these DFs have poor accuracy for BDE calculation of C−CN bond. The SD and RMSD ranges from 2.70 to 12.08 kcal/mol. Although PBE functional has less deviation (SD = 2.70 kcal/mol) and error (MAE = 2.72 kcal/mol) with significant R of 0.97 among DFs of its own class, yet the deviations and error values are larger than DFs of above DFT classes. The LSDA functional of local density approximation class shows least accuracy for BDE of C−CN bond with 6-311G(d,p) basis set as evident from the statistical analysis results. SD, RMSD, R and MAE values are 4.27 kcal/mol, 16.32 kcal/mol, 0.93 and 15.80 kcal/mol, respectively. Similar results are observed previously for the BDE of C−X bond (X = Cl and Br) [206], where deviations and error of LSDA are the highest.

The comparative analysis of DFT classes with 6-311G(d,p) basis set shows that CAM- B3LYP of RS H-GGA class efficiently reproduces experimentally reported BDE values for homolytic cleavage of C−CN bond.

Initially, 6-31G(d) and 6-31G(d,p) basis sets are modelled for simulation at HF level while 6-311G(d,p) basis set is designed at unrestricted MP2 level to give lowest possible energy of a system. The addition of UMP2 in later basis set also results in correction for valence electron correlation. Polarization functions (d,p) are added for further lowering the possible energy of atomic ground state. Latterly, these basis sets are being used along with all ab initio and DFT methods and give satisfactory results. 6-311G(d,p) basis set has six Gaussian functions for inner shell electrons and three split Gaussian functions for outer valence shells. Polarization functions are added to give additional flexibility for the proper description of molecular orbitals. 6-311G(d,p) basis set has a better compromise between speed and accuracy [179, 163]. The addition of polarization function p for hydrogen atom in 6-311G(d,p) basis set has greater impact on stabilization energies of compounds and their resultant radicals in comparison to d function on heavy atoms. Because intermolecular hydrogen bonding is present between hydrogen (electropositive) and nitrogen (electronegative) atoms due to which the compounds are more stable. It is important to use p function for proper description of lighter atoms and to obtain real minima structures of all selected compounds. The addition of diffuse functions (++) provide additional space for the description of anions where lone pair of electrons are present far away from the nucleus. In current study,

116 these basis sets have no significant impact on lowering the stabilization energies of respective compounds in comparison to 6-311G(d,p) basis set. To further elucidate the accuracy of 6-311G(d,p) basis set, its efficiency is compared with higher basis sets for the BDE calculation of C−CN bond. 3.3.4.2 Evaluation of DFs with 6-31G(d) Basis Set

Statistical analyses based on 6-31G(d) basis set are tabulated in Table 3.11, and detailed observations are as following:

CAM-B3LYP/6-311G(d,p) method shows consistency of high accuracy for BDE calculations of C−X bond similar to its performance. However, SD, RMSD and MAE are increased with 6-31G(d) basis set (except R). For current study, it gives the best results for the homolytic cleavage of C−CN bond compared to other DFs. ωB97X-D with 6-31G(d) basis set competes with B3LYP-D2 functional and gets second place for the BDE measurement of C−CN bond of nitriles with respect to accuracy. Compared to its performance with 6-311G(d,p) basis set, SD and R are similar while RMSD and MAE are increased by almost 1 kcal/mol. The results illustrate the medium accuracy of B3LYP-D2 with 6-31G(d) basis set for BDE measurement of C−CN bond. Compared to 6-311G(d,p) basis set, SD and RMSD are increased by 1 kcal/mol. On the other side, MAE is lowered but R is decreased by 3 times.

Table 3.11 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 3.55 18.10 0.96 17.78 BP86 3.04 5.09 0.96 4.18 BLYP 3.15 8.69 0.96 8.15 G96LYP 3.14 10.51 0.96 10.07 GGA HCTH 3.11 8.25 0.96 7.70 MPWLYP1W 3.17 6.87 0.96 6.16 PBE 3.02 3.52 0.96 2.01 휏-HCTH 2.82 3.19 0.97 1.70 meta-GGA TPSS 2.80 7.76 0.97 7.28

117

TPSSLYP1W 2.95 9.15 0.97 8.71 B1LYP 2.96 4.86 0.96 3.95 BH and HLYP 2.80 2.74 0.97 0.55 B3LYP 3.41 3.84 0.96 2.02 B3PW91 2.84 3.28 0.96 1.84 B98 2.81 2.75 0.97 0.59

B97-1 3.39 3.56 0.96 1.47 Hybrid GGA B97-2 2.92 2.80 0.97 0.12 MPW1K 3.00 4.82 0.96 3.87 MPW1PW91 2.74 2.68 0.97 0.55 MPW3LYP 3.17 6.87 0.96 6.16 X3LYP 3.00 3.53 0.96 2.04 BMK 2.68 5.51 0.97 4.87 B1B95 2.83 3.01 0.97 1.29 BB95 3.25 4.91 0.96 3.80

MPW1B95 3.16 3.48 0.96 1.71 GH meta-GGA MPW1KCIS 3.16 4.31 0.96 3.06 PBE1KCIS 3.19 3.36 0.96 1.39 TPSS1KCIS 2.90 6.30 0.97 5.65 RS H-GGA CAM-B3LYP 2.87 3.52 0.97 2.21 RS H-GGA-D ꞷB97X-D 2.62 3.53 0.97 2.48 H-GGA-D B3LYP-D2 3.82 3.66 0.95 0.12

All DFs of H-GGA class with 6-31G(d) basis set have improvement in accuracy for BDE calculations of C−CN bond compared to their performance at 6-311G(d,p) basis set. GH meta-GGA class when compared with H-GGA class, the accuracy becomes lower due to higher SD, RMSD and MAE (whereas R is in the range of 0.96-0.97). Although, the results (deviations and errors with R are improved) of this class are satisfactory compared to its efficiency with 6-311G(d,p) basis set. But overall, this class is designated as an ordinary performer for BDE measurement of C−CN bond. 휏-HCTH functional retains its good accuracy as is previously observed with 6-311G(d,p) basis set. The accuracy of remaining two functionals has improved with reduction of SD, RMSD and significant R value. Still, MAE is high which designate this class as less accurate for BDE measurement of C−CN bond. These functionals of exact exchange

118 lack electron correlation results in convex energy curve and give less accurate results for BDE of C−CN bond.

From GGA class with 6-31G(d) basis set, only PBE shows better accuracy for desired data set and shows consistency with 6-311G(d,p) basis set. All remaining functionals with 6-31G(d) basis set have similar trend of accuracy as that of their performance with 6-311G(d,p) basis set. MAE and RMSD are higher with moderate SD and the R of all functionals is 0.96. LSDA with 6-31G(d) basis set again shows poor accuracy for BDE calculations of C−CN bond of nitriles. The SD and R are improved but the RMSD and MAE are higher (Table 3.11). DFs of LDA and GGA classes suffer from self- interaction error leading to energetic preference for unrealistically delocalized electronic densities which leads to too low energies. This self-interaction error (SIE) over stabilize the molecules which leads to chemical reaction without any reaction barrier and results in wrong BDE curve. Previously, this self-interaction error is identified during thermodynamic studies of charged compounds [260], due to similar error the poorer results are obtained at these methods for BDE measurement of homolytic cleavage of nitriles in current study.

All the results indicate that CAM-B3LYP functional of RS H-GGA class with 6-31G(d) basis set has good accuracy for BDE calculations of C−CN bond.

3.3.4.3 Evaluation of DFs with 6-31G(d,p) Basis Set

B3LYP-D2 functional of H-GGA-D class has improved accuracy for homolytic cleavage of C−CN bond with 6-31G(d,p) basis set among all DFs of selected DFT classes. This means that polarization function (p) in 6-31G(d,p) basis set has impact on improving efficiency of B3LYP-D2. The results indicate that SD and RMSD are decreased by 1 kcal/mol and R is improved from 0.95 to 0.97 except MAE is increased up to 1 kcal/mol, compared to 6-31G(d) basis set. In comparison to 6-311G(d,p) basis set, the accuracy of B3LYP-D2 with 6-31G(d,p) basis set is poor. SD is higher with less significant R except lower MAE and RMSD values (Table 3.12). The accuracy of CAM-B3LYP is better with 6-31G(d,p) basis set, having less deviations and error with a significant correlation. Compared to 6-31G(d) basis set, SD is increased, and R is decreased between experimental and theoretical data. MAE and RMSD are also decreased. In comparison to 6-311G(d,p) basis set, more deviations and errors are

119 observed, although R is similar. ωB97X-D functional of RS H-GGA-D class has good accuracy among DFT classes with significant R of 0.97. The results are similar as obtained with 6-31G(d) basis set, but compared to 6-311G(d,p) the deviations and error are increased.

The accuracy of DFs of GH meta-GGA with 6-31G(d,p) basis set is moderate for BDE calculation of C−CN bond. The results of this class with 6-31G(d,p) basis set are similar to its results with 6-31G(d) basis set. The DFs of this class are less accurate at 6- 31G(d,p) compared to 6-311G(d,p) basis set and have more deviations and errors but R is consistent, which signify this class as an average performer for BDE of C−CN bond calculations. From the class of H-GGA, MPW1PW91 functional has the highest accuracy among all functionals. Other DFs (from B1LYP to X3LYP) have ordinary performance for BDE of C−CN bond calculation. Compared to above basis sets (6- 31G(d) and 6-311G(d,p)) the accuracy of DFs of hybrid GGA class with 6-31G(d,p) is lower due to more deviations and errors with less significant R.

휏-HCTH functional in meta-GGA class gives good accuracy with MAE of 0.93 kcal/mol. But SD, RMSD and R are less significant at 6-31G(d,p) basis set. TPSS and TPSSLYP1W functionals have more deviations and errors with less significant R compared to 6-31G(d). Interestingly, the statistical values are enhanced compared to 6- 311G(d,p) basis sets. So, this class is signified as moderately accurate for BDE calculations of C−CN bond. LSDA functional of LDA class has less accuracy among all DFs of DFT classes because of high MAE. However, SD, RMSD and R are good. The results are almost in consistent to the results obtained at 6-31G(d) basis set. Compared to 6-311G(d,p) basis set, MAE is increased up to 2 kcal/mol and RMSD is increased up to 1 kcal/mol. However, SD is decreased by 1 kcal/mol with improvement in R from 0.93 to 0.96. So, these results justify this class as a least efficient for BDE calculation of C−CN bond.

Table 3.12 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-31G(d,p) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 3.59 17.91 0.96 17.58

120

BP86 3.81 5.33 0.95 3.88

3.60 10.14 0.93 9.54 BLYP GGA G96LYP 4.32 10.92 0.94 10.11

HCTH 3.17 8.46 0.95 7.89 MPWLYP1W 3.19 6.99 0.96 6.29 PBE 3.06 3.65 0.96 2.17 휏-HCTH 3.73 3.69 0.95 0.93 meta-GGA TPSS 2.82 7.90 0.96 7.42 TPSSLYP1W 2.97 9.27 0.96 8.82 B1LYP 2.61 4.78 0.97 4.08 BH and HLYP 7.01 6.82 0.86 1.23 B3LYP 3.05 4.32 0.96 3.19 B3PW91 2.89 3.42 0.96 2.01 B98 5.61 6.14 0.80 2.98

B97-1 4.09 4.07 0.95 1.11 Hybrid GGA B97-2 7.14 7.30 0.68 2.55 MPW1K 13.98 13.46 0.79 1.49 MPW1PW91 2.82 2.80 0.96 0.74 MPW3LYP 3.19 6.99 0.96 6.29 X3LYP 3.04 3.64 0.96 2.19 BMK 2.74 5.38 0.97 4.70 B1B95 3.68 3.84 0.96 1.53 BB95 7.13 7.04 0.86 1.74

MPW1B95 3.15 3.57 0.97 1.91 GH meta-GGA MPW1KCIS 2.86 4.00 0.97 2.92 PBE1KCIS 3.25 3.48 0.96 1.58 TPSS1KCIS 2.96 6.47 0.96 5.81 RS H-GGA CAM-B3LYP 2.95 3.12 0.96 1.32 RS H-GGA-D ꞷB97X-D 2.67 3.43 0.97 2.28 H-GGA-D B3LYP-D2 2.73 2.85 0.97 1.13

B3LYP-D2 functional of H-GGA-D class shows the highest accuracy for BDE calculation of C−CN bond, when executed with 6-31G(d,p) basis set. Among Pople

121 basis sets with polarization functions, 6-311G(d,p) basis set gives the most accurate results for BDE calculation of C−CN bond. Among all above discussed DFs of different DFT classes, CAM-B3LYP, shows excellent performance. To further confirm the accuracy of Pople basis set, we have chosen two Pople basis set (with polarization function) with diffuse functions (6-31+G(d) and 6-311++G(d)).

3.3.4.4 Evaluation of DFs with 6-31+G(d) Basis Set

The mean absolute error (MAE) of CAM-B3LYP functional from standard experimental data is 0.73 kcal/mol. Although, R is lower, but deviations are less compared to other DFs at 6-31+G(d) basis set. So, among all DFs, CAM-B3LYP maintains its first position for reproducing accurate results for BDE calculations of C−CN bond with 6-31+G(d) basis set. But, compared to 6-311G(d,p) basis sets, the deviations and error are more, which categorize CAM-B3LYP/6-31+G(d) as less accurate compared to CAM-B3LYP/6-311G(d,p) method. The B3LYP-D2 functional of H-GGA-D class with 6-31+G(d) basis set has shown good accuracy for BDE calculation of C−CN bond, these results are similar to its previous performance at 6- 311G(d,p) basis set. But the SD and RMSD are increased up to 1 kcal/mol with MAE of 1.65 kcal/mol, R is also less significant (Table 3.13). ωB97X-D functional of RS H- GGA-D class with 6-31+G(d) basis set has average performance for the desired data. The reason for its less efficiency is its high deviations and error with less significant R compared to CAM-B3LYP and B3LYP-D2 functionals. In addition, deviations and error of ωB97X-D/6-31+G(d) are high with lower R, in comparison to performance with 6-311G(d,p) basis set.

SD and MAE for the selected DFs of GH meta-GGA with 6-31+G(d) basis set are also increased compared to 6-311G(d,p) basis set (although R is better with lower RMSD values) which make this class less efficient for BDE calculation of C−CN bond. DFs of H-GGA class have almost similar accuracy for BDE calculation of C−CN bond with 6- 31+G(d) basis set. Therefore, this class is designated as moderate performer for desired data set. The results (errors and deviations) are improved up to 2 kcal/mol compared to 6-311G(d,p) but the deviations are still more with less significant R. Although, 휏-HCTH is consistent in its good performance among three selected functionals of meta-GGA class but compared to the DFs of other classes (vide supra), its performance is less significant with 6-31+G(d) basis set. Two DFs, TPSS and TPSSLYP1W have same

122 trend of less accuracy for BDE of C−CN bond. Conclusively, based on these results this class is less efficient for desired data set. DFs of GGA class shows poor performance with 6-31+G(d) basis set; besides high deviations and error, the R is less significant. More pronounced results are observed in BP86 where MAE is decreased by 2 kcal/mol compared to 6-311G(d,p) basis set. Local density approximation functional (LSDA) has still worse performance for BDE calculations of C−CN bond among all DFs. The MAE, RMSD and SD of LSDA/6-31+G(d) are increased by 1 kcal/mol with a consistent R of 0.93 in comparison to LSDA/6-311G(d,p) basis set.

Conclusively, among all DFs, again the CAM-B3LYP functional of RS H-GGA class gives the highest accuracy with 6-31+G(d) basis set for desired property calculation. But, the accuracy is still less in comparison to 6-311G(d,p) basis set. It also supports the earlier work that diffuse function does not significantly improve the efficiency of Pople basis set for BDE measurement [198].

Table 3.13 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-31+G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 3.74 17.29 0.93 16.91 BP86 3.57 5.99 0.93 4.92 BLYP 3.89 9.19 0.92 8.40 GGA G96LYP 3.62 11.42 0.93 10.88 HCTH 3.45 9.21 0.92 8.60 MPWLYP1W 3.47 8.71 0.92 8.05 PBE 3.53 4.52 0.93 3.01 휏-HCTH 3.45 3.93 0.93 2.12 meta-GGA TPSS 3.47 8.61 0.93 7.94

TPSSLYP1W 3.53 10.45 0.93 9.89 B1LYP 3.47 5.87 0.93 4.84 BH and HLYP 7.54 7.30 0.80 1.04

B3LYP 3.50 5.17 0.93 3.94 Hybrid GGA B3PW91 3.46 4.08 0.93 2.38 B98 3.98 3.84 0.89 0.51

123

B97-1 3.42 3.28 0.93 0.05 B97-2 3.49 3.38 0.93 0.50 MPW1K 3.54 5.94 0.93 4.88 MPW1PW91 6.36 6.55 0.79 2.42 MPW3LYP 3.58 8.46 0.93 7.73 X3LYP 3.48 4.46 0.93 2.96 BMK 3.31 5.62 0.93 4.64 B1B95 3.57 3.48 0.92 0.67 BB95 3.71 5.99 0.92 4.82

MPW1B95 3.70 4.66 0.92 3.04 GH meta-GGA MPW1KCIS 3.60 5.56 0.93 4.37 PBE1KCIS 3.80 4.42 0.93 2.51 TPSS1KCIS 3.51 7.42 0.93 6.61 RS H-GGA CAM-B3LYP 3.36 3.29 0.93 0.73 RS H-GGA-D ꞷB97X-D 3.35 3.98 0.93 2.35 H-GGA-D B3LYP-D2 3.49 3.73 0.92 1.65

3.3.4.5 Evaluation of DFs with 6-311++G(d) Basis Set

6-311++G(d) is a type of Pople basis set that includes diffuse with polarization functions. This basis set is also compared to 6-311G(d,p) basis set and the results are tabulated in Table 3.14.

B3LYP-D2/6-311++G(d) method is analyzed for BDE calculation of C−CN bond and interestingly it gives good accuracy among all DFs of selected DFT classes. R is the best having value of 0.98. Compared to its efficiency with 6-31+G(d) basis set; MAE is large with almost similar RMSD (SD is decreased up to 1 kcal/mol) which makes it a good choice for desired data set with 6-31+G(d) basis set. On the other side, in comparison to 6-311G(d,p) basis set, R is more significant, but deviation and error are high. These results reveal that B3LYP-D2 is less accurate compared to its performance with 6-311G(d,p) basis set. The accuracy of ωB97X-D of RS H-GGA class with 6- 311++G(d) basis set is considerable due to the lowest MAE of 0.18 kcal/mol. Deviations are up to 4 kcal/mol and less R (0.89) make this class less efficient for BDE calculation of C−CN bond.

124

CAM-B3LYP of RS H-GGA class is observed as efficient performer with previous Pople basis sets but less accuracy is observed with 6-311++G(d) basis set. Compared to 6-311G(d,p) and 6-31+G(d) basis sets, deviations and error are increased up to 2 kcal/mol with the least significant R value of 0.88. Hybrid GGA class functional, BH and HLYP with 6-311++G(d) basis set has improved accuracy compared to 6- 311G(d,p) basis set and 6-31+G(d) basis set. MPW3LYP shows least accuracy compared to 6-311G(d,p) and 6-31+G(d) basis sets due to more deviations and error (more than 5-10 kcal/mol). Rest of the functionals of this class have less efficient performance and increased deviations and errors by 1-2 kcal/mol compared to previous basis sets. R is also less significant and justifies this class as less accurate for BDE calculation of C−CN bond.

The trend of accuracy of selected functionals of meta-GGA class with 6-311++G(d) basis set is similar to 6-311G(d,p) and 6-31+G(d) basis sets; 휏-HCTH performed better than TPSS and TPSSLYP1W. But, the statistical values (SD, RMSD and MAE) of these three DFs with 6-311++G(d) basis set are more (than the former basis sets) with less significant R due to which this class is signify as the less efficient. Selected DFs of GGA class have SD, RMSD and MAE of 2-14 kcal/mol. R is less significant except BP86 which has R of 0.97. But, this class is signified as poorer performer for BDE calculation of C−CN bond. LSDA functional has the least performance for BDE calculation of C−CN bond with 6-311++G(d) basis set. Results indicate increase in SD with lower R compared to previous basis sets (6-311G(d,p), 6-31+G(d)). B3LYP-D2 functional shows good accuracy for BDE calculation of desired data set. In case of Pople basis set with diffuse functions are better for anionic species as discussed in literature. In the current study, the radical species are examined where these basis set show less accuracy for desired data [261].

Table 3.14 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using 6-311G++(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 13.77 17.03 0.68 10.78 GGA BP86 2.82 6.07 0.97 5.43 BLYP 4.30 12.27 0.88 11.56

125

G96LYP 4.19 13.57 0.89 12.97 HCTH 4.03 11.17 0.89 10.48 MPWLYP1W 4.37 10.69 0.88 9.83 PBE 4.04 6.25 0.89 4.91 휏-HCTH 4.01 5.39 0.90 3.79 meta-GGA TPSS 3.99 10.54 0.89 9.83 TPSSLYP1W 4.20 12.65 0.88 11.99 B1LYP 3.95 7.93 0.90 6.96 BH and HLYP 2.57 3.24 0.97 2.10 B3LYP 3.99 7.12 0.89 6.01 B3PW91 4.04 5.28 0.90 3.59 B98 3.95 4.87 0.90 3.08

B97-1 3.93 4.31 0.90 2.11 Hybrid GGA B97-2 3.83 4.39 0.91 2.41 MPW1K 4.11 7.78 0.89 6.72 MPW1PW91 3.83 4.79 0.90 3.07 MPW3LYP 4.60 10.11 0.87 9.10 X3LYP 4.00 6.36 0.89 5.08 BMK 3.90 4.36 0.90 2.25 B1B95 3.95 4.05 0.90 1.44 BB95 4.24 7.97 0.89 6.86

MPW1B95 4.30 6.58 0.88 5.12 GH meta-GGA MPW1KCIS 4.17 7.47 0.89 6.31 PBE1KCIS 4.12 6.16 0.89 4.73 TPSS1KCIS 4.00 9.39 0.89 8.58 RS H-GGA CAM-B3LYP 4.20 4.32 0.88 1.57 RS H-GGA- D ꞷB97X-D 4.04 3.87 0.89 0.18 H-GGA-D B3LYP-D2 2.59 3.66 0.98 2.69

All the five types of Pople basis sets i.e. 6-31G(d), 6-31G(d,p), 6-311G(d,p), 6-31+G(d) and 6-311++G(d) are tested for BDE measurement of C−CN bond of nitrile compounds. These selected Pople basis sets showed almost similar results. However, 6-311G(d,p) basis set with all selected DFs reproduced accurate results, specifically with CAM-B3LYP functional. The much improvement in the results of 6-311G(d,p) basis set is may be due to the addition of polarization function (p for hydrogen atom

126 and d for atoms other than hydrogen in respective compounds). Results indicate that the intermolecular hydrogen bonding through space has a pronounced effect in such compounds due to which the 6-311G(d,p) basis set is better than other selected Pople basis sets.

3.3.5 Efficiency of Dunning Basis Sets

Two basis sets including aug-cc-pVDZ and aug-cc-pVTZ basis sets are selected from Dunning basis sets for BDE measurement of C−CN bond.

3.3.5.1 Evaluation of DFs with Aug-cc-pVDZ Basis Set

To evaluate BDEs of C−CN bond in organo-nitrile compounds, aug-cc-pVDZ basis set is also studied with all selected DFs of DFT classes (Table 3.15).

The combination of CAM-B3LYP with aug-cc-pVDZ basis set gives the highest accuracy among all selected DFs of DFT classes for BDE calculations of C−CN bond. Compared to 6-311G(d,p) basis set, deviations and MAE are higher with an efficient R value of 0.97. B3LYP-D2 (with aug-cc-pVDZ basis set) also shows good performance for dissociation of respective bond as indicated from its significant R, similar to CAM- B3LYP. However, deviations and error are 1-2 kcal/mol higher than those for CAM- B3LYP. Similarly, these deviations and error of B3LYP-D2 with aug-cc-pVDZ basis set are also higher than for 6-311G(d,p) basis set. The R is less significant than with Pople basis set. A decrease of MAE is noticed when ωB97X-D is used with aug-cc- pVDZ basis set. But, the deviation is up to 3 kcal/mol and R value is 0.95. Based on statistical analyses, ωB97X-D can be classified as an ordinary performer for desired property calculation.

B1LYP, B3LYP, B3PW91, B98, B97-1 and MPW1PW91 functionals have SD and MAE in the range of 2-5 kcal/mol and R in the range of 0.97-0.98. Remaining DFs, BH and HLYP, B97-2, MPW1K, MPW3LYP and X3LYP have deviations of 7 to 12 kcal/mol, with R below 0.80. The MAE is between 1-8 kcal/mol. All these results justify this class as less efficient with aug-cc-pVDZ basis set for BDE calculation of C−CN bond. The trend is almost similar to previous 6-311G(d,p) basis set. B1B95 functional shows good performance in GH meta-GGA class but other DFs of this class have deviation and error above 3 kcal/mol, with less significant R except BB95 (R =

127

0.97) and BMK (MAE = 1.45 kcal/mol). The trend is almost similar to 6-311G(d,p) basis set. As a whole, this class is an average performer with aug-cc-pVDZ basis set for BDE calculation of C−CN bond.

Table 3.15 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using Aug-cc-pVDZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 9.19 9.89 0.89 4.53 BP86 10.12 6.24 4.36 5.68 BLYP 2.71 10.41 0.97 10.08 G96LYP 3.13 12.14 0.96 11.77 GGA HCTH 7.99 10.40 0.79 7.04 MPWLYP1W 2.59 8.48 0.98 8.11 PBE 3.35 4.39 0.95 3.00 휏-HCTH 2.51 3.51 0.98 2.55 meta-GGA TPSS 2.61 8.99 0.97 8.63

TPSSLYP1W 2.54 11.05 0.97 10.77 B1LYP 2.53 5.96 0.97 5.45 BH and HLYP 12.25 11.77 0.80 0.89 B3LYP 2.68 5.40 0.97 4.75 B3PW91 2.64 4.05 0.97 3.17 B98 4.38 4.99 0.92 2.71 Hybrid GGA B97-1 2.52 2.51 0.98 0.70 B97-2 10.17 10.44 0.69 3.76 MPW1K 7.45 7.83 0.81 3.22 MPW1PW91 2.62 3.28 0.97 2.12 MPW3LYP 2.60 8.46 0.98 8.08 X3LYP 8.62 8.35 0.77 1.23 BMK 8.43 8.20 0.75 1.45 B1B95 2.69 2.58 0.97 0.15 BB95 2.84 6.05 0.97 5.41 GH meta-GGA MPW1B95 4.52 4.94 0.92 2.38 MPW1KCIS 3.44 5.31 0.95 4.16 PBE1KCIS 3.21 4.68 0.96 3.53

128

TPSS1KCIS 4.03 7.35 0.93 6.25 RS H-GGA CAM-B3LYP 2.66 2.57 0.97 0.31 RS H-GGA-D ꞷB97X-D 3.24 3.20 0.95 0.78 H-GGA-D B3LYP-D2 2.56 3.66 0.97 2.72

The trend of higher accuracy of 휏-HCTH with aug-cc-pVDZ basis set is consistent, as is observed with 6-311G(d,p) basis set. Besides 휏-HCTH, SD and R of TPSS and TPSSLYP1W functionals are in satisfactory range, but their MAE and RMSD are more than 8 kcal/mol. This makes these functionals less accurate for the desired data. The results obtained with aug-cc-pVDZ basis set signify this class less efficient for BDE calculation of C−CN bond. From GGA class, all five DFs have SD of 3 kcal/mol with a good R of 0.97, but MAE and RMSD are very high (up to 12 kcal/mol) which suppress the accuracy of this class compared to all above DFT classes with aug-cc-pVDZ basis set. However, HCTH functional shows worse performance due to higher deviations, error and lower R. Although LSDA shows the highest deviation and error with least significant R (0.89) compared to the above DFs, however, its performance can be regarded better when compared with6-311G(d,p) basis set. As a whole, CAM-B3LYP with aug-cc-pVDZ basis set shows good accuracy for homolytic cleavage of C−CN bond.

3.3.5.2 Evaluation of DFs with Aug-cc-pVTZ Basis Set

Augmented correlation consistent polarized valence-only triple zeta basis set is also studied with all selected DFs of DFT classes to evaluate BDE calculation of C−CN bond in twelve nitriles compounds. SD, RMSD, R and MAE are graphically represented in Figs. 3.12-3.14.

CAM-B3LYP shows the highest accuracy with aug-cc-pVTZ basis set. RMSD, SD, R and MAE are 2.61 kcal/mol, 2.71 kcal/mol, 0.97 and 0.23 kcal/mol, respectively. Compared to aug-cc-pVDZ basis set, standard deviations (SD and RMSD) of CAM- B3LYP are increased by 10% with aug-cc-pVTZ basis set. The deviations are also minimized by 5% in comparison to 6-311G(d,p) basis set but the R is improved from 0.96 to 0.97 with more MAE. The ωB97X-D functional gives the more accurate results for BDE measurement of C−CN bond with aug-cc-pVTZ basis set. MAE is smaller i.e.

129

1.09 kcal/mol with R value of 0.97. However, SD and RMSD values are increased up to 3 kcal/mol. Deviations and MAE are high with less R compared to aug-cc-pVDZ basis set. The errors and deviations are decreased in comparison to 6-311G(d,p) basis set with similar R (0.97).

6-311G(d,p) 15 Aug-cc-pVTZ Def2-SVP 10

5 ) 0

-5

kcal/mol (

-10 MAE

-15

-20

B98

PBE

BP86

TPSS

BB95

BMK

B97-1 B97-2

LSDA BLYP

B1B95

HCTH

B1LYP B3LYP

X3LYP

t-HCTH

G96LYP

B3PW91

MPW1K

WB97X-D

MPW1B95 B3LYP-D2

PBE1KCIS

MPW3LYP

TPSS1KCIS

MPW1KCIS

MPW1PW91

TPSSLYP1W

CAM-B3LYP

MPWLYP1W BH and HLYP and BH Density Functionals

Fig. 3.14 Mean Absolute Error (MAE) of Different Density Functionals with Three Basis Sets for C−CN BDEs

B3LYP-D2 functional of H-GGA-D class gives SD of 2.62 kcal/mol and the highest R of 0.98. On the other side, MAE is up to 4 kcal/mol and RMSD is 4.42 kcal/mol. SD is similar to that of aug-cc-pVDZ basis set but the R and MAE are increased. Compared to 6-311G(d,p) basis set, MAE is increased by 2 kcal/mol while SD and R are almost similar. Based on these results, this class is classified as moderate performer for BDE calculation of C−CN bond. A similar trend of accuracy is observed for the DFs of GH meta-GGA class with aug-cc-pVTZ basis set. B1B95 has good performance, while all other DFs have average performance (deviation is in the range of 2.71-7.90 kcal/mol

130 and R is 0.97). Compared to the results of BMK with aug-cc-pVDZ basis set, the performance of BMK is improved with aug-cc-pVTZ basis set due to less deviations and error, while the rest of DFs have almost similar results. Thus, GH meta-GGA class is signified as an average performer for desired data calculation.

Table 3.16 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using Aug-cc-pVTZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

DFT classes DFs SD RMSD R MAE LDA LSDA 3.94 16.17 0.95 15.73 BP86 3.26 3.25 0.96 0.94 BLYP 2.85 10.55 0.97 10.19 G96LYP 2.99 11.95 0.97 11.60 GGA HCTH 2.87 9.72 0.97 9.32 MPWLYP1W 2.84 8.88 0.97 8.45 PBE 2.65 10.92 0.97 10.62 휏-HCTH 15.00 14.45 0.58 1.59 meta-GGA TPSS 2.63 9.14 0.97 8.78

TPSSLYP1W 2.65 10.92 0.97 10.62 B1LYP 2.82 5.25 0.98 4.51 BH and HLYP 2.76 5.11 0.97 4.37 B3LYP 2.77 5.40 0.97 4.71 B3PW91 2.73 4.17 0.97 3.24 B98 4.19 4.78 0.93 2.61 Hybrid GGA B97-1 2.72 2.68 0.97 0.64

B97-2 11.32 11.60 0.64 4.14 MPW1K 2.68 6.09 0.97 5.52 MPW1PW91 2.88 3.67 0.97 2.42 MPW3LYP 2.84 8.88 0.97 8.45 X3LYP 2.76 4.60 0.97 3.77 BMK 2.71 4.12 0.97 3.20 B1B95 2.74 2.64 0.97 0.32 GH meta-GGA BB95 3.02 6.44 0.97 5.76 MPW1B95 2.87 4.80 0.97 3.94

131

MPW1KCIS 2.95 5.90 0.97 5.18 PBE1KCIS 2.95 4.53 0.97 3.55 TPSS1KCIS 2.71 7.90 0.97 7.46 RS H-GGA CAM-B3LYP 2.71 2.61 0.97 0.23 RS-H-GGA-D ωB97X-D 2.57 2.69 0.97 1.09 H-GGA-D B3LYP-D2 2.62 4.42 0.98 3.64

The performance of all DFs of H-GGA class is improved with aug-cc-pVTZ basis set as compared to their performance with 6-311G(d,p) and aug-cc-pVDZ basis sets. However, accuracy with experimental data is still less due to the highest values of SD, RMSD and MAE (Only B97-1 has the lowest MAE = 0.64 kcal/mol). Therefore, this class is designated as less efficient for BDE calculation of C−CN bond. 휏-HCTH functional, previously observed as good performer with 6-311G(d,p) and aug-cc-pVDZ basis sets, showed poor performance with aug-cc-pVTZ basis set for BDE measurement of C−CN bond. SD and RMSD are 15 kcal/mol with good R (0.58), and MAE is 1.59 kcal/mol. The performance of TPSS and TPSSLYP1W is improved with aug-cc-pVTZ basis set, due to less SD (< 3 kcal/mol) and R is 0.97, but MAE values are high (> 8 kcal/mol). The trend of accuracy of these latter DFs is similar to as observed with aug- cc-pVDZ basis set. In comparison to 6-311G(d,p) basis set, these deviations and errors are increased up to 1 kcal/mol. Decisively, this class shows less accuracy for desired property calculation.

GGA class functionals with aug-cc-pVTZ basis set have less accuracy, as observed with 6-311G(d,p) and aug-cc-pVDZ basis sets. SD, RMSD and MAE are in the range of 2- 12 kcal/mol. These results entail this class as a poor performer for desired data calculation. The LSDA functional with aug-cc-pVTZ basis set gives poor performance, as is observed previously with aug-cc-pVDZ basis set. The SD is up to 4 kcal/mol with R of 0.95. RMSD and MAE reach to the highest value of 16 kcal/mol. The trends of SD and MAE are almost similar to 6-311G(d,p) basis set but R is improved from 0.93 to 0.95. GGA class with aug-cc-pVTZ basis set is the poorest for BDE calculation of C−CN bond.

CAM-B3LYP functional of RS H-GGA class gives better accuracy among all DFT classes. Overall, by comparing the Dunning and Pople basis sets, Pople basis sets give

132 better accuracy for desired property calculation. A possible reason is that Dunning basis sets are parametrized with correlation while Pople basis sets are parametrized in HF. Furthermore, Pople basis sets are preferred because they require less computational cost than the Dunning basis sets. Pople basis sets give high accuracy for geometry optimization and thermodynamic properties of compounds. DFT also shows compatibility with faster Pople basis sets and shows preference to these basis sets compared to Dunning basis sets.

Equation y = a + b*x Weight Instrumental Residual Sum 9.8225 of Squares 3 Pearson's r 0.9643 140 Adj. R-Square 0.9228 Value Standard Err C1 Interce -12.446 11.44818 C1 Slope 1.10507 0.09596

120

100

Theoretical C-CN BDE (CAM-B3LYP/6-311G(d,p)) BDE C-CN Theoretical 105 110 115 120 125 130 135 Experimental C-CN BDE (kcal/mol)

Fig. 3.15 Pearson’s Correlation (R) of CAM-B3LYP with 6-311G(d,p) Basis Set for BDE Calculations of C−CN bond

3.3.6 Efficiency of Karlsruhe Basis Sets

Def2-SVP basis set is selected from Karlsruhe basis sets for BDE calculations of C−CN bond.

133

3.3.6.1 Evaluation of DFs with Def2-SVP Basis Set

B3LYP-D2 functional of hybrid GGA-D class functional has better accuracy with def2- SVP basis set among all DFs of selected DFT classes. SD, RMSD, R and MAE are 2.71, 2.83 kcal/mol, 0.97 and 0.06 kcal/mol, respectively (Table 3.17). Compared to the best Pople basis set (6-311G(d,p)) in the current study, B3LYP-D2 shows less deviations (SD and RMSD) but the MAE is decreased by 1 kcal/mol with decrease in R from 0.98 to 0.97. Performance of CAM-B3LYP becomes inferior with def2-SVP basis set for BDE measurement of C−CN bond but still it secures second position. Compared to 6-311G(d,p) basis set, SD and RMSD are increased by 1 kcal/mol and MAE is > 2 kcal/mol with similar R of 0.96. ωB97X-D functional of H-GGA-D class with def2-SVP basis set has a similar R as CAM-B3LYP functional with reduction in SD but RMSD and MAE is increased by 1 kcal/mol. Therefore, CAM-B3LYP is a moderate performer for BDE calculation of C−CN bond. SD, RMSD, R and MAE values of CAM-B3LYP with def2-SVP basis set are increased in comparison to 6- 311G(d,p) basis set. BH and HLYP functional of hybrid GGA class shows good performance. All remaining functionals except a few have average accuracy for desired property (BDE) measurement of C−CN bond. B97-2, MPW1K and X3LYP functionals have least accuracy (due to high deviations, errors and less R), although MAE of B97- 2 and MPW1K is good. B97-2, which shows good performance with 6-311G(d,p) basis set, becomes least efficient with def2-SVP basis set. Overall, this class is an average performer for BDE calculation of desired data set. BMK is observed as a good performer with 6-311G(d,p) basis set but becomes less accurate performer with def2- SVP basis set. BMK has deviation up to 8 kcal/mol with less significant R of 0.73 and MAE is more than 3 kcal/mol. MAE of MPW1B95 functional is 0.09 kcal/mol. However, its deviations are up to 4 kcal/mol, similar to other DFs of this class and R of all these DFs is 0.94. Compared to 6-311G(d,p) basis set, deviations and errors are enhanced up to 4 kcal/mol but R is decreased from 0.96 to 0.73. So, this class is less efficient with def2-SVP basis set for BDE calculation of C−CN bond. When 휏-HCTH functional of meta-GGA class is treated with def2-SVP basis set, it has more deviation (SD) with least significant R of 0.91. MAE is good as observed with 6-311G(d,p) basis set. Other two functionals (TPSS and TPSSLYP1W) have reduction in deviations and error by 1 kcal/mol at def2-SVP basis set, whereas R is increased to 0.96 compared to 0.94 for TPSS and 0.95 for TPSSLYP1W, respectively at 6-311G(d,p) basis set. Still,

134

MAE are large compared to other DFs which signify this class as least efficient for desired property measurement of C−CN bond (Table 3.17).

Table 3.17 SD, RMSD, R and MAE of C−CN BDEs Calculated with Different DFs While Using Def2-SVP Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE LDA LSDA 3.70 19.38 0.96 19.05 BP86 3.35 4.10 0.96 2.55 BLYP 3.35 8.22 0.95 7.56 G96LYP 3.68 10.16 0.94 9.53 GGA HCTH 3.41 7.50 0.95 6.75 MPWLYP1W 3.78 6.97 0.94 5.95 PBE 3.27 3.40 0.95 1.34 휏-HCTH 4.83 4.69 0.91 0.77 meta-GGA TPSS 3.03 7.22 0.96 6.61

TPSSLYP1W 3.13 8.68 0.96 8.14 B1LYP 3.13 4.27 0.96 3.04 BH and HLYP 2.93 2.83 0.96 0.39 B3LYP 3.17 3.70 0.96 2.11 B3PW91 3.05 3.04 0.96 0.87 B98 4.60 4.41 0.91 0.22 Hybrid GGA B97-1 2.90 3.26 0.97 1.70

B97-2 10.53 10.21 0.68 1.60 MPW1K 7.22 7.00 0.82 1.09 MPW1PW91 3.11 2.99 0.95 0.23 MPW3LYP 3.24 6.35 0.96 5.54 X3LYP 9.47 9.23 0.74 1.77 BMK 8.79 9.31 0.73 3.98 B1B95 3.01 3.79 0.96 2.45 BB95 3.38 4.40 0.96 2.97 GH GGA MPW1B95 4.06 3.89 0.94 0.09 MPW1KCIS 3.50 4.23 0.95 2.58 PBE1KCIS 8.74 9.96 0.74 3.61 TPSS1KCIS 3.94 5.50 0.94 4.00

135

RS H-GGA CAM-B3LYP 3.05 3.79 0.96 2.41 RS H-GGA-D ωB97X-D 2.76 4.35 0.96 3.45 H-GGA-D B3LYP-D2 2.83 2.71 0.97 0.06

Selected functionals of GGA class have deviations and errors in the range of 3-10 kcal/mol with R in the range of 0.94-0.96. Compared to 6-311G(d,p) basis set, all these statistical values are largely increased. Based on these results, this class is designated as poorly accurate for BDE calculation of C−CN bond. Similarly, LSDA functional is a poor performer as MAE and RMSD are above 19 kcal/mol, and SD is 3.70 kcal/mol.

Although B3LYP-D2 functional of hybrid-GGA-D class has better performance with def2-SVP basis set, yet the results are still inferior to the CAM-B3LYP results with 6- 311G(d,p) basis set.

Comparative analyses of DFT classes with a variety of basis sets indicate that CAM- B3LYP/6-311G(d,p) is a better performer for BDE of C−CN bond in nitriles. Further, we compared our work with previously reported data. A huge experimental data set is present in literature on the activation and dissociation of C−CN bond for the formation of novel chemical bonds. However, a limited theoretical work is done on C−X bond. Jones and his co-workers studied the activation of C–H and C−CN bond of succinonitrile and acetonitrile and analyzed the thermodynamics and transition states geometries for the cleavage of C−CN bond of acetonitrile using DFT method. B3LYP/6-31G(d,p) level of theory is used in their studies with effective core potential basis set (SDD) for rhodium (Rh) and phosphorus (P) atoms. They determined the kinetics and thermodynamic feasibility of C–C bond cleavage over C–H bond in gaseous and solution phases, and the results are comparable to experimental data [262]. Hirotaka et al., theoretically studied the Rh catalyzed borylation of nitrile via C−CN bond dissociation. They optimized all structures at B3LYP/6-31G(d) for all atoms (3- 21G basis set for phenyl ring on P atom and SDD basis set for Rh). For single point calculation, their selected level of theory is B3LYP-D3/ 6-311+G(d,p) and SDD basis set for Rh. They concluded that oxidative addition pathway (dissociation of C−CN bond) had higher activation barrier compared to iminoacyl pathway [263]. Our suggested method can be used for the mechanistic studies of such reactions where

136 dissociation of C−CN bond is involved. Besides C−CN bond of nitriles, Nakazawa group theoretically studied the dissociative mechanism of X–CN (X = C, O, and N) bonds using different transition metal complexes (Mo, Fe etc.).They used B3LYP with 6-31G(d) basis set (for all atoms except Mo and Fe) and specify LANL2DZ for Mo and Fe atoms They concluded that C−CN bond cleaved more easily as compared to O−CN and N−CN [264–266]. B3LYP/6-31G(d) method is used for the dissociation of F−CN bond silyl-migration-induced reaction in the presence of iron silyl complexes by AbdelRahman et al., For Fe atom LANL2DZ basis sets is used [267]. These are only mechanistic studies (not a proper benchmark study) therefore only one functional is used. B3LYP method is better for geometry optimization however a number of DFs are developed in last few years with better performance compared to B3LYP. The benchmark phenomenon is used to obtain a better DF for a particular system through comparative analysis of these DFs. Along with DF, a compatible basis set is also required. In our studies, for the first-time, computational benchmark studies are executed for the homolytic dissociation of the C−CN of nitriles compounds. For that purpose, a number of DFs are selected from the eight different DFT classes, including LDA, GGA, meta-GGA, H-GGA, GH meta-GGA, GH GGA-DC, RS-H-GGA and RS- H GGA-DC. Besides DFs, three types of basis sets (Pople, Dunning and Karlsrhu basis sets) are used. For most of nitrogen containing compounds, 6-311G(d,p) basis set is used which provides a better compromise between cost and accuracy. Similarly, for our selected compounds it provides, comparable results to experimental data. These outcomes are guideline for chemist to work on C−CN bond.

3.4 Benchmark Study for BDE of C−Mg Bond of Grignard Reagents

Grignard reagents was discovered by Victor Grignard in 1900 and he was awarded Nobel Prize (1912) in Chemistry. Grignard reagents (alkyl, vinyl, or aryl- magnesium halide) reacts with carbonyl carbon of aldehyde or ketone and results in carbon-carbon bond formation [268].

3.4.1 Importance of Grignard Reagents

Since from the discovery of Grignard reagents, it has been proved as a valuable precursor used for the synthesis of organic, natural [269], medicinal, industrial and agrochemicals. The synthesis of a variety of fluoroquinolone antibiotic drugs

137

(Ciprofloxacin, norfloxacin and pefloxacin) and nonsteroidal anti-inflammatory analgesics or cyclooxygenase inhibitors have involvement of various types of Grignard reagents [270]. Grignard reaction is a key step for the industrial production of Tamoxifen which is currently used for the treatment of estrogen receptor-positive breast cancer in women [99]. Nowadays, degradation of Grignard reagents is used for the chemical analysis of certain triacylglycerols. Besides, it is also employed for the synthesis of p-type dopants having interesting photophysical and electronic properties [271]. Grignard reagents is an attractive coupling partner of various reactions due to its ease of synthesis, low economic cost and commercial availability. Furthermore, many other organometallic coupling partners are also prepared from the corresponding Grignard reagents [272]. Grignard reagents play important role for activation of C- hetero atom (N, S, Cl, etc.) bond, otherwise the cleavage of such bonds is very difficult [273].

3.4.2 Cross Coupling Reaction using Grignard Reagents

Kumada Corriu, Negishi, Stille, or Suzuki reactions are classical cross-coupling processes of utmost importance. Among catalytic cross coupling reactions, Kumada coupling is an excellent method for the formation of carbon-carbon bonds via transition-metal catalyzed cross-coupling of Grignard reagents with organic electrophiles under mild reaction conditions [274].

3.4.3 Ionic versus Radical Pathways

Scientist have noticed ionic as well as radical pathways during the course of reactivity of Grignard reagents. Kambe and his co-workers synthesized sterically congested carbon compounds via an ionic pathway of alkyl (pseudo)halides with tertiary alkyl Grignard reagents. Terao and Kambe converted alkyl halides into hydrocarbons by Cu- catalyzed cross coupling reaction in the presence of Grignard reagents and they also proposed anionic pathway of the reaction [275]. Masayoshi et. al. reported enantioselective iron catalyzed cross coupling reaction of aryl Grignard reagents with α-chloro- and α-bromoalkanoates for the formation of α-arylalkanoic acids. They observed that the reaction proceeds through radical based mechanism, where alkyl radical is formed as an intermediate which further undergoes arylation via intermolecular rearrangement [276]. Nakamura and his co-workers introduced an

138 efficient, simple and high-yielding cross-coupling reaction of low-cost primary, secondary, and tertiary alkyl chlorides with aryl Grignard reagents by using catalytic amounts of N-heterocyclic carbene ligands and iron salts. Radical based mechanism is observed for this reaction. Hu and his co-workers performed Ni catalyzed Kumada- Corriu-Tamao cross coupling of aryl and heteroaryl Grignard reagents with nonactivated alkyl halides and observed a radical rebound mechanism for the activation of alkyl halide [231].

3.4.4 Experimental Study on BDE of C−Mg Bond of Grignard Reagents

Torkil holm studied the bond dissociation energy of C−Mg bond of alkyl magnesium halide for the estimation of enthalpies of formation of alkyl bromide via reaction of hydrogen bromide with Grignard reagents. Bond dissociation energies of C−Mg bond in alkylmagnesium bromides increase in the order of tert-butyl, allyl, benzyl, sec-butyl ⁓ isopropyl, ethyl, isobutyl, n-butyl, methyl, and phenyl groups [277]. Similarly, Holm implemented the phenomena of homolytic bond dissociation energy of RMgX (R = alkyl and aryl, X = Cl and Br etc.) for understanding the mechanism of the benzophenone reaction with Grignard reagents using calorimetric method. Three pathways are studied including electron transfer (ET), concerted and polar concerted mechanism. It was concluded that a homolytic cleavage of C−Mg bond is the rate determining step of all transition states where radicals are produced as intermediates [278].

3.4.5 Benchmark DFT Study on BDE of C−Mg Bond of Grignard Reagents

The emergence of computational methods put forward the efforts for the understanding of hidden truths of BDE of C−Mg bond of Grignard reagents. Still a reliable methodology is lacking for better result accuracy. In continuation to our previous benchmark reports on BDE of C−X (X = Br, Cl) [211] and C−Sn bond [212], we are intending to implement the benchmark approach to have an accurate DFT method for BDE calculation of C−Mg bond of Grignard reagents. In the current study, 31 density functionals with four basis sets are used to study the BDE of C−Mg bond.

In search of an accurate and cost-effective method, fifteen structurally diverse Grignard reagents (39-53) are selected and their structures are listed in Fig. 3.16. The optimized

139 geometries of all compounds are in singlet spin state and their resultant radicals are in doublet spin states. Computational methodologies include twenty-nine density functionals from thirteen classes of DFT with four different basis sets. Theoretical results are compared with already reported experimental values [221] [279]. The SD, RMSD, R and MAE values are given in Tables 3.18-3.21, and graphically represented in Figs. 3.17-3.20.

Fig. 3.16 The Structures of Grignard Reagents with Known Experimental BDEs of C−Mg Bond

3.4.6 Efficiency of Pople Basis Sets

Among Pople basis sets, 6-31G(d) and 6-311G(d) are selected for current study.

3.4.6.1 Evaluation of DFs with 6-31G(d) Basis Set

All selected DFs from different DFT classes are compared at 6-31G(d) basis set. TPSS shows outstanding performance for the measurement of BDE of C−Mg bond. SD, RMSD, R and MAE are 2.36 kcal/mol, 2.33 kcal/mol, 0.95 and 0.46 kcal/mol,

140 respectively. Other selected DFs of this class have less accurate results e.g. SD, RMSD, R and MAE of Minnesota functional (M06-L) are 2.79 kcal/mol, 5.56 kcal/mol, 0.93 and 4.86 kcal/mol, respectively (Table 3.18). Another DF, M11-L, has SD and R of 3.05 kcal/mol and 0.92, respectively. On the other side, its RMSD and MAE are more than 10 kcal/mol. The efficiency of M11-L is the least among selected DFs of respective class.

Table 3.18 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE BP86 3.02 3.34 0.92 1.64 GGA SOGGA-11 2.74 6.08 0.94 5.47 M06-L 2.79 5.56 0.93 4.86 meta-GGA TPSS 2.36 2.33 0.95 0.46 M11-L 3.05 10.58 0.92 10.16 NGA N12 2.73 3.57 0.94 2.40 meta-NGA MN12-L 2.43 7.77 0.95 7.41 B3LYP 2.64 2.55 0.94 0.11 B3P86 2.44 2.99 0.95 1.85 H-GGA MPW1K 2.58 2.63 0.94 0.84 PBE0 2.42 2.53 0.95 0.97 B3PW91 2.42 2.44 0.95 0.70 M05 2.67 8.62 0.94 8.23 M05-2X 2.74 9.29 0.94 8.91 M06 3.05 8.34 0.92 7.81 M06-2X 2.53 5.52 0.95 4.95 GH meta-GGA M06-HF 3.17 8.85 0.92 8.30 MPW1B95 2.92 5.61 0.93 4.86 MPW1KCIS 2.79 3.20 0.93 1.73 BMK 3.04 4.99 0.92 4.04 RS H-GGA CAM-B3LYP 2.61 3.08 0.95 1.77 ωB97 2.77 10.94 0.94 10.60 RS H-meta GGA M11 2.39 7.15 0.95 6.77

141

RS H-NGA MN12-SX 2.47 15.58 0.95 15.40 B97-D 3.31 8.21 0.90 7.56 GGA-D B97-D3 2.96 7.22 0.93 6.63 BP86-D3 3.15 5.18 0.91 4.20 H-GGA-D B3LYP-D3 2.97 4.16 0.93 3.01 RS H-GGA-D ωB97X-D 2.54 5.83 0.95 5.29

From H-GGA class, B3LYP, B3P86, MPW1K, PBE0 and B3PW91 functionals are chosen for this study. Among different DFs, B3PW91 functional gives better results. SD, RMSD, R and MAE of this DF are 2.42 kcal/mol, 2.44 kcal/mol, 0.95 and 0.70 kcal/mol, respectively. After B3PW91, the PBE0 has better performance for the BDE measurement of C−Mg bond. SD, RMSD, R and MAE are 2.42 kcal/mol, 2.53 kcal/mol, 0.95 and 0.97 kcal/mol, respectively. The SD, RMSD, R and MAE of B3P86 are 2.44 kcal/mol, 2.99 kcal/mol, 0.95 and 0.11 kcal/mol, respectively. SD and R of B3P86 are almost similar to those of B3PW91, but its RMSD and MAE are high. B3LYP and MPW1K have deviations in the range of 2.55 to 2.64 kcal/mol. MAEs of both are 0.11 kcal/mol and 0.84 kcal/mol, respectively. R values of both functionals are similar. The efficiency of RS H-GGA class is lower than the above two classes for BDE measurement of C−Mg bond. CAM-B3LYP functional has SD, RMSD, R and MAE of 2.61 kcal/mol, 3.08 kcal/mol, 0.95 and 1.77 kcal/mol, respectively which represents good accuracy of this class but still less efficient as compared to meta-GGA and H- GGA classes. N12 of NGA class shows moderate performance and its SD, RMSD, R and MAE are 2.73 kcal/mol, 3.57 kcal/mol, 0.94 and 2.40 kcal/mol, respectively. Two functionals (BP86 and SOGGA-11) are selected from GGA. SD, RMSD, R and MAE of BP86 functional of GGA class are 3.02 kcal/mol, 3.34 kcal/mol, 0.92 and 1.64 kcal/mol, respectively. During comparative analyses of both DFs, BP86-D3 showed lower deviations and error than SOGGA-11. Hybrid DFs are modified with Grimme dispersion correction, to encounter the dispersion forces. One of the most famous DF, B3LYP-D3 of this class is selected for current study. Its SD, RMSD, R and MAE are 2.97 kcal/mol, 4.16 kcal/mol, 0.93 and 3.01 kcal/mol, respectively. ꞷB97X-D functional has SD of 2.54 kcal/mol and R is 0.95. RMSD and MAE above 5 kcal/mol which indicates the moderate efficiency of this functional for BDE measurement of

142

C−Mg bond. SD and R of MN12-L of meta NGA class are 2.43 kcal/mol and 0.95, respectively. On the other side, its RMSD and MAE are more than 7 kcal/mol.

6-31G (d) 6-311G (d) Def2-SVP 16 Aug-cc-pVDZ

) 12

kcal/mol (

6 RMSD

N12

M05

M06

M11

BP86

TPSS

wB97

PBE0

BMK

B97-D

B3P86

M06-L

M11-L

B3LYP

B97-D3

M05-2X

M06-2X

M06-HF

MN12-L

B3PW91

MPW1K

BP86-D3

wB97X-D

MN12-SX

MPW1B95

B3LYP-D3

SOGGA-11

MPW1KCIS CAM-B3LYP Density Functionals

Fig. 3.17 Root Mean Square Deviation (RMSD) of Different Density Functionals with Four Basis Sets for C−Mg BDEs

The Pearson’s R between experimental and theoretical values is 0.91. Based on high RMSD, MAE and lower R, this DF is suggested as less efficient for the required data. This class is designated as an average performer for the BDE measurement of C−Mg bond of Grignard reagents. Eight DFs including M05, M05-2X, M06, M06-2X, M06- HF, MPW1B95, MPW1KCIS and BMK are selected from the GH meta-GGA class. MPW1KCIS/6-31G(d) method performed better compared to other DFS of this class. The SD, RMSD, R and MAE are 2.79 kcal/mol, 3.20 kcal/mol, 0.93 and 1.73 kcal/mol, respectively. All other DFs of this class have SD in the range of 2-3 kcal/mol, and R in the range of 0.92-0.94. But the RMSD and MAE are higher (from 4 kcal/mol to 9 kcal/mol). These results illustrate the least efficiency of this class for the BDE measurement of C−Mg bond of Grignard reagents. M11 functional of RS H-meta-GGA

143 class has SD, RMSD, R and MAE of 2.39 kcal/mol, 7.15 kcal/mol, 0.95 and 6.77 kcal/mol, respectively. SD is less with good Pearson’s R. But the RMSD and MAE are lerger which illustrate the lower efficiency of this DF for BDE measurement of the desired bond. MN12-SX [136] functional of RS H-NGA has less SD (2.47) and significant R (0.95) between experimental and theoretical data but their RMSD and MAE are above 15 kcal/mol. These results designated this class as a poor performer for reproducing BDE values of C−Mg bond. B97-D and B97-D3 functionals are selected from GGA with dispersion correction class. SD, RMSD, R and MAE of B97-D are 3.31 kcal/mol, 8.21 kcal/mol, 0.90 and 7.56 kcal/mol, respectively whereas for B97-D3 functional has SD, RMSD, R and MAE are 2.96 kcal/mol, 7.22 kcal/mol, 0.93 and 6.63 kcal/mol, respectively. The results based on correlation, error and deviations of B97- D3 are better than B97-D. BP86-D3 is also selected from GGA class with dispersion correction. BP86-D3 showed lower SD (3.15 kcal/mol) but the RMSD and MAE are more than 4 kcal/mol. However, the performance of both these functionals (B97-D3 and B97-D) is poorest when compared to DFs of above DFT classes. So, GGA class is designated as least efficient for reproducing BDEs of C−Mg bond of Grignard reagents.

Herein, TPSS has the best results for required data set among DFs of chosen DFT classes. TPSS is a controlled interpolation between slowly varying limits and 1 or 2 electrons limits as compared to other functionals, which may be the reason for its better performance [191]. Along with 6-31G(d) basis set, TPSS has lower deviations and error with significant Pearson’s correlation and signified as the best performer for desired data set.

3.4.6.2 Evaluation of DFs with 6-311G(d) Basis Set

All selected DFs of DFT classes are also compared with 6-311G(d) basis set. PBE0 of H-GGA class has more accurate results and its efficiency is better with 6-311G(d) basis set among all DFs of its own class. Deviations are up to 2.80 kcal/mol whereas R and MAE are 0.93 and 0.46 kcal/mol, respectively (Table 3.19). Upon comparison with 6- 31G(d) basis set, deviations and errors are increased to 0.45 kcal/mol and the R is decreased from 0.95 to 0.93. The deviations and errors for rest of the DFs are even further increased up to 3 kcal/mol. The R is in the range of 0.89-0.93 which is also lower as compared to 6-31G(d) basis set. Over bound MAE becomes under bound in case of B3LYP and MPW1K functionals, but again lower than 6-31G(d) basis set.

144

B3LYP and B3PW91 functionals have larger error (up to 1 kcal/mol) with 6-311G(d) basis set compared to the 6-31G(d) basis set. Based on the results, this class is suggested as moderate performer for BDE measurement of C−Mg bond.

Table 3.19 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using 6-311G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE BP86 3.56 3.45 0.90 0.25 GGA SOGGA-11 2.98 5.58 0.93 4.79 M06-L 2.95 3.91 0.93 2.67 meta-GGA TPSS 2.84 2.86 0.93 0.81 M11-L 3.83 8.42 0.87 7.56 NGA N12 3.97 4.03 0.88 1.22 meta-NGA MN12-L 3.10 7.20 0.92 6.55 B3LYP 3.70 3.78 0.89 1.24 B3P86 3.27 3.20 0.91 0.50 Hybrid GGA MPW1K 3.38 3.32 0.90 0.55 PBE0 2.80 2.75 0.93 0.46 B3PW91 3.10 3.61 0.92 2.02 M05 2.80 6.91 0.94 6.36 M05-2X 3.17 8.00 0.91 7.39 M06 4.43 7.67 0.83 6.36 M06-2X 3.42 4.40 0.90 2.90 GH meta-GGA M06-HF 3.56 6.28 0.89 5.26 MPW1B95 3.76 4.76 0.88 3.08 MPW1KCIS 3.45 3.34 0.90 0.13 BMK 3.83 4.64 0.88 2.80 CAM-B3LYP 3.81 3.71 0.89 0.46 RS H-GGA ωB97 3.16 9.38 0.92 8.87 RS H-meta GGA M11 3.48 6.63 0.90 5.71 RS H-NGA MN12-SX 2.51 13.41 0.95 13.19 B97-D 3.46 6.72 0.90 5.83 GGA-D B97-D3 3.36 6.13 0.91 5.20 BP86-D3 3.96 4.75 0.86 2.82

145

H-GGA-D B3LYP-D3 3.84 3.99 0.88 1.48 RS H-GGA-D ωB97X-D 3.22 4.97 0.91 3.87

Among three selected DFs of meta-GGA class, TPSS functional has good performance with 6-311G(d) basis set for reproducing BDE of C−Mg bond. Again here, this DF is suggested as a better performer for the respective study. Compared to 6-31G(d) basis set, the efficiency of 6-311G(d) is lower. Deviations are increased, and the R is also decreased from 0.95 to 0.93. Over bound MAE of TPSS/6-31G(d) method becomes under bound at TPSS/6-311G(d) method and its numerical value is also increased. Deviations and error of M06-L are decreased up to1-2 kcal/mol, while R is almost similar (0.93). The efficiency of M11L with 6-311G(d) basis set is less because the value of R is 0.87. Although, the deviation (RMSD) and error of M11L are decreased up to 3 kcal/mol, compared to 6-31G(d) basis set. The efficiency of BP86 is enhanced with 6-311G(d) basis set. Deviations (SD and RMSD) are increased to some extent, but the error is decreased to 0.25 kcal/mol. The R is also decreased from 0.92 to 0.90. Deviations of B3LYP-D3 with 6-311G(d) basis set are more enhanced, and R is lower as compared to 6-31G(d) basis set. However, MAE is decreased up to 2 kcal/mol. Error and deviations of CAM-B3LYP are decreased/increased up to 1 kcal/mol, respectively. R is decreased from 0.95 to 0.89 respectively. N12 functional with 6-311G(d) basis set is proved less efficient as compared to its performance with 6-31G(d) basis set. The deviations are increased up to 1 kcal/mol and R is lowered from 0.94 to 0.88. The MAE is decreased approximately by 1 kcal/mol. Due to high deviations and lower R, this DF is signified as moderate performer for current study.

146

6-31G(d) 6-311G(d) Def2-SVP 8 Aug-cc-pVDZ

6 )

kcal/mol 4

( SD

N12

M05

M06

M11

BP86

TPSS

wB97

PBE0

BMK

B97-D

B3P86

M06-L

M11-L

B3LYP

B97-D3

M05-2X

M06-2X

M06-HF

MN12-L

B3PW91

MPW1K

BP86-D3

wB97X-D

MN12-SX

MPW1B95

B3LYP-D3

SOGGA-11 MPW1KCIS Density Functionals CAM-B3LYP

Fig. 3.18 Standard Deviation (SD) of Different Density Functionals with Four Basis Sets for C−Mg BDEs

Two DFs (BP86 [148, 153] and SOGGA-11) are analyzed with 6-311G(d) basis set and lower efficiency is observed for the required data. Upon comparison with previous basis set, the deviations (RMSD) and errors are decreased by 1 kcal/mol. SD values are increased up to 1 kcal/mol. R values of BP86-D3 and SOGGA-11 are 0.90 and 0.93 respectively, which are also low as compared with 6-31G(d) basis set. BP86-D3 [148, 153] is analyzed with 6-311G(d) basis set and lower efficiency is observed for the required data. Upon comparison with previous basis set, the deviations and errors are decreased by 1 kcal/mol. R values of BP86-D3 is 0.86 which are also low as compared with 6-31G(d) basis set. B97-D and B97-D3 with 6-311G(d) basis set have lower deviations and errors (up to 1 kcal/mol). However, the R of B97-D/6-311G(d) is almost similar to B97-D/6-31G(d) method. But the R of B97-D3 is decreased from 0.93 to 0.91. MPW1KCIS/6-311G(d) method performs better with basis set and sustains its good efficiency as is observed with 6-31G(d) basis set. Here, the deviations are decreased up to1 kcal/mol while the error is decreased to 0.13 kcal/mol. The R is

147 decreased from 0.93 to 0.90. The deviations and errors of the rest of all selected DFs of same class are decreased approximately 1-2 kcal/mol, while R is also lowered up to 0.83. Thus, the statistical analyses of the DFs of this class designated it as less efficient performer for BDE measurement of C−Mg bond. ωB97X-D of RS H-GGA-D is proved less efficient with 6-311G(d) basis set. Deviations and errors are decreased up to 1-2 kcal/mol and R is also decreased from 0.95 to 0.91. M11 functional of RS H-meta-GGA class has enhanced its efficiency with 6-311G(d) basis set as compared to 6-31G(d) basis set, its deviations and error are decreased up to 1 kcal/mol but the significant R is decreased from 0.95 to 0.91 which makes it less efficient class for BDE measurement of C−Mg bond. RMSD and MAE of MN12-L functional of meta-GGA class are lower while its SD is increased (up to 1 kcal/mol). The R is decreased from 0.95 to 0.92. These results justify the less efficient performance of the meta-GGA class for desired data. MN12-SX functional of RS H-NGA class has lower SD (2.51 kcal/mol) with a significant R (0.95). But RMSD and MAE are above 13 kcal/mol proving that MN12- SX is the poorest functional among all, and also signifies this class as the least efficient performer for desired data set.

In short, PBE0 form H-GGA class with 6-311G(d) basis set has shown the best performance for BDE measurement of C−Mg bond.

3.4.7 Efficiency of Karlsruhe Basis Sets

Def2-SVP among Karlsruhe basis sets is selected to explore the BDE measurement of C−Mg bond.

3.4.7.1 Evaluation of DFs with Def2-SVP basis set

All selected DFs of different DFT classes are tested with def2-SVP basis set, and the results are as following:

PBE0 of H-GGA class sustains its high efficiency for BDE measurement of C−Mg bond. SD, RMSD, R and MAE are 2.59 kcal/mol, 2.55 kcal/mol, 0.95 and 0.50 kcal/mol, respectively (Table 3.20). Compared to the results obtained with 6-31G(d) basis set, SD is increased to 0.17 kcal/mol and error is decreased to 0.50 kcal/mol. Both, RMSD and R are almost similar as obtained with 6-31G(d) basis set. Compared to the results obtained with 6-311G(d) basis set, deviations of PBE0/def2-SVP are decreased

148 with lower MAE and R. Efficiency of B3P86 is enhanced with def2-SVP basis set as compared to 6-31G(d) basis set. Over bound MAE is also decreased to 0.47 kcal/mol, with almost similar R value of 0.95. SD and RMSD are increased (2.61 kcal/mol) and decreased (2.57 kcal/mol), respectively. While compared to 6-311G(d) basis set, deviations and error are decreased by 1 kcal/mol and R is increased from 0.91 to 0.95. On the other side, deviations (SD and RMSD) of MPW1K are increased and under bound MAE (0.49 kcal/mol) is observed with def2-SVP basis set in comparison to 6- 31G(d) basis set (R is similar). RMSD, SD and MAE are decreased compared to 6- 311G(d) basis set having significant R (0.94). These results illustrate the moderate efficiency of these three functionals with def2-SVP as compared to Pople basis sets. In case of remaining two functionals, B3LYP and B3PW91 with def2-SVP basis set, deviations (SD and RMSD) and errors are increased up to 1 kcal/mol. R of the B3LYP is increased from 0.94 to 0.95 and of B3PW91 remains similar (0.95) in comparison to 6-31G(d) basis set. MAE of these two DFs with def2-SVP basis set is increased, deviations and R are decreased compared to 6-311G(d) basis set. Overall, this class has good performance with def2-SVP basis set for the respective bond study.

Table 3.20 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using Def2-SVP Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE BP86 2.75 2.66 0.94 0.22 GGA SOGGA-11 3.58 5.54 0.90 4.33 M06-L 2.85 3.80 0.93 2.61 meta-GGA TPSS 2.55 2.72 0.95 1.16 M11-L 3.05 10.25 0.92 9.82 NGA N12 2.96 3.29 0.93 1.64 meta- NGA MN12-L 2.71 7.36 0.94 6.88 B3LYP 2.72 2.83 0.95 1.05 B3P86 2.61 2.57 0.95 0.47 Hybrid GGA MPW1K 2.73 2.68 0.94 0.49 PBE0 2.59 2.55 0.95 0.50 B3PW91 2.58 3.24 0.95 2.06 GH meta-GGA M05 3.04 7.55 0.94 6.96

149

M05-2X 3.60 6.74 0.91 5.78 M06 2.99 3.15 0.93 1.25 M06-2X 2.78 4.45 0.93 3.56 M06-HF 3.55 5.44 0.92 3.26 MPW1B95 3.09 4.53 0.92 3.41 MPW1KCIS 2.94 2.88 0.93 0.47 BMK 2.75 4.02 0.94 3.01 CAM-B3LYP 2.68 2.66 0.95 0.61 RS H-GGA ωB97 2.89 9.70 0.94 9.29 RS H-meta-GGA M11 2.32 6.08 0.96 5.65 RS H-NGA MN12-SX 2.51 14.31 0.95 14.10 B97-D 3.22 7.27 0.91 6.57 GGA-D B97-D3 2.76 6.12 0.94 5.51 BP86-D3 4.04 5.34 0.86 3.64 H-GGA-D B3LYP-D3 2.85 3.29 0.93 1.80 RS H-GGA-D ωB97X-D 2.84 4.79 0.93 3.93

CAM-B3LYP of RS H-GGA class with def2-SVP basis set shows enhanced efficiency as compared to Pople basis sets. SD and R are almost similar to 6-31G(d) basis set, whereas RMSD and MAE are decreased up to 1 kcal/mol. Compared to 6-311G(d) basis set, deviations with def2-SVP basis set are decreased by 1 kcal/mol with significant R (0.95) between experimental and theoretical values. Besides R, a small increase in MAE (0.15 kcal/mol) is observed. Deviations and error of TPSS/def2-SVP are increased up to 1 kcal/mol, but the R is almost similar as is observed with TPSS/6-31G(d). In comparison to 6-311G(d) basis set, SD, RMSD and MAE are decreased up to 1 kcal/mol. On the other hand, R is improved and increased from 0.93 to 0.95. Deviations and errors of M06-L are increased up to 1-2 kcal/mol and R is similar to as obtained with 6-31G(d) basis set. Compared to 6-311G(d) basis set, the results of M06-L with def2-SVP basis set are almost similar. M11-L has almost similar SD and R, RMSD and MAE are decreased up to 1 kcal/mol as are observed with 6-31G(d) basis set. On the other side, comparing with 6-311G(d) basis set. SD, RMSD and MAE are decreased up to 1-2 kcal/mol but R is improved to 0.92. Thus, based on these results meta-GGA class is observed as an efficient class for the current study.

150

Aug-cc-PVDZ Def2-SVP 6-311G(d) wB97X-D 6-31G(d) B3LYP-D3 SOGGA-11 BP86-D3 B97-D3 B97-D MN12-SX M11 wB97 CAM-B3LYP BMK MPW1KCIS MPW1B95 M06-HF M06-2X M06 M05-2X M05 B3PW91 PBE0 MPW1K

B3P86 DensityFunctionals B3LYP MN12-L N12 M11-L TPSS M06-L BP86

-15 -10 -5 0 5 MAE (kcal/mol)

Fig. 3.19 Mean Absolute Error (MAE) of Different Density Functionals with Four Basis Sets for C−Mg BDEs

Performance of BP86 functional of GGA class is improved with def2-SVP basis set, deviations and error are decreased by 1-2 kcal/mol, R is improved from 0.92 to 0.94 compared to 6-31G(d) basis set. In comparison to 6-311G(d) basis set, the deviations and errors are decreased to same extent as discussed above but R is enhanced from 0.90 to 0.94. SD and MAE of SOGGA-11 with def2-SVP basis set are decreased up to 1 kcal/mol in comparison to 6-31G(d) basis set. R is reduced from 0.94 to 0.90. Besides these, RMSD is increased up to 1 kcal/mol. These results justify the SOGGA-11 with def2-SVP, as moderate performer for desired data. These results indicate the good efficiency of M06-L for BDE measurement of C−Mg bond. SD and MAE of N12 functional of NGA class are increased (0.28 kcal/mol of decrease is observed in RMSD) and R is decreased from 0.94 to 0.93 compared to 6-31G(d) basis set. The results indicate that efficiency of N12 is decreased with def2-SVP basis set. B3LYP-D3 functional of H-GGA-D class has deviations and errors up to 1 kcal/mol whereas R is almost similar, compared to 6-31G(d) basis set. On the other side, deviations are decreased up to 1 kcal/mol while MAE is increased with almost similar R when

151 compared to 6-311G(d) basis set. This represents the moderate performance of H-GGA- D class with def2-SVP basis set. ωB97X-D with def2-SVP basis set has average performance for reproducing BDE of C−Mg bond, among all selected DFs. Compared to 6-31G(d) basis set, deviations (SD and RMSD) are decreased/increased up to 1 kcal/mol respectively, with reduction of R between theoretical and experimental data. R and deviations are reduced while MAE are increased to some extent, in comparison to 6-311G(d) basis set. All these results elucidate the average performance of ωB97X- D/def2-SVP method for the current study.

All selected DFs from GH meta-GGA class have similar performance with def2-SVP basis set, as is observed with Pople basis sets with approximate difference of 1-2 kcal/mol. The R is improved in comparison to 6-311G(d) basis set but still the deviation and errors are more which classify this class as moderate performer for desired data. Among all DFs, MN12-L functional shows least efficiency for desired data measurement. Compared to 6-31G(d) basis set, the RMSD, MAE and R are reduced while SD is increased. Contradictory results are observed upon comparison to 6- 311G(d) basis set. SD is decreased, whereas RMSD, MAE and R are increased (R is increased from 0.92 to 0.94). These results depict the less efficiency of this class for the BDE measurement of C−Mg bond. M11 functional of RS H-meta-GGA class has very poor performance. Although deviation, and error are decreased up to 1 kcal/mol and R is increased to 0.96, compared to Pople basis sets, yet these values are high. Both DFs of GGA-D class have lower values of SD, and also R is good but RMSD and MAE are above 5.5 kcal/mol. These results classify this class as a poor performer for the BDE measurement of C−Mg bond. Upon comparing with Pople basis sets, deviations and error are decreased whereas significant R is observed. This class is designated as less efficient performer for desired data due to high RMSD and MAE. MN12-SX functional of RS H-NGA class is analyzed with def2-SVP basis set. Its SD is 2.51 kcal/mol and significant R is 0.95 but the RMSD and MAE are above 14 kcal/mol which indicate the least efficiency for desired data. Compared to Pople basis sets, the RMSD and MAE are increased/decreased up to 1 kcal/mol respectively, with similar R and SD. These high deviation (RMSD) and error signify this functional as the poorest DF which ultimately proves this class as least efficient for the BDE measurement of C−Mg bond.

152

Collectively, the results of all selected DFs of DFT classes show that H-GGA class shows good performance and PBE0 functional of this class has good efficiency among all selected DFs.

3.4.8 Efficiency of Dunning Basis Sets

Among Dunning basis sets, aug-cc-pVDZ is selected for exploring the BDE measurement of C−Mg bond.

3.4.8.1 Evaluation of DFs with Aug-cc-pVDZ Basis Set

PBE0 maintains its high efficiency among all DFs of its own class and other chosen DFT classes. SD, RMSD, R and MAE are 3.49 kcal/mol, 4.12 kcal/mol, 0.91 and 2.36 kcal/mol, respectively. Deviations and error are increased (more than 1 kcal/mol) compared to Pople and Karlsruhe basis sets although R is decreased to 0.91. B3P86 performs well in achieving good results as is also observed with def2-SVP basis set. Here, the deviations are increased to 1 kcal/mol with less significant R of 0.91 (Table 3.21). Over bound MAE becomes under bound and is high in magnitude (1.38 kcal/mol) as compared to previous basis sets. B3PW91 is observed as good performer with Pople basis sets but moving towards higher Karlsruhe and Dunning basis sets, the efficiency is lower. R becomes less significant with aug-cc-pVDZ basis set, deviations and error are decreased up to 1-2 kcal/mol. Similar results are observed for B3LYP with aug-cc- pVDZ basis set. The least efficient DF of this class is MPW1K, which has better performance with previous basis sets. RMSD, SD and MAE of MPW1K are above 7 kcal/mol and R is reduced to 0.67.

Table 3.21 SD, RMSD, R and MAE of C−Mg BDEs Calculated with Different DFs While Using Aug-cc-pVDZ Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs SD RMSD R MAE BP86 4.29 4.81 0.88 2.43 GGA SOGGA-11 3.59 4.44 0.90 2.78 M06-L 4.13 4.79 0.88 2.65 meta-GGA TPSS 3.60 4.52 0.91 2.89 M11-L 4.85 9.49 0.84 8.26

153

NGA N12 4.79 4.89 0.86 1.58 meta- NGA MN12-L 2.34 6.31 0.95 5.90 B3LYP 4.58 5.65 0.88 3.51 B3P86 3.99 4.09 0.90 1.38 Hybrid GGA MPW1K 7.46 7.29 0.67 1.08 PBE0 3.49 4.12 0.91 2.36 B3PW91 3.80 5.38 0.90 3.94 M05 3.40 6.00 0.93 5.02 M05-2X 3.76 6.47 0.90 5.35 M06 5.22 6.63 0.83 4.30 M06-2X 3.91 3.94 0.88 1.13 GH meta-GGA M06-HF 3.45 4.87 0.90 3.55 MPW1B95 4.33 4.23 0.87 0.66 MPW1KCIS 4.14 4.55 0.88 2.16 BMK 4.79 4.74 0.85 1.03 RS H-GGA CAM-B3LYP 4.79 5.00 0.87 1.89 ωB97 4.02 8.40 0.90 7.45 RS H-meta-GGA M11 4.58 6.00 0.86 4.06 RS H-NGA MN12-SX 3.75 14.15 0.88 13.68 B97-D 4.16 6.02 0.87 4.48 GGA-D B97-D3 7.51 8.70 0.70 4.81 BP86-D3 4.36 4.27 0.86 0.74 H-GGA-D B3LYP-D3 4.41 4.34 0.87 0.78 RS H-GGA-D B97XD 3.88 4.31 0.89 2.14

TPSS is proved as the second best functional for reproducing BDE of C−Mg bond with aug-cc-pVDZ basis set. Deviations and error are 0.5 kcal/mol more than PBE0, while R of both DFs is similar. Compared to previous basis sets the deviations and error of TPSS are increased (up to 1-2 kcal/mol) with a lower R (0.91). Compared to Pople and Karlsruhe basis sets, the RMSD and MAE of M06-L with aug-cc-pVDZ basis set are decreased up to 1-2 kcal/mol (except RMSD value with 6-311G(d) basis set). RMSD and SD are reduced to 1 kcal/mol and also R value is decreased to 0.88. M11-L with aug-cc-pVDZ basis set shows less efficiency among three DFs of meta-GGA class. Although, the deviations and error are reduced up to 2 kcal/mol as compared to 6-31 G

154

(d) and Karlsruhe basis sets, but RMSD and MAE are increased to 1 kcal/mol when compared to 6-311G(d) basis set. SD is high in comparison to all previous basis sets and R is further decreased to 0.84. Based on these results, this class is designated as an average performer for the BDE measurement of C−Mg bond. ꞷB97X-D has deviations and error above 4 kcal/mol and R is below 0.90 which indicate its moderate performance for the BDE measurement of C−Mg bond. SD is increased up to 1 kcal/mol in comparison to Pople and def2-SVP basis sets. The reduction in RMSD and MAE is more than 1 kcal/mol but R is decreased between experimental and theoretical data.

Equation y = a + b*x Weight Instrumental Residual Sum of 13.0125 Squares 7 70 Pearson's r 0.95283 Adj. R-Square 0.90079 Value Standard Error C1 Intercept 4.6555 4.26994 C1 Slope 0.9193 0.08122

50 Theoretical C-Mg BDE (TPSS/6-31G(d)) BDE C-Mg Theoretical

40 45 50 55 60 65 70

Experimental C-Mg BDE (kcal/mol)

Fig. 3.20 Pearson's Correlation (R) of TPSS with 6-31G(d) Basis Set for BDE Calculations of C−Mg Bond

Selected DFs of H-GGA-D, GGA, GH meta-GGA and RS H-GGA classes with aug- cc-pVDZ basis set have deviations up to 4 kcal/mol and errors in the range of 0.66 kcal/mol to 5.35 kcal/mol while R in the range of 0.87 to 0.93. Except M05, M05-2X and M06-HF, all other DFs have low R value. Compared to previous basis sets, deviations and error are increased with low R value. Although, MAE is decreased for some DFs but still other factors (SD, RMSD and R) are high. Based on the results of

155 these DFs, all these DFT classes are classified as moderate performer for C−Mg BDE measurement. Previously, it has been reported that hybrid GGA and GGA classes are less efficient for Ru-based organometallic complexes [280] and similar results are observed in the current study as well. Some selected DFs of RS H-meta-GGA, NGA, meta-NGA and GGA-D classes with aug-cc-pVDZ basis set are also analyzed. Deviations and errors are in the range of 4-9 kcal/mol (except MAE of N12) and R is in the range of 0.77-0.95. The difference from previous basis sets is in the range of 1-3 kcal/mol which is still high (except R). Thus, these results reflect the less efficiency of respective classes for the BDE measurement of C−Mg bond of Grignard reagents. These findings are similar to previously reported results of NGA and GGA classes used for BDE measurement of organic singlet diradicals compounds [281].

MN12-SX functional with aug-cc-pVDZ basis set shows worst performance in current study. Although SD is 3.75 kcal/mol but RMSD and MAE are above 13 kcal/mol. Besides high deviation (RMSD) and error, the low R (0.88) value is observed for MN12-SX/aug-cc-pVDZ method. Results of MN12-SX with Dunning basis set are almost similar to Pople and Karlsruhe basis sets.

Among all, TPSS of meta-GGA class with 6-31G(d) basis set is observed as the best methodology for homolytic BDE measurement of C−Mg bond of Grignard reagents. Previously, the literature provides important information about the utilization of Grignard reagent from its synthesis to further applicability towards organic synthesis but limited theoretical work is done on the dissociation C−X bond. Shoko et al., used B3LYP/6-31G(d) method for the insertion of carbonyl group to Grignard reagent. They observed that SET (single electron transfer) mechanism is preferable, which involves the homolytic cleavage of C−Mg bond over polar mechanism [282]. However, with the advancement in field of computational chemistry, a huge variety of methods are developed specifically a number of DFs are developed which provides more reliable results than B3LYP with low computational cost. In our study, we selected DFs from thirteen DFT classes with three types of basis set to provide a better DF with a proper molecular orbitals descriptor for BDE measurement of C−Mg bond of Grignard reagents through benchmark studies. The method can be used for the mechanistic studies of C−Mg bond dissociation which can further broadens the area of Grignard reagents applications in field of organic synthesis.

156

3.5 Benchmark Study for BDE of M−O2 Bond for Water Splitting

3.5.1 Introduction

Petroleum, coal and natural gas are classical non-renewable sources of energy and cause environmental pollution by emitting CO2 to atmosphere [283]. Therefore, the scientific community has been struggling over the past few decades to develop new and alternative renewable sources of energy and fuel [284]. The search for renewable sources of energies is further demonstrated by the facts that the power consumption requirement would be doubled by 2050, whereas the fossil fuels are depleting rapidly

[285]. H2 is a green fuel which produces water as by-product after combusion. Water constitutes two third of the earth surface. It would be ideal if we can use water to produce hydrogen and then combust hydrogen to regenerate water [286]. In this perspective, catalytic water splitting using sunlight provides an attractive solution for a renewable energy source as well as a cleaner and greener future. Water splitting includes water oxidation and reduction. Water oxidation produces protons and electrons required to make renewable fuels [287]. Water is a plentiful and attractive candidate to be used as raw material. In this perspective, establishing a simple and superior catalytic system for efficient water oxidation is a challenging task.

3.5.2 Literature Review

A number of systems including metal oxides to composite materials, noble metal complexes to transition metal organometallics, mono to multinuclear site catalysts and various water oxidation complexes (WOC) have been investigated in a homogeneous environment and on the surfaces of photo or electrochemical conditions for water splitting [288]. This true catalytic system for efficient water splitting operates with four consecutive proton coupled electron transfer (PCET) steps to generate oxygen and hydrogen [289].

157

Fig. 3.21 Catalytic Water Oxidation and Dioxygen Evolution Mechanism by Ru- Complexes

Naturally some of the examples are present for production of clean fuel generation e.g. during photosynthesis tetra manganese oxygen evolving complex (OEC) undergoes water oxidation. OEC is involved in the four-electrons water oxidation process which results in the generation of dioxygen with the release of four protons. The whole process is completed through four step consecutive proton coupled electron transfer cycle. Synthetic chemists are using this idea for water splitting in artificial solar energy conversion complexes for fuel production [289].

3.5.3 Experimental Study

Philipp Kurz screened a set of six multinuclear manganese complexes for catalyzing the oxygen evolving reactions under coherent experimental condition [290]. Zong and Thummel synthesized a series of three well-organized mono- and di-nuclear Ru- complexes and added acetonitrile solution of Ru-catalyst to an aqueous (Ce (IV)-

CF3SO3H) solution at 24 °C. Oxygen evolution is observed for both mono- and di- nuclear Ru-systems [291]. Experimental study is based on hit and trial, many times it didn’t give the desired products. Recently, theoretical studies are being used for water splitting employing various transition metal complexes.

158

3.5.4 Theoretical Study

Theoretically, water oxidation has been investigated by using different methods and softwares [292]. Baran and Hellman analyzed metal hangman-Porphyrines as catalysts for electrochemical reduction of O2 and oxidation of H2O [293]. The literature reveals that rate-determining step in water oxidation involves the formation of dioxygen. So far, a well-established theoretical method which accurately predicts new catalytic system for water oxidation is missing. In this study, we aim to search an accurate method for rate determination step. Literature reveal that the best way to explore an accurate method is benchmark study (cost-effective and quality evolving study).

Five compounds are selected from literature to search an accurate and cost-effective method for M−O2 bond cleavage (for water splitting). The experimental BDEs of M−O2 bond of transition metal complexes with oxygen are already reported in literature [294– 296] and their structural representations (54-58) are given in Fig. 3.22. As already discussed, computational methodology includes fourteen DFs from seven classes of DFT with two series of basis sets. Theoretically calculated results are compared with already reported experimental results. For further validations of results, SD, RMSD, R and MAE are calculated. These results are given in Tables 3.22-3.23 and graphically represented in Figs. 3.23-3.25.

159

Fig. 3.22 The Structures of Transition Metal Complexes having Oxygen Molecule with

Known Experimental BDEs of M−O2 Bond

The spin multiplicity of optimized geometry of compound 54 is sextet and its resultant transition metal containing specie (-Co) has quartet spin state. The stable optimized geometry of Compound 55 has triplet spin state, and after dissociation resultant copper containing specie (-Cu) is stable in singlet state. The most stable geometries of compound 56, 57 and 58 and their resultant transition metals species (-TM (TM = Ir and Rh)) are in singlet spin sate. On the other hand, the oxygen (O2) is more stable in triplet state in each case. The dissociation asymptote is -M−O2 -M + O2 (M = Co, Cu, Ir and Rh) in each case. All these complexes favor the homolytic pathway according to octet rule to fulfil their electronic configuration.

3.5.5 Efficiency of Effective Core Potential and Pople Basis Sets

Two series of basis set are implemented, including LANL2DZ and SDD of ECP basis sets and 6-31G(d) and 6-31+G(d) basis sets of Pople basis sets.

160

3.5.5.1 Evaluation of DFs with SDD & 6-31+G(d) Basis Sets

SDD & 6-31+G(d) series of mixed basis sets is selected with fourteen DFs, and their results are statistically analyzed.

From the GH meta-GGA class, the best performance is observed for M06 functional with SDD & 6-31+G(d) basis sets. The results indicate that the deviations i.e. RMSD and SD are 1.04 kcal/mol and 2.47 kcal/mol, respectively. Over bound MAE is 1.80 kcal/mol and R between experimental and theoretical data is 0.98. RMSD, SD, R and MAE of M05-2X functional are 2.06 kcal/mol, 3.67 kcal/mol, 0.94 and 3.44 kcal/mol, respectively. These results illustrate the moderate efficiency of M05-2X with SDD &

6-31+G(d) basis sets for M−O2 BDE measurement. M05 and M06-2X functionals have above 3 kcal/mol of deviations whereas R values are 0.82 and 0.86, respectively. The underestimated error of M06-2X is high (5.49 kcal/mol) while the underestimated error of M05 is low (0.05 kcal/mol) with SDD & 6-31+G(d) basis sets. Due to high deviations and low R value, M05 is less efficient than M06 and M05-2X functionals. The results suggest that M06 functional is a better choice for BDE measurements of M−O2 bond and GH meta-GGA class is observed as better one among all selected DFT classes.

161

LANL2DZ & 6-31G(d) SDD & 6-31+G(d) 6

0 )

-2 cal/mol

k -4 (

-6 MAE -8

-10

-12

B97

M05

M06

BP86

TPSS

PBE0

LSDA

B3LYP

M05-2X

M06-2X

B3PW91

MPW1K B3LYP-D

Density Functionals CAM-B3LYP

Fig. 3.23 Mean Absolute Error (MAE) of Different Density Functionals with Two

Series of Basis Sets for M−O2 BDEs

BP86 from the GGA class is the second good performer observed for BDE measurement of M−O2 bond. RMSD, SD, R and MAE of BP86/SDD & 6-31+G(d) are 2.36 kcal/mol, 2.53 kcal/mol, 0.95 and 1.37 kcal/mol, respectively. Deviations and error are low with good R value. CAM-B3LYP from RS H-GGA class has low RMSD value of 1.70 kcal/mol but SD and MAE are above 3 kcal/mol. Pearson’s correlation is also lower between experimental and theoretical data. On the basis of results, CAM-B3LYP functional is designated as moderate performer for the current study. Two functionals are selected from H-GGA-D and meta-GGA classes (B3LYP-D from H-GGA-D and TPSS from meta-GGA). Lower efficiency is observed for both DFs with SDD & 6- 31+G(d) basis sets. The deviations are up to 5 kcal/mol. The low R value classifies these DFs as an average performer for respective bond cleavage. From H-GGA class B97, B3PW91, B3LYP, PBE0 and MPW1K functionals are selected for current study. The deviations and errors of all these DFs are high, and on the other side R values are

162 low. Their RMSD, SD and MAE values range from 2.73 kcal/mol to 11.92 kcal/mol. Whereas, R values range from 0.76 to 0.94. The addition of exchange-correlation (푥푐) decreases the self- interaction error (SIE) but static correlation error is appeared due to 푥푐 inclusion. Due to this reason the efficiency of this class is low for the treatment of transition metal complexes [296]. These analyses reflect the less efficiency of H-GGA class for required bond dissociation energy measurements.

LANL2DZ & 6-31G(d) SDD & 6-31+G(d)

4 SD(kcal/mol)

B97

M05 M06

BP86

TPSS

PBE0

LSDA

B3LYP

M05-2X M06-2X

B3PW91

MPW1K B3LYP-D

Density functionals CAM-B3LYP

Fig. 3.24 Standard Deviation (SD) of Different Density Functionals with Two Series of

Basis Sets for M−O2 BDEs

LSDA functional of LDA class showed the worse performance with SDD & 6-31+G(d) basis sets. Deviations and error are in the range of 6.67-13.66 kcal/mol, whereas R value is 0.80. So, this class is signified as the least efficient performer for BDE measurements of M−O2 bond.

163

Among all selected DFs, M06 of GH meta-GGA class has the best performance. KS theory is based on single Slater determinant and calculate exchange from it (Slater determinant). Which causes static correlation error; because of which it is more difficult to obtain good approximations for multi-reference systems when HF exchange is included. Transition metals and open-shell systems have multi-reference character which are also known as “strongly correlated”. The required functional for these systems must be an exchange–correlation energy functional which includes nonlocal exchange effects and does not have static correlation error. In the current scenario, most of DFs failed to illustrate the BDE measurement of M−O2 bond. The reason for the highest efficiency of M06 functional is its better treatment of the multi-reference character and minimizes the static correlation error, as already reported in literature [297]. Thus, highest efficiency of M06 is observed with SDD & 6-31+G(d) basis sets. On the other side, Stuttgart–Dresden (SDD) basis set treats the inner core electrons with a constant pseudopotential and the valence electrons with triple zeta valence basis set. So, its efficiency is more enhanced compared to LANL2DZ & 6-31G(d) basis sets. Pople’s 6-31+G(d) basis set with SDD basis set, is used to describe atoms other than transition metals and oxygen. The inclusion of diffuse function in Pople basis set stabilizes the molecules, their resultant radicals and completely explains their nature [298]. Literature reveals that Pople basis sets are the best basis set for BDE measurements of organic and inorganic molecules [299]. Due to all these merits of M06/SDD & 6-31+G(d) method, its performance is outstanding among all selected methods.

Table 3.22 RMSD, SD, R and MAE of M−O2 BDEs Calculated with Different DFs While Using SDD & 6-31+G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs RMSD SD R MAE

LDA LSDA 13.66 6.67 0.80 11.45

GGA BP86 2.36 2.53 0.95 1.37 meta-GGA TPSS 4.85 5.89 0.85 5.59 B3LYP 7.22 5.84 0.94 3.29 H-GGA B3PW91 11.92 9.10 0.94 2.75 PBE0 3.97 5.02 0.84 5.08

164

MPW1K 3.25 5.66 0.76 4.49 B97 8.33 6.51 0.89 2.73 M05-2X 2.06 3.67 0.94 3.44 M05 3.20 4.61 0.82 0.05 GH meta-GGA M06-2X 5.65 4.95 0.86 5.49 M06 1.04 2.47 0.98 1.80 RS H-GGA CAM-B3LYP 1.70 3.81 0.91 3.08 H-GGA-D B3LYP-D 3.62 4.05 0.89 4.81

3.5.5.2 Evaluation of DFs with LANL2DZ & 6-31G(d) Basis Sets

LANL2DZ & 6-31G(d) series of mixed basis sets is also implemented with fourteen DFs, and their statistically analyzed results are given in Table 3.23 and graphically represented in Figs. 3.23-3.26.

Table 3.23 RMSD, SD, R and MAE of M−O2 BDEs Calculated with Different DFs While Using LANL2DZ & 6-31G(d) Basis Set (All Values are Given in kcal/mol, Except R Which is Presented as Fraction of 1.0)

Classes of DFT DFs RMSD SD R MAE LDA LSDA 6.99 6.40 0.66 8.09 GGA BP86 5.08 5.08 0.69 0.37 meta-GGA TPSS 4.21 4.84 0.77 1.26 B3LYP 8.69 5.20 0.84 1.95 B3PW91 10.09 4.94 0.86 3.25 H-GGA PBE0 3.67 8.17 0.76 0.70 MPW1K 4.28 5.02 0.70 0.10 B97 9.20 5.14 0.85 2.41 M05-2X 3.54 4.33 0.90 3.51 M05 4.15 3.75 0.89 3.50 GH meta-GGA M06-2X 7.50 4.69 0.83 2.37 M06 2.74 3.54 0.90 2.56 RS H-GGA CAM-B3LYP 11.79 7.98 0.84 3.37 H-GGA-D B3LYP-D 6.86 5.50 0.85 0.07

165

Among four functionals of GH meta-GGA (M05, M05-2X, M06 and M06-2X), M06 functional also sustains its efficiency with LANL2DZ & 6-31G(d)) basis sets. RMSD, SD, MAE and R values are 2.74 kcal/mol, 3.54 kcal/mol, 0.90 and 2.56 kcal/mol, respectively (over bound error is observed here with LANL2DZ & 6-31G(d)). However, the efficiency of M06/LANL2DZ & 6-31G(d)) method is less in comparison to M06/SDD & 6-31+G(d) method. This is because that deviations and error are increased up to 1 kcal/mol, and R is decreased by 8%. Efficiency of the rest of the DFs (M05-2X, M05 and M06-2X) is further decreased due to more deviation, errors and lower R compared to SDD series of basis sets. RMSD, SD and MAE are in the range of 2.37 kcal/mol to 7.50 kcal/mol, and R is in the range of 0.83-0.90. Hence, good results of M06 reflect the better performer of respective class for desired data set.

LANL2DZ & 6-31G(d) SDD & 6-31+G(d) 14

10 )

kcal/mol ( 6

RMSD 4

B97

M05 M06

BP86

TPSS

PBE0

LSDA

B3LYP

M05-2X M06-2X

B3PW91

MPW1K B3LYP-D

Density Functionals CAM-B3LYP

Fig. 3.25 Root Mean Square Deviation (RMSD) of Different Density Functionals with

Two Series of Basis Sets for M−O2 BDE

166

BP86 functional of GGA class with LANL2DZ & 6-31G(d) got second position in reproducing good results against the BDE measurements of M−O2 bond. Over bound MAE of 0.37 kcal/mol is observed for this functional. Despite, the deviations are above 5 kcal/mol with lower R value of 0.69. B3LYP, B3PW91, B97, PBE0 and MPW1K functionals from H-GGA class have lower errors. As a result, this class is designated as good performer for desired data set. However, their deviations (RMSD and SD) are high. The results of statistical analyses indicate that the efficiency of these DFs further decreases with LANL2DZ & 6-31G(d) series of basis set in comparison to SDD series of basis sets. Conclusively, all these results illustrate the average performance of this class for required data.

B3LYP-D of H-GGA-D class with LANL2DZ & 6-31G(d) series of basis sets has a moderate performance for BDE measurement of M−O2 bond. MAE is minimized for this functional (B3LYP-D) compared to SDD series of basis sets. About 1-3 kcal/mol of increase in deviations is observed and R (0.85) is further lowered compared to SDD series of basis sets (although MAE is very low). On the other side, TPSS functional from meta-GGA class at LANL2DZ & 6-31G(d) series of basis sets shows exceptional behavior. SD and MAE of TPSS are decreased up to 1 kcal/mol but its RMSD value is increased with low R value between experimental and theoretical data in comparison to SDD series of basis sets. Hence, these results illustrate the moderate performance of meta-GGA class, similar to H-GGA-D class with LANL2DZ & 6-31G(d) series of basis sets. CAM-B3LYP functional of RS-HGGA has less efficient performance because the deviations (both RMD and SD) are 11.79 kcal/mol and 7.98 kcal/mol, respectively. The R (0.84) is low, although the under bound MAE of 3.37 kcal/mol is less compared to deviations. Comparatively, there is drastic increase in the deviations of CAM- B3LYP/LANL2DZ & 6-31G(d) method compared to CAM-B3LYP/SDD & 6-31+G(d) method. LSDA functional of LDA class has the least efficiency for the BDE measurements of M−O2 bond due to high deviations, error and low R value. Results reflect the worst performance of LDA class for desired data set.

M06 functional from the GH meta-GGA has better performance among all selected DFs. The efficiency is more enhanced at SDD & 6-31+G(d) basis sets in comparison to LANL2DZ & 6-31G(d) basis sets. Overall, M06 functional with SDD & 6-31+G(d) basis set is the best methodology for BDE measurements of M−O2

167 bond for water splitting. Our study shed light on one of the most important step in water splitting reactions through transition metal complexes. In literature, a number of papers are published on the experimental and theoretical work on water splitting but a detailed theoretical study on M−O2 bond dissociation step in such cycles was missing [274]. It is observed that a better computational method is needed for the BDE measurement of

M−O2 bond for water splitting. The benchmark study is a way that can provide an efficient method for M−O2 bond dissociation. For that purpose, we selected some examples of reversible oxygen binding with transition metal complexes from literature [294]. Different DFs are chosen from DFT method with composite basis sets (a combination of Pople and ECP basis sets [300]). Our studies provide cost-effective and an efficient DF (M06) with compatible basis sets (SDD & 6-31+G(d)).

Overall, a bench of density functional with suitable basis set are explored for the theoretical study of BDEs of chemically important bonds (C−X (Cl and Br), C−Sn,

C−CN, C−Mg and M−O2). Although in previous literature, some of these bonds (C−X and C−Sn) are studied but the computational methodology used is expensive. Besides, computational methodology, statistical analyses methods are limited. For better results, different combination of density functionals with basis sets are used for the respective bonds. For the validation and summarization of these results a variety of statistical methods are implemented. For each of the selected bond, a suitable method having best compromise between cost and accuracy is explored.

168

Chapter 4

4 Conclusions

169

4.1 Conclusions

In conclusion, the performance of various DFT methods has been examined for accurate measurement of dissociation energies measurements of chemically important bonds via benchmark approach. Selected chemical bonds are C−X (X = Cl and Br), C−CN, C−Sn,

C−Mg and M−O2 (M = Cu, Co, Ir and Rh) bonds. All these bonds have an important role in the rate determination steps of various chemical reactions.

For the BDE measurements of C−X (X = Cl and Br) bond of sixteen halogen-containing compounds, thirty-three DFs from eight DFT classes with four basis sets are implemented. The results obtained from this study illustrate that among all chosen DFT classes, RS H-GGA-D class excellently reproduces the BDE of C−X (X = Cl and Br) bond. Four long range DFT-D and DFT-D3 functionals show varying degree of accuracy with different basis sets, but consistent accuracy is found in case of ωB97X- D. ωB97X-D with Pople basis set 6-311G(d) has RMSD, SD, R and MAE of 3.14 kcal/mol, 3.05 kcal/mol, 0.97 and 1.07 kcal/mol, respectively. However, LSDA functional of LDA class has worst performance as compared to all selected DFs and the trend is consistent in all basis sets. The trend of increasing efficiency of selected DFT classes with 6-311G(d) basis set is as follows:

RS H-GGA-D > GGA-D > GGA > RS H-GGA > H-GGA > GH meta-GGA ~ meta- GGA > LDA

RS H-GGA-D class shows a similar trend of efficient performance with rest of the three basis sets. The order of preferred basis set based on performance is as follows: 6-311G(d) > 6-31G(d) > aug-ccpVTZ > aug-cc-pVDZ Nineteen DFs from eight different classes of DFT with four basis sets are selected and analyzed, in search of an accurate method for homolytic cleavage of C−Sn bond of ten organotin compounds. These comparative studies conclude that GGA-D class is proved as the best class for reproducing BDE of C−Sn bond among all classes of DFT. Overall, it is observed that BLYP-D3/SDD method shows the remarkable performance in reproducing BDE of C−Sn bond with more accuracy. The SD is 4.11 kcal/mol, RMSD is 3.9 kcal/mol, R is 0.963 and MAE is 0.01 kcal/mol, respectively. BMK-D3BJ functional of GH meta-GGA-D class has the poorest performance among chosen DFs

170 due to high RMSD and MAE. The trend of DFT classes with SDD basis set for BDE calculations of C−Sn bond is as follows:

GGA-D > GH meta-GGA > GH meta-GGA-D > DH-GGA-D > H-GGA-D > RS H- GGA-D > meta-GGA > LC-GGA-D

GGA-D class maintained its good performance with remaining three basis sets while all other classes show varying degree of efficiency. The increasing trend of four selected basis sets for BDE calculations of C−Sn bond is as follows:

SDD > def2-SVP > def2-TZVP > LANL2DZ

For the homolytic cleavage of C−CN bond of twelve organo-nitrile compounds, eight different DFT classes with 3 different classes of basis sets are analyzed. Statistical outcomes indicate that among the selected DFT classes, RS H-GGA class showed outstanding performance. CAM-B3LYP of RS H-GGA class shows the best performance among all selected DFs of DFT classes. The RMSD, SD, R and MAE of CAM-B3LYP/6-311G(d,p) method are 2.67 kcal/mol, 2.79 kcal/mol and 0.96 and 0.06 kcal/mol, respectively when compared with experimental data. LSDA functional of LDA class shows the poorest performance for desired data set. Efficiency of different DFT classes with 6-311G(d,p) basis set is described below in the increasing order of performance:

RS H-GGA > hybrid GGA > RS H-GGA-D > GH meta-GGA > H-GGA > meta-GGA > GGA > LDA RS H-GGA class of DFT maintained its good performance with remaining selected basis sets. The trend (improved efficiency) of selected basis sets for BDE measurements of C−CN bond is as follows:

6-311G(d,p) > 6-31G(d) > 6-31G(d,p) > 6-31+G(d) > 6-311++G(d,p) > aug-cc-pVTZ > aug-cc-pVDZ > def2-SVP

For BDE measurements of C−Mg bond of fifteen Grignard reagents; twenty-nine DFs from thirteen different DFT classes with three types basis sets are selected. Among all selected DFs, TPSS of meta-GGA class with 6-31G(d) basis set from Pople basis sets gave outstanding results and its SD, RMSD, R and MAE are 2.36 kcal/mol, 2.33

171 kcal/mol, 0.95 and 0.46 kcal/mol, respectively. B97-D functional of GGA-D class has the least efficiency based on high deviations and error with low R between experimental and theoretical data. Remarkable results are obtained for meta-GGA class of DFT, the increasing order of selected DFT classes are as follows: meta-GGA > H-GGA > GGA > RS H-GGA > NGA > H-GGA-D > RS H-GGA-D > meta-NGA > meta-GGA-D > GH meta-GGA > RS H-meta-GGA > RS H-NGA > GGA-D

Highest efficiency of meta-GGA class is observed with remaining basis sets. The trend of selected basis sets for BDE measurements of C−Mg bond is as follows:

6-31G(d) > 6-311G(d) > def2-SVP > aug-cc-PVDZ

For BDE measurements of M−O2 bond in five metal complexes with dioxygen; fourteen DFs from seven different DFT classes with two series of mixed basis sets are selected. These series of basis sets include LAN2DZ & 6-31G(d) and SDD & 6- 31+G(d) basis sets. GH meta-GGA class shows better performance. Among all selected DFs, M06 functional with SDD and 6-31+G(d) method shows outstanding results due to lower deviation, error and the best R between experimental and theoretical data. RMSD, SD, R and over bound MAE of M06/SDD & 6-31+G(d) method are 1.04 kcal/mol, 2.47 kcal/mol, 0.98 and 1.80 kcal/mol, respectively. This level of theory is considered as excellent for the BDE measurements of M−O2 bond in metal complexes. LSDA functional of LDA class is observed as least efficient performer for desired data. The proficiency of selected DFT classes with SDD series of basis sets is described below:

GH meta-GGA > GGA > RS H-GGA > H-GGA-D > meta-GGA > H-GGA > LDA GH meta-GGA class of DFT sustains its better performance with remaining basis sets.

The trend of selected basis sets for BDE measurements of M−O2 bond is as follows:

SDD & 6-31+G(d) > LANL2DZ & 6-31G(d)

These theoretical benchmark studies not only justify the already reported experimental results but are also fruitful for experimentalists and theoreticians working on the reactivity of these important bonds and predicting new chemical pathways.

172

Chapter 5

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