Computational Mechanistic Studies of Decamethyldizincocene Formation and the Enantioselective Reactive Nature of a Chiral Neutral Zirconium Amidate Complex By

Computational Mechanistic Studies of Decamethyldizincocene Formation and the Enantioselective Reactive Nature of a Chiral Neutral Zirconium Amidate Complex by

Steven Scott Hepperle B.Sc. Honours, University of Regina, Canada, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

The Faculty of Graduate Studies

(Chemistry)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2011 c Steven Scott Hepperle 2011 Abstract

Computational methods were employed to study the surprising 2004 synthesis of de- 5 camethyldizincocene, Zn2(η −C5Me5)2, which was the first molecule to have a direct, unbridged bond between two first-row transition metals. The computational re- 5 1 sults show that the methyl groups of decamethylzincocene, Zn(η −C5Me5)(η −C5Me5), affect the transition-state stability of its reaction with ZnEt2 (or ZnPh2) through steric hindrance, and this allows a counter-reaction, the homolytic dissociation of 5 1 5 1 Zn(η −C5Me5)(η −C5Me5) into Zn(η −C5Me5)• and (η −C5Me5)• radicals to occur, 5 1 and since no such steric hindrance exists when zincocene, Zn(η −C5H5)(η −C5H5), is used as a reactant, its dissociation never occurs. Experimentally, it was found that forming decamethyldizincocene is more efficient when using a reducing agent (e.g., KH) and ZnCl2 as opposed to a ZnR2 reagent. The computational results show that the methyl groups of decamethylzincocene have a similar indirect effect on the reaction. When zincocene is used, the reaction with + 1 − KH favours the route that results in the formation of the zincate, K [Zn(η −C5H5)3] . + 1 − However, the path of formation for the zincate K [Zn(η −C5Me5)3] is simply not favourable kinetically or thermodynamically, so the formation of decamethyldizincocene 5 1 is the only option when Zn(η −C5Me5)(η −C5Me5) is used. Finally, it had been found that a particular chiral neutral zirconium amidate complex makes an effective catalyst for cyclizing primary aminoalkenes in a highly enantioselective fashion. The computational analysis indicates that the reason why one enantiomer is favoured is because of steric interference with the catalytic backbone that is non-existent with the other enantiomer, and this affects the major transition states throughout the cycle. This finding agrees with the experimental hypothesis.

ii Preface

An early version of Chapter 2 (the reactions of zincocene and decamethylzincocene with

ZnEt2) was published in the Journal of Physical Chemistry (2008) with my research supervisor (Y.A. Wang) as the principal investigator (Hepperle, S.S.; Wang, Y.A. J. Phys. Chem. A 2008, 112, 9619-9622).1 The remainder of Chapter 2 as well as Chapter 3 is currently unpublished pending ﬁnal data. Again, Dr. Wang is the principal investigator on this work. The research in Chapter 4 was performed in collaboration with a Ph. D. student, Yue Chen, at Beijing Normal University in China. Yue Chen performed all calculations on the σ−bond insertion mechanism while I worked on the [2+2] cycloaddition mechanism. Before the collaboration began, Chen performed many calculations on the [2+2] mechanism as well which served as a check against my data. Due to time constraints, Chen provided me with the sterically correct starting geometries for the ﬁnal transition state (‡regeneration) for both the R and S conformers. Otherwise, the [2+2] data presented in this thesis (and the upcoming publication) were compiled by me. Yue Chen will be the primary author of this work while I will be the secondary author. Principal Investigators in this work include Guofu Zi and Weihei Fang (Beijing Normal University) with Laurel L. Schafer and Yan Alexander Wang (University of British Columbia).

iii Table of Contents

Abstract ...... ii

Preface ...... iii

Table of Contents ...... iv

List of Tables ...... vii

List of Figures ...... x

List of Abbreviations ...... xiii

Acknowledgements ...... xiv

Dedication ...... xvi

1 Introduction ...... 1 1.1 Discussion of Computational Chemistry ...... 1 1.1.1 Simplifying an Impossibly-Sized System ...... 1 1.1.2 Introducing the Electronic Wave Function ...... 2 1.1.3 The Born-Oppenheimer Approximation ...... 3 1.1.4 The Orbital ...... 4 1.1.5 Building the Electronic Wave Function ...... 7 1.1.6 Hartree-Fock Self-Consistent Field Method ...... 9

iv Table of Contents

1.1.7 Geometry Optimizations ...... 13 1.1.8 Density Functional Theory ...... 13 1.1.9 Conﬁguration Interaction ...... 16 1.1.10 Møller-Plesset Second-order Perturbation Theory ...... 17 1.1.11 Solvent Eﬀects ...... 18 1.1.12 Vibrational Frequencies of Molecules ...... 19 1.1.13 Equilibrium Statistical Mechanics ...... 21 1.2 Metallocene Chemistry and Decamethyldizincocene ...... 24 1.3 Enantioselective Catalysis with Primary Aminoalkenes ...... 27

2 Formation of Decamethyldizincocene via

Decamethylzincocene and ZnR2 ...... 32 2.1 Strategy and Methods ...... 32 2.2 Results and Discussion ...... 34

2.2.1 Neutral-Charge Electrostatic Dizincocene Formation via ZnEt2 5 1 and Zn(η −C5H5)(η −C5H5) ...... 35 2.2.2 Neutral-Charge Electrostatic Decamethyldizincocene Formation 5 1 via ZnEt2 and Zn(η −C5Me5)(η −C5Me5) ...... 35 2.2.3 Radical Dissociation of Parent Zincocenes ...... 38

2.2.4 Reaction of Parent Zincocenes with Other ZnR2 Reagents . . . 38 2.3 Summary and Further Work ...... 42

3 Formation of Decamethyldizincocene via

KH and ZnCl2 ...... 43 3.1 Strategy and Methods ...... 43 3.2 Results and Discussion ...... 47 3.2.1 Reaction of 1 and 5 with KH ...... 47

3.2.2 Reaction of 1 and 5 with KH and ZnCl2 ...... 50

v Table of Contents

3.3 Summary and Future Work ...... 58

4 Enantioselective Catalysis of Primary Aminoalkenes via a Chiral Neu- tral Zirconium Amidate Complex ...... 59 4.1 Strategy and Methods ...... 59 4.2 Results and Discussion ...... 60 4.3 Summary and Future Work ...... 67

5 Conclusion ...... 68

Bibliography ...... 71

Appendices

Appendix A - Calculated Energies for Chapter 2 ...... 77

Appendix B - Calculated Energies for Chapter 3 ...... 83

Appendix C - Calculated Energies for Chapter 4 ...... 88

vi List of Tables

2.1 Energy compilation of ‡symmetric and the energy diﬀerence (in kcal/mol) between the symmetric process and radical dissociation of the parent zincocene...... 41

3.1 Reaction energies (in kcal/mol) of 1 and 5 reacting with KH (from Fig- ures 3.2 and 3.3, respectively). Energies are relative to separated reactants. 50

3.2 Reaction energies (in kcal/mol) of 1 and 5 reacting with KH and ZnCl2 (from Figures 3.11 and 3.12, respectively). Energies are relative to separated reactants...... 57

A.1 Reaction energies (in hartrees) of 5 reacting with ZnEt2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.) ...... 77

A.2 Reaction energies (in hartrees) of 1 reacting with ZnEt2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.) ...... 78

A.3 Reaction energies (in hartrees) of 5 reacting with ZnMe2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF- PCM for diethyl ether.) ...... 78

A.4 Reaction energies (in hartrees) of 1 reacting with ZnMe2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF- PCM for diethyl ether.) ...... 79

A.5 Reaction energies (in hartrees) of 5 reacting with ZniPr2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF- PCM for diethyl ether.) ...... 79

vii List of Tables

A.6 Reaction energies (in hartrees) of 1 reacting with ZniPr2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF- PCM for diethyl ether.) ...... 80

A.7 Reaction energies (in hartrees) of 5 reacting with ZnPh2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.) ...... 80

A.8 Reaction energies (in hartrees) of 1 reacting with ZnPh2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.) ...... 81 A.9 Homolytic dissociation energy components for (in hartrees) for 5, 6, 1, and 3. Raw ROB3LYP complex energies include counterpoise corrections. B3LYP Thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether. ROMP2 SP Counterpoise corrections computed separately using counterpoise optimized B3LYP

geometries. zc = CpZnCp, dz = CpZnZnCp, dmz = Me5CpZnCpMe5,

dmdz = Me5CpZnZnCpMe5...... 82

B.1 Homolytic dissociation energy components for (in hartrees) for 3 and

6 (Step 3a), and CpZnK and Me5CpZnK (Step 2a). Raw ROB3LYP complex energies include counterpoise corrections. B3LYP Thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for Tetrahydrofuran. ROMP2 SP Counterpoise corrections computed separately using counterpoise optimized B3LYP geometries. dz = CpZnZnCp,

dmdz = Me5CpZnZnCpMe5...... 83 B.2 Reaction energies (in hartrees) of 5 reacting with KH. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran...... 84 B.3 Additional reaction energies (in hartrees) of 5 reacting with KH as well

as ZnCl2. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran...... 85 B.4 Reaction energies (in hartrees) of 1 reacting with KH. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran...... 86

viii List of Tables

B.5 Additional reaction energies (in hartrees) of 1 reacting with KH as well

as ZnCl2. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran...... 87

C.1 Reaction energies (in hartrees) for the catalytic cycle for the S product. B3LYP thermal corrections were at 383 K and MP2 single point energies used IEF-PCM for toluene...... 88 C.2 Reaction energies (in hartrees) for the catalytic cycle for the R product. B3LYP thermal corrections were at 383 K and MP2 single point energies used IEF-PCM for toluene...... 89

ix List of Figures

1.1 Plots of φ(r) vs. r for a Gaussian function compared with a Slater function (α=1, left) and a Slater function (α=1) compared with STO- 1G, STO-2G, and STO-3G functions (right)...... 6 1.2 Structure of ferrocene...... 24

5 1 5 1.3 Reaction of Zn(η −C5Me5)(η −C5Me5) with ZnEt2 to form Zn2(η −C5Me5)2 5 and Zn(η −C5Me5)Et...... 25

5 1 1.4 Reaction of Zn(η −C5Me5)(η −C5Me5) with KH or KH and ZnCl2. . . 26

5 1 1.5 Reaction of Zn(η −C5H5)(η −C5H5) with KH or KH and ZnCl2. . . . . 26 1.6 Zr precatalyst and its reaction with 5-amino-4,4-dimethylpent-1-ene to produce 2,4,4-trimethylpyrrolidine...... 28 1.7 Activated Zr catalyst forming 2,4,4-trimethylpyrrolidine via σ−bond insertion (left) or [2+2] cycloaddition mechanisms (right)...... 29

5 1 5 2.1 Reactions of Zn(η −C5Me5)(η −C5Me5) with ZnR2 to form Zn2(η −C5Me5)2 5 5 1 and Zn(η -C5Me5)R (top) and Zn(η −C5H5)(η −C5H5) with ZnR2 to 5 5 form Zn2(η −C5H5)2 and Zn(η −C5H5)R (bottom)...... 33

5 1 2.2 The HOMO of ZnEt2 (top right) and the LUMO of Zn(η −C5H5)(η −C5H5) (bottom right)...... 34

2.3 Reaction pathway of ZnEt2 reacting with zincocene (zinc atoms appear in red and hydrogens have been removed for clarity)...... 36

2.4 Reaction pathway of ZnEt2 reacting with decamethylzincocene. . . . . 37

5 1 5 2.5 Dissociation of Zn(η −C5R5)(η −C5R5) into Zn(η −C5R5)• and (C5R5)• radicals...... 38

x List of Figures

2.6 Simpliﬁed reaction pathways of ZnMe2 (top), ZniPr2 (middle), and ZnPh2 (bottom) reacting with zincocene...... 39

2.7 Simpliﬁed reaction pathways of ZnMe2 (top left), ZniPr2 (top right), and

ZnPh2 (bottom) reacting with decamethylzincocene...... 40

5 1 5 1 3.1 Reactions of Zn(η −C5Me5)(η −C5Me5) (top) and Zn(η −C5H5)(η −C5H5)

(bottom) with KH or KH and ZnCl2...... 44 3.2 Reaction scheme for 1 reacting with KH...... 45 3.3 Reaction scheme for 5 reacting with KH...... 46 3.4 Reaction complex for KH reacting with 5 (left) and 1 (right). Zinc atoms appear in red. Potassium atoms are purple. Hydrogens have been removed for clarity...... 47 3.5 Top: Transition states (Step 1a) for KH reacting with 5 to form CpZnK

and CpH (top left) and with 1 to form Me5CpZnK and Me5CpH (top right). Bottom: Transition states (Step 1b) for KH reacting with 5 to

form CpZnH and CpK (left) and with 1 to form Me5CpZnH and Me5CpK (right)...... 48 3.6 Top: Reactant complexes (pre-Step 2d) for KH reacting with CpZnH to

form CpZnK and H2 (top left) and with Me5CpZnH to form Me5CpZnK

and H2 (top right). Bottom: Transition states (Step 2d) for KH reacting

with CpZnH to form CpZnK and H2 (left) and with Me5CpZnH to form

Me5CpZnK and H2 (right)...... 49

3.7 Top: Reactant complexes (pre-Step 2e) for ZnCl2 reacting with CpZnK

to form CpZnZnCl and KCl (top left) and with Me5CpZnK to form

Me5CpZnZnCl and KCl (top right). Bottom: Transition states (Step

2e) for ZnCl2 reacting with CpZnK to form CpZnZnCl and KCl (bottom

left) and with Me5CpZnK to form Me5CpZnZnCl and KCl (bottom right). 51

3.8 Top: Reactant complexes (pre-Step 2f) for ZnCl2 reacting with CpZnH

to form CpZnZnCl and HCl (top left) and with Me5CpZnH to form

Me5CpZnZnCl and HCl (top right). Bottom: Transition states (Step

2f) for ZnCl2 reacting with CpZnH to form CpZnZnCl and HCl (bottom

left) and with Me5CpZnH to form Me5CpZnZnCl and HCl (bottom right). 52

xi List of Figures

3.9 Top: Reactant complexes (pre-Step 3b) for CpK reacting with CpZnZnCl

to form 6 and KCl (top left) and Me5CpK reacting with Me5CpZnZnCl to form 3 and KCl (top right). Bottom: Transition states (Step 3b) for CpK reacting with CpZnZnCl to form 6 and KCl (bottom left) and

Me5CpK reacting with Me5CpK to form 3 and KCl (bottom right). . . 53 3.10 Reactant complex (top left), transition state (top middle), and zincate product (top right) for CpK reacting with 5 to form 9, and the reactant complex (bottom left) transition state (bottom middle) and zincate

product (bottom right) for Me5CpK reacting with 1 to form 8 (Step 2c). 54

3.11 Reaction scheme for 1 reacting with KH and ZnCl2...... 55

3.12 Reaction scheme for 5 reacting with KH and ZnCl2...... 56

4.1 Precatalyst activation process to form activated bis-amido complex. . . 60 4.2 Proton transfer from tethered bis-amido to form tethered imido (catalytic backbone removed for clarity)...... 61 4.3 [2+2] cycloaddition mechanism for R (red) and S (green) products starting from activated bis-amido...... 62 4.4 Detailed catalytic cycle for S product starting from the tethered imido. 63 4.5 Detailed catalytic cycle for R product starting from the tethered imido. 64 4.6 Proposed metallacycles for R (left) and S cycles (right) product from (-) precatalyst...... 65 4.7 Computed metallacycles for R (left) and S cycles (right). Nitrogens appear in blue while zirconiums are light green...... 66

xii List of Abbreviations

B3LYP Becke-3 Lee Yang Parr CI Conﬁguration Interaction CC-PVDZ Correlation-Consistent Polarized Valence Double Zeta DFT Density Functional Theory IEF Integral Equation Formalism (Solvent Model) IRC Intrinsic Reaction Coordinate LCAO-MO Linear Combination of Atomic Orbitals (to form) Molecular Orbitals LUMO Lowest Unoccupied Molecular Orbital HETS Highest Energy Transition State HF Hartree-Fock HOMO Highest Occupied Molecular Orbital KH Potassium Hydride KS Kohn-Sham LANL2DZ Los Alamos National Laboratory 2-Double Zeta MARI Most Abundant Reactive Intermediate MP2 Møller-Plesset Second-order Perturbation Theory PCM Polarizable Continuum Model SCF Self-Consistent Field SCRF Self-Consistent Reaction Field STO Slater-Type Orbital TDI Turnover Determining Intermediate TDTS Turnover Determining Transition State THF Tetrahydrofuran TOF Turnover Frequency UEG Uniform Electron Gas XC Exchange Correlation

xiii Acknowledgements

I would first like to acknowledge the 14 wonderful chemistry professors that I have had the pleasure of learning from during my 10-year career as a university student: Johannes V. Barth, W. David Chandler, Donna L. Draper, Allan L. L. East, Brian D. Kybett, Donald G. Lee, Mark J. MacLachlan, Lynn M. Mihichuk, R. Scott Murphy, Renata J. Raina, R. Reid Robinson, Mark E. Thachuk, Yan Alexander Wang, and Andrew G. H. Wee. With respect to my research, I would first like to acknowledge Dr. Allan L. L. East from the University of Regina for providing a somewhat aimless second-year un- dergraduate with a career focus. I would also like to thank my supervisor, Dr. Yan Alexander Wang, for putting up with my “all or nothing” approach to my projects over the years. I would also like to acknowledge the guidance of Dr. Laurel L. Schafer. Since my research involves studying chemical reactions, having a strong experimentalist for consultation was invaluable (especially for Chapters 3 and 4 of this thesis). The members of the Wang Group: Lei Vincent Liu, Dr. Yakun Chen, Dr. Yu Adam Zhang, Dr. Baojing Zhou, and Cindy Li all encouraged me to better myself over the years and provided me with an introduction to Chinese culture. The first three in this list were also extremely helpful during my studies for their advice regarding my Comprehensive Exam and my first publication. I would like to especially acknowledge Dr. Yu Adam Zhang for being the smartest person I have ever met, for always going the extra mile to help me understand various quantum mechanical concepts, and for his advice on how to better use the Gaussian software package. I would like thank Dr. Roman V. Krems for allowing me to monopolize his group’s compute cluster at times over the last three years. Although WestGrid also provided computing resources, Dr. Krems’ cluster has far less traffic and usage is much less restricted. Also, Drs. Mark E. Thachuk and Roman Baranowski were extremely generous with

xiv Acknowledgements their time helping me grasp Linux and doing most of the work with compiling the various software packages. Dr. Baranowski also was extremely helpful (in his capacity as UBC’s WestGrid Technical Site Lead) with making sure all of my jobs were able to run normally and as eﬃciently as possible. I would like to gratefully acknowledge the Gladys Estella Laird Foundation for providing me with funding to begin my graduate program, the Natural Sciences and Engineering Research Council of Canada for providing me with four years of PGS funding (2005-2009), and the Izaak Walton Killam Memorial Foundation for awarding me a pre-Doctoral scholarship unexpectedly in 2006 (2006-2008). Finally, I would like to acknowledge the guidance of my Grade 10 English Teacher, Mr. Wayne Hunter, for without his encouragement during my high school days, my academic career would have turned out very diﬀerently (most certainly not as posi- tively).

xv Dedication

I would like to dedicate this thesis to my parents: Valerie, whose advice and guidance over the years has been invaluable and direct, and Ronald, who never let the prospect of long odds hamper his life choices. Also, I would like to dedicate this thesis to the residents (past and present) of St. John’s College at UBC, who have proved that social growth is just as important as academic growth.

xvi Chapter 1

Introduction

1.1 Discussion of Computational Chemistry

1.1.1 Simplifying an Impossibly-Sized System

The theoretical study of chemical reactions involves calculating the behaviours of molar quantities of molecules randomly colliding in space. To completely understand a problem like this on paper (or even with a supercomputer) would simply be impossible, and it needs to be simpliﬁed. Audio encoders like MP3 and MPEG4-AAC compress sound recordings by taking into account that human ears:2

1. cannot distinguish between two sounds that have very similar frequencies.

2. cannot resolve two sounds that diﬀer greatly in frequency but also in volume (in most cases).

3. have an auditory perception range that is typically between 20Hz and 20kHz, but the most sensitive (and therefore the most important) range is between 2kHz and 4kHz.

Using these guidelines, a sound file can be greatly simplified (typically by an order of magnitude), and the result is a compressed file which is a fraction of the size of the source but one that preserves the majority of the source quality. All computational chemistry methods more or less use a similar ideology. Because of the many simplifica- tions involved in these calculations, it can be argued that these “real” systems are being studied at a basic level, but the majority of the vital information about the systems is captured.

1 1.1. Discussion of Computational Chemistry

Consider the reaction:

A + B −→ C + D. (1.1)

Regardless of whether a reaction occurs in the liquid or gas phase, enormous quantities of reactant atoms (or molecules) are involved, and there are going to be an uncountable amount of random collisions amongst them. One simpliﬁcation involves completely ignoring most of these collisions, and therefore an ideal gas approach to studying reactions is taken. Homo-collisions between A’s and between B’s are ignored since these will have little to no eﬀect on the reaction, and hetero-collisions between A’s and B’s are also ignored unless those collisions happen to correspond to transition-state (or saddle-point) geometries,

[A + B]‡. (1.2)

The system has now been colossally simplified by considering only single molecules, or, in the cases of transition states and their corresponding intermediates, single super- molecules. Computing thermal properties of these so-called non-interacting (or ideal) systems can be done trivially using Fermi-Dirac statistics (discussed in Section 1.1.13). Still, one major problem remains: even though the problem has been simplified to deal- ing with single molecules, how does one begin to understand these molecules? Also, even though this ideal approach works very well, solvent effects can also be very important in a reaction, and these unfortunately are ignored as well. So, in Section 1.1.11, implicit solvent models and their effect on electronic calculations will be discussed.

1.1.2 Introducing the Electronic Wave Function

The ﬁrst postulate of quantum mechanics conveys that “the state of a quantum mechanical system is completely speciﬁed by a function Ψ(r1, ···, rN , t) that depends on the coordinates of the N particle(s) and on the time. This function, called the wave ∗ function, has the important property that Ψ (ri, t)Ψ(ri, t)dxdydz is the probability that th the i particle lies in the volume element dxdydz, located at ri, at the time t.” Since the vast majority of computational calculations are time-independent (or time-averaged), the time portion of the wave function is irrelevant, and only the spatial coordinates, r,

2 1.1. Discussion of Computational Chemistry are the issue.3 The wave function therefore has the form:

Ψ(x1, x2, ···xN, R1, R2, ···, RM), (1.3)

where xi and RA refer to the three-dimensional coordinates of the electrons and nuclei, respectively. In order to calculate the energy of this wave function (and therefore the system), the time-independent Schr¨odingerequation is used:

Hˆ Ψ(r) = EΨ(r), (1.4) where Hˆ is the Hamiltonian operator, and E is the total energy. From this, it is apparent that Ψ must be an eigenfunction of Hˆ . For a molecular system where the nuclei and electrons are moving about in space, Hˆ has the form:

n M N M N N M N ˆ 1 X 2 1 X 1 2 X X ZA X X 1 X X ZAZB H = − ∇i − ∇A − + + . (1.5) 2 2 MA riA rij RAB i=1 A=1 i=1 A=1 i=1 j>i A=1 B>A

The indices i and j correspond to the ith and jth electrons, respectively while indices A and B correspond to the Ath and Bth nuclei, respectively. The ﬁrst two terms of the Hamiltonian operator concern the one-particle kinetic energies of the electrons and nuclei, respectively. The third term concerns the attraction between the electrons and the nuclei, while the last two terms concern the two-particle interactions between the electrons and nuclei, respectively. This is a very complicated problem not simply because there are a lot of sub-operators, but because nuclear motion is coupled with electronic motion within the wave function, and this becomes a mathematical nightmare even for the smallest of molecules. As will be discussed in Section 1.1.3, it turns out that in most cases, only the electronic motion needs to considered rather than the motion of all particles in the system.

1.1.3 The Born-Oppenheimer Approximation

When calculations on molecules are being performed, the objective is finding the orbital and geometrical configurations which produce the lowest possible electronic energy. Ge- ometrical configurations will be discussed in Sections 1.1.7 and 1.1.12. There are four

3 1.1. Discussion of Computational Chemistry main components of the electronic energy: 1) the kinetic energy of the electrons, 2) the attraction of the electrons to the nuclei, 3) the repulsion between electrons, and 4) the exchange energy between electrons of like spin. It is the second term which highlights the difficulty in finding the electronic energy since the electrons and the nuclei are “coupled” to each other. As the nuclei move, so do the electrons. However, since electrons have approximately 1/2000th the mass of protons and neutrons, the nuclei move very slowly and on a completely different time scale than the electrons. The nuclear and electronic wave functions are therefore decoupled in this approximation, and hence only the electronic wave function needs to be considered. The result of this approximation is a multi-dimensional potential energy surface (the number of dimensions is equal to either 3N−6 or 3N−5 (for linear molecules) where N is the number of atoms) where the nuclei are free to move about and the electrons instantaneously adjust to the nuclear positions.4 There are some minor short-comings of this approximation (as well as the similar adiabatic approximation), but since this whole chapter is meant to be an introduction, they will not be discussed.

1.1.4 The Orbital

Though not a physical observable, the orbital is one of the first concepts that students studying chemistry learn about. Molecules react with each other via orbital interactions, so coming up with an accurate orbital picture is important if reactions are to be correctly understood. In 1924, Wolfgang Pauli proposed that no two fermionic particles (including electrons) can occupy the same quantum state simultaneously. More simply, this resulted in the proposal of a unique set of four quantum numbers for each electron, {n, l, ml, ms}. These terms describe the principal quantum number (n), the orbital angular momentum quantum number (l), the magnetic quantum number (ml, a projection of the orbital angular momentum along a specified axis), and the intrinsic spin angular momentum quantum number (ms) related to each individual electron. 1 1 Each orbital can contain only two electrons with m values of either + or − . s 2 2 In 1930, John Slater proposed that orbital functions have the basic form rn−1 e−αr where α is an orbital exponent and r is the distance from the nuclei.5 This choice comes as no surprise because both the Hˆ and fˆ operators (Eqns. 1.5 and 1.10, respectively) contain differential (d/dr-type) components, and er -type functions are the only ones known to be eigenfunctions of differential operators. There is a roadblock to using Slater

4 1.1. Discussion of Computational Chemistry

(or e−αr ) functions though: two electron integrals containing them do not contain simple analytical antiderivatives!6,7 Therefore, the millions of Coulomb and exchange integrals that need to be evaluated in an electronic calculation would have to be evaluated numerically, and this would be very taxing on modern supercomputers except for the smallest of systems. In 1950, Samuel Francis Boys proposed constructing the orbitals as linear combinations of Gaussian (e−αr2 ) basis functions.8 Coulomb and exchange integrals, which are four-centre, two electron integrals, are easily integrable analytically when they are constructed from Gaussian-type orbitals (GTOs), and this is because the Gaussian Product Theorem in this context states that four-centre integrals can be reduced to a sum of two-centre integrals, and finally, a sum of one-centre integrals. Using Gaussian functions greatly simplifies electronic structure calculations. However, as a side effect of this simplification, more than one Gaussian function needs to be used to represent an atomic orbital since Gaussian functions dissipate too quickly from the nucleus due to the r2 factor, and they also lack a cusp at the nucleus (when r = 0, Figure 1.1). The normalized, radial form of a Slater function for an atomic 1s orbital is:

3 −αr φSlater(r) = (α /π)e . (1.6)

Conversely, the normalized, radial form of a Gaussian function for an atomic 1s orbital is: −αr2 φGaussian(r) = (2α/π)e . (1.7)

In electronic structure calculations, in order to mimic the characteristics of Slater- type functions, atomic orbital basis functions are made up of linear combinations of Gaussian functions. The simplest example of this process is to consider Slater-type orbitals (STOs).9,10 The STO-1G set uses only one Gaussian function, STO-2G uses two, and STO-3G three. Each set has predetermined orbital exponents, α, and contrac- tion coeﬃcients, d, for each Gaussian function. Figure 1.1 shows that as more Gaussian functions are added, the closer the overall function exhibits Slater-like behaviour. There is still a lack of a cusp at the nucleus with STO-3G, but the behaviour of φ(r) at large distances from the nucleus is largely corrected for. Since valence interactions between atoms are mainly responsible for chemical bonding, having φ(r) behave properly in the valence region is the most important quality of a basis function.

5 1.1. Discussion of Computational Chemistry

Figure 1.1: Plots of φ(r) vs. r for a Gaussian function compared with a Slater function (α=1, left) and a Slater function (α=1) compared with STO-1G, STO-2G, and STO-3G functions (right).

Each spin orbital is composed of a Gaussian-type function multiplied by a spin function (α or β) :

χ(r) = φ(r)σ(s), σ = α, β. (1.8)

The individual spin-orbital wave functions χa must be eigenfunctions of the Fock operator, fˆ:11 ˆ f(1)χa(1) = aχa(1), (1.9)

th where a is the orbital energy of electron 1 (the reference electron) existing in the a spin-orbital.

The full form of Eq. (1.9) is:

  M n Z ∗ 1 2 X ZA X χb(2) χb(2)dr2 − ∇1χa(1) − χa(1) +   χa(1) 2 riA r12 A=1 b6=a   n Z ∗ X χb(2) χa(2)dr2 −   χb(1) = aχa(1). (1.10) r12 b6=a

6 1.1. Discussion of Computational Chemistry

The math can be tricky to follow, but what Eq. (1.10) means is that to compute the orbital energy of the ath orbital, the reference electron (1) is placed inside it. Computing the ﬁrst two terms (the one electron energies) is trivial, but the last two terms require the computation of every possible Coulombic and exchange interaction that electron 1 in the ath orbital can have with electron 2, which can exist in any one of the other occupied bth orbitals. The ﬁnal term of the expression requires electrons 1 and 2 to switch places (hence why electron 1 exists in the bth orbital).

1.1.5 Building the Electronic Wave Function

The overall ground-state electronic wave function Ψ0 is not simply a multiplicative product of spin orbitals:

Ψ0 = χ1(r1)χ2(r2) ··· χN (rN). (1.11)

This type of wave function is called a Hartree product. The problem with the Hartree product is two-fold: 1) Each orbital is completely independent of the others and they each contain a speciﬁc electron, but the antisymmetry principle states that electrons are indistinguishable. More importantly, 2) the concept of a “Coulomb hole” is ignored. Because of Coulombic repulsion, two electrons cannot exist in the same place at the same time. However, with a Hartree product, since each spin orbital is independent of the others, the probability density distribution for each orbital is also independent, which means that in theory, two electrons could actually co-exist in space-time (which is completely unphysical, i.e., it cannot possibly occur in reality). One form of an electronic wave function is a Slater determinant, Eq. (1.12). These determinants satisfy the criteria that electronic wave functions are antisymmetric with respect to the exchange of two electrons (changing the positions of two electrons switches the sign of the wave function):12

χ1(r1) χ2(r1) ··· χN (r1)

1 χ1(r2) χ2(r2) ··· χN (r2)

Ψ0 = √ . . . . , (1.12) N! . . . .

χ1(rN) χ2(rN) ··· χN (rN)

7 1.1. Discussion of Computational Chemistry or, more simply,

1 Ψ0 = √ Det[χ1(r1) χ2(r2) ··· χN (rN)]. (1.13) N!

To compute the energy of a Slater determinant (i.e., the electronic wave function), the variational theorem is employed. It states that any trial wave function will have an energy that is greater than or equal to the true ground-state energy:

R ∗ ˆ Ψ0HΨ0dr R ∗ ≥ E0, (1.14) Ψ0Ψ0dr

∗ where the function Ψ0 represents the complex conjugate of the wave function. In order to compute the expectation value of any operator, the complex conjugate must be considered. If a function has an imaginary component, then that component must be multiplied by −1 (e.g., the complex conjugate of a − i is a + i). Also, when ﬁnding the complex conjugate of a matrix, all of the matrix elements must have their imaginary components multiplied by −1.

R ∗ If the wave function is normalized, i.e., Ψ0Ψ0dr = 1, then the expression simpliﬁes R ∗ ˆ to Ψ0HΨ0dr ≥ E0. Throughout the rest of this chapter, the simpler Dirac notation will be used exclusively, and therefore Eq. (1.14) becomes

ˆ hΨ0|H|Ψ0i ≥ E0. (1.15)

If a Slater determinant is used as the trial wave function (as is typical) and it is plugged into Eq. (1.15), the ground-state energy is expressed as:

This simpliﬁed expression is similar to the expression for computing orbital energies, Eq. (1.10). Each integral ha|hˆ|ai represents a one-electron energy (the kinetic energy of the electron in orbital a and the nuclear-electron attraction energy of the electron in the orbital a to all of the nuclei). The integrals haa|bbi represent the two-electron repulsion integrals, while the hab|bai integrals represent two-electron exchange integrals.

8 1.1. Discussion of Computational Chemistry

However, the energy of the wave function (Slater determinant) is not simply equal to the sum of the orbital energies. There are N−1 Coulombic repulsion terms involved for each orbital energy (Eq.

(1.10)), but for the overall electronic energy there are nC2 repulsion terms, so simply summing the orbital energies together would mean that many repulsion terms are counted twice.

1.1.6 Hartree-Fock Self-Consistent Field Method

So, it has been discussed that the electronic wave function for a molecule has the form of a Slater determinant of spin orbitals, and each orbital is constructed with Gaussian- type atomic orbital functions (with a spin component). Even the energy of this wave function can be computed using the concepts learned thus far, but the question is: how does one go about optimizing a wave function to get the lowest possible energy (i.e., the ground-state energy). To do this, the self-consistent ﬁeld (SCF) method is employed.11 It is extremely important to note that with this method, the wave function is not optimized directly, rather the orbitals themselves are optimized. More speciﬁcally, the orbitals best for a single Slater determinant wave function are optimized. The hierarchy of molecular orbitals should be restated:

1. basis functions are linear combinations of m Gaussian functions (if m > 1, these are called contracted basis functions).

2. atomic orbitals are linear combinations of solo/contracted basis functions.

3. molecular orbitals are linear combinations of atomic orbitals (LCAO-MO).

With the SCF method, molecular orbitals are delocalized throughout the entire molecule, so each molecular orbital can use the full spectrum of basis functions in its description. A typical basis set (like 6-31G) involves using one contracted basis function for each core orbital on the atom (m = 6) and two contracted functions for each valence orbital: the ﬁrst function is a combination of three Gaussian functions (m = 3) while the second is a solo basis function (m = 1). These basis functions have coeﬃcients (Cµi) (where µ is the spatial molecular orbital index and i is the basis function index) that describe

9 1.1. Discussion of Computational Chemistry the algebraic contribution of the ith basis function to the µth orbital. For example, in a 6-31G description of methane, the carbon 1s basis function will obviously have a large contribution ( Cµi ≈ 1) to the description of the lowest energy a1 molecular orbital (which represents the carbon 1s atomic orbital). However, the same 1s basis function will have a negligible contribution to the description of any of the four σC−H bonding orbitals.13 Although each molecular orbital has access to all of the basis functions, the molecular orbitals must be orthonormal to each other. That is, overlap integrals between basis functions hφµ|φνi are either 1 or 0 (1 if µ = ν or 0 otherwise). A simple way to demonstrate the necessity of orthonormality is to arbitrarily consider two orbitals in a system (such as the 1s and 2s orbitals on an argon atom). There could not be any overlap between the 1s or 2s orbitals (hence why the 2s has a node) or else the 1s electrons could exist in the 2s orbital and vice versa (which goes against the Aufbau and Pauli Exclusion Principles). In order to initiate the SCF procedure, the user must define the geometrical coordinates for the atoms in the molecule as well as a set of basis functions for each atom. The software package (e.g., Gaussian) will then use this input to make an initial orbital guess, resulting in the initial coefficient matrix, Cµi (with elements µi). During each SCF cycle, all of the orbitals have their energies optimized, but since orbitals interact with each other, it is likely the procedure will require many steps. Effectively, each orbital eigenvalue equation, Eq. (1.9), is optimized individually while the other orbitals are frozen. It can be said that each orbital is optimized against the average, or “field” of the other occupied orbitals.

Each spatial orbital is represented in the basis φµ by the linear expansion:

K X ψi = Cµiφµ. (1.17) µ=1

The matrix (Cµi) has dimensions K × K where K is equal to the number of basis functions. The eigenvalue equations are assembled into the Roothaan matrix equation:14,15

FC = SC. (1.18)

10 1.1. Discussion of Computational Chemistry

The matrix elements of the Fock matrix, F , are deﬁned as:

N K K nucl X X X ∗ Fµν = Tµν + Vµν + CλaCσa[hµν|λσi − hµλ|σνi] a λ σ

N K K core X X X ∗ = Hµν + CλaCσa[hµν|λσi − hµλ|σνi]. (1.19) a λ σ

The Fock matrix also has dimensions K × K. A natural consequence of using a large basis set is that there will be a lot of unoccupied orbitals (a.k.a. virtual orbitals). core The ﬁrst term, Hµν , is the one-electron Hamiltonian matrix which contains the kinetic energy terms and nuclear-electron attraction terms for all of the electrons. The notation 1 of integrals is further simpliﬁed (e.g., hµν|λσi = hφµ(1)φν(1)| |φλ(2)φσ(2)i) to increase r12 comprehension.

The coeﬃcient matrix C, as discussed previously, has elements Cλa in this equation, where λa represents the contribution of the λth basis function to the ath molecular orbital. Eq. (1.19) can be further manipulated into:

K K core X X Fµν = Hµν + Pλσ[hµν|σλi − hµλ|σνi] λ σ

core = Hµν + Gµν. (1.20)

During the SCF procedure, the matrix Gµν (more specifically the density matrix Pλσ) is updated at every iteration, but all of the four-index integrals between the basis functions (i.e., the hµν|λσi and hµλ|σνi integrals) are computed and stored on the computer’s hard disk only once. These integrals do not change during the SCF procedure because they are based on the pre-defined basis functions. The matrix elements of the overlap matrix S are defined as:

Sµν = hµ|νi. (1.21)

If the basis functions were all orthonormal to each other, then the matrix S would be

11 1.1. Discussion of Computational Chemistry

equal to I (the identity matrix) since all of the oﬀ-diagonal elements, Sµν (µ 6= ν), would be zero. In general, the basis functions are not orthonormal initially, so the atomic orbitals must be transformed such that S = I. This is a mathematically involved procedure, but it will be sidestepped it for brevity, so Eq. (1.18) then becomes:

F 0C0 = C0, (1.22) where C0 is the coeﬃcient matrix in the new orthonormal basis (creating a new density 0 0 0 matrix, Pλσ). The Fock matrix F has been transformed similarly (since C is an element of F 0). is the energy eigenvalue matrix, which once diagonalized at the end of the SCF procedure, will have diagonal elements equal to the orbital energies of the molecule in increasing order. The transformed Fock matrix F 0 is now partitioned into one- and two-electron matrices similar to Eq. (1.20):

0 0 0 Fµν = Hµν + Gµν. (1.23)

0 The one-electron Fock matrix elements, Hµν, are computed at the beginning of the SCF procedure and at the end, but they are fixed during each cycle. The two-electron 0 0 0 0 Fock matrix elements, Gµν, are added to Hµν during each cycle to form Fµν, and F is then diagonalized. Since the transformed density matrix P 0 is an element of F 0, diagonalizing F 0 will result in a new P 0 matrix. The new P 0 is plugged into F 0 and it is again diagonalized. This will result in yet another P 0 matrix. This procedure will continue until the P 0 matrix at the beginning of the nth cycle is equal to the P 0 matrix at the end of the nth cycle (within a certain tolerance). Then, the orbitals are said to have reached “self-consistency” after the nth cycle. The diagonalized F 0 matrix, with respect to Eq. (1.22), is equivalent to the orbital eigenvalue matrix, : the diagonal elements are equal to the orbital energies (in increasing order). One of the glaring holes in the SCF process is that it only uses a single Slater determinant to describe the ground state of an electronic system. During each SCF cycle, each orbital is optimized against the average field of the other elections. In other words, when orbital a is being optimized, all of the other orbitals are frozen. The problem is that electronic motion is correlated (i.e., each electron’s motion affects the

12 1.1. Discussion of Computational Chemistry

others). This issue brings about a phenomenon called the correlation energy, Ecorr,

Ecorr = Etrue − EHF. (1.24)

Etrue is equal to the electronic energy of a molecule if a complete basis set were used in the calculation along with a variational method that uses all possible excited state

Slater determinants, and not just the ground-state Slater determinant, |Ψ0i. On a relative scale, the correlation energy is often less than 1% of the total electronic energy of a system, but it is a very important 1% because it is mostly concerned with the valence electrons.16 Further, most of the electronic energy of a molecule is contained in the core orbitals, since the orbital probability densities are situated close to the nucleus and have large (negative) nuclear-electron attraction energies. Hartree-Fock (with a single Slater determinant) makes no attempt to address Ecorr and therefore it is almost never used for any high-level research purposes.

1.1.7 Geometry Optimizations

Even after the optimized orbitals are assembled, the geometry of the molecule needs to be optimized as well. At each step of a geometry optimization, a new set of optimized SCF orbitals needs to be generated as well. After a step is completed, the orbitals are plugged into Eq. (1.16), and the overall electronic energy of the molecule is computed. Then, the ﬁrst derivatives of the overall energy are computed with respect to all 3N −6 geometrical variables (3N −5 for linear molecules). Each variable is adjusted simultaneously, and a new set of SCF orbitals and a new overall electronic energy are generated. The optimization is complete when all of the ﬁrst derivatives of energy become zero and intramolecular forces are negligible.

1.1.8 Density Functional Theory

The simple Hartree-Fock method is remarkable in its ﬂuidity, but it lacks the accuracy to be useful to solve chemical problems. More importantly, as the number of atoms (and consequently the number of orbitals and basis functions) increases, the computational eﬀort required to complete an SCF cycle (and a geometry optimization) can rapidly become astronomical. The size of an SCF calculation at the HF level scales with K4

13 1.1. Discussion of Computational Chemistry

(where K is equal to the number of basis functions) due to the ﬂood of new two-electron integrals that are created with the introduction of each additional basis function. On the surface, the problem seems unavoidable: each electron with its four coordinates (three spatial and the electron spin) is treated explicitly, and there does not seem to be a way to simplify it. However, an alternative method is to look at the electron probability densities as opposed to the orbitals. The electron probability density of a chemical system is deﬁned by:

N X 2 ρ(r) = |ψi(r)| . (1.25) i

Eq. (1.25) shows that the density is equal to the sum of the individual probability densities of each occupied orbital. If the density is known, the ground-state energy, E0, might be easily derived in a formation similar to Eq. (1.16). The advantage of using the density in electronic structure calculations is that the computational effort required scales less intensively with the size of the system being studied, making larger systems doable. Still, the only known way to compute the exact density is to use Eq. (1.25), which requires the orbitals. If the orbitals are needed anyway, then what is the point of DFT? Well, as early as the 1920s, physicists have worked on many models to try to approximate the density without the orbitals. The first of which is the uniform electron gas (UEG). This system is purely theoretical and involves an infinite number of electrons occupying infinite space. There are no explicit, point-like nuclei present, for the positive charge is uniformly distributed, and therefore the gas has a constant non-zero density everywhere in the system. In 1927, Thomas and Fermi were able to derive the kinetic energy of this UEG system:17,18

Z 3 2 2 5 T [ρ(r)] = (3π ) ρ(r) 3 dr. (1.26) UEG 10

Work continued in DFT for almost four decades but was mostly irrelevant due to the less than stellar accuracy that the density models produced. However, in 1965 Kohn and Sham hypothesized a “non-interacting” system of electrons (i.e., the conventional non-local exchange operator is replaced by a local exchange potential), whose density

14 1.1. Discussion of Computational Chemistry is equal to the density of a real system where the electrons do interact in the normal way.19 So, in other words, the purpose of Kohn-Sham DFT is to optimize the kinetic energy of the non-interacting system, but under the constraint that the density of this system is equal to the density of the real (interacting) system. Kohn and Sham proposed building the density using a set of Kohn-Sham orbitals (which are linear combinations of basis functions), but since the density of the real (interacting) system is required, the objective is to ﬁnd that density, and to make a long story short, the task here is similar to Hartree-Fock since Kohn-Sham DFT is an SCF method:

KC = SC. (1.27)

The diﬀerence between Eq. (1.27) and Eq. (1.18) is that there is now a matrix K instead of F . K, the Kohn-Sham Fock matrix, is more complex than the conventional Fock matrix for the Hartree-Fock method. The Kohn-Sham Fock operator, fˆKS, for each electron is analogous to Eq. (1.10) except that the exchange portion of the operator is deleted and replaced with VXC, the local exchange-correlation potential. VXC is purely a construct of DFT and successfully removes two-electron exchange integrals from the Fock matrix (though not the repulsion integrals). Also, Kohn-Sham DFT has a distinct advantage over the Hartree-Fock method since it attempts to deal with the correlation issue that the Hartree-Fock method completely ignores. The art of developing exchange-correlation functionals, EXC[ρ(r)], continues to be a hot topic in theoretical chemistry even to this day. Typically, theorists design EXC[ρ(r)] functionals so that the Kohn-Sham electronic energies agree (as close as possible) with known experimental data (heats of formation, activation barriers, etc.) from well-known thermochemical databases such as Gaussian-2 or Gaussian-3.20,21 Essentially, Kohn-Sham methods are semi-empirical since they are inﬂuenced by real chemical data. The most well-known 22 EXC[ρ(r)] functional, B3LYP, is still used in the majority of computational studies, and it is used as the exclusive DFT method in this thesis.

23,24 The exchange portion of the B3LYP EXC[ρ(r)] functional, Becke-3, combines the local density of the crude uniform electron gas (UEG) model (10%) with the gradient of the density (70%) constructed from four noble gases: helium, argon, neon, and krypton (from electron diﬀraction experiments). The exact Hartee-Fock exchange makes up the ﬁnal 20%. The LYP correlation functional25 uses the Colle-Salvetti formula for the

15 1.1. Discussion of Computational Chemistry correlation energy.26,27 The four empirical parameters used in the LYP functional were determined from ﬁtting using the Hartree-Fock density of the helium atom. The only real shortcoming of Kohn-Sham DFT is that the exchange energy is (for the most part) not computed using conventional two-electron integrals, and the lack of explicit orbital deﬁnition makes it possible for an arbitrarily chosen electron to interact with itself, which is obviously unphysical. Part of the art of designing EXC[ρ(r)] functionals is correcting for such self-interaction.

1.1.9 Conﬁguration Interaction

Both the Hartree-Fock method and Kohn-Sham DFT are referred to as single-reference methods. That is, they use a single Slater determinant (the ground-state Slater determinant) to describe the electronic wave function. In order to calculate the true ground-state energy, however, excited state Slater determinants need to be included in the wave function. The ideal computational method for chemical research is conﬁgura- tion interaction (CI) theory, with the electronic wave function described as:

occ. vir. occ. occ. vir. vir. X X r r X X X X rs rs |ΦCIi = c0|Ψ0i + ca|Ψai + cab|Ψabi a r a b6=a r s6=r

occ. occ. occ. vir. vir. vir. X X X X X X rst rst + cabc|Ψabci + ··· . (1.28) a b6=a c6=b6=a r s6=r t6=s6=r

The CI wave function, |ΦCIi, uses the Hartree-Fock wave function, |Ψ0i, as its reference (or zeroth-order wave function) and then using the SCF orbitals of |Ψ0i, pro- 28 r ceeds to assemble excited-state Slater determinants. The determinants |Ψai represent singly-excited determinants (i.e., an electron is promoted from the occupied orbital a to rs the virtual orbital r). Also, determinants |Ψabi represent doubly-excited determinants (i.e., two electrons are promoted from occupied orbitals a and b to virtual orbitals r r rs and s, respectively). The coeﬃcients ca, cab, etc. represent the algebraic contribution of a particular excited determinant to the CI wave function. If one examines Eq. (1.28) closely, it becomes apparent that the CI wave function is a linear combination of Slater determinants. At the Hartree-Fock level, the orbitals are linear combinations of basis functions (Eq. (1.17)). CI, a variational method, em-

16 1.1. Discussion of Computational Chemistry ploys a diagonalization procedure much the same as Hartree-Fock. However, instead of diagonalizing a Fock matrix of orbital basis functions, in CI a Hamiltonian matrix of Slater determinants (which are based on the optimized and now frozen SCF orbitals) is diagonalized. The CI equation becomes:

HC = C. (1.29)

Many of the details are analogous to Hartree-Fock, however, the Hamiltonian matrix, H, contains Slater determinants as the basis functions, the coefficient matrix, C, contains the coefficients of the Slater determinants (not the basis function contributions to the orbitals), and the energy matrix, (once diagonalized), contains the state energies of the molecule (ground, first-excited, second-excited state energy, etc.) as its diagonal elements in increasing order (and not the orbital energies). Unless one is using a small basis set, studying a small molecule, or both, however, there are simply way too many Slater determinants to make this method applicable to chemical research with modern supercomputers. One countermeasure is to define an active space: low-lying occupied orbitals (specifically core orbitals) and high-energy virtual orbitals are frozen (i.e., they are not participating in the excitation process of forming configurations), and this reduces the number of excitation possibilities, but the computational effort of diagonalizing the Hamiltonian matrix H has still doomed CI to be used sparingly in chemical research efforts.

1.1.10 Møller-Plesset Second-order Perturbation Theory

The excited Slater determinants are useful, but diagonalizing a very large Hamilto- nian matrix in CI (even with a relatively small active space) is not feasible. An alternative method that uses excited Slater determinants, Møller-Plesset Second-order Perturbation Theory (MP2), can avoid the diagonalization bottleneck. Rather than diagonalizing a Hamiltonian matrix of Slater determinants, MP2 involves directly adding the doubly-excited Slater determinant energies to the zeroth-order energy.29 In MP2, the zeroth-order energy is redefined as being the sum of the HF orbital energies from Pocc. Eq. (1.9), a a. The first-order correction eliminates all of the double counting of two-electron integrals in the zeroth-order energy. Therefore, E0 + E1 = EHF. The second-order correction to energy is defined by:

17 1.1. Discussion of Computational Chemistry

occ. occ. vir. vir. X X X X ˆ rs E2 = hΨ0|H|Ψabi a b6=a r s6=r occ. occ. vir. vir. X X X X |hab|rsi − har|bsi|2 = . (1.30) r + s − a − b a b6=a r s6=r

It is worth noting that E2 is only dependent on the two-electron integrals, and that is because all one-electron integral contributions (kinetic energy and nuclear- electron attraction terms) will be zero since regardless of which overlap energy integral ˆ rs (hΨ0|H|Ψabi) is chosen, two orbitals will be diﬀerent making all one electron terms (e.g., ha|hˆ|ri) zero by symmetry. This result makes up part of the Condon-Slater rules.12,30

The MP2 energy is then defined as EMP2 = EHF + E2, and given that it is much less computationally expensive to compute than the CI energy, MP2 continues to be an attractive method that is widely used in chemical research. An interesting effect of using MP2 is that MP2 energies are generally lower than HF energies even though higher-order (excited) Slater determinants are used when computing an MP2 energy. Well, the reason why excited states are less stable energetically than the ground state is because nuclear-electron attraction energies are less stable (electrons in the diffuse virtual orbitals are further from the nuclei). The two-electron energies of excited states are actually lower in energy overall relative to the ground state since virtual orbitals are more diffuse and allow for lower Coulombic repulsion, but the less stable nuclear-electron attraction energies overtake this effect. Since only two-electron integrals contribute to MP2 energies, that is why they are lower relative to HF. It should be noted that configuration interaction (CI) energies are also lower than HF in general.

1.1.11 Solvent Eﬀects

The computation of electronic energies has been discussed at great length so far, but most chemical reactions take place in solution, and ignoring the effects that a solvent has on a reaction can be a grave error (especially so if a molecule is highly polar). Obviously, adding hundreds of solvent molecules explicitly to a calculation would astronomically ramp up the computational effort, so the most effective way to incorporate solvent effects is to use an implicit model. In this context, computational methods generally

18 1.1. Discussion of Computational Chemistry employ a continuum of overlapping solvent spheres which contain “united atoms” within them. An example of how this would be applied is to consider benzene (C6H6). While hydrogens are permitted to have their own sphere, by default, each CH group would be contained within one solvent sphere. The spheres typically have radii that are slightly larger than the van der Waals radii of the united atoms, and consequently the spheres attempt to simulate a continuous electric field on the molecule being studied which would be exerted by an explicit solvent.31 Introducing solvent into a calculation will affect the calculation of the orbitals and the equilibrium geometry of the molecule since the atoms will reorient themselves ac- cordingly to maximize the interactions between solute and solvent. The orbitals are computed in an SCF fashion, but the Fock operator, fˆ (Eq. 1.10) now contains a solvent interaction component (much like the Kohn-Sham fˆ operator adds an exchange- correlation component). In order to converge the SCF orbitals in a solvent calculation, not only does the density matrix, Pλσ (Eq. (1.19)) need to reach “self-consistency”, the dipole moment of the molecule needs to reach this stage as well. Therefore, solvent energy calculations are referred to as “self-consistent reaction field” (SCRF) calculations. The solvation model employed throughout this thesis is the integral equation formalism polarizable continuum model (PCM) initially devised by Tomasi and coworkers.32–36

1.1.12 Vibrational Frequencies of Molecules

When studying reactions, there exists a potential energy surface (PES) that has 3N−6 dimensions (or 3N−5 for linear molecules). Conventionally, each of these 3N−6/3N−5 degrees of freedom (where N is equal to the number of atoms in a molecule) is assigned to a particular geometrical variable (i.e., a bond length, a bond angle, or a dihedral angle). In the case of a homo- or heteronuclear diatomic molecule, the PES can be described by a truncated Taylor series:3

dV 1 d2V V (r) ≈ V (r ) + (r − r ) + (r − r )2 + ··· 0 dr 0 2 dr2 0 r=r0 r=r0

1 d2V V (r) ≈ (r − r )2. (1.31) 2 dr2 0 r=r0

19 1.1. Discussion of Computational Chemistry

The potential energy is arbitrarily set to zero at the energy minimum, and the ﬁrst derivative of energy with respect to the bond length (r) is zero at this point, so the ﬁrst two terms vanish in Eq. (1.31). Still, in order to compute the remaining second derivative term, the potential function about the minimum must be known, so the 1 2 formula 2 k(r − r0) , which is Hooke’s law for a spring with k being the force constant, 1 2 is used, and if it is plugged this into Eq. (1.31), V (r) ≈ 2 k(r − r0) . The potential function V (r) is treated as a harmonic oscillator, and therefore the energy eigenvalues become:

s 1 1 k E(n) = n + ω, ω = . (1.32) 2 ~ 2π µ

So Eq. (1.32) shows that the zero-point vibrational energy of a harmonic oscillator 1 d2 is 2 ~ω. Since µ is the reduced mass and k can be computed easily via dr2 V (r) = k, all of the necessary information is seemingly in hand, but unfortunately this problem becomes more complicated when polyatomic molecules are dealt with. If a triatomic molecule is considered, the PES has three (or four) dimensions. If one assumes three, then Eq. (1.31) becomes:

1 d2V 1 d2V V (x , x , x ) ≈ (x − x )2 + (x − x )2 1 2 3 2 1 10 2 2 20 2 dx1 2 dx2 x1=x10 x2=x20

1 d2V 1 ∂2V + (x − x )2 + (x − x )(x − x ) 2 3 30 1 10 2 20 2 dx3 2 ∂x1∂x2 x =x x =x 1 10 3 30 x2=x20

1 ∂2V + (x − x )(x − x ) 2 ∂x ∂x 1 10 3 30 1 3 x1=x10 x3=x30

1 ∂2V + (x − x )(x − x ). (1.33) 2 ∂x ∂x 2 20 3 30 2 3 x2=x20 x3=x30

Since the three bond modes (x1, x2, x3) are coupled to each other via the three cross derivative terms at the end of Eq. (1.33), each degree of freedom cannot simply be partioned in order to evaluate the individual potential functions. Therefore, the modes need to be transformed into a new set of coordinates such that the cross derivative

20 1.1. Discussion of Computational Chemistry terms disappear. As with the transformation of the Fock matrix in Section 1.6, the procedure will be ignored here, but realize that the PES with 3N−6/3N−5 dependent degrees of freedom has been transformed into a PES with 3N−6/3N−5 independent, one dimensional potential functions. These functions are each called normal modes, and once their force constants k are evaluated, all of their vibrational frequencies can be computed. At the end of a geometry optimization, an evaluation of all of the normal mode force constants will reveal the status of the stationary point, i.e., whether it is a minimum, a transition state (saddle point), or a multi-dimensional saddle point. If all of the force constants are real, this implies a local minimum and that all of the 3N−6/3N−5 potential functions are concave up. If one force constant is imaginary (i.e., the potential function is concave down), then the stationary point is a transition state along the reaction coordinate defined by the imaginary frequency. Finally, if more than one force constant is imaginary, then the stationary point is a multi-dimensional saddle point with no useful chemical meaning. If one’s intention at the beginning of the optimization is to find a transition state, the force constants must also be computed at the beginning of the calculation so that the program (e.g., Gaussian) can know which mode is concave down and maximize that mode while minimizing all of the others. Since the concavity is irrelevant for a minimization calculation, only estimates of the force constants are required, and these approximate force constants are improved at each step of the optimization with the computed first derivatives.37,38

1.1.13 Equilibrium Statistical Mechanics

So far this discussion has covered how to calculate the electronic energy of single molecules, and further the incorporation of solvent eﬀects and the computation of the vibrational force constants has been addressed, but the last question remains: how can one compute thermodynamic properties like H, S, and G on a molar scale? In order to understand this, the canonical partition function is now introduced:

1 X − [1(V )+2(V )+···+N (V )]i Z(N,V,T ) = e kbT . (1.34) N! i

21 1.1. Discussion of Computational Chemistry

The partition function, Z(N,V ,T ) assumes that the number of molecules in the system, the volume, and the temperature all remain constant at equilibrium. The function is a sum of all of the probabilities of the molecule existing in each type of energy state (mainly the electronic, vibrational, rotational, or translational energy states) at all levels (ground or any of the excited states). The N! factor results from the indistinguishability of electrons. Since the assumption is that the molecules do not interact (i.e. they behave like an ideal gas), Eq. 1.34 becomes:

N  elec   vib   rot  trans  1  X − i X − j X − k X − l  Z(N,V,T ) = g e kbT g e kbT g e kbT g e kbT N!  i   j   k   l   i j k l 

1 n oN = [z ] z (T ) z (T ) z (V,T ) (1.35) N! elec vib rot trans

ln[Z(N,V,T )] ≈N(ln [zelec] + ln zvib(T ) + ln zrot(T ) + ln ztrans(V,T ) ) −N ln N + N. (1.36)

The partition function can be neatly separated into its electronic, vibrational, rotational, and translational components. zelec is easy to calculate since at room temperature the chances of a molecule existing in any of the excited states are negligible, and the degeneracy of the ground state (gi) is equal to 2S+1 (where S = the spin multiplicity). So if the ground-state energy is arbitrarily set to zero, zelec = 2S + 1.

ztrans(V,T ) is also simple under the ideal gas assumption:

3 2πMk T 2 z (V,T ) = b V. (1.37) trans h2

Eq. (1.37) is derived from the particle in a box solution, which is a model for translational motion in quantum mechanics.3

zrot(V,T ) is not too complicated either as long the three principal moments of inertia,

22 1.1. Discussion of Computational Chemistry

IA,IB,IC , for the molecule are known at the stationary point:

3 √ ! 2 πI I I 8π2k T z (T ) = A B C b . (1.38) rot σ h2

Eq. (1.38) is derived from the rigid rotator solution, which is a model for rotational motion in quantum mechanics.3 σ is the degeneracy associated with the rotational partition function.

In order to compute zvib(T ), all of the force constants for the 3N−6/3N−5 normal modes are needed, and since these are all computed at the end of the optimization anyway, they can simply be plugged into Eq. (1.39):

3N−6   Y 1 z (T ) = . (1.39) vib  −~ωj  j 1 − e kbT

All of the components for the overall partition function, Z(N,V,T ) (or rather ln Z), have been derived, so how does one compute thermodynamic properties? These thermodynamic deﬁnitions can now be employed:3

∂ ln Z U = k T 2 b ∂T N,V H = U + PV (1.40)

∂ ln Z S = k ln Z + k T b b ∂T N,V G = H − TS.

The necessary understanding of computational methods has now been suﬃciently developed to aﬀord an appreciation of how they are applied in this thesis.

23 1.2. Metallocene Chemistry and Decamethyldizincocene

1.2 Metallocene Chemistry and Decamethyldizincocene

5 In 1951, the synthesis of ferrocene, Fe(η −C5H5)2 (Figure 1.2), created a brand new sub-field of organometallic chemistry.39 It did not take long before metallocenes were discovered for other first-row transition metals including Va, Ti, Cr, Mn, Co, and Ni. These compounds have a variety of uses. Ferrocene derivatives have been shown to be useful in the treatment of breast cancer,40,41 and they have also been touted as effective fuel additives to reduce engine-knock much like the discontinued (and toxic) tetra-ethyl lead.42 Also, vanadocene and titanocene derivatives have been shown to be cytotoxic against selected types of cancer cells.43,44 The concept of multiple bonds between metal atoms was first introduced by Cotton and coworkers in 1964 when they proposed that a quadruple bond exists between the rhenium atoms of the K2[Re2Cl8]•2H2O complex.45,46 Until this breakthrough, a triple bond was Figure 1.2: Structure of ferrocene. assumed to be the highest multiplicity that a chemical bond could have, and since then compounds containing direct single and multiple bonds between two transition-metal atoms have been reported for much of the d-block.47 Though many of these dimetal compounds have no practical applications, they are interesting due to their unique bonding structures, which consist of σ, π, and δ bonds. Intriguingly, until the 2004 discovery of decamethyldizincocene (3, Figure 1.3), 5 Zn2(η −C5Me5)2 (the focus of Chapters 2 and 3 of this thesis), there had never been a compound with a direct, unbridged metal−metal bond between two first-row transition metals.48 This dizinc compound contains a double bond between two zinc atoms. The Carmona group accidently synthesized decamethyldizincocene (3) while at- 5 tempting to synthesize the half-sandwich Zn complex with the formula Zn(η −C5Me5)Et 5 1 (4) by reacting decamethylzincocene, Zn(η −C5Me5)(η −C5Me5)(1), with ZnEt2 (2) (Figure 1.3). The desired half-sandwich compound (4) does form, but it is only the minor product. The authors have noted that this process seldom produces yields of 3 higher than 30%.49 More often, yields are much less, and sometimes 3 does not form in any detectable amount at all (despite careful efforts to reproduce the reaction conditions).50

24 1.2. Metallocene Chemistry and Decamethyldizincocene

5 1 5 Figure 1.3: Reaction of Zn(η −C5Me5)(η −C5Me5) with ZnEt2 to form Zn2(η −C5Me5)2 5 and Zn(η −C5Me5)Et.

The discovery of 3 came at a time when modern supercomputers had developed to a point where energy, geometry, and frequency calculations on molecules of this size (with or without symmetry) could be done at the DFT level in a matter of a few days, and not surprisingly, many theoretical groups jumped at the opportunity to research and discuss the theoretical properties of this interesting new compound. H. F. Schaefer and coworkers quickly published four papers on various structural properties of decamethyldizincocene and other potential ﬁrst-row dimetallocene compounds.51–54 Xie and Fang contributed a thorough study on the bonding structure (and dissociation energies) of the [Zn−Zn] unit of 3.55 Kress gave an in-depth analysis of the molecular orbitals and vibrational structure of 3.56 Later, Liu et al. studied the aromaticity of dizinc and zinc half-sandwich complexes.57 Carmona and coworkers published two 58 follow-up papers that included a more eﬃcient synthesis of 3 involving KH and ZnCl2 (Figure 1.4) as well as an in-depth discussion of more related experimental work.49 They also noted that in the original synthesis of 3 (Figure 1.3), ZnPh2 can be used in place of 2, but no other ZnR2 reagent will result in the formation of 3, and they further 5 suggested that 3 likely forms as a result of the combination of two Zn(η −C5Me5)• radicals. Our group was able to support this hypothesis in 2008.1 Until recently, 3 was not known to have any practical applications. Presently, it is being sought as a potential catalyst for inter/intramolecular hydroamination59 (the reaction studied in Chapter 4 of this thesis). However, despite the fact that 3 can produce quantitative or near-quantitative yields of the desired products in mild conditions, it can only produce racemic products, and this severe limitation may blunt any future impact of 3 as a catalyst. Regardless, 3 continues to inspire further experimental and computational work on new metal−metal bonded transition and main group elements.60–62 Still,

25 1.2. Metallocene Chemistry and Decamethyldizincocene

5 1 Figure 1.4: Reaction of Zn(η −C5Me5)(η −C5Me5) with KH or KH and ZnCl2. even though the original synthesis of 348 has been cited over 100 times, no one outside of our group has been able to explain the most interesting aspect of its discovery: why does it form in the ﬁrst place? Equally as important an anomaly to explain is the fact 5 1 that only deca-substituted zincocenes like 1 or Zn2(η −C5Me4Et)(η −C5Me4Et) can form dizincocene products. If one strips oﬀ the methyl groups of 1 and 3, they become zincocene and dizincocene, respectively, and zincocene cannot lead to the formation of dizincocene regardless of what ZnR2 is used, and when zincocene is reacted with either + 1 − KH or KH and ZnCl2, it forms the zincate salt K [Zn(η −C5H5)3] instead (Figure 1.5).

5 1 Figure 1.5: Reaction of Zn(η −C5H5)(η −C5H5) with KH or KH and ZnCl2.

Before proceeding, it is necessary to understand the chemical nature of metallocenes and why they can be formed. If one considers ferrocene (Figure 1.2), the structure contains one Fe atom and two C5H5• radical units. Each radical unit has ﬁve π electrons. − Iron is oxidized to its +2 state and this in turn reduces the C5H5• radical units to C5H5 − ions. After reduction, the C5H5 units have six π electrons and are therefore aromatic by H¨uckel’s [4n+2] rule.63 Thus, the driving force for the formation of metallocenes is

26 1.3. Enantioselective Catalysis with Primary Aminoalkenes aromaticity (and the resulting strong hapticity between the rings and the metal atom). The theory behind the existence of dimetallocenes is analogous to this.

Chapter 2 of this thesis will focus on the formation of 3 via 1 and ZnR2 reagents.

The formation of 3 via KH and ZnCl2 will be discussed separately in Chapter 3 since both Chapters discuss completely diﬀerent reactions.

1.3 Enantioselective Catalysis with Primary Aminoalkenes

In the aftermath of the Thalidomide debacle of the early 1960s that caused over 10,000 children to be born with birth defects worldwide,64 synthetic chemists became more concerned with producing chiral organic compounds in a stereoselective fashion for use in the production of bulk chemicals, specialty chemicals, and, in this context, phar- maceutical products. Though the desired (R) enantiomer of thalidomide can racemize in the body after it is ingested (rendering an enantiopure dose just as dangerous as a racemic dose), other drugs, like citalopram (which is used to treat anxiety and depres- sion), have been shown to be more eﬀective in clinical trials when an enantiopure (S) dose is taken.65 The development of transition-metal and lanthanide catalysts with wide substrate scope capabilities, which can produce the desired stereocentre(s), is currently a popular research ﬁeld. Metal catalysts can be inexpensive, the reactions do not require a lot of energy, and the generation of waste by-products is minimized. For this thesis (Chapter 4), the main concern is being able to form C−N bonds in a highly enantioselective fashion.66 L. L. Schafer and coworkers have discovered a chiral neutral zirconium amidate precatalyst that, when activated, can cyclize primary aminoalkenes with the desired enantioselectivity (Figure 1.6).67 For computational purposes, the question is: does this reaction proceed via [2+2] cycloaddition or σ−bond insertion (Figure 1.7)? Previous experimental and computational work on similar Group-4 metal precata- lysts favour the [2+2] cycloaddition mechanism,68–71 but both pathways needed to be investigated for thoroughness. This work was done collaboratively with a research group in China. Our group focused solely on studying the [2+2] cycloaddition mechanism, and the results of that work are included in Chapter 4 of this thesis. The Chinese group

27 1.3. Enantioselective Catalysis with Primary Aminoalkenes

Figure 1.6: Zr precatalyst and its reaction with 5-amino-4,4-dimethylpent-1-ene to produce 2,4,4-trimethylpyrrolidine.

28 1.3. Enantioselective Catalysis with Primary Aminoalkenes bond insertion (left) or [2+2] cycloaddition mechanisms − σ Activated Zr catalyst forming 2,4,4-trimethylpyrrolidine via (right). Figure 1.7:

29 1.3. Enantioselective Catalysis with Primary Aminoalkenes studied the σ−bond insertion mechanism and also independently studied part of the [2+2] mechanism and oﬀered advice to ensure that our transition-state and intermediate structures were sterically correct. In order to demonstrate the enantioselectivity of the [2+2] mechanism from a computational perspective, it will be necessary to discover the immediate and transition state energies of each step of the catalytic cycle (for the S and R isomers), and use these values to compute the Turnover Frequency (TOF)72 ratio for the S/R processes. Given that the enantiomeric excess (ee) for this process is 93% in favour of S,67 the TOF ratio should be around 28:1. The formula to compute the TOF for a three-step process using the Christiansen model72 is shown in Eq. (1.41):

∆ TOF = M ∆ = e(−∆G)/RT − 1 3 3 X X ˆ M = M ab a=1 b=1   e(T1−I1)/RT e(T1−I2)/RT e(T1−I3)/RT   Mˆ =  e(T2−I1−∆G)/RT e(T2−I2)/RT e(T2−I3)/RT    e(T3−I1−∆G)/RT e(T3−I2−∆G)/RT e(T3−I3)/RT

∆(S)M(R) TOF ratio (S/R) = . (1.41) ∆(R)M(S)

th th The values Ti and Ij indicate the energies of the i transition state and the j intermediate, respectively, relative to the start of the catalytic cycle (which is usually set to zero). The value ∆ is the “net molecular flux” or driving force of the cycle. In order to compute ∆ (∆ = e−∆G/RT −1), ∆G must be known, and it is simply equal to the total thermodynamic benefit (or cost) of the entire cycle. Unfortunately, many catalytic cycles have a lot of steps, and therefore the number of matrix elements increases rapidly as the Christiansen matrix, Mˆ (Eq. 1.41), is an N × N matrix (where N is the number of steps in the cycle). In 2006, Kozuch and Shaik proposed finding the degree of TOF control of each state, XTOF,i (i indicates an Intermediate or Transition State within the

30 1.3. Enantioselective Catalysis with Primary Aminoalkenes cycle) to determine how small changes in i aﬀect the TOF:73–76

1 ∂TOF XTOF,i = . (1.42) TOF ∂Ei

Since intermediates and transition states will inﬂuence the TOF in opposite di- rections, Eq. (1.42) requires an absolute value sign. Typically, the highest energy transition state (HETS) and the most abundant reactive intermediate (MARI) are the two most important values since they maximize the energy span and will have the greatest inﬂuence on the TOF. The HETS and MARI can also be referred to as the turnover determining transition state (TDTS) and turnover determining intermediate (TDI), respectively. Since each matrix element of Mˆ requires an exponential calculation, a consequence of this way of thinking is that many of the terms can be ignored because of their negligible values. In fact, calculating the TOF for most catalytic cycles requires only one or two terms to be considered, and this approach will be explored in Chapter 4.

31 Chapter 2

Formation of Decamethyldizincocene via Decamethylzincocene and ZnR2

2.1 Strategy and Methods

In order to fully understand why and how decamethyldizincocene (3) forms, it is necessary to figure out why dizincocene (6) does not form (Figure 2.1). Calculations on the reactants, products, intermediates, and transition states along the reaction paths of formation of 6 and 3 were performed, and the resultant geometries for the formation of 6 were used (with methyl groups added) to find the intermediates and transition states along the formation path of 3. This facilitates an efficient and thorough comparison of these two processes. The formation reactions of these dizinc products were studied from two hypothetical perspectives: (1) neutral-charge electrostatic attraction and rearrangement of the reactants, and (2) radical dissociation and recombination of the reactants. Gaussian 09 77 was used for all calculations. Geometry optimizations were performed using the popular Kohn-Sham method B3LYP22–25 with Dunning’s cc-pVDZ basis set78 while the Pople 6-31G(d)79 basis set was used exclusively for the Zn atoms. This basis set combination produces a geometry for 3 that agrees very well with the experimental neutron diffraction geometry, in which the Zn−Zn bond length is 2.292(±1) A.˚ 80 The cc-pVDZ (with 6-31G(d) for Zn) basis set produces a length of 2.293 A.˚ However, a more conventional selection like cc-pVDZ (with LANL2DZ pseudopoten- tial basis set for Zn) gives an elongated bond length of 2.469 A.˚ All transition states were analyzed and confirmed by multiplying the imaginary mode Cartesian displace- ments by a scale factor (which varied depending on the transition state), adding them

32 2.1. Strategy and Methods to the saddle-point geometries, and performing geometry optimizations to connect the transition states to their corresponding minima.

5 1 5 Figure 2.1: Reactions of Zn(η −C5Me5)(η −C5Me5) with ZnR2 to form Zn2(η −C5Me5)2 5 5 1 5 and Zn(η -C5Me5)R (top) and Zn(η −C5H5)(η −C5H5) with ZnR2 to form Zn2(η −C5H5)2 5 and Zn(η −C5H5)R (bottom).

All thermodynamic energies (G263) reported in this Chapter are free energies at the temperature (263 K) of the initial experiment.81 Møller-Plesset second-order perturbation theory (MP2)29 single-point calculations were performed using the same basis set with the B3LYP optimized geometries. Solvent eﬀects were further incorporated by employing an integral equation formalism polarizable continuum model (PCM), a solvent model appropriate for diethyl ether, with the MP2 single-point energies.32–36 The ﬁnal energies were derived by adding the B3LYP free energy corrections to the PCM-MP2 single-point energies. In the context of the homolytic dissociation energies 5 1 5 of Zn(η −C5R5)(η −C5R5) into Zn(η −C5R5)• and (C5R5)• radicals (R = H, Me), the counterpoise optimization method proposed by Simon et al.82 was employed. These restricted open-shell B3LYP energies were also improved using PCM-MP2. The counterpoise correction was obtained by computing counterpoise-corrected single-point MP2

33 2.2. Results and Discussion energies with the counterpoise optimized geometries of the parent zincocenes. The exact same procedure was employed to compute the association energy of 3 and 6 from 5 5 Zn(η −C5Me5)• and Zn(η −C5H5)• radicals, respectively. Because current computational resources do not allow for the computation of MP2 Hessians for molecules of this size, the B3LYP thermal data and optimized geometries are our best alternative. Ball and stick molecular images throughout this thesis were generated using Gaussview 5.83

2.2 Results and Discussion

When reacting 1 or 5 with a ZnR2 reagent, literally any ZnR2 reagent could be used, so for the purposes of this study we have selected ZnMe2, ZnEt2, ZniPr2, and ZnPh2.

Only the reaction of ZnEt2 will be analyzed extensively, but all necessary kinetic and thermodynamic data will be shown for the other three ZnR2 reagents.

5 1 Figure 2.2: The HOMO of ZnEt2 (top right) and the LUMO of Zn(η −C5H5)(η −C5H5) (bottom right).

34 2.2. Results and Discussion

2.2.1 Neutral-Charge Electrostatic Dizincocene Formation 5 1 via ZnEt2 and Zn(η −C5H5)(η −C5H5)

84–86 A Natural Bond Orbital (NBO) analysis of 5 and ZnEt2 (Figure 2.2) shows that the HOMO of ZnEt2, the Zn−C σ-bonding orbital, can donate electron density to the LUMO of 5, the π∗ orbital on the bottom (η1) Cp ring of 5. Consequently, we have discovered that ZnEt2 can attack zincocene in a sideways fashion, cutting open the sandwich structure while forming an intermediate (Figure 2.3). The activation barrier for this process is 4.7 kcal/mol (from the complex to ‡associative). At this point in the reaction, the intermediate can do one of two things. It can overcome a symmetric barrier (‡symmetric) of 3.1 kcal/mol to form two equivalents of symmetric products, 5 Zn(η −C5H5)Et (7, R = Et), which are considerably more stable than the reactants (−16.7 kcal/mol). Based on their experimental data, the Carmona group hypothesized a mechanism like this in 2007.49 Alternatively, the intermediate can overcome an asymmetric barrier (‡asymmetric) with a steep 62.9 kcal/mol activation barrier to form the asymmetric products, 6 and butane. Obviously, the activation barrier for the asymmetric process is much too high relative to the symmetric process to form 6, even though the asymmetric products are much more thermodynamically stable than the reactants (−35.8 kcal/mol).

2.2.2 Neutral-Charge Electrostatic Decamethyldizincocene 5 1 Formation via ZnEt2 and Zn(η −C5Me5)(η −C5Me5)

As one might assume, the reactive species in the analogous formation of 3 are similar in geometry (excluding the methyl groups) compared to the pathway for 6 (Figure 2.4). Note that the MP2 single-point energies predict that the reactants will immediately reach the intermediate without an associative barrier. However, the absolutely crucial diﬀerence when using 1 in this synthesis is that the symmetric barrier is much higher (13.2 vs. 3.1 kcal/mol when using 5). However, this diﬀerence is not enough to suggest that 3 forms via the asymmetric barrier. Based on these results, it is reasonable to conclude that 3 does not form via simple neutral-charge electrostatic attraction and rearrangement of 1 and 2.

35 2.2. Results and Discussion reacting with zincocene (zinc atoms appear in red and hydrogens have been removed for 2 Reaction pathway of ZnEt Figure 2.3: clarity).

36 2.2. Results and Discussion reacting with decamethylzincocene. 2 Reaction pathway of ZnEt Figure 2.4:

37 2.2. Results and Discussion

5 1 5 Figure 2.5: Dissociation of Zn(η −C5R5)(η −C5R5) into Zn(η −C5R5)• and (C5R5)• radicals.

2.2.3 Radical Dissociation of Parent Zincocenes

There apparently is a moderate difference in the homolytic dissociation energies of 5 (43.0 kcal/mol) and 1 (36.1 kcal/mol) due to the methyl groups stabilizing the resulting 5 1 radicals from the cleavage of Zn(η −C5R5)(η −C5R5) (Figure 2.5). This data alone is not significant enough to suggest that the formation of 6 will not occur, as it does not experimentally. On the other hand, if this difference in dissociation is coupled with the large difference in the symmetric activation barriers in the formation of 4 (R = Et) and 7 (R = Et) (13.1 vs. 3.1 kcal/mol), it is possible that during the formation of 3, the homolytic dissociation of 1 competes with the formation of the symmetric products (36.1 vs. 13.2 kcal/mol). Further, the formation of 6 would not be possible due to the prohibitive energy difference between these two processes (43.0 vs. 3.1 kcal/mol).1

2.2.4 Reaction of Parent Zincocenes with Other ZnR2 Reagents

Though it has been hypothesized that 3 forms because the homolytic dissociation of 1 competes with the formation of 4 (R = Et),1 the Carmona group has reported that 3 only forms when 1 is reacted with ZnR2 (R = Et,Ph) and 6 will never form regardless of 81 what other ZnR2 reagent is used. So the question now becomes, can this “competition” hypothesis be applied to this expanded list of reagents? The reactions of 1 and 5 with

ZnR2 (R = Me, iPr, Ph) were investigated and compared with the results for ZnEt2. For each reaction, only (‡symmetric) will be discussed since it is the rate limiting step when

1 reacts with ZnEt2 and ZnPh2, and (‡asymmetric) is ignored because it is irrelevant.

For each reaction of ZnR2 (R = Me,Et,iPr,Ph) with 5, the geometries of ‡symmetric

38 2.2. Results and Discussion

Figure 2.6: Simpliﬁed reaction pathways of ZnMe2 (top), ZniPr2 (middle), and ZnPh2 (bottom) reacting with zincocene. 39 2.2. Results and Discussion (bottom) reacting with decamethylz- 2 (top right), and ZnPh 2 (top left), ZniPr 2 Simpliﬁed reaction pathways of ZnMe incocene. Figure 2.7:

40 2.2. Results and Discussion

Table 2.1: Energy compilation of ‡symmetric and the energy diﬀerence (in kcal/mol) between the symmetric process and radical dissociation of the parent zincocene.

Reactant 1 Reactant 2 E(‡symmetric) ∆E(dissociation − ‡symmetric)

ZnMe2 3.5 39.5 ZnEt2 3.1 39.9 ZniPr2 1.7 41.3 ZnPh2 9.9 33.1 ZnMe2 5.2 30.9 ZnEt2 13.2 22.9 ZniPr2 7.3 28.8 ZnPh2 17.1 19.0

do not change all that much (Figure 2.6), and in accordance with experimental ﬁnd- ings49 and our hypothesis, the barrier heights remain low (1.7 − 9.9 kcal/mol, Table

2.1). Even though ‡symmetric for ZnPh2 is relatively high (9.9 kcal/mol), the reaction is very exothermic (−14.8 kcal/mol) with ‡symmetric being 4.5 kcal/mol below the energy of the reactants (Figure 2.6).

It appears that because of steric reasons, when ZnR2 (R = Me,Et,iPr,Ph) reacts with 1, the geometries of ‡symmetric are relatively consistent (Figure 2.7), but the energies vary quite dramatically. Predictably, ‡symmetric for ZnMe2 is the lowest (5.8 kcal/mol) due to minimal steric interactions between the methyl groups of ZnMe2 and

1. However, one would assume that ‡symmetric for ZniPr2 would be higher than ZnEt2 due to the bulkiness of the isopropyl groups, and that would contradict our hypothesis since ZniPr2 does not lead to the formation of 3 experimentally, but in reality that is not the case. The energy of ‡symmetric for ZniPr2 reacting with 1 is 7.3 kcal/mol while it is 13.2 kcal/mol for ZnEt2. ZnPh2, which produces consistently higher experimental yields of 3 when reacted with 1, has the highest ‡symmetric at 17.1 kcal/mol (Table

2.1). Intriguingly, the attraction of the right phenyl ring of ZnPh2 to the left zinc necessitates a second ‡symmetric (5.4 kcal/mol) in order to break this attraction and form the symmetric products (second ‡symmetric, Figure 2.7).

41 2.3. Summary and Further Work

2.3 Summary and Further Work

Strong kinetic arguments have been presented for why 3 forms during this reaction but 6 does not. Our results show that the most likely scenario is that experimentally, some

ZnEt2 or ZnPh2 reacts with 1 to form 4 (R = Et or Ph) through ‡symmetric, releasing 20.2 or 23.2 kcal/mol for 4 (R = Et) and 4 (R = Ph), respectively. Once enough energy has been accumulated this way, the remaining 1 in solution could begin to 5 1 5 dissociate into Zn(η −C5Me5)• and (η −C5Me5)• radicals. Two Zn(η −C5Me5)• units should readily combine to form 3, given their considerable association energy (46.5 kcal/mol). This large energy release could then further fuel the radical dissociation of 1 and consequently form more 3. This proposal1 therefore supports the experimental 5 49 hypothesis that 3 forms via combination of two Zn(η −C5Me5)• units. Whenever another ZnR2 besides ZnEt2 or ZnPh2 is reacted with 1, ‡symmetric is low enough that 1 reacts entirely (and quickly) to form symmetric products (Table 2.1), and therefore 5 1 there is no possibility for 1 to dissociate into Zn(η −C5Me5)• and (η −C5Me5)• radicals.

Further, when 5 is used, ‡symmetric for all ZnR2 reagents are low, which prohibits the homolytic dissociation of 5 and makes the formation of 6 impossible via this type of mechanism. A further piece of evidence to support this proposal has to do with the yields of 3.

It has been found that yields for 3 can be as high as 30% when ZnEt2 is used, but often they are much less (and sometimes no detectable amount of 3 forms at all).81 When

ZnPh2 is used, yields are a little higher, but the inconsistency remains. Given the lack of ability to control homolytic radical dissociations and reassociations in an experimental setting, it seems logical that the yields of these processes would lack consistency and reproducability. It should be noted that the research in this Chapter only focuses on the formation mechanisms of 3 and 6 and the corresponding half-sandwich products. No attempts were made to study the formation of other minor by-products (both identiﬁed and unidentiﬁed).

42 Chapter 3

Formation of Decamethyldizincocene via KH and ZnCl2

3.1 Strategy and Methods

As interesting as the ZnR2 reaction study may be, it is not at all efficient at forming 3. Any number of reducing agents can be reacted with 1 to produce 3 in a much higher yield than ZnEt2 or ZnPh2. Potassium hydride (KH) has been found to be the most 81 convenient. The most efficient synthesis of 3 involves reacting 1 with KH and ZnCl2 in a 1:2:1 stoichiometric ratio (Figure 3.1). At some point during the formation of 3, the Zn2+ atoms are reduced by KH to Zn+, but since 6 does not form when 5 is used (rather the zincate salt, 9), the mechanisms needed to be investigated in detail. The reactions of 1 and 5 with KH were studied first, and then the addition of ZnCl2 was taken into consideration. As with Chapter 2, Gaussian 09 77 was used for all calculations. Geometry optimizations were again performed using B3LYP,22–25 but Pople’s 6-31+G(d,p)79 basis set was used in this study, for the additional diffuse functions are more suitable for modeling the anionic character of the mechanism(s). For steps involving KH, diffuse functions were added to the hydride hydrogens by using 6-31++G(d,p).79 Because of the redox nature of this study, transition states were more rigorously confirmed using the Intrinsic Reaction Coordinate (IRC)87,88 method whenever possible since the imaginary mode displacement method resulted in some transition states being able to relax to different minima with only minor differences in the scalar value. PCM-MP2 single-point energies were again computed with the B3LYP geometries, but the basis set was upgraded to 6-31++G(d,p) (for all atoms) for these calculations. Also, since these reactions take

43 3.1. Strategy and Methods

5 1 5 1 Figure 3.1: Reactions of Zn(η −C5Me5)(η −C5Me5) (top) and Zn(η −C5H5)(η −C5H5) (bottom) with KH or KH and ZnCl2.

44 3.1. Strategy and Methods reacting with KH. 1 Reaction scheme for Figure 3.2:

45 3.1. Strategy and Methods reacting with KH. 5 Reaction scheme for Figure 3.3:

46 3.2. Results and Discussion

Figure 3.4: Reaction complex for KH reacting with 5 (left) and 1 (right). Zinc atoms appear in red. Potassium atoms are purple. Hydrogens have been removed for clarity. place in tetrahydrofuran (THF) at 293 K,81 the B3LYP thermal corrections at this temperature were computed and added to the PCM-MP2 energies in order to arrive at the G293 values, which are the exclusive energy units reported in this Chapter. For the homolytic dissociation and association energies, the counterpoise-optimized restricted open-shell B3LYP geometries were again improved by using counterpoise-corrected MP2 single-point energies.

3.2 Results and Discussion

3.2.1 Reaction of 1 and 5 with KH

By following the reaction scheme (Figure 3.2), KH reacts with 1 to form a reactant complex (−22.2 kcal/mol, Figure 3.4, Table 3.1). It appears that the complex could go through Step 1a to form Me5CpZnK and Me5CpH (∆G = −10.2 kcal/mol, Figure 3.5).

Me5CpZnK could then dissociate into Me5CpZn• and K• radicals (Step 2a, ∆G = −37.4 kcal/mol). Two Me5CpZn• radical units could then combine to form 3 (Step 3a, ∆G =

−38.5 kcal/mol). However, forming Me5CpZnK directly (Step 1a) carries a signiﬁcant energy cost (∆G‡ = 24.2 kcal/mol). Conversely, the complex could go through Step 1b

47 3.2. Results and Discussion

Figure 3.5: Top: Transition states (Step 1a) for KH reacting with 5 to form CpZnK and CpH (top left) and with 1 to form Me5CpZnK and Me5CpH (top right). Bottom: Transition states (Step 1b) for KH reacting with 5 to form CpZnH and CpK (left) and with 1 to form Me5CpZnH and Me5CpK (right).

48 3.2. Results and Discussion

‡ (Figure 3.5) to form Me5CpZnH and Me5CpK (∆G = 14.7 kcal/mol) with an energy beneﬁt of −24.4 kcal/mol. Me5CpZnH could then react with an additional equivalent of KH to form the desired Me5CpZnK (Step 2d, Figure 3.6). Although this process is exothermic by −16.7 kcal/mol, the barrier is 39.1 kcal/mol. The most likely scenario is that Step 1a prevails over Step 1b despite its higher barrier (∆G‡ = 24.2 vs. 14.7 kcal/mol, respectively). The reason for this is that the Me5CpH generated in Step

1a can react with an additional equivalent of KH to form Me5CpK and hydrogen gas (Step 2b). This step releases 28.8 kcal/mol and can be thought of as the thermodynamic driving force for Step 1a.

Figure 3.6: Top: Reactant complexes (pre-Step 2d) for KH reacting with CpZnH to form CpZnK and H2 (top left) and with Me5CpZnH to form Me5CpZnK and H2 (top right). Bottom: Transition states (Step 2d) for KH reacting with CpZnH to form CpZnK and H2 (left) and with Me5CpZnH to form Me5CpZnK and H2 (right).

If 5 is used in place of 1, the reaction scheme (Figure 3.3, Table 3.1) is analogous, but a new variable to the process is introduced. Step 1a has an even higher barrier relative to Step 1b (∆G‡ = 37.8 vs. 11.1 kcal/mol, respectively), and the resulting CpK

49 3.2. Results and Discussion

Table 3.1: Reaction energies (in kcal/mol) of 1 and 5 reacting with KH (from Figures 3.2 and 3.3, respectively). Energies are relative to separated reactants.

Zincocene Step E(complex) E(‡activation) E(products) 1a −22.2 2.0 −10.2 1b −22.2 −7.5 −24.4 2a n/a n/a 37.4 2b n/a n/a −28.8 2c −1.9 12.8 +1.7 2d −16.8 22.3 −16.7 3a n/a n/a −38.5 1a −30.0 7.8 −16.2 1b −30.0 −18.9 −31.1 2a n/a n/a 35.2 2b n/a n/a −38.7 2c 1.6 5.3 −1.0 2d −18.9 20.5 −23.9 3a n/a n/a −44.5

product can react with more 5 (Step 2c) to form a zincate salt, 9. The thermodynamic stability of 9 (−1.0 kcal/mol) does not exactly make it a strong driving force, but considering that Step 1a has a prohibitive barrier in this case (∆G‡ = 37.8 kcal/mol) and the formation of 9 has a barrier of just 3.7 kcal/mol, it is understandable that when 5 reacts with KH, the major product will be 9 as opposed to dizincocene, 6. Although theoretically the zincate salt 8 could form when 1 is reacted with KH, the steric interactions between the methyl groups make the formation of 8 endothermic (+1.7 kcal/mol) with a larger activation barrier (14.7 kcal/mol), making its formation unfavourable. Figure 3.10 shows the geometries for zincate formation from 5 and 1.

3.2.2 Reaction of 1 and 5 with KH and ZnCl2

Our proposed formation route of 3 from 1 and KH is fairly straightforward to understand, but it still involves free radicals, and radical processes are diﬃcult to control and yields can be inconsistent. The addition of one equivalent of ZnCl2 into the reaction mixture does not change the basic reaction scheme all that much since Step 1a and 1b remain, but there is one major diﬀerence: the radical dissociation and reassociation

50 3.2. Results and Discussion

Figure 3.7: Top: Reactant complexes (pre-Step 2e) for ZnCl2 reacting with CpZnK to form CpZnZnCl and KCl (top left) and with Me5CpZnK to form Me5CpZnZnCl and KCl (top right). Bottom: Transition states (Step 2e) for ZnCl2 reacting with CpZnK to form CpZnZnCl and KCl (bottom left) and with Me5CpZnK to form Me5CpZnZnCl and KCl (bottom right).

51 3.2. Results and Discussion

Figure 3.8: Top: Reactant complexes (pre-Step 2f) for ZnCl2 reacting with CpZnH to form CpZnZnCl and HCl (top left) and with Me5CpZnH to form Me5CpZnZnCl and HCl (top right). Bottom: Transition states (Step 2f) for ZnCl2 reacting with CpZnH to form CpZnZnCl and HCl (bottom left) and with Me5CpZnH to form Me5CpZnZnCl and HCl (bottom right).

52 3.2. Results and Discussion

Figure 3.9: Top: Reactant complexes (pre-Step 3b) for CpK reacting with CpZnZnCl to form 6 and KCl (top left) and Me5CpK reacting with Me5CpZnZnCl to form 3 and KCl (top right). Bottom: Transition states (Step 3b) for CpK reacting with CpZnZnCl to form 6 and KCl (bottom left) and Me5CpK reacting with Me5CpK to form 3 and KCl (bottom right).

53 3.2. Results and Discussion 5 CpK 5 (Step 2c). 8 to form Reactant complex (top left), transition state (top middle), and zincate product (top right) for CpK reacting with 1 , and the reactant complex (bottom left) transition state (bottom middle) and zincate product (bottom right) for Me 9 Figure 3.10: to form reacting with

54 3.2. Results and Discussion . 2 reacting with KH and ZnCl 1 Reaction scheme for Figure 3.11:

55 3.2. Results and Discussion . 2 reacting with KH and ZnCl 5 Reaction scheme for Figure 3.12:

56 3.2. Results and Discussion

Table 3.2: Reaction energies (in kcal/mol) of 1 and 5 reacting with KH and ZnCl2 (from Figures 3.11 and 3.12, respectively). Energies are relative to separated reactants.

Zincocene Step E(complex) E(‡activation) E(products) 1a −22.2 2.0 −10.2 1b −22.2 −7.5 −24.4 2b n/a n/a −28.8 2c −1.9 12.8 +1.7 2e −27.9 −16.8 −33.9 2f −2.3 31.9 12.0 3b −9.1 −6.6 −3.8 1a −30.0 7.8 −16.2 1b −30.0 −18.9 −31.1 2b n/a n/a −38.7 2c 1.6 5.3 −1.0 2e −28.2 −10.5 −25.8 2f −0.1 33.9 12.9 3b 1.3 3.9 −4.4

steps (Steps 2a and 3a, respectively, in Figure 3.2) are eliminated and replaced with the reaction of Me5CpZnK with ZnCl2 to form Me5CpZnZnCl and KCl (Step 2e). The complex (Figure 3.7) is extremely stable (∆G = −29.2 kcal/mol), and Step 2e overall is exothermic by −33.9 kcal/mol (Table 3.2), but most importantly, the activation barrier is a mere 11.1 kcal/mol (Figure 3.7). The resulting Me5CpZnZnCl can react with Me5CpK from Step 1b to form 3 and KCl (Step 3b). This final step is slightly exothermic (−3.8 kcal/mol) with a barrier of 2.5 kcal/mol. Although this final ∆G value (−3.8 kcal/mol) for Step 3b (Figure 3.9 may seem low, it should be noted that most of the thermodynamic benefit of forming 3 is realized in the preceding steps. Conversely, the desired Me5CpZnZnCl could also form by reacting Me5CpZnH with ZnCl2, but Step 2f (Figure 3.8) carries a barrier of 34.2 kcal/mol and the reaction is endothermic (∆G = 12.0 kcal/mol), so the formation of 3 is unlikely to proceed through this route. It is still theoretically possible to form 6 from 5 since steps 2e (−25.8 kcal/mol) and 3b (−4.4 kcal/mol) are still relatively exothermic. However, the formation of CpZnK via Step 1a is still prohibitive since ZnCl2 in this proposed reaction mechanism does not affect either Step 1a or 1b, so it is not surprising that 9 is still the dominate product.

57 3.3. Summary and Future Work

3.3 Summary and Future Work

It appears that the formation of 3 can be achieved much more eﬃciently through the use of KH and ZnCl2. Although 3 can be formed without the use of ZnCl2, the use of the latter seems to allow for the elimination of the free radical steps and therefore creates a more stable (and controllable) reaction. The formation of 6 is not possible because the production of the necessary precursor, CpZnK (Step 1a, ∆G‡ = 37.8 kcal/mol), is not favourable.

58 Chapter 4

Enantioselective Catalysis of Primary Aminoalkenes via a Chiral Neutral Zirconium Amidate Complex

4.1 Strategy and Methods

Before the zirconium amidate precatalyst can cyclize a primary aminoalkene, it must first be activated.67 This is done by placing it in solution (toluene, 383 K) and adding an excess amount of the aminoalkene (Figure 4.1). The resulting catalyst, 10 (in bis- amido form), can then participate in either a σ−bond insertion or a [2+2] cycloaddition mechanism (Figure 1.6). The latter will be discussed in this Chapter. Since the activation process has no effect on the enantioselectivity, it will be ignored computationally, and this study will begin from the activated bis-amido catalyst, 10 (with 5-amino-4,4- dimethylpent-1-ene as the aminoalkene). Despite the impressive advancements in microprocessor technology over the last 25 years, the ability to study large molecular systems (with 50+ carbons) using modern Kohn-Sham and other advanced methods with acceptably-sized basis sets has only been available since very recently. In this Chapter, these advancements will be put to full use. B3LYP22–25 was used yet again with Dunning’s cc-pVDZ78 basis set. For the Zr atom, the Los Alamos LANL2DZ89 basis set was used with an effective core potential for the core electrons. PCM-MP2 solvent effects32–36 were again incorporated with the B3LYP geometries while the B3LYP thermal corrections (383 K) were added to arrive at the (G383) values (in kcal/mol).

59 4.2. Results and Discussion

Figure 4.1: Precatalyst activation process to form activated bis-amido complex.

4.2 Results and Discussion

The catalytic cycle for the [2+2] cycloaddition mechanism starts from the tethered bis- amido stage for both the S and R products (10, Figure 4.2), and the production of the tethered imido is endothermic by 2.0 kcal/mol with a barrier of 29.9 kcal/mol. The production of the S product will now be discussed: the tethered imido must coil itself into a premetallacycle (Figure 4.4). This process would require many minor steps due to all of the dihedral angles involved. However, most of the necessary reaction information can be captured in one major transition state, ‡premetallacycle (5.3 kcal/mol), so this will be the only step considered and discussed. The premetallacycle will then overcome another barrier (‡metallacycle, 1.0 kcal/mol, Figure 4.3) to form the metallacycle. The coordinated aminoalkene can then donate a proton to the terminal carbon of the metallacycle (‡protonolysis, 21.2 kcal/mol). This step serves the dual purpose of bonding the extra aminoalkene to the zirconium (as an amido) and completing the cyclization of the initial aminoalkene. Yet another aminoalkene coordinates itself to

60 4.2. Results and Discussion

Figure 4.2: Proton transfer from tethered bis-amido to form tethered imido (catalytic backbone removed for clarity). the zirconium (no barrier), and after it donates a proton to the cyclized amido (16.5 kcal/mol, ‡regeneration), the now cyclized aminoalkene can dissociate into solution leaving us back at the beginning with a bis-amido, 10. In order to explain the penchant of the (-) precatalyst to form the S product (93% ee), a comparison of the kinetic and thermodynamic energies for the S and R cycles would need to show at least some energetic favourtism towards the S product, and somewhat surprisingly, every single step throughout the S cycle has a lower barrier and is more thermodynamically stable than its R counterpart up until the protonized minimum (Figure 4.5). This can be easily explained by the steric repulsions between the methyl groups of the active aminoalkene with the methyl group on the backbone of the catalyst during an R cycle, Figure 4.6. Due to these repulsions, the methyl groups prefer to point up instead of out, and this results in the formation of a strained ﬁve-

61 4.2. Results and Discussion [2+2] cycloaddition mechanism for R (red) and S (green) products starting from activated bis-amido. Figure 4.3:

62 4.2. Results and Discussion Detailed catalytic cycle for S product starting from the tethered imido. Figure 4.4:

63 4.2. Results and Discussion Detailed catalytic cycle for R product starting from the tethered imido. Figure 4.5:

64 4.2. Results and Discussion

Figure 4.6: Proposed metallacycles for R (left) and S cycles (right) product from (-) precatalyst. Figure courtesy Schafer and coworkers.67 membered ring on top of the metallacycle instead of the boat-shaped five-membered ring which would form during an S cycle when the methyl groups point out. In order to further demonstrate this, Figure 4.7 shows that the dihedral angle (φ1−2−3−4) for the metallacycle (minima) is −22.9◦ for R and 39.3◦ for S. This ring strain is present throughout most of the cycle and causes the turnover of the R cycle to be less favourable and slower than S, so therefore the S product will obviously be favoured, but with these numbers, the question is by how much? Since the S product is produced experimentally at a yield of ca. 28:1, the Turnover ˆ ˆ 72,75,76 Frequency ratio for S/R using the Christiansen matrices, MS and MR, (Eq. 1.41) should theoretically be the same. For our purposes, we will only consider the energy profiles for the S and R processes up to the protonized minimum. At this point of the cycle, the product has essentially been formed and there is a negligible chance of the reaction reversing. If the energy profiles for the S and R processes (in kcal/mol, Figures 4.4 and 4.5) are plugged into the model for assembling Christiansen matrices (Eq. 1.41), we get the following matrices:

  e7.1/RT e9.7/RT e25.0/RT ˆ  20.7/RT 3.6/RT 20.5/RT  M S =  e e e  (4.1)   e38.3/RT e40.9/RT e36.5/RT

  e13.7/RT e10.5/RT e28.9/RT ˆ  26.8/RT 4.3/RT 23.3/RT  M R =  e e e  (4.2)   e41.2/RT e37.4/RT e37.7/RT

65 4.2. Results and Discussion

Figure 4.7: Computed metallacycles for R (left) and S cycles (right). Nitrogens appear in blue while zirconiums are light green.

Each matrix has three elements (in bold) that cannot be ignored. This is not surprising since the third row of each matrix concerns the third transition state (‡protonolysis), which is the TDTS for each cycle. With these values, computing the TOF ratio is trivial:

∆(S)M(R) TOF ratio (S/R) = ∆(R)M(S)

(e19.7/RT )(e41.2/RT + e37.4/RT + e37.7/RT ) TOF ratio (S/R) ≈ (4.3) (e18.7/RT )(e38.3/RT + e40.9/RT + e36.5/RT ) ≈ 5.41 : 1

If the logic of Kozuch and Shaik is followed,75,76 we need only consider the TDTS and the TDI since they will have the greatest eﬀect on the TOF. With this knowledge in hand, M for S and R can be simpliﬁed down to just one element. For the S cycle, that element is e40.9/RT (Eq. 4.1, which corresponds to e(T3−I2−∆G)/RT in Eq. 1.41) since ‡protonolysis is the TDTS and the metallacycle is the TDI. For R, it is e41.2/RT (Eq. 4.2, which corresponds to e(T3−I1−∆G)/RT in Eq. 1.41) since the premetallacycle is

66 4.3. Summary and Future Work actually the TDI in this case. If these values are considered, the new simpliﬁed TOF ratio is:

(e19.7/RT )(e41.2/RT ) TOF ratio (S/R) ≈ (e18.7/RT )(e40.9/RT )

≈ (e1.3/RT ) (4.4)

≈ 5.51 : 1

The original TOF ratio value of ca. 5.4:1 (Eq. 4.3) corresponds to an ee% value for the S product of ca. 69%. Although this ratio does not come anywhere near the experimental value of ca. 28:1 (ee of 93%), it is rare for computational methods to produce both qualitative and (highly) quantitative accuracy for a large system such as this, so given that the S cycle has been shown to have a TOF that is signiﬁcantly higher than R, this result is about as good as can be expected.

4.3 Summary and Future Work

It has been shown that the (-) precatalyst favours the formation of the S product over the R at every stage of the catalytic cycle due to the methyl groups of the substrate not having any steric interactions with the catalytic backbone. Although the calculated TOF ratio using the Christiansen matrices72 is ca. 5.4:1 relative to the experimental ratio of ca. 28:1, the calculated TOF ratio still corresponds to a high ee% (ca. 69%). As discussed earlier, it is often diﬃcult to achieve high quantitative accuracy with computational methods for such a complex system, so this value should be lauded, but it should also be taken with a grain of salt. L. L. Schafer and coworkers have used a variety of primary aminoalkenes in their work with a wide range of ee% values. They have also shown that using an unsubstituted aminoalkene (5-aminopent-1-ene) does not form any of the cyclized products at all. These mechanisms should all be calculated and their TOF ratios compared, and if all of the calculated values trend with the experimental values, then perhaps more praise is deserved.

67 Chapter 5

Conclusion

The main purpose of this thesis was obviously to study real chemical reactions using computational methods, but the underlying theme was an attempt to bridge the gap between experiment and theory. Due to the incredible advancements in microprocessor technology over the last decade, there has since existed the ability to use computational methods to study some very large and practical systems, and there has been a large effort by computational chemists around the globe to try and convert experimentally trained chemistry students to study theory in order to have the best of both worlds. Unfortunately, quantum mechanics is often found to be an intimidating topic for most chemists, and due largely to teaching methods that are geared towards people with a penchant for physics, this conversion effort has been mostly unsuccessful. The first chapter of this dissertation attempts to bridge this rift by providing a slightly watered- down, but abstract view of quantum mechanics, and in less than 25 pages, hopefully a non-specialist can more effectively grasp concepts as simple as the Schrödingerequation (Equation 1.4) all the way up to the computation of MP2 energies (Equation 1.30) and how MP2 energies differ from HF energies. Chemists typically think in three dimensions and often picture reactions taking place, and therefore the design and flow of Chapter 1 took great care and attention to this by trying to explain the purpose of all of the major equations and giving an abstract picture of most concepts. Perhaps instructors of quantum mechanics can use these ideas to better teach their students in this field. With any luck, in 10 years or so, every chemistry Ph.D. student who graduates will have a strong computational background in their repertoire. After all, with the never ending advancements in computers, it is important now more than ever for groups to incorporate computational methods into their research. In Chapter 2, the groundbreaking discovery of decamethyldizincocene was studied at great length. For some bizarre reason, despite the original discovery of decamethyldiz- incocene48 being cited over 100 times, no one else has been able to explain how and why this product formed. Perhaps most computational groups figured this study would

68 Chapter 5. Conclusion be immensely diﬃcult to complete. They were correct, but someone had to do it. Af- ter exhaustive eﬀorts to carve out a reaction pathway, it was reported in Chapter 2 that decamethylzincocene can react with ZnEt2 to form the half-sandwich products,

Me5CpZnEt, but because the barrier for this process is 13.2 kcal/mol, this reaction can compete with the radical dissociation of decamethylzincocene, resulting in the formation of decamethyldizincocene.1 If zincocene is used as the starting material, the barrier to form the half-sandwich products is so low (3.1 kcal/mol) that assumably all of the zincocene is reacted without offering an opportunity for the zincocene to dissociate. This hypothesis was expanded to include a wider breadth of ZnR2 reagents to see if it was consistent with the experimental finding that only ZnEt2 and ZnPh2 can form a dizincocene-type molecule (when reacted specifically with a decasubstituted zincocene), and given that the barriers for forming the half-sandwich products when

ZnPh2 and ZnEt2 react with decamethylzincocene are the highest out of the eight test cases (17.1 and 13.2 kcal/mol, respectively), it appears that experimental and theory are in agreement with this reaction.1 Since it was found by Carmona and coworkers that decamethyldizincocene forms 49 more efficiently when using KH and ZnCl2, Chapter 3 focused on the reactions of zincocene and decamethylzincocene with two equivalents of KH first, and then one equivalent of ZnCl2 was considered. It was reported that both zincocene and decamethylzincocene can theoretically react with KH to form either the dizinc product or a zincate. However, due to the methyl groups of decamethylzincocene, the formation route for the zincate product is not energetically favourable, and the formation of the dizinc product (via free radical dissociation of Me5CpZnK into Me5CpZn• and K• radicals) is the only option. When zincocene is the reactant, the zincate, while not extraordinar- ily stable, becomes the favoured product because forming the desired CpZnK species (which could dissociate into CpZn• radicals) is not feasible kinetically thermodynamically or kinetically. When ZnCl2 is added, the two free radical steps are eliminated and the formation of decamethyldizincocene becomes a two-step neutral-charge electrostatic attraction and rearrangement of Me5CpZnK and ZnCl2, and since neutral electrostatic reactions are easier to control, assumably yields would be higher and more consistent, and this agrees with experimental findings.49 Although dizincocene formation is still a possibility when zincocene reacts with KH and ZnCl2, the formation of the the CpZnK species is still a necessary precursor, and its formation is unfavourable. Despite the success of the research reported in Chapters 2 and 3, decamethyldizincocene has only

69 Chapter 5. Conclusion recently found a practical application (as a catalyst for hydroamination), and even with this new revelation, it cannot produce the desired products (only racemic mixtures), so that could be another reason why no other theory groups attempted to study its formation since it is essentially a “token” compound. In Chapter 4, the enantioselective reactive properties of a chiral neutral zirconium amidate complex were explored and explained by focusing on the [2+2] cycloaddition mechanism. Although only one primary aminoalkene was studied, 5-amino-4,4- dimethylpent-1-ene, the calculated TOF ratio for the catalytic cycle (S product relative to R) when using Christiansen matrices72 was ca. 5.4:1, and although numerically this value still diﬀers greatly from the experimental value of ca. 28:1,67 the fact that both experiment and theory suggest that the S product is heavily favoured with this cycle is as good a result as can be hoped for, and it provides strong evidence that the [2+2] cycloaddition mechanism is an appropriate description for how this catalyst reacts, but as suggested in Chapter 4, more primary aminoalkenes need to be studied to see if their theoretical TOF ratios agree qualitatively with experiment. Only then can the [2+2] mechanism be truly solidiﬁed as being correct. Also, the [2+2] mechanism needs to explain why the catalytic cycle for the unsubstituted aminoalkene, 5-amino-pent-1-ene, does not turnover at all.

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76 Appendix A - Calculated Energies for Chapter 2

Table A.1: Reaction energies (in hartrees) of 5 reacting with ZnEt2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZnEt2 −1937.55558 0.09742 −1935.64742 −1935.54999 CpZnCp −2166.19054 0.13092 −2163.61369 −2163.48277 Total −4103.74613 0.22835 −4099.26111 −4099.03277 Reactant Complex −4103.76027 0.25023 −4099.28671 −4099.03648 ‡associative −4103.75513 0.24970 −4099.27859 −4099.02889 Intermediate −4103.76175 0.25051 −4099.28878 −4099.03827 ‡symmetric −4103.75714 0.25031 −4099.28366 −4099.03335 CpZnEt −2051.88663 0.11473 −2049.64444 −2049.52971 CpZnEt −2051.88663 0.11473 −2049.64444 −2049.52971 Total −4103.77325 0.22945 −4099.28888 −4099.05942 ‡asymmetric −4103.66958 0.24864 −4099.18677 −4098.93813 CpZnZnCp −3945.35103 0.12879 −3941.42662 −3941.29783 Et2 −158.45877 0.10708 −157.89907 −157.79199 Total −4103.80980 0.23586 −4099.32569 −4099.08983

77 Appendix A - Calculated Energies for Chapter 2

Table A.2: Reaction energies (in hartrees) of 1 reacting with ZnEt2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZnEt2 −1937.55558 0.09742 −1935.64742 −1935.54999 Me5CpZnCpMe5 −2559.35755 0.39071 −2555.47056 −2555.07985 Total −4496.91313 0.48813 −4491.11797 −4490.62984 Reactant Complex −4496.91661 0.50872 −4491.14035 −4490.63163 ‡associative −4496.91041 0.51027 −4491.14074 −4490.63047 Intermediate −4496.91299 0.51014 −4491.14422 −4490.63408 ‡symmetric −4496.89906 0.50786 −4491.12099 −4490.61313 Me5CpZnEt −2248.47007 0.24401 −2245.57103 −2245.32702 Me5CpZnEt −2248.47007 0.24401 −2245.57103 −2245.32702 Total −4496.94013 0.48802 −4491.14206 −4490.65403 ‡asymmetric −4496.83536 0.50956 −4491.05204 −4490.54248 Me5CpZnZnCpMe5 −4338.51871 0.38526 −4333.28334 −4332.89808 Et2 −158.45877 0.10708 −157.89907 −157.79199 Total −4496.97748 0.49234 −4491.18241 −4490.69007

Table A.3: Reaction energies (in hartrees) of 5 reacting with ZnMe2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZnMe2 −1858.93701 0.04471 −1857.29773 −1857.25303 CpZnCp −2166.19054 0.13092 −2163.61369 −2163.48277 Total −4025.12755 0.17563 −4020.91143 −4020.73580 Intermediate −4025.14424 0.19539 −4020.93556 −4020.74018 ‡symmetric −4025.13767 0.19508 −4020.92959 −4020.73451 CpZnMe −2012.57707 0.08807 −2010.46912 −2010.38105 CpZnMe −2012.57707 0.08807 −2010.46912 −2010.38105 Total −4025.15414 0.17614 −4020.93824 −4020.76211

78 Appendix A - Calculated Energies for Chapter 2

Table A.4: Reaction energies (in hartrees) of 1 reacting with ZnMe2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZnMe2 −1858.93701 0.04471 −1857.29773 −1857.25303 Me5CpZnCpMe5 −2559.35755 0.39071 −2555.47056 −2555.07985 Total −4418.29456 0.43542 −4412.76829 −4412.33287 Intermediate −4418.30039 0.45407 −4412.79400 −4412.33993 ‡symmetric −4418.29594 0.45409 −4412.78569 −4412.33161 Me5CpZnMe −2209.16071 0.21570 −2206.39544 −2206.17974 Me5CpZnMe −2209.16071 0.21570 −2206.39544 −2206.17974 Total −4418.32142 0.43140 −4412.79088 −4412.35948

Table A.5: Reaction energies (in hartrees) of 5 reacting with ZniPr2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZniPr2 −2016.18062 0.15078 −2014.00753 −2013.85674 CpZnCp −2166.19054 0.13092 −2163.61369 −2163.48277 Total −4182.37116 0.28171 −4177.62122 −4177.33952 Intermediate −4182.38181 0.30414 −4177.64819 −4177.34406 ‡symmetric −4182.38030 0.30293 −4177.64428 −4177.34134 CpZniPr −2091.19908 0.14126 −2088.82508 −2088.68382 CpZniPr −2091.19908 0.14126 −2088.82508 −2088.68382 Total −4182.39815 0.28253 −4177.65016 −4177.36763

79 Appendix A - Calculated Energies for Chapter 2

Table A.6: Reaction energies (in hartrees) of 1 reacting with ZniPr2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZniPr2 −2016.18062 0.15078 −2014.00753 −2013.85674 Me5CpZnCpMe5 −2559.35755 0.39071 −2555.47056 −2555.07985 Total −4575.53817 0.54149 −4569.47808 −4568.93659 Intermediate −4575.52918 0.56530 −4569.49994 −4568.93464 ‡symmetric −4575.52099 0.56242 −4569.48544 −4568.92303 Me5CpZniPr −2287.78230 0.26879 −2284.75202 −2284.48322 Me5CpZniPr −2287.78230 0.26879 −2284.75202 −2284.48322 Total −4575.56460 0.53758 −4569.50403 −4568.96645

Table A.7: Reaction energies (in hartrees) of 5 reacting with ZnPh2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZnPh2 −2242.44068 0.14540 −2239.61769 −2239.47229 CpZnCp −2166.19054 0.13092 −2163.61369 −2163.48277 Total −4408.63122 0.27633 −4403.23139 −4402.95506 Intermediate −4408.65753 0.29773 −4403.27580 −4402.97806 ‡symmetric −4408.63949 0.29601 -4403.25819 −4402.96217 CpZnPh −2204.32666 0.13835 −2201.62769 −2201.48935 CpZnPh −2204.32666 0.13835 −2201.62769 −2201.48935 Total −4408.65332 0.27669 −4403.25539 −4402.97870

80 Appendix A - Calculated Energies for Chapter 2

Table A.8: Reaction energies (in hartrees) of 1 reacting with ZnPh2. B3LYP thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether.)

Species Raw B3LYP Thermal MP2 SP Total

ZnPh2 −2242.44068 0.14540 −2239.61769 −2239.47229 Me5CpZnCpMe5 −2559.35755 0.39071 −2555.47056 −2555.07985 Total −4801.79823 0.53611 −4795.08825 −4794.55213 1st Intermediate −4801.8050 0.55534 −4795.12918 −4794.57384 ‡1stsymmetric −4801.7872 0.55626 −4795.10276 −4794.54650 2nd Intermediate −4801.8059 0.55567 −4795.12522 −4794.56954 ‡2nd symmetric −4801.8021 0.55538 −4795.11628 −4794.56090 Me5CpZnPh −2400.91067 0.26540 −2397.55531 -2397.28992 Me5CpZnPh −2400.91067 0.26540 −2397.55531 -2397.28992 Total −4801.82133 0.53079 −4795.11062 −4794.57983

81 Appendix A - Calculated Energies for Chapter 2

Table A.9: Homolytic dissociation energy components for (in hartrees) for 5, 6, 1, and 3. Raw ROB3LYP complex energies include counterpoise corrections. B3LYP Ther- mal corrections were at 263 K and MP2 single point energies used IEF-PCM for diethyl ether. ROMP2 SP Counterpoise corrections computed separately using counterpoise opti- 82 mized B3LYP geometries. zc = CpZnCp, dz = CpZnZnCp, dmz = Me5CpZnCpMe5, dmdz = Me5CpZnZnCpMe5.

Species Raw ROB3LYP ROMP2 SP Thermal Counterpoise Total zc −2166.1761 −2163.6141 0.1307 0.0200 −2163.4634 dz −3945.3406 −3941.4276 0.1299 0.0160 −3941.2818 dmz −2559.3407 −2555.4700 0.3897 0.0261 −2555.0542 dmdz −4338.5090 −4333.2824 0.3853 0.018 −4332.8805 CpZn• −1972.6188 −1970.6618 0.0563 n/a −1970.6055 Cp• −193.47491 −192.8436 0.0542 n/a −192.7894 Total −2166.0937 −2163.5054 0.1105 n/a −2163.3949 CpZn• −1972.6188 −1970.6618 0.0563 n/a −1970.6055 CpZn• −1972.6188 −1970.6618 0.0563 n/a −1970.6055 Total −3945.2376 −3941.3236 0.1126 n/a −3941.2110 Me5CpZn• −2169.2035 −2166.5873 0.1841 n/a −2166.4032 Me5Cp• −390.0730 −388.7760 0.1826 n/a −388.5935 Total −2559.2765 −2555.3633 0.3667 n/a −2554.9967 Me5CpZn• −2169.20351 −2166.5873 0.18415 n/a −2166.4032 Me5CpZn• −2169.20351 −2166.5873 0.18415 n/a −2166.4032 Total −4338.40702 −4333.1746 0.36830 n/a −4332.8064

82 Appendix B - Calculated Energies for Chapter 3

Table B.1: Homolytic dissociation energy components for (in hartrees) for 3 and 6 (Step 3a), and CpZnK and Me5CpZnK (Step 2a). Raw ROB3LYP complex energies include counterpoise corrections. B3LYP Thermal corrections were at 263 K and MP2 single point energies used IEF-PCM for Tetrahydrofuran. ROMP2 SP Counterpoise corrections computed separately using counterpoise optimized B3LYP geometries.82 dz = CpZnZnCp, dmdz = Me5CpZnZnCpMe5.

Species Raw ROB3LYP ROMP2 SP Thermal Counterpoise Total dz −3945.4449 −3941.5749 0.1239 0.0120 −3941.4390 dmdz −4338.6389 −4333.4551 0.3933 0.0120∗ −4333.0497 CpZnK −2572.6406 −2569.9669 0.05813 0.0068 −2569.9020 ∗ Me5CpZnK −2769.2267 −2765.8950 0.1726 0.0068 −2765.7156 CpZn• −1972.6701 −1970.7349 0.0508 n/a −1970.6841 CpZn• −1972.6701 −1970.7349 0.0508 n/a −1970.6841 Total −3945.3403 −3941.4698 0.1016 n/a −3941.3682 Me5CpZn• −2169.2673 −1970.6720 0.1778 n/a −2166.4942 Me5CpZn• −2169.2673 −1970.6720 0.1778 n/a −2166.4942 Total −4338.5346 −3941.3440 0.3556 n/a −4332.9884 K• −599.8904 −599.1619 −0.0155 n/a −599.1619 CpZn• −1972.6701 −1970.7349 0.0508 n/a −1970.6841 Total −2572.5606 −3941.4698 0.1016 n/a −3941.3682 K• −599.8904 −599.1619 −0.0155 n/a −599.1619 Me5CpZn• −2169.2673 −1970.6720 0.1778 n/a −2166.4942 Total −2769.1578 −3941.3440 0.3556 n/a −4332.9884

∗ Counterpoise corrections for Me5CpZnZnCpMe5 and Me5CpZnK were unavailable due to “excessive mixing of core and valence orbitals” with Frozen Core MP2 calculation. Corrections for CpZnZnCp and CpZnK, respectively, were used instead.

83 Appendix B - Calculated Energies for Chapter 3

Table B.2: Reaction energies (in hartrees) of 5 reacting with KH. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran.

Species Raw B3LYP Thermal MP2 SP Total CpZnCp −2166.23427 0.12346 −2163.68884 −2163.56538 KH −600.45733 −0.01656 −599.73101 −599.74757 Total −2766.69160 0.10690 −2763.41985 −2763.31295 Reactant Complex −2766.76085 0.12785 −2763.48860 −2763.36074 (pre-Steps 1a,1b) ‡Step 1a −2766.70602 0.12318 −2763.42377 −2763.30059 CpZnK −2572.64139 0.04387 −2569.96221 −2569.91834 CpH −194.11593 0.06637 −193.48685 −193.42049 Total −2766.75732 0.11024 −2763.44907 −2763.33883 ‡Step 1b −2766.73262 0.12653 −2763.46965 −2763.34311 CpZnH −1973.29282 0.05819 −1971.34983 −1971.29165 CpK −793.45799 0.05213 −792.12293 −792.07080 Total −2766.75081 0.11032 −2763.47277 −2763.36245 CpZnH −1973.29282 0.05819 −1971.34983 −1971.29165 KH −600.45733 −0.01656 −599.73101 −599.74757 Total −2573.75015 0.04163 −2571.08084 −2571.03922 Reactant Complex −2573.80732 0.05813 −2571.12745 −2571.06932 ‡Step 2d −2573.74007 0.05387 −2571.06050 −2571.00663 CpZnK −2572.64139 0.04387 −2569.96221 −2569.91834 H2 −1.17897 −0.00111 −1.15782 −1.15892 Total −2573.82037 0.04277 −2571.12003 −2571.07727 CpK −793.45799 0.05213 −792.12293 −792.07080 CpZnCp −2166.23427 0.12346 −2163.68884 −2163.56538 Total −2959.69226 0.17559 −2955.81177 −2955.63618 Reactant Complex −2959.70271 0.19229 −2955.82586 −2955.63358 ‡Step 2c −2959.70181 0.19545 −2955.82319 −2955.62774 K[ZnCp3] −2959.71744 0.19743 −2955.83516 −2955.63773 CpH −194.11593 0.06637 −193.48685 −193.42049 KH −600.45733 −0.01656 −599.73101 −599.74757 Total −794.57326 0.04981 −793.21786 −793.16806 CpK −793.45799 0.05213 −792.12293 −792.07080 H2 −1.17897 −0.00111 −1.15782 −1.15892 Total −794.63697 0.05103 −793.28075 −793.22973

84 Appendix B - Calculated Energies for Chapter 3

Table B.3: Additional reaction energies (in hartrees) of 5 reacting with KH as well as ZnCl2. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran.

Species Raw B3LYP Thermal MP2 SP Total CpZnK −2572.64139 0.04387 −2569.96221 −2569.91834 ZnCl2 −2699.68323 −0.02322 −2697.21050 −2697.23372 Total −5272.32462 0.02065 −5297.17271 −5267.15206 Reactant Complex −5272.37970 0.03813 −5267.22461 −5267.18648 ‡Step 2e −5272.36664 0.03995 −5267.20871 −5267.16876 CpZnZnCl −4212.16989 0.04658 −4208.32998 −4208.28340 KCl −1060.17809 −0.02237 −1058.88748 −1058.90985 Total −5272.34798 0.02421 −5267.21746 −5267.19325 CpZnH −1973.29282 0.05819 −1971.34983 −1971.29165 ZnCl2 −2699.68323 −0.02322 −2697.21050 −2697.23372 Total −4672.97605 0.03497 −4668.56033 −4668.52536 Reactant Complex −4672.97934 0.04838 −4668.57389 −4668.52551 ‡Step 2f −4672.93073 0.04804 −4668.51941 −4668.47137 CpZnZnCl −4212.16989 0.04658 −4208.32998 −4208.28340 HCl −460.80327 −0.01083 −460.21059 −460.22143 Total −4672.97316 0.03575 −4668.54058 −4688.50483 CpZnZnCl −4212.16989 0.04658 −4208.32998 −4208.28340 CpK −793.45799 0.05213 −792.12293 −792.07080 Total −5005.62788 0.09871 −5000.45292 −5000.35420 Reactant Complex −5005.65454 0.11455 −5000.46674 −5000.35219 ‡Step 3b −5005.64358 0.11748 −5000.46544 −5000.34796 CpZnZnCp −3945.44755 0.12374 −3941.57518 −3941.45144 KCl −1060.17809 −0.02237 −1058.88748 −1058.90985 Total −5005.62564 0.10137 −5000.46266 −5000.36129

85 Appendix B - Calculated Energies for Chapter 3

Table B.4: Reaction energies (in hartrees) of 1 reacting with KH. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran.

Species Raw B3LYP Thermal MP2 SP Total

Me5CpZnCpMe5 −2559.42735 0.38308 −2555.57249 −2555.18941 KH −600.45733 −0.01656 −599.73101 −599.74757 Total −3159.88468 0.36652 −3155.30350 −3154.93699 Reactant Complex −3159.92737 0.38309 −3155.35552 −3154.97243 (pre-Steps 1a,1b) ‡Step 1a −3159.88746 0.38068 −3155.31441 −3154.93373 Me5CpZnK −2769.22804 0.17142 −2765.88939 −2765.71797 Me5CpH −390.71796 0.19556 −389.43082 −389.23526 Total −3159.94601 0.36698 −3155.32021 −3154.95323 ‡Step 1b −3159.88667 0.38636 −3155.33536 −3154.94900 Me5CpZnH −2169.88687 0.18608 −2167.28883 −2167.10275 Me5CpK −990.04483 0.17793 −988.04778 −987.86985 Total −3159.93170 0.36401 −3155.33661 −3154.97260

Me5CpZnH −2169.88687 0.18608 −2167.28883 −2167.10275 KH −600.45733 −0.01656 −599.73101 −599.74757 Total −2770.34420 0.16952 −2767.01984 −2766.85032 Reactant Complex −2770.38827 0.18455 −2767.06159 −2766.87704 ‡Step 2d −2770.33166 0.18143 −2766.99628 −2766.81485 Me5CpZnK −2769.22804 0.17142 −2765.88939 −2765.71797 H2 −1.17897 −0.00111 −1.15782 −1.15892 Total −2770.40702 0.17032 −2767.04721 −2766.87689

Me5CpK −990.04483 0.17793 −988.04778 −987.86985 Me5CpZnCpMe5 −2559.42735 0.38308 −2555.57249 −2555.18941 Total −3549.47218 0.56101 −3543.62027 −3543.05926 Reactant Complex −3549.48115 0.58063 −3543.64293 −3543.06230 ‡Step 2c −3549.44987 0.58676 −3543.62558 −3543.03882 K[ZnMe5Cp3] −3549.45979 0.59037 −3543.64699 −3543.05663

Me5CpH −390.71796 0.19556 −389.43082 −389.23526 KH −600.45733 −0.01656 −599.73101 −599.74757 Total −991.17530 0.17900 −989.16183 −988.98283 Me5CpK −990.04483 0.17793 −988.04778 −987.86985 H2 −1.17897 −0.00111 −1.15782 −1.15892 Total −991.22381 0.17683 −989.20560 −989.02877

86 Appendix B - Calculated Energies for Chapter 3

Table B.5: Additional reaction energies (in hartrees) of 1 reacting with KH as well as ZnCl2. B3LYP thermal corrections were at 293 K and MP2 single point energies used IEF-PCM for tetrahydrofuran.

Species Raw B3LYP Thermal MP2 SP Total

Me5CpZnK −2769.22804 0.17142 −2765.88939 −2765.71797 ZnCl2 −2699.68323 −0.02322 −2697.21050 −2697.23372 Total −5468.91127 0.14820 −5463.09988 −5462.95168 Reactant Complex −5468.97620 0.16753 −5463.16374 −5463.99621 ‡Step 2e −5468.95841 0.16870 −5463.14710 −5463.97840 Me5CpZnZnCl −4408.76869 0.17474 −4404.27066 −4404.09592 KCl −1060.17809 −0.02237 −1058.88748 −1058.90985 Total −5468.94678 0.15237 −5463.15815 −5463.00578

Me5CpZnH −2169.88687 0.18608 −2167.28883 −2167.10275 ZnCl2 −2699.68323 −0.02322 −2697.21050 −2697.23372 Total −4869.57010 0.16286 −4864.49932 −4864.33646 Reactant Complex −4869.57646 0.17893 −4864.51904 −4864.34011 ‡Step 2f −4869.52825 0.17600 −4864.46166 −4864.28567 Me5CpZnZnCl −4408.76869 0.17474 −4404.27066 −4404.09592 HCl −460.80327 −0.01083 −460.21059 −460.22143 Total −4869.57196 0.16391 −4864.48126 −4864.31735

Me5CpZnZnCl −4408.76869 0.17474 −4404.27066 −4404.09592 Me5CpK −990.04483 0.17793 −988.04778 −987.86985 Total −5398.81352 0.35267 −5392.31844 −5391.96577 Reactant Complex −5398.83922 0.37117 −5392.35140 −5391.98023 ‡Step 3b −5398.83244 0.37494 −5392.35127 −5391.97632 Me5CpZnZnCpMe5 −4338.64183 0.39333 −4333.45527 −4333.06194 KCl −1060.17809 −0.02237 −1058.88748 −1058.90985 Total −5398.81992 0.37096 −5392.34275 −5391.97179

87 Appendix C - Calculated Energies for Chapter 4

Table C.1: Reaction energies (in hartrees) for the catalytic cycle for the S product. B3LYP thermal corrections were at 383 K and MP2 single point energies used IEF-PCM for toluene.

Species Raw B3LYP Thermal MP2 SP Total Bis-Amido −2282.7998 0.8310 −2275.5207 −2274.6896 ‡Imido −2282.7442 0.8301 −2275.4721 −2274.6420 Tethered Imido −2282.7883 0.8323 −2275.5187 −2774.6863 ‡Premetallacycle −2282.7738 0.8374 −2275.5153 −2274.6778 Premetallacycle −2822.7812 0.8402 −2275.5294 −2274.6891 ‡Metallacycle −2282.7768 0.8409 −2275.5284 −2274.6875 Metallacycle −2282.7846 0.8351 −2275.5268 −2274.6917 ‡Protonolysis −2282.7554 0.8363 −2275.4942 −2274.6579 Protonized −2282.8218 0.8399 −2275.5576 −2274.7177 Extra Substrate −330.5147 0.1607 −329.4264 −329.2657 Total −2613.3365 1.0006 −2604.9840 −2603.9834 Triple Protonized −2613.3471 1.0282 −2605.0228 −2603.9946 ‡Regeneration −2613.3202 1.0256 −2604.9945 −2603.9689 Coordinated Product −-2613.3434 1.0312 −2605.0173 −2603.9860

88 Appendix C - Calculated Energies for Chapter 4

Table C.2: Reaction energies (in hartrees) for the catalytic cycle for the R product. B3LYP thermal corrections were at 383 K and MP2 single point energies used IEF-PCM for toluene.

Species Raw B3LYP Thermal MP2 SP Total Bis-Amido −2282.7998 0.8310 −2275.5207 −2274.6896 ‡Imido −2282.7442 0.8301 −2275.4721 −2274.6420 Tethered Imido −2282.7883 0.8323 −2275.5187 −2774.6863 ‡Premetallacycle −2282.7709 0.8387 −2275.5088 −2274.6700 Premetallacycle −2282.7837 0.8387 −2275.5306 −2274.6919 ‡Metallacycle −2282.7742 0.8405 −2275.5195 −2274.6790 Metallacycle −2282.7830 0.8386 −2275.5245 −2274.6858 ‡Protonolysis −2282.7554 0.8363 −2275.4923 −2274.6560 Protonized −2282.8197 0.8380 −2275.5540 −2274.7161 Extra Substrate −330.5147 0.1607 −329.4264 −329.2657 Total −2613.3344 0.9986 −2604.9804 −2603.9818 Triple Protonized −-2613.3503 1.0282 −2605.0277 −2603.9995 ‡Regeneration −2613.3181 1.0238 −2604.9937 −2603.9699 Coordinated Product −2613.3369 1.0281 −2605.0091 −2603.9809