The Pennsylvania State University the Graduate School QUANTUM

The Pennsylvania State University The Graduate School

QUANTUM MECHANICAL METHODS FOR CALCULATING

PROTON TUNNELING SPLITTINGS AND PROTON-COUPLED

ELECTRON TRANSFER VIBRONIC COUPLINGS

A Dissertation in Chemistry by Jonathan H. Skone

c 2008 Jonathan H. Skone

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

May 2008 The dissertation of Jonathan H. Skone was reviewed and approved∗ by the following:

Sharon Hammes-Schiﬀer Eberly Professor of Biotechnology and Professor of Chemistry Thesis Advisor, Chair of Committee

James Bernhard Anderson Evan Pugh Professor of Chemistry

Mark Maroncelli Professor of Chemistry

Kristen Fichthorn Merrel R. Fenske Professor of Chemical Engineering

Ayusman Sen Professor of Chemistry Chemistry Department Head

∗Signatures are on ﬁle in the Graduate School. Abstract

Development of quantum mechanical methods for the calculation of proton tunneling splittings and proton-coupled electron transfer vibronic couplings is presented in this thesis. The fundamental physical principles underlying proton transfer in the electronically adiabatic and nonadiabatic limits are illustrated by applying the quantum mechanical methods we developed to chemical systems exemplary of the electronically adiabatic and nonadiabatic proton-tunneling regimes. Overall, this thesis emphasizes the need for quantum chemical methods that avoid the adiabatic separation of the quantum proton and electron, are computationally tractable, and treat all quantum particles three-dimensionally. The nuclear-electronic orbital nonorthogonal configuration interaction (NEO- NOCI) approach is presented for calculating proton tunneling splittings and vibronic couplings. The NEO approach is a molecular orbital based method that avoids the Born-Oppenheimer separation of the select protons and electrons, thereby making methods developed within this scheme, such as NEO-NOCI, applicable to electronically nonadiabatic proton transfer. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wavefunctions are expressed as linear combinations of two nonorthogonal localized nuclear- electronic wavefunctions obtained at the NEO-Hartree-Fock level. The advantages of the NEO-NOCI approach are the removal of the adiabatic separation between the electrons and the quantum nuclei, the computational efficiency, the potential for systematic improvement by enhancing the basis sets and number of configurations, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between these two delocalized vibronic states. The proton tunneling splittings calculated with the NEO-NOCI approach for the [He–H–He]+ model system with a range of fixed He–He distances are shown to be in excellent agreement with NEO-full CI and Fourier grid calculations.

iii The vibronic couplings for the phenoxyl/phenol and the benzyl/toluene self- exchange reactions are calculated with a semiclassical approach. The magnitude of the vibronic coupling and its dependence on the proton donor-acceptor distance can significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of proton-coupled electron transfer reactions. Both of these self-exchange reactions are vibronically nonadiabatic with respect to a solvent environment at room temperature, but the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. For both systems, the vibronic coupling decreases exponentially with the proton donor-acceptor distance for the range of distances studied. In addition to providing insights into the fundamental physical differences between these two types of reactions, the present analysis provides a new diagnostic for differentiating between the conventionally defined hydrogen atom transfer and proton-coupled electron transfer reactions. The impact of substituents on the vibronic coupling for the phenoxyl/phenol self-exchange reaction, which occurs by a proton-coupled electron transfer mechanism, is also investigated. The vibronic couplings are calculated with a grid-based nonadiabatic method and the NEO-NOCI method. The quantitative agreement between these two methods for the unsubstituted phenoxyl/phenol system and the qualitative agreement in the predicted trends for the substituted phenoxyl/phenol systems provides a level of validation for both methods. Correlations between the vibronic coupling and physical properties of the phenol are also analyzed. The observed trends enable the prediction of the impact of general substituents on the vibronic coupling, and hence the rate, for the phenoxyl/phenol self-exchange reaction.

iv Table of Contents

List of Figures vii

List of Tables xi

Acknowledgments xvii

Chapter 1 Introduction 1

Chapter 2 Nuclear-Electronic Orbital Nonorthogonal Conﬁguration Inter- action Approach 6 2.1 Introduction ...... 6 2.2 Theory ...... 10 2.2.1 Background ...... 10 2.2.2 NEO nonorthogonal CI ...... 12 2.2.3 Calculation of tunneling splittings ...... 15 2.3 Results ...... 17 2.4 Conclusions ...... 22

Chapter 3 Calculation of Vibronic Couplings for Proton-Coupled Electron Transfer Reactions 25 3.1 Introduction ...... 25 3.2 Theory and methods ...... 28 3.2.1 Analytical expressions for vibronic couplings ...... 28 3.2.2 Computational methodology ...... 30 3.3 Results ...... 33

v 3.4 Conclusions ...... 44

Chapter 4 Substituent Eﬀects on the Proton-Coupled Electron Transfer Vibronic Coupling 46 4.1 Introduction ...... 46 4.2 Theory and methods ...... 48 4.2.1 Grid-based nonadiabatic method for calculating vibronic couplings ...... 51 4.2.2 Nuclear-electronic orbital method for calculating vibronic couplings ...... 54 4.3 Results and discussion ...... 57 4.4 Conclusions ...... 65

Chapter 5 Conclusions 66 5.1 Summary and concluding remarks ...... 66

Appendix A NEO-NOCI Additional Equations and Data 70 A.1 Expression for the tunneling splitting ∆ and vibronic coupling VDA for asymmetric systems ...... 70 A.2 Additional data collected from the NEO-NOCI tunneling splitting calculations ...... 73 A.3 Memory demands and timing for the NEO-NOCI method ...... 79

Appendix B Valence Bond Models 81

Appendix C Substituted Phenoxyl-Phenol Optimized Geometries and Vi- bronic Couplings 84 C.1 Transition state optimized geometries ...... 84 C.2 Vibronic couplings of transition state optimized substituted phenoxyl/phenol systems ...... 100

Appendix D Multi-Dimensional Malonaldehyde Tunneling Splitting Calcu- lations 101

Bibliography 108

vi List of Figures

2.1 [X–H–X] model system with the hydrogen nuclear wavefunction depicted at (a) the NEO-HF level (localized) and (b) the NEO-FCI level (delocalized). Here X represents a general donor and acceptor group. For the [He–H–He]+ model system, X=He...... 8 2.2 Schematic illustration of the one-dimensional hydrogen potential energy curve along the He–He axis for the [He–H–He]+ model system with (a) the corresponding NEO-HF localized wavefunctions ΨI and ΨII and (b) the corresponding ground and excited state NEO-NOCI wavefunctions Ψ0 and Ψ1, which are linear combinations of the localized wavefunctions shown in (a) ...... 9

3.1 State-averaged CASSCF ground and excited state electronically adiabatic potential energy curves along the transferring hydrogen coordinate for (a) the phenoxyl/phenol and (b) the benzyl/toluene system. The coordinates of all nuclei except the transferring hydrogen correspond to the transition state geometry. The proton donor-acceptor distances are 2.40A˚ and 2.72A,˚ respectively, for the phenoxyl/phenol and the benzyl/ toluene system. The CASSCF results are depicted as open circles that are blue for the ground state and red for the excited state. The black dashed lines represent the diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II. The mixing of these two diabatic states with the electronic coupling V ET leads to the CASSCF ground and excited state electronically adiabatic curves depicted with solid colored lines following the colored open circles. For the phenoxyl/phenol system, the solid colored lines and the black dashed lines are nearly indistinguishable because the adiabatic and diabatic potential energy curves are virtually identical except in the transition state region...... 34

vii 3.2 The two highest-energy occupied electronic molecular orbitals for (a) the phenoxyl/phenol and (b) the benzyl/toluene system. The electronic wavefunctions for diabatic states I and II are calculated at the minima of the ground state electronically adiabatic potential energy curves shown in Figure 3.1, and the electronic wavefunctions for the transition states (TS) are calculated at the maxima of these potential energy curves. For both systems, the ground state electronic wavefunction is predominantly single configurational, and the lower molecular orbital is doubly occupied, while the upper molecular orbital is singly occupied. A contour value of 0.08 was used for all molecular orbital pictures. These figures were generated with MacMolPlot [95]...... 35 3.3 (a) Diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II and the correspond- (I) (II) ing proton vibrational wavefunctions ϕD (blue) and ϕA (red) for the phenoxyl/phenol system. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling V ET and the overlap of the reactant and product proton D (I) (II)E vibrational wavefunctions ϕD |ϕA . (b) Electronically adiabatic ground state potential energy curve and the corresponding proton vibrational wavefunctions for the benzyl/toluene system. Since this reaction is electronically adiabatic, the vibronic coupling is equal to half of the energy splitting ∆ between the symmetric (cyan) and antisymmetric (magenta) proton vibrational states for the electronic ground state potential energy surface. For illustrative purposes, the excited vibrational state is shifted up in energy by 0.8 kcal/mol. . . 39 (sc) 3.4 The dependence of the semiclassical vibronic couplings VDA on the proton donor-acceptor distance R for (a) the phenoxyl/phenol and (b) the benzyl/toluene system. The results for hydrogen transfer are shown as open circles, and the results for deuterium transfer are shown as open triangles. The dashed and solid lines correspond to a fit to the functional form exp [−αR] for hydrogen (solid red) and deuterium (dashed blue) transfer. The corresponding values of α (sc) −1 are given in Table 3.4. The couplings VDA are in units of cm . . . 41

viii 4.1 Optimized geometry for the unsubstituted phenoxyl/phenol system. (a) The proton donor-acceptor O–O distance R is indicated. The pairs of substitution positions maintaining C2 symmetry are also indicated, where m, p, and o refer to meta, para, and ortho, respectively. (b) The COOC dihedral angle θ is indicated...... 50 4.2 (a) State-averaged CASSCF ground and excited state electronically adiabatic potential energy curves along the transferring hydrogen coordinate for the unsubstituted phenoxyl/phenol system. The coordinates of all nuclei except the transferring hydrogen correspond to the transition state geometry. The proton donor-acceptor distance is R = 2.40 A.˚ The CASSCF results are depicted as open circles that are blue for the ground state and red for the excited state. The black dashed lines represent the diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II. The mixing of these two diabatic states with the electronic coupling V ET leads to the CASSCF ground and excited state electronically adiabatic curves depicted with solid colored lines following the colored open circles. The solid colored lines and the black dashed lines are nearly indistinguishable because the adiabatic and diabatic potential energy curves are virtually identical except in the transition state region. (b) Diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II and the corresponding proton vibrational (I) (II) wavefunctions ϕD (blue) and ϕA (red) for the phenoxyl/phenol system. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling and the overlap of the reactant and product proton vibrational wavefunc- D (I) (II)E tions ϕD |ϕA ...... 53 4.3 Mixed electronic-nuclear wavefunctions calculated with the NEO approach for the unsubstituted phenoxyl/phenol system. (a) The two localized NEO-HF nuclear-electronic wavefunctions, where the highest-energy doubly occupied electronic molecular orbital and the proton molecular orbital are depicted. (b) The delocalized NEO- NOCI ground and ﬁrst excited vibronic state nuclear-electronic wavefunctions, where the highest-energy doubly occupied electronic molecular orbital and the proton molecular orbital are depicted. These ﬁgures were generated with MacMolPlot [95]...... 56

ix 4.4 Vibronic couplings of substituted phenoxyl/phenol systems (a) arranged from highest to lowest in magnitude, (b) as functions of Hammett constant σm,p [104], (c) as functions of the O–H bond dissociation enthalpy (BDE) [105], (d) as functions of ionization potential (IP) [105], (e) as functions of redox potential (E) at pH = 7 [106], (f) as functions of pKa [107, 108]. The vibronic couplings are calculated with the one-dimensional grid-based nonadiabatic method at R = 2.30 A.˚ Electron donating substituents are denoted with open diamonds, electron withdrawing substituents are denoted with ﬁlled diamonds, and the unsubstituted phenoxyl/phenol system is denoted with a red ﬁlled square. The m-OH substituent is considered to be electron withdrawing because of its positive Ham- mett constant...... 63

A.1 The memory footprint and timing information for the NEO-NOCI method using either the two step (black) or three-step (red) AO to MO transformation...... 80

x List of Tables

2.1 Exponents for the QZSPNB and QZSPDN0 nuclear basis sets. The QZSPNB nuclear basis set includes four each of s- and p-type Gaus- sians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians...... 17 2.2 Vibrational energy level splittings in cm−1 calculated with the one- dimensional and three-dimensional grid methods. The potential was generated at the full CI level with 128 grid points for the one- dimensional grid and 64 grid points per dimension for the three- dimensional grid. The 6-31G electronic basis set [72] was used for all calculations...... 18 2.3 Splittings in cm−1 calculated with the NEO-NOCI and NEO-FCI methods for two diﬀerent nuclear basis sets. The DZSPNB nuclear basis set includes two each of s- and p-type Gaussians. The DZSPDN0 nuclear basis set is the same as DZSPNB with the addition of one set of d-type Gaussians. The nuclear basis function center positions were optimized at the NEO-NOCI level for both the ground and excited vibronic states. The 6-31G electronic basis set was used for all calculations. Due to the large memory requirements and computational expense of the NEO-FCI/DZSPDN0 calculations, only two representative He–He distances were studied at this level of theory...... 19

xi 2.4 Tunneling splittings in cm−1 calculated with the NEO-NOCI method for two diﬀerent nuclear basis sets and with the one-dimensional and three-dimensional grid methods. The QZSPNB nuclear basis set includes four each of s- and p-type Gaussians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians. For the grid calculations, the potential was generated at the full CI level with 128 grid points for the one- dimensional grid and 64 grid points per dimension for the three- dimensional grid. The nuclear basis function centers were optimized at the NEO-HF level for the localized states and were the same for the ground and excited vibronic states calculated with the NEO- NOCI method. The 6-31G electronic basis set was used for all calculations...... 21 2.5 Tunneling splittings in cm−1 calculated with the NEO-NOCI method for two diﬀerent nuclear basis sets and with the one-dimensional and three-dimensional grid methods. The QZSPNB nuclear basis set includes four each of s- and p-type Gaussians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians. For the grid calculations, the potential was generated at the full CI level with 128 grid points for the one- dimensional grid and 64 grid points per dimension for the three- dimensional grid. The nuclear basis function centers were optimized at the NEO-HF level for the localized states and were the same for the ground and excited vibronic states calculated with the NEO- NOCI method. The 6-311G(p) electronic basis set was used for all calculations...... 22

3.1 The electronic coupling V ET , adiabaticity parameter p, prefactor κ, proton tunneling time τp, and electronic state transition time τe for the phenoxyl/phenol and benzyl/toluene systems...... 37 3.2 Vibronic couplings in cm−1 for the phenoxyl/phenol system calcu- (sc) lated with the semiclassical method (VDA ) and for the adiabatic (ad) (na) (VDA ) and nonadiabatic (VDA ) limits. The values in parentheses are the vibronic couplings calculated with the transferring hydrogen nucleus represented by a three-dimensional vibrational wavefunction. For all other values of the vibronic coupling given, the transferring hydrogen nucleus is represented by a one-dimensional vibrational wavefunction. The vibronic couplings are given for both hydrogen and deuterium transfer...... 37

xii 3.3 Vibronic couplings in cm−1 for the benzyl/toluene system calculated (sc) (ad) with the semiclassical method (VDA ) and for the adiabatic (VDA ) (na) and nonadiabatic (VDA ) limits. The values in parentheses are the vibronic couplings calculated with the transferring hydrogen nucleus represented by a three-dimensional vibrational wavefunction. For all other values of the vibronic coupling given, the transferring hydrogen nucleus is represented by a one-dimensional vibrational wavefunction. The vibronic couplings are given for both hydrogen and deuterium transfer...... 38 3.4 The values of α, which indicate the dependence of the semiclassical (sc) vibronic coupling VDA on the proton donor-acceptor distance R, for the phenoxyl/phenol and benzyl/toluene systems. The distance (sc) dependence of VDA is ﬁt to the functional form exp [−αR]. The α values are given for both the one-dimensional (1D) and three- dimensional (3D) treatment of hydrogen (H) and deuterium (D). The values of α are given in A˚−1...... 42

4.1 Proton donor-acceptor O–O distance R and COOC dihedral angle θ for the series of substituted phenoxyl/phenol systems...... 58 4.2 Vibronic couplings for the unsubstituted phenoxyl/phenol system. . 60 4.3 Vibronic couplings and associated quantities for the series of substituted phenoxyl/phenol systems at R = 2.30 A.˚ ...... 61

A.1 Additional information for the 2×2 NEO-NOCI tunneling splitting calculations of the [He–H–He]+ system. The additional data provided are the nuclear basis function center (nuc b.f.c.) separation, the mixed state unorthogonalized Hamiltonian matrix element HI,II , the orthogonalized mixed state Hamiltonian matrix element 0 p e HI,II , the proton overlap S , the electronic overlap S , the total overlap ST = Se ∗ Sp, and the corresponding NEO-NOCI tunneling splittings. ∆(HF) refers to the NEO-NOCI tunneling splitting evaluated at the NEO-HF minimum and ∆(NOCI) refers to the NEO-NOCI tunneling splitting between the variationally optimized NEO-NOCI ground and excited state energies. The 6-31G electronic basis set and QZSPDN0 nuclear basis set were used for all calculations...... 75

xiii A.2 Additional information for the 2×2 NEO-NOCI tunneling splitting calculations of the phenoxyl/phenol system. The additional data provided are the nuclear basis function center (nuc b.f.c.) separation, the mixed state unorthogonalized Hamiltonian matrix element HI,II , the orthogonalized mixed state Hamiltonian matrix element 0 p e HI,II , the proton overlap S , the electronic overlap S , the total overlap ST = Se ∗ Sp, and the corresponding NEO-NOCI tunneling splittings. ∆(HF) refers to the NEO-NOCI tunneling splitting evaluated at the NEO-HF minimum and ∆(NOCI) refers to the NEO-NOCI tunneling splitting between the variationally optimized NEO-NOCI ground and excited state energies. The 6-31G electronic basis set and QZSPDN0 nuclear basis set were used for all calculations...... 76 A.3 Additional information for the 2×2 NEO-NOCI tunneling splitting calculations of the benzyl/toluene system. The additional data provided are the nuclear basis function center (nuc b.f.c.) separation, the mixed state unorthogonalized Hamiltonian matrix element HI,II , the orthogonalized mixed state Hamiltonian matrix 0 p e element HI,II , the proton overlap S , the electronic overlap S , the total overlap ST = Se ∗ Sp, and the corresponding NEO-NOCI tunneling splittings. ∆(HF) refers to the NEO-NOCI tunneling splitting evaluated at the NEO-HF minimum and ∆(NOCI) refers to the NEO-NOCI tunneling splitting between the variationally optimized NEO-NOCI ground and excited state energies. The 6-31G electronic basis set and QZSPDN0 nuclear basis set were used for all calculations...... 77

xiv A.4 Tunneling splittings, in cm−1, for the [He–H–He]+ system calculated with the NEO-NOCI method, the NEO-FCI method, the one- dimensional grid-based method, and the three-dimensional grid- based method are presented for ﬁxed He–He distances of 1.86 and 2.30 A.˚ For the NEO-NOCI and the NEO-FCI calculations the nuclear basis set used is indicated in brackets. The QZSPNB nuclear basis set includes four each of s- and p-type Gaussians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians. For the grid calculations, the potential was generated at the full CI level with 128 grid points for the one-dimensional grid and 64 grid points per dimension for the three-dimensional grid. For the NEO calculations two nuclear basis function centers were used to represent the quantum nucleus. The nuclear basis function centers contain both electronic and nuclear basis functions and by construction are always equally spaced from the midpoint between the donor and acceptor. This separation is minimized for the ground and excited state energies of the corresponding NEO level of theory (NEO-NOCI or NEO-FCI). The 6-31G electronic basis set was used for all calculations...... 78

C.1 p-F phenoxyl/phenol transition state structure...... 85 C.2 m-F phenoxyl/phenol transition state structure...... 86 C.3 o-F phenoxyl/phenol transition state structure...... 87 C.4 p-CN phenoxyl/phenol transition state structure...... 88 C.5 m-CN phenoxyl/phenol transition state structure...... 89 C.6 o-CN phenoxyl/phenol transition state structure...... 90 C.7 p-NO2 phenoxyl/phenol transition state structure...... 91 C.8 m-NO2 phenoxyl/phenol transition state structure...... 92 C.9 o-NO2 phenoxyl/phenol transition state structure...... 93 C.10 p-OH phenoxyl/phenol transition state structure...... 94 C.11 m-OH phenoxyl/phenol transition state structure...... 95 C.12 o-OH phenoxyl/phenol transition state structure...... 96 C.13 p-NH2 phenoxyl/phenol transition state structure...... 97 C.14 m-NH2 phenoxyl/phenol transition state structure...... 98 C.15 o-NH2 phenoxyl/phenol transition state structure...... 99 C.16 Vibronic couplings and associated quantities for the optimized transition structures of the substituted phenoxyl/phenol systems. . . . . 100

xv D.1 Malonaldehyde DFT(B3LYP)/6-31G** reactant-product averaged geometry. The geometry is the average of the reactant and product optimized structure’s internal coordinates. The tunneling hydrogen atom that is treated quantum mechanically with the 3D grid method is marked with an asterik. The position of this hydrogen atom is not used in the grid calculation, but gives some idea of where to build the hydrogen grid box...... 105 D.2 Malonaldehyde MP2/6-31G** reactant-product averaged geometry. The geometry is the average of the reactant and product optimized structure’s internal coordinates. The tunneling hydrogen atom that is treated quantum mechanically with the 3D grid method is marked with an asterik. The position of this hydrogen atom is not used in the grid calculation, but gives some idea of where to build the grid box...... 106 D.3 Results for the approximate multi-dimensional tunneling splitting calculation of malonaldehyde at the DFT(B3LYP)/6-31G** and the MP2/6-31G** levels of theory. ∆ep here is the 3D grid proton tunneling splitting calculated at the reactant-product averaged geometry. The full multi-dimensional tunneling splitting (Full ∆) is the product of the grid proton tunneling splitting and the Franck- Condon Overlap of reactant and product normal modes (F.C. Over- lap) as expressed in Eq. D.14. 32 points per dimension were used for the 3D grid tunneling splitting calculation. ∆exp is the experimental proton tunneling splitting for malonaldehyde [15]. All splittings are in units of cm−1...... 107

xvi Acknowledgments

I first express my sincere gratitude to my advisor, Professor Sharon Hammes- Schiffer, for her patience, direction, encouragement, and optimism, which has shaped my development as a scientist and as a professional. I attribute much of the success of my graduate career to the bright, talented, and incredibly re- sourceful group of postdocs and research scientists Sharon has assembled in our group. Many thanks are given to the Hammes-Schiffer group, whom I have had the pleasure of knowing and the honor of being colleagues with. I must individually thank a few members for their extra help and their direct contributions to this manuscript. I am grateful to Mike Pak for many of the early discussions on NEO and general electronic structure theory. Many of his intellectual insights paved part of the course of my graduate studies. I especially thank both Chet Swalina and Arindam Chakraborty for the many discussions pertaining to my research and their assistance with technical issues. They both acted as role models for how to successfully eradicate the most daunting programming bugs. I must also acknowledge Anirban Hazra for his inquisitive nature that spurred me to look further into my own research, yielding a new level of understanding to it. I thank Alexander Soudackov for kindly sharing his knowledge and experience in the field and for helping my research branch into new areas. I must also thank Michelle Ludlow for the contributions she has made to some of this work. A lot of credit to my present pursuit of academics must be given to my under- graduate advisor, Professor Emanuele Curotto, who introduced me to computational chemistry and encouraged me to further pursue my interests in chemistry. His passion for science and love of his labor was contagious. I am very fortunate to have had such an enthusiastic scientist and teacher as a mentor. Lastly, I thank my friends and family who have both supported and encouraged me throughout my graduate studies. I especially thank my parents, Mark and Antoinette, who made many sacrifices in order for me to receive an uncompromised high quality education throughout my life.

xvii Dedication

For my parents, Mark and Antoinette.

xviii Chapter 1

Introduction

Processes that involve the transfer of protons are ubiquitous in nature and ac- countable for much of the chemistry of life. The energetic cycles of respiration and photosynthesis, both responsible for sustaining most living organisms, are prime examples of processes where proton transfer is present [1, 2, 3, 4, 5, 6, 7, 8]. Be- cause of the small mass of the proton, nuclear quantum effects such as tunneling, zero point energy (ZPE), and vibrational excitations can be significant [9]. Two experimentally observable manifestations of the quantum behavior of the proton are proton tunneling splittings and kinetic isotope effects (KIE). Many experimental results exist for these phenomena [10, 11, 12, 13, 14, 15, 16, 17, 18, 19], and theoretical tools can be helpful for interpretation and direction of future studies [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Accordingly, improvements on old theories and development of new approaches are essential to continue the pace of discovery and comprehension. Often the proton transfer is accompanied by a concerted electron transfer. This type of coupled motion is referred to as proton-coupled electron transfer (PCET). Many of the aforementioned proton transfer processes, which are important to nature, involve the coupling of electron motion to the proton motion and can thus be described as PCET. A classification scheme that reflects the origin and destination of the transferring particles has been suggested [32]. Although not rigorous, since both the electron and proton behave quantum mechanically, it provides a basis for a theoretically formulated distinction between mechanisms. In the case where the proton and the electron are transferred between the same 2 donor and acceptor, the process is referred to as hydrogen atom transfer (HAT). The other case, in which the electron and proton transfer between different donors and acceptors, is categorized as PCET. We choose to use these designations and will refer to the general case, which encompasses both classifications, as general PCET. Given the importance of general PCET reactions, several theoretical methods have been developed to study them [33, 34, 35, 36, 37, 38, 39, 40]. Within the formulation developed by Soudackov and Hammes-Schiffer [37, 38, 39], the general PCET reaction occurs between two diabatic electronic states, denoted I and II, representing the localized electron transfer states. The transferring electron is localized on the donor for diabatic state I and on the acceptor for diabatic state II. The proton vibrational wavefunctions are calculated for each diabatic electronic state, leading to a set of reactant and product proton vibrational wavefunc- (I) (II) tions denoted ϕD and ϕA , respectively. The general PCET reaction is described in terms of transitions between the resulting electronic-proton vibrational states, which are referred to as vibronic states. The coupling between the reactant and product vibronic states or, more simply, the vibronic coupling, determines whether the general PCET reaction is vibronically adiabatic or nonadiabatic with respect to a solvent or protein environment. When the vibronic coupling is much less than the thermal energy kBT , the general PCET reaction is vibronically nonadiabatic and the reaction rate is proportional to the square of the vibronic coupling [33, 34, 38, 39]. As a result, the vibronic coupling plays an important role in determining the rates, kinetic isotope effects, and temperature dependences of general PCET reactions [29, 41, 31]. Analogous to the general PCET reaction being vibronically nonadiabatic with respect to the environment, the tunneling of the quantum proton may be electronically nonadiabatic [40]. This occurs when the electrons do not have adequate time to respond to the motion of the proton, resulting in the contributions of higher electronic states. If the electrons have enough time to respond to the proton motion, the proton will move on an electronically adiabatic surface and the proton tunneling will thus be electronically adiabatic. It is important to recognize that the proton tunneling can be either electronically adiabatic or electronically nonadiabatic. To calculate quantitatively accurate vibronic couplings and obtain meaningful in- 3 sights, it is necessary to treat the proton tunneling in the correct limit of electronic adiabaticity. A comprehensive method capable of calculating vibronic couplings continuously over the range of electronically adiabatic to electronically nonadiabatic proton tunneling requires methods that avoid making the Born-Oppenheimer separation between transferring protons and electrons. One promising approach is the nuclear electronic orbital method (NEO) [42, 43, 44, 45, 46, 47, 48, 49, 50]. The NEO approach provides a consistent way of dealing with nuclear quantum effects by directly incorporating them into the electronic structure calculation, treating a subset of nuclei quantum mechanically on the same level as the electrons. The NEO approach is a molecular orbital based method that avoids the Born-Oppenheimer separation of the select protons and electrons, thereby making methods developed within this scheme applicable to electronically nonadiabatic proton transfer. Several methods have been developed within the NEO framework, including Hartree-Fock (NEO-HF) [42], configuration interaction (NEO-CI) [42], multiconfigurational self-consistent-field (NEO-MCSCF) [42, 44], second-order many-body perturbation theory (NEO-MP2) [47], nonorthogonal configuration interaction (NEO-NOCI) [48], density functional theory (NEO-DFT) [50], and explicitly correlated Hartree-Fock (NEO-XCHF) [49]. The NEO approach has several advantages over other approaches. These include the straightforward treatment of the three-dimensional quantum proton, computational tractability for larger chemical systems, and the potential for systematic improvement. Similar nuclear-electronic molecular orbital methods [51, 52, 53, 54, 55, 56, 57, 58, 59] have been developed by other groups, but none of these methods have been used to evaluate proton tunneling splittings or vibronic couplings. For most levels of theory within the NEO approach, lack of electron-proton correlation leads to localization of the nuclear wavefunction [44, 45, 46]. This is problematic for symmetric transition state structures, where the nuclear density should be both bilobal and delocalized. The explicitly correlated Hartree-Fock method overcomes this problem [49], but it is computationally prohibitive for most systems of chemical interest. The NEO-NOCI method [48] takes advantage of the localization by combining two nonorthogonal localized NEO wavefunctions in a configuration interaction scheme. The resulting wavefunctions are bilobal and delocalized, and the difference between the ground NEO-NOCI state and first excited 4

NEO-NOCI state that corresponds to the proton vibration is the tunneling splitting. The vibronic coupling in this implementation is half the tunneling splitting for a symmetric system. In general, the vibronic coupling can be calculated from the coupling between the symmetrically orthogonalized localized states [60]. This thesis has two objectives. The first is to develop and implement reliable quantum chemical methods for the purpose of calculating proton tunneling splittings and vibronic couplings. To this end, a novel approach for calculating proton tunneling splittings and vibronic couplings, developed within the nuclear electronic orbital (NEO) scheme, is established, and an alternative semiclassical grid-based method for calculating vibronic couplings is implemented. The second objective is to illustrate the fundamental physical principles underlying proton transfer in the electronically adiabatic and nonadiabatic limits by applying both of the aforementioned methods to chemical systems that are exemplary of the adiabatic and nonadiabatic proton-tunneling regimes. Furthermore, our analysis using these methods provides a more rigorous diagnostic for classifying a general PCET reaction as HAT or PCET. Overall, this thesis emphasizes the need for quantum chemical methods that avoid the adiabatic separation of the quantum proton and electron, are computationally tractable, and treat all quantum particles three-dimensionally. By achiev- ing the two objectives discussed above, it will be possible to investigate the proton transfer mechanism in many chemical systems, thus providing new physical insights and offering experimentally testable predictions. An outline of this thesis is as follows. Chapter 2 presents the details of the NEO nonorthogonal CI (NEO-NOCI) method and showcases its computational feasibility and accuracy in calculating tunneling splittings for a model [He–H–He]+ system. Chapter 3 illustrates the importance of treating the proton transfer in the correct limit of electronic adiabaticity by using a semiclassical method to evaluate the vibronic coupling. For two exemplary chemical systems, the phenoxyl/phenol and benzyl/toluene self-exchange reactions, these calculations further show the necessity of treating the quantum proton three-dimensionally in order to obtain quantitatively accurate results. Chapter 3 also discusses the use of the adiabaticity parameter as a more rigorous diagnostic for identifying general PCET reactions as HAT or PCET. Chapter 4 contains the results from a study of substituent 5 effects on the vibronic coupling for the phenoxyl/phenol self-exchange reaction using both a grid-based method and the NEO-NOCI method. The results from these two methods are in qualitative agreement, which gives a level of validation to the NEO-NOCI method. An analysis of the correlations between the vibronic coupling and the physical properties of the system provides useful insights that are applicable to other general PCET systems. Chapter 2

Nuclear-Electronic Orbital Nonorthogonal Conﬁguration Interaction Approach

Reproduced in part with permission from: J. H. Skone, M. V. Pak, and S. Hammes-Schiﬀer, “Nuclear-electronic orbital nonorthogonal conﬁguration interaction approach”, J. Chem. Phys. 123, 134108 (2005). c 2005 American Institue of Physics.

2.1 Introduction

Hydrogen tunneling plays a central role in a wide range of chemical and biological processes, including photosynthesis, respiration, and enzyme reactions [9, 61]. A variety of approaches have been developed to calculate tunneling splittings in multidimensional systems. One strategy is to perform fully quantum dynamical calculations on a reduced-dimensional potential energy surface [21]. For relatively small systems, numerically exact quantum dynamical calculations using the full- dimensional potential energy surface are possible [24]. Another strategy is to use a semiclassical treatment based on classical trajectory simulations [62, 23, 25]. Re- cently, an instanton theory has been developed for the calculation of tunneling splittings in multidimensional systems [63, 22]. Although these approaches have successfully predicted tunneling splittings for molecules such as malonaldehyde, they will be computationally expensive for larger systems of chemical and biological interest. Furthermore, these approaches invoke the adiabatic separation be- 7 tween the electrons and the nuclei. This adiabatic approximation may not be valid for reactions involving low-lying excited electronic states, such as proton-coupled electron transfer reactions [34, 36]. In this chapter, a computationally efficient approach for calculating delocalized, bilobal hydrogen wavefunctions and the corresponding hydrogen tunneling splittings in the framework of the nuclear-electronic orbital (NEO) approach [42, 43, 44, 45, 46, 47] is presented. In this framework, specified nuclei are treated quantum mechanically on the same level as the electrons, and a mixed nuclear- electronic time-independent Schrödingerequation is solved with molecular orbital techniques. Both electronic and nuclear molecular orbitals are expressed as linear combinations of Gaussian basis functions. When the NEO approach is applied to hydrogen transfer systems, the transferring hydrogen nucleus and all electrons are treated quantum mechanically. This approach removes the adiabatic separation between the electrons and the quantum nuclei. A variety of molecular orbital methods have been implemented, including Hartree-Fock (HF) [42], configuration interaction (CI), multiconfigurational self-consistent-field (MCSCF) [42, 44, 45], and second-order perturbation theory (MP2) [47] within the NEO framework. A number of related mixed nuclear-electronic molecular orbital methods have been developed [51, 54, 64, 55, 56, 57, 59, 65] but have not been applied to hydrogen tunneling systems. The calculation of delocalized, bilobal hydrogen wavefunctions and the corresponding tunneling splittings within the NEO framework is challenging due to the importance of electron-proton correlation [44, 45]. Consider the symmetric [X-H- X] system, where X is a general donor and acceptor group. For sufficiently large separations between the donor and acceptor, the hydrogen moves in a symmetric double well potential, and the ground state nuclear wavefunction is delocalized over both wells. In the NEO approach, the hydrogen atom for this type of system is represented by two basis function centers to allow delocalization of the hydrogen vibrational wavefunction. The positions of the hydrogen basis function centers may be optimized variationally during the self-consistent-field procedure. Based on the symmetry of this model system, the exact nuclear wavefunction is expected to be delocalized equally over both basis function centers, as depicted in Figure 2.1a. The variational NEO-HF solution, however, corresponds to a nuclear wavefunc- 8 tion localized on one of the basis function centers, as depicted in Figure 2.1b. As shown in Ref. [45], the localization of the nuclear density at the NEO-HF level is a consequence of the neglect of electron-proton correlation, which is particularly important due to the attractive electron-proton Coulomb interaction.

Figure 2.1. [X–H–X] model system with the hydrogen nuclear wavefunction depicted at (a) the NEO-HF level (localized) and (b) the NEO-FCI level (delocalized). Here X represents a general donor and acceptor group. For the [He–H–He]+ model system, X=He.

The inclusion of sufficient electron-proton correlation with multiconfigurational methods enables the calculation of delocalized, symmetric nuclear wavefunctions [44, 45]. The NEO-full CI (NEO-FCI) approach allows the calculation of hydrogen tunneling splittings, but this approach is not computationally practical for most chemical systems due to the large number of configurations in the CI expansion. The NEO-MCSCF approach, in which the electronic and nuclear molecular orbitals as well as the CI coefficients are optimized variationally, is more computationally tractable because a smaller number of configurations is included in the CI expansion. As shown previously [44, 45], however, the variational NEO-MCSCF solution is still localized for the accessible active spaces of most chemical systems. The state-averaged NEO-MCSCF method, in which the molecular orbitals are optimized to minimize the energy of an equally weighted linear combination of the lowest two vibronic states, provides qualitatively reasonable, delocalized wave- 9 functions for symmetric hydrogen tunneling systems [44, 45]. Unfortunately, this approach is not easily applied to asymmetric systems due to difficulties in obtaining the delocalized molecular orbitals in the active space. Thus, these previous NEO methods do not enable the incorporation of sufficient electron-proton correlation for the calculation of tunneling splittings in a computationally tractable manner for general hydrogen tunneling systems.

Figure 2.2. Schematic illustration of the one-dimensional hydrogen potential energy curve along the He–He axis for the [He–H–He]+ model system with (a) the corresponding NEO-HF localized wavefunctions ΨI and ΨII and (b) the corresponding ground and excited state NEO-NOCI wavefunctions Ψ0 and Ψ1, which are linear combinations of the localized wavefunctions shown in (a)

For the remainder of the chapter, the NEO nonorthogonal configuration interaction (NEO-NOCI) approach for calculating delocalized hydrogen wavefunctions and the corresponding tunneling splittings is presented. The basic scheme is anal- 10 ogous to the nonorthogonal CI method used previously in conventional electronic structure calculations [66, 67, 68]. Here we use the localization of the variational NEO-HF wavefunctions to our advantage. In the simplest implementation, the delocalized nuclear-electronic wavefunctions are expressed as linear combinations of the two NEO-HF localized nuclear-electronic wavefunctions, as depicted in Fig- ure 2.2. The generation of the localized wavefunctions is straightforward because the variational NEO-HF solution is localized. The physically motivated choice of these two localized states ensures that the NEO-NOCI wavefunctions will be delocalized, in contrast to the localization previously observed for NEO-MCSCF calculations with relatively large active spaces. NEO-NOCI calculations are computationally practical because the main expense is the relatively fast calculation of the off-diagonal Hamiltonian matrix element between the two localized NEO-HF states. In addition, the accuracy of this approach can be improved systematically by enhancing the electronic and nuclear basis sets and including more localized states in the NOCI procedure. Thus, the NEO-NOCI method is robust and computationally efficient, and it can be applied to asymmetric as well as symmetric systems. The organization of this chapter is as follows. In Section 2.2, the NEO approach is summarized, and the theoretical basis of the NEO-NOCI is presented. In Section 2.3, the NEO-NOCI approach is applied to a model hydrogen transfer system for fixed geometries and is evaluated by comparison to NEO-FCI and Fourier grid calculations. Section 2.4 provides conclusions and future directions for this approach.

2.2 Theory

2.2.1 Background

In the NEO approach [42, 43, 44, 45, 46, 47], the system is divided into three parts: Ne electrons, Np quantum nuclei, and Nc classical nuclei. The total NEO 11

Hamiltonian is expressed as:

Ne Np Nc Ne Nc Np ˆ X 1 2 X 1 2 X X ZA X X ZAZi0 Htot = − ∇i − ∇i0 − + 2 2Mi0 riA ri0A i i0 A i A i0

Ne Ne Np Np Ne Np Nc Nc X X 1 X X 1 X X Zi0 X X ZAZB + + − + (2.1) rij ri0j0 ri0i rAB i j>i i0 j0>i0 i i0 A B>A

Here the unprimed indices i, j refer to electrons, the primed indices i0, j0 refer to quantum nuclei, and the indices A, B refer to classical nuclei. The subscripts e, p, and c denote electrons, quantum nuclei, and classical nuclei, respectively. The masses, charges, and distances, respectively, are denoted by M, Z, and r with the appropriate subscripts. The Hartree-Fock (NEO-HF) [42], full CI (NEO- FCI) [42], multiconfigurational self-consistent-field (NEO-MCSCF) [42, 44, 45], and second order perturbation theory (NEO-MP2) [47] approaches within the NEO framework have been described in detail elsewhere. At the Hartree-Fock level, the total nuclear-electronic wavefunction is approximated as a product of single configurational electronic and nuclear wavefunctions:

e p Ψtot(re, rp) = Φ0(re)Φ0(rp) (2.2)

e p where Φ0(re) and Φ0(rp), respectively, are antisymmetrized determinants of spin orbitals representing the electrons and fermionic nuclei such as protons. (Here re and rp denote the spatial coordinates of the electrons and quantum nuclei, respectively.) The spatial orbitals for the electrons and the quantum nuclei are expanded in Gaussian basis sets, and the variational method is used to minimize the total energy with respect to both the electronic and nuclear molecular orbitals. The multiconﬁgurational NEO wavefunction has the form

e p NCI NCI X X e p Ψtot(re, rp) = C(II0)ΦI (re)ΦI0 (rp) (2.3) I I0

e p where ΦI (re) and ΦI (rp) are determinants of spin orbitals representing the electrons and quantum nuclei, respectively, and C(II0) are conﬁgurational interaction e p (CI) coeﬃcients. Here there are NCI electronic determinants and NCI quantum 12

e p nuclear determinants, leading to a total of NCI = NCI ∗ NCI nuclear-electronic configurations. For the NEO-CI approach, the variational method is used to minimize the total energy with respect to the CI coefficients. In the NEO-FCI method, a complete active space is used for both the electrons and the quantum nuclei. In the NEO-MCSCF approach, smaller active spaces are used, and the energy is minimized with respect to the electronic and nuclear molecular orbitals as well as the CI coefficients. All possible CI configurations that result from the chosen electronic and nuclear active spaces are included. The NEO methodology has been implemented in the GAMESS electronic structure program [69].

2.2.2 NEO nonorthogonal CI

In the NEO-NOCI method, the total wavefunction is expressed as a linear combination of nonorthogonal products of electronic and nuclear determinants. For the two-state implementation described here, the total wavefunction is expressed as a linear combination of two localized NEO-HF wavefunctions. Speciﬁcally, e p e p |ΨIi = ΦI (re)ΦI (rp) is localized in the left well, and |ΨIIi = ΦII (re)ΦII (rp) is localized in the right well. The ground and excited state NEO-NOCI wavefunctions are

0 0 Ψ0 = CI |ΨIi + CII |ΨIIi 1 1 Ψ1 = CI |ΨIi + CII |ΨIIi (2.4)

The coeﬃcients of these wavefunctions are determined by solving a 2×2 NEO- NOCI matrix equation: " #" # " #" #" # H H C0 C1 S S C0 C1 E 0 I,I I,II I I = I,I I,II I I 0 (2.5) 0 1 0 1 HII,I HII,II CII CII SII,I SII,II CII CII 0 E1 where HI,I = hΨI| H |ΨIi and HII,II = hΨII| H |ΨIIi are the energies of the localized

NEO-HF solutions, and HI,II = hΨI| H |ΨIIi = HII,I is the mixed state Hamiltonian matrix element. The diagonal elements of the overlap matrix are unity, and SI,II = hΨI|ΨIIi = SII,I . To solve the NEO-NOCI matrix equation, the oﬀ-diagonal overlap and Hamiltonian matrix elements must be evaluated, and both matrices must be diagonalized. 13

We have derived an expression for the off-diagonal Hamiltonian matrix elements between two nonorthogonal nuclear-electronic wavefunctions. Our derivation is based on the previous derivation of Löwdin[66] for calculating the matrix element of an operator with respect to two nonorthogonal electronic determinants. This type of nonorthogonal CI scheme has been used in conventional electronic structure calculations as an alternative to using large active space MCSCF calculations for unpaired electron systems [67, 68]. In this framework, the off-diagonal overlap matrix elements are

e p SI,II = SII,I = hΨI|ΨIIi = Det(S )Det(S ) (2.6) and the oﬀ-diagonal Hamiltonian matrix elements are expressed as

HI,II = HII,I = hΨI| H |ΨIIi =

Ne Ne X X e e e e p χI,i |h | χII,j CijDet(S ) i j N N N N 1 Xe Xe Xe Xe + χe χe |χe χe Ce Det(Sp) 2 I,i II,j I,k II,l ij,kl i j k l

Np Np X X p p p p e + χI,i0 |h | χII,j0 Ci0j0 Det(S ) i0 j0

Np Np Np Np 1 X X X X p p p p p e + χ 0 χ 0 |χ 0 χ 0 C 0 0 0 0 Det(S ) 2 I,i II,j I,k II,l i j ,k l i0 j0 k0 l0

Np Np Ne Ne Nc Nc X X X X p p e e p e X X ZAZB p e − χI,i0 χII,j0 |χI,iχII,j Ci0j0 Cij + Det(S )Det(S ) rAB i0 j0 i j A B>A (2.7)

e e p where χI,i denotes the spin orbital i for the electronic determinant ΦI , χI,i0 0 p denotes the spin orbital i for the nuclear determinant ΦI , and the analogous spin orbitals are defined for the localized nuclear-electronic wavefunction ΨII. The one- particle operators he and hp for the electrons and protons, respectively, are defined 14 in Ref. [42]. The two-electron operator terms are defined as

Z Z e e e e e∗ e −1 e∗ e χi χj|χkχl = dx1 dx2χi (x1) χj (x1) r12 χk (x2) χl (x2) (2.8) and the two-proton and mixed electron-proton operator terms are defined anal- e ogously. The electronic spatial orbitals can be expressed in terms of the Nbf e electronic basis functions ϕµ, and the nuclear spatial orbitals can be expressed p p in terms of the Nbf nuclear basis functions ϕµ0 . Specifically, the spatial orbitals corresponding to the localized solution ΨI are defined as:

e Nbf e X e e ψI,i = cI,µiϕµ µ p Nbf p X p p ψI,i0 = cI,µ0i0 ϕµ0 (2.9) µ0 and the spatial orbitals corresponding to the localized solution ΨII are deﬁned analogously. In Eqs. 2.6 and 2.7, Se and Sp are the electronic and nuclear overlap matrices of the occupied molecular orbitals. The matrix elements of these overlap matrices can be expressed in terms of the overlaps and of the electronic and nuclear basis functions:

e e Nbf Nbf e X X e e e Sij = cI,µicII,νjσµν µ ν p p Nbf Nbf p X X p p p Si0j0 = cI,µ0i0 cII,ν0j0 σµ0ν0 (2.10) µ0 ν0

e p The terms Cij and Ci0j0 are the cofactors of the electronic and nuclear occupied overlap matrices and are deﬁned as:

e i+j e e −1 e Cij = (−1) Det sij = (S )ij Det [S ] p i0+j0 p p −1 p Ci0j0 = (−1) Det si0j0 = (S )ij Det [S ] (2.11) 15

e e where sij is the matrix minor of S obtained by deleting row i and column j of e p p 0 0 S , and si0j0 is the matrix minor of S obtained by deleting row i and column j p e p of S . Similarly, Cij,kl and Ci0j0,k0l0 are the second-order cofactors of the electronic and nuclear occupied overlap matrices and are deﬁned as:

e i+j+k+l e Cij,kl = (−1) Det sij,kl h e −1 e −1 e −1 e −1i e = (S )ji (S )lk − (S )li (S )jk Det [S ]

p i0+j0+k0+l0 p Ci0j0,k0l0 = (−1) Det si0j0,k0l0 h p −1 p −1 p −1 p −1 i p = (S )j0i0 (S )l0k0 − (S )l0i0 (S )j0k0 Det [S ] (2.12)

e e where sij,kl is the second-order minor of S obtained by deleting rows i and k and e p p columns j and l of S , and si0j0,k0l0 is the second-order minor of S obtained by deleting rows i0 and k0 and columns j0 and l0 of Sp. The extension of the NEO- NOCI approach to include more than two nonorthogonal states is straightforward. Inclusion of excited states corresponding to stretching or bending modes will improve the quantitative accuracy. Furthermore, the localized states are general and do not need to be NEO-HF states.

2.2.3 Calculation of tunneling splittings

Within the NEO framework, the tunneling splitting at a fixed geometry is E1 −E0, the difference between the ground and first excited vibronic state energies. For a symmetric system, HI,II = HII,I, and the tunneling splitting can be expressed as

2H − 2S H E − E = I,II I,II I,I (2.13) 1 0 2 1 − SI,II

In general, the tunneling splitting is sensitive to the geometry of the molecule. For the model system [He-H-He]+, however, the only relevant geometrical parameter is the He–He distance, RHe–He. To test the NEO-NOCI method, we calculated the tunneling splittings as a function of this distance and compared to the results from NEO-FCI and Fourier grid calculations. The calculated tunneling splitting is sensitive to the nuclear basis set. The localized states may be described with a single hydrogen basis function center 16 located on the side of localization or with multiple hydrogen basis function centers. We have found that the utilization of two hydrogen basis function centers to describe the localized states leads to quantitatively accurate tunneling splittings. Each center contains both nuclear and electronic basis functions, so two hydrogen basis function centers provide a consistent electronic basis set for the two localized states. The two centers are constrained to be equidistant from the midpoint of the symmetric system. The basis function center separation can be optimized variationally at the NEO-NOCI level for both the ground and excited states. Alternatively, the positions of the hydrogen basis function centers can be optimized variationally at the NEO-HF level for the localized states and held fixed for the NEO-NOCI calculation of the ground and excited vibronic states. When the hydrogen is moving in a double well potential with a relatively high barrier, the results are virtually identical for the optimization of the hydrogen basis function center positions at the NEO-NOCI or NEO-HF levels, and the positions are nearly identical for the ground and first excited vibronic states when optimized at the NEO-NOCI level. As the barrier to the double well potential decreases, the results become more sensitive to the specific hydrogen basis function center optimization scheme chosen. Enhancement of the nuclear basis set with diffuse nuclear basis functions also improves the accuracy. The nuclear basis set previously developed for hydrogen nuclei within the NEO framework was optimized for a set of five simple diatomic molecules that each contained one hydrogen atom [42]. This double-ζ s, p, d nuclear basis set (DZSPDN) includes two each of s-, p-, and d-type Gaussians, resulting in a total of 20 nuclear basis functions per hydrogen center. The DZSPDN0 basis set is the same as the DZSPDN basis set with the removal of the second set of d-type Gaussians (i.e., double-ζ s, p, single-ζ d). For the calculations presented here, we developed two larger nuclear basis sets. The quadruple-ζ s, p nuclear basis set (QZSPNB) includes four each of s- and p-type Gaussians, resulting in a total of 16 nuclear basis functions per hydrogen center. This basis set was optimized using the same scheme described previously for the development of the DZSPDN basis set [42]. We also constructed the quadruple-ζ s, p, double-ζ d nucleabasis set (QZSPDN0), which is the same as the QZSPNB but is augmented with two d-type Gaussians, resulting in a total of 28 nuclear basis functions per hydrogen center. 17

The exponents for these basis sets are given in Table 2.1.

Table 2.1. Exponents for the QZSPNB and QZSPDN0 nuclear basis sets. The QZSPNB nuclear basis set includes four each of s- and p-type Gaussians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians.

Gaussian Exponent Function s-type 34.63529 s-type 23.53723 s-type 16.15828 s-type 2.88060 p-type 33.60131 p-type 22.91237 p-type 15.80526 p-type 6.89795 d-type 23.70662 d-type 16.73036

2.3 Results

In this section, we use the [He–H–He]+ model proton transfer system to analyze and benchmark the NEO-NOCI method. This model system was used previously to investigate a number of fundamental issues arising in applications to hydrogen transfer processes [46]. We chose this system as a model because it has few enough electrons to be treated at the NEO-FCI level, which includes all electron-electron and electron-proton correlation. In our NEO calculations on this system, only the transferring hydrogen nucleus is treated quantum mechanically. The quantum mechanical treatment of the helium nuclei as well as the hydrogen nucleus would be appropriate for quantitative studies of the dynamical and spectroscopic aspects of this chemical system. Unfortunately, we are unable to include the He motions in our calculations of tunneling splittings because the equilibrium geometry for this molecule corresponds to a single well hydrogen potential along the He–He axis. Our objective, however, is to use this system as a model to illustrate the capabilities of the NEO-NOCI approach for the calculation of tunneling splittings at ﬁxed 18 geometries. Future work will focus on the inclusion of the eﬀects of the modes corresponding to the nuclei treated classically for systems such as malonaldehyde. As discussed in Ref. [46], the hydrogen potential along the He–He axis is single well at short He–He distances and double well at long He–He distances for this model system. Since the electronic density is localized predominantly on the helium nuclei, this model system may be viewed as a proton moving between two He atoms, and the adiabatic separation between the electrons and the hydrogen nucleus is a reasonable approximation for this model system. Thus, the Born- Oppenheimer grid method can be used to calculate accurate vibrational energy level splittings for benchmarking purposes. In these grid calculations, the Born- Oppenheimer electronic potential energy surface is calculated at the full CI level of electronic structure theory, and the Fourier grid Hamiltonian method [70, 71] is used to calculate the hydrogen vibrational wavefunctions for this potential energy surface. The resulting vibrational energy level splittings are given in Table 2.2. At the conventional FCI/6-31G level of theory, the hydrogen potential energy curve along the He–He axis is a single well for distances less than ∼2.1 A˚ and becomes double well for larger distances. At distances greater than ∼2.2 A,˚ the lowest

Table 2.2. Vibrational energy level splittings in cm−1 calculated with the one- dimensional and three-dimensional grid methods. The potential was generated at the full CI level with 128 grid points for the one-dimensional grid and 64 grid points per dimension for the three-dimensional grid. The 6-31G electronic basis set [72] was used for all calculations.

RHe–He (A)˚ 1D grid 3D grid 1.80 1991 1541 1.90 1400 1084 2.00 884 644 2.10 456 285 2.20 165 79 2.25 83 35 2.30 38 15 2.35 16 5.5

two vibrational states are below the barrier for the one-dimensional grid calculations, as illustrated schematically in Figure 2.2b. The diﬀerences between the 19 one-dimensional and three-dimensional grid splittings are due mainly to the mixing of the low-frequency bending modes with the asymmetric hydrogen stretching mode in the three-dimensional grid calculations. The one-dimensional grid method does not include the bending modes, which correspond to motions of the hydrogen nucleus away from the He–He axis. Table 2.3 compares the vibrational energy level splittings calculated for a range of He–He distances with the NEO-NOCI and NEO-FCI methods with two diﬀerent nuclear basis sets. For this system, the conventional HF and FCI potential energy surfaces are similar with the 6-31G electronic basis set [72]. The agreement between the NEO-NOCI and NEO-FCI methods is excellent for the entire range of He–He distances. Note that the range of He–He distances studied spans both single well and double well hydrogen potential energy surfaces. The NEO-NOCI vibrational

Table 2.3. Splittings in cm−1 calculated with the NEO-NOCI and NEO-FCI methods for two diﬀerent nuclear basis sets. The DZSPNB nuclear basis set includes two each of s- and p-type Gaussians. The DZSPDN0 nuclear basis set is the same as DZSPNB with the addition of one set of d-type Gaussians. The nuclear basis function center positions were optimized at the NEO-NOCI level for both the ground and excited vibronic states. The 6-31G electronic basis set was used for all calculations. Due to the large memory requirements and computational expense of the NEO-FCI/DZSPDN0 calculations, only two representative He–He distances were studied at this level of theory.

RHe-He (A)˚ NEO-NOCI NEO-FCI NEO-NOCI NEO-FCI DZSPNB DZSPNB DZSPDN0 DZSPDN0 1.70 2796 2920 1.80 2129 2245 1.90 1555 1658 1459 1579 2.00 1049 1132 2.10 587 653 2.20 114 160 2.25 0.49 0.46 0.25 0.25

energy level splittings in the single well regime (i.e., 1.8–2.0 A)˚ agree well with the one-dimensional grid splittings given in Table 2.2. Thus, the NEO-NOCI method is a promising approach for the calculation of accurate nuclear wavefunctions and frequencies for large amplitude asymmetric stretch modes, as well as hydrogen 20 tunneling splittings. The NEO-NOCI results agree better with the one-dimensional grid results than with the three-dimensional grid results for the single well regime because the low-frequency bending modes are not described accurately with the DZSPNB and DZSPDN0 nuclear basis sets. The remainder of this chapter focuses on the calculation of tunneling splittings in the double well regime. The agreement between the NEO-NOCI and NEO-FCI methods illustrates that the NEO-NOCI approach captures the majority of non-dynamical electron-proton correlation required to calculate accurate tunneling splittings for these types of systems. Comparison of these tunneling splittings to the Born-Oppenheimer grid results in Table 2.2, however, indicates that the DZSPNB and DZSPDN0 nuclear basis sets are not adequate for the calculation of quantitatively accurate tunneling splittings in the double well regime (i.e., for distances larger than ∼2.2 A).˚ The deﬁciencies of these nuclear basis sets arise from the lack of diﬀuse nuclear basis functions that are required for an accurate description of the overlap region between the two localized wavefunctions. Unfortunately, the NEO-FCI calculations are too expensive to perform with larger nuclear and electronic basis sets. Nevertheless, the agreement between NEO-NOCI and NEO-FCI for the smaller basis sets provides validation of the NEO-NOCI approach for the calculation of tunneling splittings. Table 2.4 provides a comparison of the tunneling splittings calculated with the NEO-NOCI method in conjunction with larger nuclear basis sets to tunneling splittings calculated with one-dimensional and three-dimensional grid methods. The splittings calculated with the NEO-NOCI/QZSPNB method are in remarkable agreement with the splittings calculated with the one-dimensional grid method, and the splittings calculated with the NEO-NOCI/QZSPDN0 method are in remarkable agreement with the splittings calculated with the three-dimensional grid method. As mentioned above, the three-dimensional grid method includes the mixing of the low-frequency bending modes with the asymmetric hydrogen stretching mode. For this combination of electronic and nuclear basis sets, the NEO-NOCI calculations describe only the stretching modes accurately and lead to tunneling splittings that agree with the one-dimensional grid calculations. When d basis functions are added to the nuclear basis set, the NEO-HF localized states use the d basis functions to describe the bending modes. As a result, the NEO-NOCI/ QZSPDN0 calculations provide reasonably accurate descriptions of the bending 21

Table 2.4. Tunneling splittings in cm−1 calculated with the NEO-NOCI method for two diﬀerent nuclear basis sets and with the one-dimensional and three-dimensional grid methods. The QZSPNB nuclear basis set includes four each of s- and p-type Gaussians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians. For the grid calculations, the potential was generated at the full CI level with 128 grid points for the one-dimensional grid and 64 grid points per dimension for the three-dimensional grid. The nuclear basis function centers were optimized at the NEO-HF level for the localized states and were the same for the ground and excited vibronic states calculated with the NEO-NOCI method. The 6-31G electronic basis set was used for all calculations.

RHe-He (A)˚ 1D Grid NEO-NOCI NEO-NOCI 3D Grid QZSPNB QZSPDN0 2.20 165 183 105 79 2.25 83 83 40 35 2.30 38 37 15 15 2.35 16 16 5.6 5.5

modes and hence agree with the three-dimensional grid calculations. Note that the agreement is not quite as good for the He–He separation of 2.2 A˚ because the vibronic states are closer to the top of the barrier, and the nuclear wavefunctions have more nuclear density in the middle regions, leading to greater technical dif- ﬁculties in the accurate description of the hydrogen vibrational wavefunctions in both the overlap region and the maximum density region simultaneously. The NEO-NOCI approach performs the best when the ground and excited vibronic states are reasonably well below the barrier, and the corresponding nuclear wavefunctions are distinctly bilobal. The data in Table 2.5 were obtained with the larger 6-311G(p) electronic basis set [73] and illustrate the impact of the electronic basis set on the tunneling splittings. The Born-Oppenheimer potential energy surfaces calculated with the 6-311G(p) electronic basis set exhibit lower barriers than those calculated with the 6-31G electronic basis set for the same He–He distance. This eﬀect is due to intramolecular basis set superposition error. Thus, for the same He–He distance, the tunneling splittings are larger with the 6-311G(p) electronic basis set than with the 6-31G electronic basis set. As with the smaller electronic basis set, the 22

Table 2.5. Tunneling splittings in cm−1 calculated with the NEO-NOCI method for two diﬀerent nuclear basis sets and with the one-dimensional and three-dimensional grid methods. The QZSPNB nuclear basis set includes four each of s- and p-type Gaussians. The QZSPDN0 nuclear basis set is the same as QZSPNB with the addition of two sets of d-type Gaussians. For the grid calculations, the potential was generated at the full CI level with 128 grid points for the one-dimensional grid and 64 grid points per dimension for the three-dimensional grid. The nuclear basis function centers were optimized at the NEO-HF level for the localized states and were the same for the ground and excited vibronic states calculated with the NEO-NOCI method. The 6-311G(p) electronic basis set was used for all calculations.

RHe-He (A)˚ 1D Grid NEO-NOCI NEO-NOCI 3D Grid QZSPNB QZSPDN0 2.30 145 100 100 72 2.35 67 32 26 28 2.40 26∗ 10 7.1 9.8 ∗ Indicates value is a correction that diﬀers from the original publication.

agreement between the NEO-NOCI/QZSPDN0 and three-dimensional grid results is excellent. For this electronic basis set, the NEO-NOCI/QZSPNB results agree with the three-dimensional grid results as well. The excellent agreement between the NEO-NOCI and grid results with two diﬀerent electronic basis sets provides further validation for the NEO-NOCI approach.

2.4 Conclusions

In this chapter, we presented the NEO-NOCI approach for calculating delocalized, bilobal hydrogen wavefunctions and the corresponding hydrogen tunneling splittings. In the NEO framework, the transferring hydrogen nuclei are treated quantum mechanically on the same level as the electrons with molecular orbital techniques. For hydrogen transfer systems, the transferring hydrogen is represented by two basis function centers to allow delocalization of the nuclear wavefunction. For a symmetric hydrogen transfer system, the variational NEO-HF solution corresponds to a nuclear wavefunction localized predominantly on one of the basis 23 function centers. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wavefunctions are expressed as linear combinations of the two nonorthogonal NEO-HF localized nuclear-electronic wavefunctions. The tunneling splitting is determined by the energy difference between these two delocalized vibronic states. We applied the NEO-NOCI approach to the [He–H–He]+ model system for fixed He–He distances and compared the results to NEO-FCI and Fourier grid calculations. The NEO-NOCI and NEO-FCI splittings calculated with a double-ζ nuclear basis set are in excellent agreement for a wide range of He–He distances spanning both single well and double well hydrogen potential energy surfaces. A larger nuclear basis set that includes diffuse basis functions was required to obtain a high level of quantitative agreement between the tunneling splittings calculated with the NEO-NOCI and the grid methods. We found that the NEO-NOCI approach performs the best when the ground and excited vibronic states are reasonably well below the barrier, and the corresponding nuclear wavefunctions are distinctly bilobal. In this regime, the NEO-NOCI and multidimensional grid tunneling splittings differ by an average of ∼2 cm−1. These benchmarking calculations indicate that NEO-NOCI is a promising approach for the calculation of delocalized, bilobal hydrogen wavefunctions and the corresponding hydrogen tunneling splittings. The NEO-NOCI approach can be generalized and extended in a variety of directions. In the implementation presented here, we use the localization of the NEO-HF wavefunctions to our advantage in order to generate the nonorthogonal states. In general, however, the NEO-NOCI approach can be used in conjunction with any choice of nonorthogonal localized states. For example, the application of the NEO-NOCI approach to asymmetric hydrogen tunneling systems may require different approaches for determining the nonorthogonal localized states. Additional flexibility is provided through the choice of the electronic and nuclear basis sets, the number of hydrogen basis function centers, the scheme for optimization of hydrogen basis function center positions, and the number of localized states included in the NOCI procedure. The calculation of quantitatively accurate tunneling splittings for comparison to experiment will require the inclusion of electron-electron correlation into the NEO-NOCI approach with a method such as perturbation theory [68]. Furthermore, the NEO-NOCI approach is applicable to asymmetric 24 as well as symmetric hydrogen tunneling systems. Even for asymmetric systems, typically hydrogen tunneling occurs at a geometry in which the hydrogen moves in a virtually symmetric double well potential, so the nuclear wavefunctions will be bilobal and delocalized. Finally, the NEO-NOCI approach will be useful for the generation of minimum energy paths for hydrogen transfer reactions described in terms of a collective, heavy-atom reaction coordinate [74, 75, 76]. The utilization of the NEO-NOCI approach to calculate hydrogen tunneling splittings in multidimensional systems such as malonaldehyde will require the inclusion of the modes corresponding to the nuclei treated classically in the NEO framework. In these applications, the tunneling splitting will be averaged over reactant and product multidimensional vibrational wavefunctions corresponding to the nuclei treated classically in the NEO framework. For simplicity, the majority of these vibrational modes will be treated harmonically, and the frequencies will be determined by the calculation of a NEO Hessian [43]. Additional approximations will enable the treatment of many of these vibrational modes as multiplicative Franck-Condon factors. The potential advantages of the NEO-NOCI approach over semiclassical approaches for calculating tunneling splittings are the computational efficiency and the removal of the adiabatic separation between the electrons and quantum nuclei. In addition to the calculation of tunneling splittings, the NEO-NOCI approach will enable the calculation of nonadiabatic couplings for hydrogen transfer reactions, particularly proton-coupled electron transfer reactions [34, 36]. Chapter 3

Calculation of Vibronic Couplings for Proton-Coupled Electron Transfer Reactions

Reproduced in part with permission from: J. H. Skone, A. V. Soudackov, and S. Hammes-Schiﬀer, “Calculation of Vibronic Couplings for Phenoxyl/Phenol and Benzyl/Toluene Self-Exchange Reactions: Analysis of Proton-Coupled Electron Transfer Mechanisms”, J. Am. Chem. Soc. 128, 16655 (2006). c 2006 American Chemical Society

3.1 Introduction

The coupling of electron and proton transfer reactions plays a vital role in a wide range of chemical and biological processes, including photosynthesis [4, 5, 8, 6, 7, 77], respiration [2, 3], and enzyme reactions [78, 79, 80, 81, 18]. A general term for reactions in which an electron and a proton are transferred in a single step is proton-coupled electron transfer (PCET). Traditionally, reactions in which the electron and proton transfer between the same donor and acceptor are denoted hydrogen atom transfer (HAT), and the term PCET is often reserved for reactions in which the electron and proton transfer between different donors and acceptors [32, 82, 36, 83, 84]. This distinction, however, is not rigorous because the electron 26 and proton behave quantum mechanically. Nevertheless, understanding the fundamental differences between these two types of reactions is important for the study of many chemical and biological processes. Recently, Mayer, Borden, and coworkers used density functional theory to investigate the self-exchange reactions of the phenoxyl radical with phenol and the benzyl radical with toluene [32]. These authors identified the former as a PCET reaction and the latter as an HAT reaction. This identification was based on an analysis of the singly occupied molecular orbital (SOMO) at the transition state geometry. For the phenoxyl/phenol system, the SOMO is dominated by 2p orbitals on the donor and acceptor oxygen atoms that are perpendicular to the proton donor-acceptor (O–H–O) axis, while the proton participates in a hydrogen bond involving σ orbitals. Since the electron and proton are transferred between different sets of orbitals, the authors describe this reaction as PCET. For the benzyl/toluene system, the SOMO is dominated by atomic orbitals oriented along the donor-acceptor (C–H–C) axis, and the authors describe this reaction as HAT. This analysis serves as a useful way to distinguish between these two types of reactions. For convenience, we will use these definitions of HAT and PCET and will use the term general PCET to encompass all reactions involving electron and proton transfer in a single step. Note that the distinction between sequential and concerted transfer within this single step is not well defined because the electron and proton behave quantum mechanically. In this chapter, a different analysis of the phenoxyl/phenol and benzyl/toluene systems is presented and additional insight into the fundamental differences between these two types of reactions is gained. Since hydrogen tunneling is often important in these types of reactions, we treat the transferring hydrogen nucleus quantum mechanically and calculate the vibronic coupling between the mixed electronic-proton vibrational wavefunctions corresponding to the reactant and the product states. Even when the splitting between the ground and excited electronic states is much larger than the thermal energy kBT , these types of reactions are often vibronically nonadiabatic with respect to the solvent and protein environment because the vibronic coupling is much less than kBT . In this case, the rate of the reaction is proportional to the square of the vibronic coupling [38, 39, 33, 34]. As a result, the magnitude of the vibronic coupling and its dependence on the 27 proton donor-acceptor distance can significantly impact the rates, kinetic isotope effects, and temperature dependences of general PCET reactions [29, 41]. The impact of the vibronic coupling on the rates and kinetic isotope effects has been illustrated for PCET reactions in iron bi-imidazoline complexes [85], oxoruthenium polypyridyl complexes [30], ruthenium polypyridyl-tyrosine systems [86], and the enzyme lipoyxgenase [87]. Recently the impact of the vibronic coupling on the temperature dependence of the kinetic isotope effect has been elucidated for the PCET reaction catalyzed by the enzyme lipoxygenase [31]. Thus, the calculation of the vibronic coupling is critical for a complete understanding of general PCET reactions. In addition to calculating the vibronic coupling for these two systems, we identify a new diagnostic for differentiating between the two types of reactions. Our analysis utilizes the semiclassical analytical expression for the vibronic coupling derived by Georgievskii and Stuchebrukhov [40], as well as analytical expressions in the limits of electronically adiabatic and nonadiabatic proton tunneling. Even when the overall reaction is vibronically nonadiabatic, the proton tunneling can be in the electronically adiabatic or electronically nonadiabatic limits or in the intermediate regime. Here the proton tunneling is defined to be electronically adiabatic when the electronic transition time is much shorter than the proton tunneling time, so the electrons are able to respond virtually instantaneously to the proton motion, and the reaction proceeds on the electronically adiabatic ground state. The proton tunneling is defined to be electronically nonadiabatic when the electronic transition time is much longer than the proton tunneling time, so the electrons are unable to rearrange fast enough to follow the proton motion, and the excited electronic state is involved in the reaction. Our analysis indicates that the phenoxyl/phenol reaction, which was previously identified to be PCET, is electronically nonadiabatic, while the benzyl/toluene reaction, which was previously identified to be HAT, is electronically adiabatic. These links between PCET and electronic nonadiabaticity and between HAT and electronic adiabaticity provide insights into the fundamental physical differences between these two types of reactions. The remainder of this chapter is organized as follows. Section 3.2 presents the theoretical framework and the computational methodology. Section 3.3 presents the results and an extensive analysis. Conclusions are summarized in Section 3.4. 28

3.2 Theory and methods

3.2.1 Analytical expressions for vibronic couplings

Previously, a theoretical formulation for general PCET reactions was developed and vibronically nonadiabatic rate expressions were derived [38, 39, 37]. In this formulation, the PCET reaction occurs between two diabatic electronic states, denoted I and II, representing the localized electron transfer states. The transferring electron is localized on the donor for diabatic state I and on the acceptor for diabatic state II. The proton vibrational wavefunctions are calculated for each diabatic electronic state, leading to a set of reactant and product proton vibrational wave- (I) (II) functions denoted φD and φA , respectively. For simplicity, here we consider the tunneling between only the ground state reactant and product mixed electronic- proton vibrational states. In this case, the rate of reaction is proportional to the square of the vibronic coupling, which is deﬁned to be the Hamiltonian matrix element between the reactant and product mixed electronic-proton vibrational wavefunctions. The overall reaction is vibronically nonadiabatic with respect to the solvent or protein environment when this vibronic coupling is much less than kBT . As mentioned above, even for vibronically nonadiabatic PCET reactions, the proton tunneling can be electronically nonadiabatic, electronically adiabatic, or in the intermediate regime. The previously derived rate expressions for general PCET reactions are valid in all of these regimes [38, 39]. The electronically nonadiabatic and adiabatic limits for general PCET reactions are deﬁned in terms of small and large values, respectively, of the electronic coupling V ET between the reactant and product diabatic electronic states. In this chapter, electronically nonadiabatic and adiabatic refer to the relative timescales of the electrons and the transferring proton. The electrons respond instantaneously to the proton motion in the electronically adiabatic limit but not in the electronically nonadiabatic limit. In the electronically nonadiabatic limit, the vibronic (na) coupling VDA can be expressed as the product of the electronic coupling and the Franck-Condon overlap of the reactant and product proton vibrational wavefunctions:

(na) ET D (I) (II)E VDA = V ϕD |ϕA (3.1) 29

In the electronically adiabatic limit, the proton dynamics occur on the electronically adiabatic ground state potential energy surface, and the vibronic coupling (ad) VDA can be calculated by standard semiclassical methods [88, 89]. For a symmet- (ad) ric system, the vibronic coupling VDA is half the splitting between the symmetric and antisymmetric proton vibrational states for the electronic ground state potential energy surface. Many general PCET reactions are in between the electronically nonadiabatic and adiabatic limits. Georgievskii and Stuchebrukhov [40] derived a (sc) semiclassical expression for the general vibronic coupling VDA :

(sc) (ad) VDA = κVDA (3.2) where the factor κ is deﬁned as

ep ln p−p κ = p2πp (3.3) Γ(p + 1)

In Eq. 3.3, Γ(x) is the gamma-function and p is the proton adiabaticity parameter deﬁned as

2 V ET p = (3.4) ~ |∆F | vt where vt is the tunneling velocity of the proton at the crossing point of the two proton potential energy curves and |∆F | is the diﬀerence between the slopes of the proton potential energy curves at the crossing point. The tunneling velocity vt can be expressed in terms of the energy Vc at which the potential energy curves cross, the tunneling energy E, and the mass m of the proton:

r 2 (V − E) v = c (3.5) t m

In the electronically adiabatic limit, p >> 1, κ = 1, and the vibronic coupling (ad) √ simpliﬁes to VDA . In the electronically nonadiabatic limit, p << 1, κ = 2πp, (na) and the vibronic coupling reduces to VDA , as given in Eq. 3.1. The adiabaticity of a general PCET reaction can be viewed in terms of the relative times of the proton tunneling and the electronic transition. Within the semiclassical framework, the time spent by the tunneling proton in the crossing region (i.e., the proton tunneling 30 time) is

V ET τp ∼ (3.6) |∆F | νt and the time required to change the electronic state (i.e., the electronic transition time) is

τ ∼ ~ (3.7) e V ET

The adiabaticity parameter is simply the ratio of these two times:

τ p = p (3.8) τe

When the proton tunneling time is much longer than the electronic transition time, the electronic states have enough time to mix completely and the proton transfer occurs on the electronically adiabatic ground state surface (i.e., the reaction is electronically adiabatic). When the proton tunneling time is much less than the electronic transition time, the reaction is electronically nonadiabatic because the electronic states no longer have enough time to mix completely during the proton tunneling process.

3.2.2 Computational methodology

We calculated the input quantities for the vibronic coupling expressions with conventional electronic structure methods. We emphasize that our goal is not to provide quantitatively accurate results for these specific systems, but rather to enable a qualitative comparison of the fundamental nature of these two types of systems. As a result, we utilize moderate levels of electronic structure theory that provide physically reasonable results. The quantitative accuracy of the results can be improved by using a larger basis set and including dynamical electron correlation. Unless otherwise specified, we used the Gaussian03 package [90] for these electronic structure calculations. The transition state geometries were optimized with density functional theory (DFT) using the B3LYP functional [91, 92] and the 6-31G* basis set [93]. The 31 qualitative dependence of the vibronic coupling on the donor-acceptor distance was determined by translating the rigid donor and acceptor molecules along the donor-acceptor axis for the transition state geometry. For each donor-acceptor distance, all nuclei were fixed except for the transferring hydrogen, which was treated quantum mechanically. The reactant and product states refer to the mixed electronic-proton vibrational quantum states in which the electron and proton are localized on the donor in the reactant state and on the acceptor in the product state for fixed geometry of all other nuclei. In general, the Franck-Condon overlap factor from the other nuclei could contribute to the vibronic coupling, but this contribution is not directly relevant to the analysis presented in this chapter. We obtained the electronically adiabatic ground and excited state potential energy curves along the hydrogen coordinate by calculating the state-averaged CASSCF(3,6) energy for the hydrogen positioned at discrete grid points along the axis connecting the donor and acceptor atoms. The active space was chosen to ensure that the character of the orbitals in the active space was conserved along the hydrogen coordinate and the electronic ground state was qualitatively similar to the ROHF ground state. As discussed below, we also calculated three-dimensional potential energy surfaces for the hydrogen at the ROHF level. The 6-31G basis set [94] was used for all ROHF and CASSCF calculations to enable the efficient calculation of the three-dimensional potential energy surfaces. We determined that the ground and excited state potential energy surfaces are qualitatively similar for the 6-31G and 6-31G* basis sets. The quantities in the expressions for the vibronic couplings given in section 3.2.1 were determined from the CASSCF potential energy curves. The electronic coupling V ET is half the splitting between the two electronically adiabatic CASSCF potential energy curves at the midpoint between the donor and acceptor atoms. A two-state valence bond model was used to fit the state-averaged CASSCF potential energy curves for the purpose of obtaining the two localized electronically diabatic potential energy curves for the phenoxyl/phenol system. A four-state valence bond model was used for the benzyl/toluene system. The details of both of these valence bond models are given in Appendix B . The quantities |∆F | and Vc in Eq. 3.4 and Eq. 3.5, respectively, were determined from these electronically diabatic potential energy curves. The one-dimensional hydrogen vibrational wavefunctions were cal- 32 culated for the diabatic and adiabatic potential energy curves using the Fourier grid Hamiltonian method [71, 70] with 128 grid points. The Franck-Condon overlap in Eq. 3.1 is the overlap between the proton vibrational wavefunctions for the two diabatic potential energy curves. The electronically adiabatic vibronic coupling (ad) VDA is half the splitting between the ground and excited hydrogen vibrational states for the electronically adiabatic ground state potential energy curve. The tunneling energy E in Eq. 3.5 is the hydrogen vibrational ground state energy for the electronically diabatic potential energy curve. To study the impact of the three-dimensional character of the hydrogen vibrational wavefunction, we also calculated three-dimensional potential energy surfaces for the hydrogen at the ROHF level. For the electronically nonadiabatic system, we obtained the electronically diabatic potential energy curves by calculating the ROHF energy for a three-dimensional grid with 32 grid points per dimension spanning half of the proton donor-acceptor axis, fitting the data points to an analytical functional form (i.e., a fourth-order polynomial), and using the analytical functional form to generate the potential energy surface for a grid with 64 points per dimension. For the electronically adiabatic system, we obtained the electronically adiabatic ground state potential energy surface by calculating the ROHF energy for a three-dimensional grid with 32 points per dimension. The three-dimensional hydrogen vibrational wavefunctions were calculated for the ROHF potential energy surfaces using the Fourier grid Hamiltonian method [71, 70]. We used the GAMESS electronic structure program [69] for the three-dimensional calculations. This analysis has two main practical advantages over the orbital-based analysis in Ref. [32]. First, the present analysis may be more reliable because the molecular orbitals corresponding to the electronically adiabatic ground state at the transition state geometry may not be meaningful for electronically nonadiabatic reactions, and the molecular orbitals are not uniquely defined. Second, the present analysis provides the vibronic couplings, which are required for calculating rates and kinetic isotope effects. 33

3.3 Results

As discussed previously, the transition state geometries of the phenoxyl/phenol and the benzyl/toluene systems are qualitatively different. The phenoxyl/phenol transition state has C2 symmetry, and the O–H–O bond is approximately planar with the phenol rings. The proton donor-acceptor distance is 2.40 A,˚ and the system forms a strong hydrogen bond along the O–H–O axis. The benzyl/toluene transition state has C2h symmetry, and the C–H–C bond is orthogonal to the planes of the benzene rings. The proton donor-acceptor distance is 2.72 A,˚ and the system does not form a strong hydrogen bond along the C–H–C axis because of the lack of lone pairs of electrons on benzyl and toluene. Figure 3.1 depicts the potential energy curves along the transferring hydrogen coordinate for the phenoxyl/phenol and the benzyl/toluene systems. The CASSCF electronically adiabatic ground and excited state curves are depicted. The electronically diabatic curves corresponding to the two electron transfer states, denoted I and II, are also depicted. These diabatic states correspond to fixed electronic wavefunctions associated with the hydrogen bonded to the donor oxygen (I) or to the acceptor oxygen (II). By construction, the mixing of these two diabatic states with the appropriate coupling V ET leads to the CASSCF electronically adiabatic ground and excited state curves. Note that the splitting between the electronically adiabatic ground and excited states is more than an order of magnitude larger for the benzyl/toluene system than for the phenoxyl/phenol system. As a result, the diabatic curves are very similar to the adiabatic curves for the phenoxyl/phenol system but are significantly different from the adiabatic curves for the benzyl/toluene system. The two highest-energy occupied electronic molecular orbitals are depicted in Figure 3.2 for the phenoxyl/phenol and the benzyl/toluene systems. The diabatic states I and II are represented by the electronic wavefunctions corresponding to the minima of the potential energy curves. For both systems, the ground state electronic wavefunction is predominantly single configurational in the regions near the minima. For diabatic state I, the highest doubly occupied molecular orbital is localized mainly on the conjugated π system of the donor ring, and the singly occupied molecular orbital is localized mainly on the acceptor oxygen or carbon. 34

Figure 3.1. State-averaged CASSCF ground and excited state electronically adiabatic potential energy curves along the transferring hydrogen coordinate for (a) the phenoxyl/phenol and (b) the benzyl/toluene system. The coordinates of all nuclei except the transferring hydrogen correspond to the transition state geometry. The proton donor-acceptor distances are 2.40 A˚ and 2.72 A,˚ respectively, for the phenoxyl/phenol and the benzyl/ toluene system. The CASSCF results are depicted as open circles that are blue for the ground state and red for the excited state. The black dashed lines represent the diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II. The mixing of these two diabatic states with the electronic coupling V ET leads to the CASSCF ground and excited state electronically adiabatic curves depicted with solid colored lines following the colored open circles. For the phenoxyl/phenol system, the solid colored lines and the black dashed lines are nearly indistinguishable because the adiabatic and diabatic potential energy curves are virtually identical except in the transition state region. 35

Figure 3.2. The two highest-energy occupied electronic molecular orbitals for (a) the phenoxyl/phenol and (b) the benzyl/toluene system. The electronic wavefunctions for diabatic states I and II are calculated at the minima of the ground state electronically adiabatic potential energy curves shown in Figure 3.1, and the electronic wavefunctions for the transition states (TS) are calculated at the maxima of these potential energy curves. For both systems, the ground state electronic wavefunction is predominantly single conﬁgurational, and the lower molecular orbital is doubly occupied, while the upper molecular orbital is singly occupied. A contour value of 0.08 was used for all molecular orbital pictures. These ﬁgures were generated with MacMolPlot [95].

The opposite configuration is observed for diabatic state II. For both systems, the electron transfer process is represented by the change in the electronic wavefunction as the system proceeds from diabatic state I to diabatic state II. The change in the electronic wavefunction involves the shifting of electronic density of the doubly occupied molecular orbital from the conjugated π orbital on the donor ring to 36 the conjugated π orbital on the acceptor ring, as well as the shifting of electronic density of the singly occupied molecular orbital from the acceptor oxygen to the donor oxygen. The electronic wavefunctions for the two systems are qualitatively different at the transition state. For both systems, the ground state electronic wavefunction remains predominantly single configurational at the transition state. In the phenoxyl/phenol system, the two highest-energy occupied molecular orbitals are dominated by 2p orbitals on the donor and acceptor oxygen atoms that are perpendicular to the hydrogen donor-acceptor axis. In the ground state, the doubly occupied molecular orbital corresponds to π-bonding, and the singly occupied molecular orbital corresponds to π-antibonding. In the benzyl/toluene system, the two highest-energy occupied molecular orbitals are dominated by σ orbitals on the donor and acceptor carbon atoms that are lying along the hydrogen donor- acceptor axis. In the ground state, the highest doubly occupied molecular orbital corresponds to σ-bonding, and the singly occupied molecular orbital corresponds to σ-antibonding. Previously Mayer, Borden, and coworkers [32] used these differences in the singly occupied molecular orbitals of the transition state wavefunctions to designate the phenoxyl/phenol and benzyl/tolune systems as PCET and HAT, respectively. Table 3.1 presents the electronic coupling V ET , the adiabaticity parameter p, the prefactor κ, the proton tunneling time τp, and the electronic transition time

τe for the two reactions. For both systems, the electronic coupling is significantly greater than the thermal energy kBT at room temperature. As will be shown below, however, the overall vibronic coupling is significantly less than kBT , leading to an overall vibronically nonadiabatic reaction with respect to a solvent environment at room temperature for both systems. The remainder of this analysis focuses on the electronic adiabaticity and nonadiabaticity of the proton tunneling process. The fundamental nature of the proton tunneling is different for the two systems. For the phenoxyl/phenol system, the adiabaticity parameter p is very small, √ κ = 2πp, and τe ≈ 80τp. In this case, the electronic transition time is significantly greater than the proton tunneling time. As a result, the electrons are not able to rearrange fast enough for the proton to move on the electronically adiabatic ground state surface, and the proton transfer reaction is electronically nonadiabatic. For 37

Table 3.1. The electronic coupling V ET , adiabaticity parameter p, prefactor κ, proton tunneling time τp, and electronic state transition time τe for the phenoxyl/phenol and benzyl/toluene systems.

a ET −1 system V (cm ) p κ τp (fs) τe (fs) phenol 700 0.0130 0.268 0.098 7.60 toluene 14300 3.45 0.976 1.28 0.370 a The proton donor-acceptor distances are 2.40 A˚ and 2.72 A,˚ respectively, for the phenoxyl/phenol and benzyl/toluene systems.

the benzyl/toluene system, the adiabaticity parameter p is larger, κ ≈ 1, and

τp ≈ 4τe. In this case, the electronic transition time is less than the proton tunneling time. Thus, the electrons can respond instantaneously to the proton motion, and the proton moves on the electronically adiabatic ground state surface. This analysis indicates that the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol system but electronically adiabatic for the benzyl/toluene system.

Table 3.2. Vibronic couplings in cm−1 for the phenoxyl/phenol system calculated with (sc) (ad) (na) the semiclassical method (VDA ) and for the adiabatic (VDA ) and nonadiabatic (VDA ) limits. The values in parentheses are the vibronic couplings calculated with the transferring hydrogen nucleus represented by a three-dimensional vibrational wavefunction. For all other values of the vibronic coupling given, the transferring hydrogen nucleus is represented by a one-dimensional vibrational wavefunction. The vibronic couplings are given for both hydrogen and deuterium transfer.

hydrogen deuterium ˚ (ad) (sc) (na) (ad) (sc) (na) ROO (A) VDA VDA VDA VDA VDA VDA 2.25 198 78.2 72.9 47.2 20.6 18.7 2.30 104 35.8 33.4 (15.3) 17.1 6.61 6.08 (2.59) 2.35 46.7 14.5 13.7 (7.23) 5.04 1.77 1.65 (0.90) 2.40 17.3 4.65 4.47 (2.86) 1.11 0.34 0.33 (0.25) 2.45 6.10 1.49 1.45 (1.11) 0.23 0.07 0.06 (0.06) 2.50 1.96 0.42 0.41 (0.36) 0.04 0.01 0.01 (0.01) 38

Table 3.3. Vibronic couplings in cm−1 for the benzyl/toluene system calculated with (sc) (ad) (na) the semiclassical method (VDA ) and for the adiabatic (VDA ) and nonadiabatic (VDA ) limits. The values in parentheses are the vibronic couplings calculated with the transferring hydrogen nucleus represented by a three-dimensional vibrational wavefunction. For all other values of the vibronic coupling given, the transferring hydrogen nucleus is represented by a one-dimensional vibrational wavefunction. The vibronic couplings are given for both hydrogen and deuterium transfer.

hydrogen deuterium ˚ (ad) (sc) (na) (ad) (sc) (na) RCC (A) VDA VDA VDA VDA VDA VDA 2.60 167 (116) 163 58.7 42.5 (26.0) 41.8 4.07 2.65 87.8 (57.0) 85.8 25.4 14.9 (8.43) 14.5 1.23 2.70 38.5 (24.8) 37.5 10.8 4.06 (2.38) 3.99 0.37 2.72 21.6 (16.1) 21.1 5.58 1.70 (1.25) 1.67 0.14 2.75 15.0 (9.89) 14.6 3.96 0.97 (0.61) 0.95 0.08 2.80 7.36 (3.64) 7.16 2.36 0.35 (0.14) 0.34 0.04

The vibronic couplings calculated with the adiabatic, nonadiabatic, and semiclassical methods are provided in Tables 3.2 and 3.3 for the phenoxyl/phenol and benzyl/toluene systems, respectively. In all cases, the adiabatic vibronic couplings are larger than the nonadiabatic vibronic couplings. The semiclassical vibronic couplings are in excellent agreement with the nonadiabatic couplings for the phenoxyl/phenol system and are in excellent agreement with the adiabatic couplings for the benzyl/toluene system. These results conﬁrm that the proton transfer is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. Tables 3.2 and 3.3 also provide the vibronic couplings for the deuterated phenoxyl/phenol and benzyl/toluene systems, respectively. For a given proton donor-acceptor distance, the vibronic couplings are signiﬁcantly smaller for deuterium than for hydrogen because of the greater localization of the deuterium wavefunction, leading to smaller overlaps between the reactant and product proton vibrational wavefunctions. Figure 3.3 illustrates the physical principles underlying the electronically nonadiabatic and adiabatic limits. For the electronically nonadiabatic phenoxyl/phenol reaction, the vibronic coupling is the product of the electronic coupling between 39

Figure 3.3. (a) Diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II and the corresponding proton vibrational wave- (I) (II) functions ϕD (blue) and ϕA (red) for the phenoxyl/phenol system. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling V ET and the overlap of the reactant and product proton vibrational wavefunctions D (I) (II)E ϕD |ϕA . (b) Electronically adiabatic ground state potential energy curve and the corresponding proton vibrational wavefunctions for the benzyl/toluene system. Since this reaction is electronically adiabatic, the vibronic coupling is equal to half of the energy splitting ∆ between the symmetric (cyan) and antisymmetric (magenta) proton vibrational states for the electronic ground state potential energy surface. For illustrative purposes, the excited vibrational state is shifted up in energy by 0.8 kcal/mol. 40 the diabatic states I and II and the overlap of the reactant and product proton vibrational wavefunctions corresponding to these diabatic states. For the electronically adiabatic benzyl/toluene reaction, the vibronic coupling is half the energy splitting between the states corresponding to the symmetric and antisymmetric proton vibrational wavefunctions for the electronically adiabatic ground state. The distance dependence of the vibronic coupling plays a key role in determining the rates and kinetic isotope eﬀects, as well as the temperature dependences, of general PCET reactions. The dependence of the vibronic couplings on the proton donor-acceptor distance is depicted in Figure 3.4 for the two systems. In the electronically nonadiabatic limit, this distance dependence is dominated by the overlap between the reactant and product proton vibrational wavefunctions. As shown in Ref. [96], in the region close to the equilibrium value R¯, the overlap can be approximated to be of the form S (R) ∝ S R¯ exp −α R − R¯. In the electronically adiabatic limit, the semiclassical tunneling matrix element for proton transfer can also be approximated to depend exponentially on the proton donor- acceptor distance [97, 98]. Thus, our recent theoretical treatment of general PCET reactions assumes an exponential dependence of the vibronic coupling [39, 41, 96]:

(0) ¯ VDA = VDA exp −α R − R (3.9)

(0) ¯ where VDA is the value of the vibronic coupling at R. The results shown in Fig- ure 3.4 validate the exponential dependence of the vibronic couplings on the proton donor-acceptor distance for both systems in the range of distances studied. The values of α for both hydrogen and deuterium transfer for both systems are given in Table 3.4. The values of α are slightly larger for the phenoxyl/phenol system than for the benzyl/toluene system because of diﬀerences in the frequencies and energy barriers. For the phenoxyl/phenol system, which is electronically nona- D (I) (II)E diabatic, the value of α is dominated by the dependence of the overlap ϕD |ϕA on the proton donor-acceptor distance. The values of α are larger for deuterium than for hydrogen because the overlap between the reactant and product deuterium wavefunctions falls oﬀ faster with distance than the corresponding overlap for the hydrogen wavefunctions due to the larger mass of deuterium. The vibronic couplings are reduced when the transferring hydrogen nucleus is 41

(sc) Figure 3.4. The dependence of the semiclassical vibronic couplings VDA on the proton donor-acceptor distance R for (a) the phenoxyl/phenol and (b) the benzyl/toluene system. The results for hydrogen transfer are shown as open circles, and the results for deuterium transfer are shown as open triangles. The dashed and solid lines correspond to a ﬁt to the functional form exp [−αR] for hydrogen (solid red) and deuterium (dashed (sc) blue) transfer. The corresponding values of α are given in Table 3.4. The couplings VDA are in units of cm−1. 42

Table 3.4. The values of α, which indicate the dependence of the semiclassical vi- (sc) bronic coupling VDA on the proton donor-acceptor distance R, for the phenoxyl/phenol (sc) and benzyl/toluene systems. The distance dependence of VDA is ﬁt to the functional form exp [−αR]. The α values are given for both the one-dimensional (1D) and three- dimensional (3D) treatment of hydrogen (H) and deuterium (D). The values of α are given in A˚−1.

system α (1D) α (3D) Phenol (H) 20 19 Phenol (D) 28 28 Toluene (H) 15 17 Toluene (D) 24 26

represented by a three-dimensional rather than a one-dimensional vibrational wavefunction. The extensions of the electronically nonadiabatic and adiabatic limits to three dimensions are straightforward. In the electronically nonadiabatic limit, we calculated the three-dimensional potential energy surface for the two diabatic states at the ROHF level and calculated the overlap between the corresponding three-dimensional hydrogen vibrational wavefunctions. In the electronically adiabatic limit, we calculated the three-dimensional potential energy surface for the electronic ground state at the ROHF level and calculated the energy splitting between the three-dimensional ground and excited state vibrational wavefunctions. The results are given in Tables 3.2 and 3.3 for both hydrogen and deuterium for the two systems. The three-dimensional treatment of the transferring hydrogen decreases the vibronic coupling by as much as a factor of two and slightly decreases the kinetic isotope effect on the magnitude of the vibronic coupling. As shown in Table 3.4, the three-dimensional treatment of the transferring hydrogen does not significantly alter the value of α, which reflects the exponential dependence of the vibronic coupling on the proton donor-acceptor distance. An alternative method for calculating the vibronic couplings with a three- dimensional treatment of the transferring hydrogen nucleus is the nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) method [48]. This method treats the electrons and transferring proton on equal footing with molecular orbital techniques and provides mixed nuclear-electronic wavefunctions. Chapter 43

4 will explore the potential of the NEO-NOCI method for calculating vibronic couplings. The magnitude and distance dependence of the vibronic couplings strongly impact the magnitudes and temperature dependence of the rates and kinetic isotope eﬀects (KIEs). As derived previously [39, 31] using a series of well-deﬁned, physically reasonable approximations, the rate of a general PCET reaction can be expressed as

2 (0) Vµν 2 r X X 2kBT αµν π k = P exp × µ MΩ2 (λ + λ ) k T µ ν ~ α B " # (∆G0 + λ + ∆ε )2 exp − µν (3.10) 4 (λ + λα) kBT where the summations are over reactant and product vibronic states, Pµ is the Boltzmann population for reactant vibronic state µ, λ is the solvent/protein re- 2 2 ~ αµν 0 organization energy, λα = 2M , ∆G is the driving force, ∆εµν is the diﬀerence between the product and reactant vibronic energy levels relative to the ground states, and M and Ω are the eﬀective mass and frequency associated with the proton donor-acceptor motion. If we consider only the nonadiabatic transition between the two ground states, the KIE can be approximated as

2 V (0) H 2kBT KIE ≈ exp α2 − α2 . (3.11) 2 2 H D (0) MΩ VD

The temperature dependence of the KIE depends on the distance dependence of the vibronic couplings, and the magnitude of the KIE also depends on the magnitude of the vibronic coupling. Thus, the calculation of the vibronic couplings is essential for predicting the magnitudes and temperature dependence of the rates and KIEs. Comparison of the calculated values to experimentally measured KIEs and their temperature dependences will be useful for validating this general approach and benchmarking the level of theory. The distinction between electronic adiabaticity and nonadiabaticity has important experimental consequences because the vibronic couplings can be substantially 44 different in the electronically adiabatic and nonadiabatic limits. These differences are clearly illustrated in Tables 3.2 and 3.3. As indicated by Eqs. 3.10 and 3.11, the magnitude and distance dependence of the vibronic couplings can significantly impact the magnitudes and temperature dependences of the rates and the KIEs. Thus, the calculation of the vibronic coupling in the correct limit, or in the intermediate regime, is critical for the interpretation of experimental data and the generation of experimentally testable predictions. Furthermore, the experimentally measured magnitude and temperature dependence of the KIE may be useful in the classification of a reaction as electronically adiabatic (i.e., HAT mechanism) or electronically nonadiabatic (i.e., PCET mechanism). For complex systems, the calculation of the semiclassical vibronic coupling and the adiabaticity parameter may not be computationally practical. For these types of systems, the vibronic coupling could be calculated for the electronically adiabatic and electronically nonadiabatic limits, and the resulting values could be used in conjunction with Eq. 3.10 to estimate the magnitude and temperature dependence of the KIE. If the KIE is different in the two limits, a comparison to the experimental data could be used to determine the mechanism.

3.4 Conclusions

In this chapter, we calculated the vibronic couplings for the phenoxyl/phenol and the benzyl/toluene self-exchange reactions. The vibronic couplings significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of general PCET reactions. Although the splittings between the ground and excited electronic states are significantly larger than the thermal energy kBT at room temperature, the vibronic couplings for both systems were found to be smaller than kBT , indicating that the reactions are vibronically nonadiabatic with respect to a solvent environment. The proton tunneling was found to be electronically nonadiabatic for the phenoxyl/phenol system and electronically adiabatic for the benzyl/toluene system. For the phenoxyl/phenol system, the electronic transition time is significantly greater than the proton tunneling time. Thus, the electrons are not able to rearrange fast enough to follow the proton motion on the electronically adiabatic state, and the proton tunneling involves the excited 45 electronic state. For the benzyl/toluene system, the electronic transition time is less than the proton tunneling time. As a result, the electrons can respond instantaneously to the proton motion, and the proton moves on the electronically adiabatic ground state surface. We also examined the dependence of the vibronic coupling on the proton donor- acceptor distance and the deuterium kinetic isotope effect on both the magnitude and the distance dependence of the vibronic coupling. The vibronic coupling decreases exponentially with the proton donor-acceptor distance for both electronically adiabatic and electronically nonadiabatic reactions. For a given proton donor-acceptor distance, the vibronic couplings are significantly smaller for deuterium than for hydrogen because of the smaller overlap between the reactant and product proton vibrational wavefunctions for deuterium. Moreover, the value of the exponential decay parameter α is larger for deuterium than for hydrogen because the overlap between the reactant and product deuterium wavefunctions falls off faster with distance than the corresponding overlap for the hydrogen wavefunctions. Furthermore, the vibronic couplings are reduced but the value of α is not significantly altered when the transferring hydrogen nucleus is represented by a three-dimensional rather than a one-dimensional vibrational wavefunction. These trends are directly relevant to the study of general PCET reactions. This type of analysis provides a new perspective on the distinction between PCET and HAT reactions. A conventional method for distinguishing PCET from HAT is that the electron and proton are transferred between different donors and acceptors (or different sets of orbitals) for PCET. Within this framework, our analysis suggests that PCET reactions are electronically nonadiabatic, whereas HAT reactions are electronically adiabatic. These two mechanisms can be differenti- ated by calculating the adiabaticity parameter, which depends on the electronic coupling and other quantities that can be determined with quantum chemistry methods. Future work will be aimed at the classification of systems as PCET or HAT based on factors such as geometry and electronic structure. Chapter 4

Substituent Eﬀects on the Proton-Coupled Electron Transfer Vibronic Coupling

Reproduced in part with permission from: M. K. Ludlow, J. H. Skone, and S. Hammes-Schiﬀer,“Substituent Eﬀects on the Vibronic Coupling for the Phenoxyl/Phenol Self-Exchange Reaction”, J. Phys. Chem. B 112, 336 (2008). c 2007 American Chemical Society.

4.1 Introduction

Proton-coupled electron transfer (PCET) reactions are essential for a wide range of chemical and biological processes, such as electrochemistry [99], photosynthesis [4, 5, 6, 7, 8, 77], respiration [2, 3], and enzyme reactions [18, 78, 79, 80, 81]. A variety of theoretical methods have been developed to study general PCET reactions in which an electron and a proton are transferred in a single step [33, 34, 35, 36, 37, 38, 39, 40, 100, 101]. The fundamental mechanism for these types of reactions involves reorganization of the solvent and protein environment, as well as the solute in some cases, to enable quantum mechanical tunneling of the hydrogen nucleus in conjunction with electron transfer. General PCET reactions are often vibronically nonadiabatic with respect to the solvent and protein environment because the vibronic coupling between the reactant and product mixed electron- proton vibronic wavefunctions is much less than the thermal energy kBT . In this case, the rate of the reaction for each pair of vibronic states is proportional to 47 the square of the vibronic coupling [33, 34, 38, 39]. As a result, the magnitude of the vibronic coupling significantly impacts the rates, kinetic isotope effects, and temperature dependences of general PCET reactions [29, 31, 41]. In the previous chapter, we implemented a semiclassical approach to calculate the vibronic couplings for the phenoxyl/phenol and benzyl/toluene self-exchange reactions [102]. Prior density functional theory studies designated the phenoxyl/ phenol reaction as PCET and the benzyl/toluene reaction as hydrogen atom transfer (HAT) because the electron and proton are transferred between different sets of orbitals for the former and between the same sets of orbitals for the latter [32]. In our approach, all electrons and the transferring hydrogen nucleus are treated quantum mechanically, and the vibronic coupling is defined as the Hamiltonian matrix element between the reactant and product mixed electron-proton vibronic wavefunctions. We calculated the vibronic couplings using a general semiclassical analytical expression [40], as well as analytical expressions in the limits of electronically adiabatic and nonadiabatic proton tunneling. Our calculations indicated that both of these self-exchange reactions are vibronically nonadiabatic with respect to a solvent environment at room temperature, but the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. Thus, our analysis provided a new diagnostic for differentiating between the conventionally defined PCET and hydrogen atom transfer reactions. For both systems, the vibronic coupling decreased with the proton donor-acceptor distance, and the replacement of hydrogen with deuterium decreased the magnitude of the vibronic coupling and enhanced the decay with distance. In Chapter 2 we developed a nuclear-electronic orbital (NEO) method for calculating the vibronic couplings corresponding to these types of systems [42, 44, 48]. In the NEO approach, the electrons and transferring proton are treated on equal footing with molecular orbital techniques. Mixed nuclear-electronic wavefunctions are calculated by solving a mixed nuclear-electronic time-independent Schrödinger equation using a variational procedure. In the two-state NEO-NOCI (nonorthogonal configuration interaction) approach [48], the vibronic coupling is calculated in terms of the Hamiltonian matrix element between two localized nuclear-electronic wavefunctions obtained at the NEO-HF (Hartree-Fock) level. The advantages of 48 the NEO-NOCI approach are the computational efficiency and the potential for systematic improvement by enhancing the basis sets and number of configurations. Benchmarking calculations have shown that the NEO-NOCI method provides accurate hydrogen tunneling splittings for the [He–H–He]+ model system for a range of He–He distances [48]. In this chapter, we apply both the semiclassical grid-based nonadiabatic method and the NEO-NOCI method to the phenoxyl/phenol self-exchange reaction and examine the impact of substituents on the vibronic coupling for this reaction. From a methodological perspective, our objective is to compare the two methods and identify the range of applicability for each method. From a chemical perspective, our objective is to understand how the vibronic coupling is influenced by the nature of the substituent (i.e., electron donating or electron withdrawing) and the position of the substituent (i.e., ortho, para, or meta). The analysis provides insight into the fundamental physical principles dictating the vibronic couplings for PCET reactions. The results provide experimentally testable predictions of the trends in the rates for substituted phenoxyl/phenol systems. This chapter is organized as follows. Section 4.2 summarizes the two methods for calculating vibronic couplings and provides the computational details. Sec- tion 4.3 presents the results and discussion. The conclusions are presented in Section 4.4.

4.2 Theory and methods

Our group has developed a theoretical formulation for general PCET reactions and derived vibronically nonadiabatic rate expressions [37, 38, 39]. In this formulation, the PCET reaction occurs between two diabatic electronic states, denoted I and II, representing the localized electron transfer states. The transferring electron is localized on the donor for diabatic state I and on the acceptor for diabatic state II. The proton vibrational wavefunctions are calculated for each diabatic electronic state, leading to a set of reactant and product proton vibrational wavefunctions (I) (II) denoted ϕD and ϕA , respectively. The vibronic coupling is defined to be the Hamiltonian matrix element between the reactant and product mixed electron- proton vibronic wavefunctions. The overall reaction is vibronically nonadiabatic 49 with respect to the solvent or protein environment when this vibronic coupling is much less than kBT . In this case, the rate of reaction for each pair of vibronic states is proportional to the square of the vibronic coupling. Consequently, the vibronic couplings significantly impact the magnitudes and temperature dependences of the rates and kinetic isotope effects (KIEs). As derived previously [31, 39] using a series of well-defined, physically reasonable approximations, the rate of a general PCET reaction can be expressed as

2 (0) Vµν 2 r X X 2kBT αµν π k = P exp × µ MΩ2 (λ + λ ) k T µ ν ~ α B " # (∆G0 + λ + ∆ε )2 exp − µν (4.1) 4 (λ + λα) kBT where the summations are over reactant and product vibronic states, Pµ is the

Boltzmann probability for the reactant state µ, λµν is the reorganization energy, 2 2 ~ αµν 0 λα = 2M , ∆G is the reaction free energy for the ground states, ∆εµν is the diﬀerence between the product and reactant vibronic energy levels relative to the ground states, and M and Ω are the eﬀective mass and frequency associated with the proton donor-acceptor motion. Here the vibronic coupling is assumed to have (0) the general form Vµν = Vµν exp [−αµν δR], where δR is the deviation of the proton donor-acceptor distance from its equilibrium value. When only the nonadiabatic transition between the two ground states is included, the KIE can be approximated as [31]

2 V (0) H 2kBT KIE ≈ exp α2 − α2 . (4.2) 2 2 H D (0) MΩ VD where the H and D subscripts refer to hydrogen and deuterium, respectively. These equations for the rate and KIE illustrate that the calculation of the vibronic coupling is essential for predicting the magnitudes and temperature dependences of the rates and KIEs. The objective of this chapter is to examine the effects of substituents on the vibronic couplings for the phenoxyl/phenol self-exchange reaction. For simplicity, 50 here we consider the tunneling between only the ground state reactant and product mixed electron-proton vibronic states. The reactant and product states refer to the mixed electron-proton vibronic quantum states in which the electron and proton are localized on the donor in the reactant state and on the acceptor in the product state for fixed geometry of all other nuclei. For the vibronic coupling calculations, all nuclei are fixed at the transition state geometries except for the transferring hydrogen, which is treated quantum mechanically. Our goal is not to provide quantitatively accurate results for these specific systems but rather to illustrate the general trends. Therefore, we utilize moderate levels of electronic structure theory that provide physically reasonable results. The quantitative accuracy of the results can be improved by using a larger basis set and including dynamical electron correlation.

Figure 4.1. Optimized geometry for the unsubstituted phenoxyl/phenol system. (a) The proton donor-acceptor O–O distance R is indicated. The pairs of substitution positions maintaining C2 symmetry are also indicated, where m, p, and o refer to meta, para, and ortho, respectively. (b) The COOC dihedral angle θ is indicated.

In our systematic study, each substituent was placed in the para, ortho, and meta positions on both rings to maintain C2 symmetry, as illustrated in Fig- ure 4.1. The transition state geometries were optimized with density functional theory (DFT) using the B3LYP functional [91, 92] and the 6-31G* basis set [93] 51 using the Gaussian03 package [90]. All transition states were confirmed to have a single imaginary frequency. For the ortho and meta substitutions, two different transition state geometries were optimized, and the lowest-energy transition state was used in the calculations except for o-NH2, where the transition state with slightly higher energy was used to avoid hydrogen bonding between the NH2 and the oxygen atoms. In all cases, the proton donor-acceptor O–O distance was R = 2.40 ± 0.01 A.˚ We found that the three-dimensional vibronic couplings were so small at this distance that the effects of the substituents were often smaller than the numerical errors in the calculations. Therefore, to examine the trends in the vibronic couplings in a numerically meaningful way, we translated the rigid donor and acceptor molecules along the donor-acceptor axis to obtain an O–O distance of R = 2.30 A˚ for each transition state structure. We confirmed that the one-dimensional vibronic couplings exhibit the same trends within the numerical accuracy at these two proton donor-acceptor O–O distances. For the calculations of the vibronic couplings, all nuclei were fixed except for the transferring hydrogen, which was treated quantum mechanically. In general, the motions of the other nuclei could contribute to the vibronic coupling, but including these effects is beyond the scope of this chapter.

4.2.1 Grid-based nonadiabatic method for calculating vibronic couplings

In Chapter 3 we calculated the vibronic couplings for the phenoxyl/phenol reaction with the semiclassical approach developed by Georgievskii and Stuchebrukhov [40]. We showed that the proton motion occurs in the electronically nonadiabatic limit, where the proton tunneling time is much less than the electronic transition time so the electronic states do not have enough time to mix completely during the proton (na) tunneling process. In this limit, the vibronic coupling VDA can be expressed as the ET D (I) (II)E product of the electronic coupling V and the Franck-Condon overlap ϕD |ϕA of the reactant and product proton vibrational wavefunctions:

(na) ET D (I) (II)E VDA = V ϕD |ϕA (4.3) 52

The input quantities for this vibronic coupling expression can be calculated with electronic structure methods. In this chapter, we present the vibronic couplings for the substituted phenoxyl/phenol systems calculated with the nonadiabatic expression given in Eq. 4.3. We obtained the electronically adiabatic ground and excited state potential energy curves along the hydrogen coordinate by calculating the state-averaged CASSCF(3,6) energy for the hydrogen positioned at discrete grid points along the axis connecting the donor and acceptor atoms. The active space was chosen to ensure that the character of the orbitals in the active space was conserved along the hydrogen coordinate and the electronic ground state was qualitatively similar to the ROHF ground state. The 6-31G basis set [72] was used for all ROHF and CASSCF calculations to enable the eﬃcient calculation of three-dimensional potential energy surfaces. We determined that the ground and excited state potential energy surfaces are qualitatively similar for the 6-31G and 6- 31G* basis sets. These CASSCF calculations were performed with the Gaussian03 package [90]. The quantities in the expression for the vibronic coupling given in Eq. 4.3 were determined from the CASSCF potential energy curves, which are depicted in Fig- ure 4.2 for the unsubstituted phenoxyl/phenol system. The electronic coupling is half the splitting between the two electronically adiabatic CASSCF potential energy curves at the midpoint between the donor and acceptor atoms. A valence bond model was used to ﬁt the state-averaged CASSCF potential energy curves for the purpose of obtaining the two localized electronically diabatic potential energy curves, which are depicted in Figure 4.2. The details of the valence bond models are given in Appendix B. The one-dimensional hydrogen vibrational wavefunctions were calculated for the diabatic potential energy curves using the Fourier grid Hamiltonian method [70, 71] with 128 grid points spaning 2.0 A.˚ The Franck-Condon overlap in Eq. 4.3 is the overlap between the proton vibrational wavefunctions for the two diabatic potential energy curves, as depicted in Fig- ure 4.2b. To study the impact of the three-dimensional character of the hydrogen vibrational wavefunction, we also calculated three-dimensional potential energy surfaces for the hydrogen at the ROHF level for the unsubstituted phenoxyl/phenol system. We obtained the electronically diabatic three-dimensional potential energy 53

Figure 4.2. (a) State-averaged CASSCF ground and excited state electronically adiabatic potential energy curves along the transferring hydrogen coordinate for the unsubstituted phenoxyl/phenol system. The coordinates of all nuclei except the transferring hydrogen correspond to the transition state geometry. The proton donor-acceptor distance is R = 2.40 A.˚ The CASSCF results are depicted as open circles that are blue for the ground state and red for the excited state. The black dashed lines represent the diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II. The mixing of these two diabatic states with the electronic coupling V ET leads to the CASSCF ground and excited state electronically adiabatic curves depicted with solid colored lines following the colored open circles. The solid colored lines and the black dashed lines are nearly indistinguishable because the adiabatic and diabatic potential energy curves are virtually identical except in the transition state region. (b) Diabatic potential energy curves corresponding to the two localized diabatic electron transfer states I and II and the corresponding proton vibrational wavefunctions (I) (II) ϕD (blue) and ϕA (red) for the phenoxyl/phenol system. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling and D (I) (II)E the overlap of the reactant and product proton vibrational wavefunctions ϕD |ϕA . 54 surfaces by calculating the ROHF energy for a three-dimensional grid with 32 grid points per dimension spanning half of the proton donor-acceptor axis, ﬁtting the data points to an analytical functional form (i.e., a fourth-order polynomial), and using the analytical functional form to generate the full potential energy surface for a grid with 64 points per dimension. The three-dimensional hydrogen vibrational wavefunctions were calculated for the ROHF potential energy surfaces using the Fourier grid Hamiltonian method [70, 71]. We used the GAMESS electronic structure program [69] for the three-dimensional calculations. Due to numerical diﬃculties and the computational expense associated with the three-dimensional grid-based nonadiabatic calculations, we did not perform them for the substituted phenoxyl/phenol systems.

4.2.2 Nuclear-electronic orbital method for calculating vibronic couplings

The vibronic couplings can also be calculated with a three-dimensional treatment of the transferring hydrogen nucleus using the nuclear-electronic orbital (NEO) method [42, 48]. This method treats specified nuclei on the same level as the electrons and provides mixed nuclear-electronic wavefunctions through the solution of a mixed nuclear-electronic time-independent Schrödingerequation with molecular orbital techniques. Both electronic and nuclear molecular orbitals are expressed as linear combinations of Gaussian basis functions, and the variational method is used to minimize the energy with respect to all molecular orbitals, as well as the centers of the nuclear basis functions. When the NEO approach is applied to hydrogen transfer systems, the transferring hydrogen nucleus and all electrons are treated quantum mechanically. The NEO approach has been implemented in the GAMESS electronic structure program [69]. The calculation of delocalized, bilobal hydrogen wavefunctions for hydrogen tunneling systems within the NEO framework is challenging due to the importance of electron-proton correlation. Typically the transferring hydrogen atom for hydrogen tunneling systems is represented by two basis function centers to allow delocalization of the hydrogen vibrational wavefunction [44]. For a symmetric system, the exact nuclear wavefunction will be delocalized equally over both basis 55 function centers. The variational NEO-HF solution, however, corresponds to a nuclear wavefunction localized on one of the basis function centers [45]. This non- physical localization of the nuclear density at the NEO-HF level arises from the neglect of electron-proton correlation [45]. Inclusion of sufficient electron-proton correlation with the NEO-full CI method enables the calculation of delocalized, symmetric nuclear wavefunctions, but this approach is not computationally practical for most chemical systems. We have developed the NEO-NOCI method for calculating delocalized hydrogen wavefunctions and the corresponding tunneling splittings [48]. The NEO- NOCI method takes advantage of the localization of the variational NEO-HF wavefunctions. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wavefunctions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wavefunctions obtained at the NEO-HF level. The tunneling splitting is determined by the energy difference between these two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He–H–He]+ model system with a range of fixed He–He distances are in excellent agreement with NEO-full CI and three-dimensional Fourier grid calculations [48]. The NEO-NOCI method is robust and computationally efficient, and it can be applied to a wide range of chemical systems. This method can also be used to calculate vibronic couplings for PCET reactions. In the application of the NEO-NOCI method to PCET systems, only the transferring hydrogen nucleus is treated quantum mechanically on the same level as the electrons, and the total wavefunction is expressed as a linear combination of two e p localized NEO-HF wavefunctions. Specifically, |ΨIi = ΦI (re)ΦI (rp) corresponds e p to the electron and proton localized on the donor, and |ΨIIi = ΦII (re)ΦII (rp) corresponds to the electron and proton localized on the acceptor. Here the e and p subscripts and superscripts denote electrons and proton, respectively. The delocalized ground and excited state NEO-NOCI wavefunctions are linear combinations of these localized states, and the coefficients of these wavefunctions are determined by solving a 2×2 matrix equation. For a symmetric system, the vibronic coupling is half the difference between the ground and first excited vibronic state energy 56

Figure 4.3. Mixed electronic-nuclear wavefunctions calculated with the NEO approach for the unsubstituted phenoxyl/phenol system. (a) The two localized NEO-HF nuclear- electronic wavefunctions, where the highest-energy doubly occupied electronic molecular orbital and the proton molecular orbital are depicted. (b) The delocalized NEO- NOCI ground and ﬁrst excited vibronic state nuclear-electronic wavefunctions, where the highest-energy doubly occupied electronic molecular orbital and the proton molecular orbital are depicted. These ﬁgures were generated with MacMolPlot [95]. and can be expressed analogously to electron transfer couplings as [60, 103]:

H − S H V (NEO) = I,II I,II I,I (4.4) DA 2 1 − SI,II where HI,I = hΨI| H |ΨIi and HII,II = hΨII| H |ΨIIi are the energies of the localized

NEO-HF solutions, HI,II = hΨI| H |ΨIIi = HII,I is the oﬀ-diagonal Hamiltonian matrix element, and SI,II = hΨI|ΨIIi = SII,I is the overlap between the localized NEO-HF solutions. This equation for the coupling can be generalized to non-symmetric systems by replacing HI,II with (HI,I + HII,II)/2, as obtained by symmetric orthogonalization of the Hamiltonian matrix [60]. 57

In this chapter, we use the NEO-NOCI method to calculate the vibronic couplings for the substituted phenoxyl/phenol systems. We used the 6-31G electronic basis set and the QZSPDN0 nuclear basis set [48], which is a quadruple-ζ, s, p, double-ζ, d nuclear basis set, for all NEO calculations. The transferring hydrogen atom was represented by two basis function centers, where each center contains both nuclear and electronic basis functions. The two centers were constrained to be equidistant from the midpoint of these symmetric systems. The basis function center separation was optimized variationally at the NEO-NOCI level for both the ground and excited states. Figure 4.3 depicts the localized NEO-HF states and the delocalized NEO-NOCI ground and excited vibronic states for the unsubstituted phenoxyl/phenol system.

4.3 Results and discussion

As shown in Figure 4.1, the phenoxyl/phenol transition state has C2 symmetry. The proton donor-acceptor O–O distance is R = 2.40 A,˚ and the COOC dihedral angle is θ = 138 ◦ for the unsubstituted phenoxyl/phenol system. Each substituent is placed in the para, ortho, and meta positions of both rings to maintain the C2 symmetry, as depicted in Figure 4.1a. Table 4.1 provides the proton donor-acceptor O–O distance, R, and the dihedral angle, θ, for all of the optimized transition states. The complete optimized structures are provided in Appendix C. Although R ≈ 2.40A˚ for all systems, the dihedral angle varies significantly. This variation in θ is due to a combination of electronic and steric effects. As discussed above, the proton donor-acceptor distance is decreased to R = 2.30 A˚ for our calculations to enhance the magnitude of the vibronic couplings and thereby enable a numerically meaningful analysis of the trends. For comparison, the grid-based nonadiabatic vibronic couplings calculated at the transition state structures with R ≈ 2.40 A˚ are provided in Appendix C. This table indicates that the qualitative trends in the substituent effects on the vibronic couplings are similar for both distances. Figure 4.2 depicts the potential energy curves along the transferring hydrogen coordinate and the overlap between the reactant and product proton vibrational wavefunctions for the unsubstituted phenoxyl/phenol system [102]. The CASSCF electronically adiabatic ground and excited state curves, as well as the electron- 58

Table 4.1. Proton donor-acceptor O–O distance R and COOC dihedral angle θ for the series of substituted phenoxyl/phenol systems.

Substituent R(A)˚ θ - 2.40 138 p-F 2.40 138 m-F 2.40 148 o-F 2.39 48 p-CN 2.40 152 m-CN 2.40 149 o-CN 2.39 180 p-NO2 2.40 156 m-NO2 2.40 144 o-NO2 2.39 154 p-OH 2.40 136 m-OH 2.40 135 o-OH 2.40 140 p-NH2 2.41 130 m-NH2 2.41 141 o-NH2 2.40 129

ically diabatic curves corresponding to the two electron transfer states I and II, are depicted in Figure 4.2a. The diabatic states correspond to fixed electronic wavefunctions associated with the hydrogen bonded to the donor atom (I) or to the acceptor atom (II), and the mixing of these two diabatic states with the appropriate coupling V ET leads to the CASSCF electronically adiabatic ground and excited state curves. Since this reaction is electronically nonadiabatic, the vibronic coupling is the product of the electronic coupling between the diabatic states I and II and the overlap of the reactant and product proton vibrational wavefunctions corresponding to these diabatic states. This overlap is depicted in Figure 4.2b. Figure 4.3 depicts the two localized NEO-HF nuclear-electronic wavefunctions and the delocalized NEO-NOCI nuclear-electronic wavefunctions for the unsubstituted phenoxyl/phenol system. For the localized NEO-HF states, the highest- energy doubly occupied electronic molecular orbital is localized mainly on the conjugated π system of the donor (I) or acceptor (II) ring, and the proton molec- 59 ular orbital is localized near the donor oxygen (I) or acceptor oxygen (II). The delocalized NEO-NOCI states are mixtures of these two localized states. For both the ground and first-excited NEO-NOCI vibronic states, the highest-energy doubly occupied electronic molecular orbital is delocalized over the conjugated π systems of both rings. The ground NEO-NOCI vibronic state corresponds to a symmetric bilobal proton molecular orbital, and the first-excited NEO-NOCI vibronic state corresponds to an antisymmetric bilobal proton molecular orbital. The NEO- NOCI vibronic coupling is half of the splitting between the energies corresponding to the delocalized NEO-NOCI ground and first-excited vibronic states, which is equivalent to the expression in Eq. 4.4 for this symmetric system. The calculated vibronic couplings for the unsubstituted phenoxyl/phenol system are given for R = 2.30A˚ and R = 2.40A˚ in Table 4.2. These vibronic couplings were calculated with the one-dimensional and three-dimensional grid-based nonadiabatic methods and the three-dimensional NEO-NOCI method. For all methods, the vibronic coupling is significantly smaller for the greater proton donor-acceptor distance. Moreover, the three-dimensional treatment of the transferring hydrogen nucleus decreases the vibronic coupling by approximately a factor of two. The vibronic couplings calculated with the three-dimensional grid-based nonadiabatic method and the NEO-NOCI method are in excellent agreement. The absolute differences are ∼2 cm−1, and the percentage difference is ∼ 10% for the shorter distance. This agreement provides a level of validation for both methods. Table 4.3 provides the vibronic couplings for the substituted phenoxyl/phenol systems calculated with the one-dimensional grid-based nonadiabatic method and the three-dimensional NEO-NOCI method. The effects of the three-dimensional versus the one-dimensional treatment of the transferring hydrogen nucleus may vary for the different substituents, so a global comparison between the two methods for all systems is not warranted. For each substituent, however, the trends in the vibronic couplings for substitutions at the para, meta, and ortho positions are the same for the one-dimensional grid-based nonadiabatic method and the three- dimensional NEO-NOCI method, with the exception of NO2 and NH2 discussed below. Note that the variation of the vibronic coupling with ring position does not exhibit any consistent general trends. Moreover, the impact of the substituents on the magnitudes of the vibronic couplings is rather modest for all substituents 60

Table 4.2. Vibronic couplings for the unsubstituted phenoxyl/phenol system.

˚ (na) a (na) b (NEO) c R(A) VDA (1D) VDA (3D) VDA 2.30 33.4 15.3 17.7 2.40 4.47 2.86 1.10 a One-dimensional grid-based nonadiabatic vibronic coupling in cm−1. b Three-dimensional grid-based nonadiabatic vibronic coupling in cm−1. c Three-dimensional NEO vibronic coupling in cm−1.

except for NH2, which signiﬁcantly increases the vibronic coupling.

The minor discrepancies between the two methods for NO2 and NH2 have been analyzed. In the case of NO2, substitutions at the meta and ortho positions lead to similar vibronic couplings, and the order is different for the two methods. This discrepancy may arise from numerical error or differences between the one- dimensional and three-dimensional treatments of the hydrogen nucleus. In the case of NH2, the order of the vibronic couplings for substitutions at the para and meta positions is interchanged. Note that the magnitude of the vibronic couplings is significantly larger and the proton transfer barriers are significantly lower for the NH2-substituted systems than for all other systems studied. We used the formalisim devised by Georgievskii and Stuchebrukhov [40] to determine that these systems are still predominantly in the electronically nonadiabatic proton tunneling regime for which Eq. 4.3 is valid, suggesting that the discrepancy may be due to limitations of the NEO-NOCI method. Previously we showed that the current implementation of the NEO-NOCI method is accurate in the relatively deep tunneling regime and becomes less accurate for systems with lower proton transfer barriers and greater vibronic couplings [48]. To enable a comparison between these two methods in a more appropriate regime for the NH2-substituted systems, we calculated the vibronic couplings at R = 2.35 A˚ for these systems. As shown in Table 4.3, the trends for the one-dimensional grid-based nonadiabatic method and the three-dimensional NEO-NOCI method are the same at this distance. 61

Table 4.3. Vibronic couplings and associated quantities for the series of substituted phenoxyl/phenol systems at R = 2.30 A.˚

a ET b c (na) d (NEO) e Substituent Barrier V Overlap VDA VDA - 12.7 836 4.00×10−2 33.4 17.7

p-F 12.7 869 3.85×10−2 33.5 17.6 m-F 13.3 674 3.85×10−2 26.0 16.1 o-F 13.2 1094 3.23×10−2 35.3 18.7

p-CN 12.0 579 5.03×10−2 29.1 18.4 m-CN 13.8 790 3.33×10−2 26.3 17.0 o-CN 11.7 551 5.55×10−2 30.6 20.4

−2 p-NO2 13.4 675 4.90×10 33.1 17.6 −2 m-NO2 14.3 790 3.08×10 26.0 16.4 −2 o-NO2 12.5 476 4.88×10 23.2 16.6

p-OH 10.9 789 5.93×10−2 46.8 20.3 m-OH 13.0 771 3.83×10−2 29.5 16.0 o-OH 12.4 696 4.61×10−2 32.1 17.3

−2 p-NH2 8.38 702 9.74×10 68.3 24.3 −2 m-NH2 7.84 436 1.37×10 59.9 32.3 −2 o-NH2 7.64 539 1.34×10 72.0 41.0

f −2 p-NH2 10.8 623 4.58×10 28.5 6.25 f −2 m-NH2 9.89 390 7.37×10 28.8 10.3 f −2 o-NH2 9.69 495 7.24×10 35.9 13.7 a barrier in kcal/mol for the CASSCF ground electronic state. b electronic coupling, which is half the splitting in cm−1 between the ground and excited CASSCF electronic states. c Franck-Condon overlap between the reactant and product proton vibrational wavefunctions for the diabatic electronic states. d one-dimensional grid-based nonadiabatic vibronic coupling in cm−1. e three-dimensional NEO vibronic coupling in cm−1. f these values were calculated for R = 2.35 A.˚ 62

Overall, the qualitative agreement between the grid-based nonadiabatic and NEO-NOCI methods provides a level of validation for both methods. The results indicate that a three-dimensional treatment of the hydrogen nucleus is desirable for the calculation of quantitatively accurate vibronic couplings. We emphasize that the NEO-NOCI method is significantly more computationally efficient and straightforward to implement than the three-dimensional grid-based nonadiabatic method. As mentioned above, however, the current implementation of the NEO- NOCI method has been shown to be accurate in the relatively deep tunneling regime but less accurate for lower proton transfer barriers. These difficulties arise mainly from the inadequate treatment of electron-proton correlation in the localized NEO-HF wavefunctions that are used as the basis states for the two-state NEO-NOCI calculations. Currently, we are extending the NEO-NOCI method to improve its accuracy and increase its range of applicability. In particular, we are developing methods that include additional electron-proton and electron-electron correlation. We analyzed the impact of substituents on the vibronic couplings in the context of the electron withdrawing and electron donating nature of the substituents. Fig- ure 4.4a illustrates that the electron donating substituents significantly increase the vibronic coupling, whereas the electron withdrawing substituents slightly decrease the vibronic coupling relative to the unsubstituted phenoxyl/phenol system. Thus, if all other aspects of the reaction are the same, then electron donating groups will tend to increase the PCET rate, while electron withdrawing groups will tend to decrease the PCET rate. As indicated by the values given in Table 4.3, the electronic couplings alone do not reflect these trends. These data illustrate that the Franck-Condon overlap between the reactant and product proton vibrational wavefunctions plays an important role in dictating these trends. For example, the NH2 substituent is the strongest electron donating group studied, and these systems exhibit the largest vibronic couplings. All of the NH2-substituted systems have larger Franck-Condon overlaps than the other systems, but the electronic couplings span a wide range. Thus, calculation of the full vibronic couplings is necessary to predict physically meaningful trends. We also analyzed the correlations between the vibronic coupling and physical properties such as the Hammett constant, bond dissociation enthalpy (BDE), ion- 63

Figure 4.4. Vibronic couplings of substituted phenoxyl/phenol systems (a) arranged from highest to lowest in magnitude, (b) as functions of Hammett constant σm,p [104], (c) as functions of the O–H bond dissociation enthalpy (BDE) [105], (d) as functions of ionization potential (IP) [105], (e) as functions of redox potential (E) at pH = 7 [106], (f) as functions of pKa [107, 108]. The vibronic couplings are calculated with the one-dimensional grid-based nonadiabatic method at R = 2.30 A.˚ Electron donating substituents are denoted with open diamonds, electron withdrawing substituents are denoted with ﬁlled diamonds, and the unsubstituted phenoxyl/phenol system is denoted with a red ﬁlled square. The m-OH substituent is considered to be electron withdrawing because of its positive Hammett constant.

ization potential, redox potential, and pKa. The correlations between the vibronic coupling and these physical quantities are depicted in Figure 4.4b-f. This ﬁgure in- 64 dicates that negative Hammett constants correspond to higher vibronic couplings, while positive Hammett constants correspond to similar or slightly lower vibronic couplings relative to the unsubsituted phenoxyl/phenol system. Moreover, lower

BDEs, ionization potentials, and redox potentials, as well as higher pKa values, tend to correspond to higher vibronic couplings relative to the unsubstituted phenoxyl/phenol system. The impact of electron donating groups on these physical properties of the phenol molecule is well understood. Electron donating groups hinder the removal of the proton from the phenol, therefore decreasing the Hammett constant and increasing the pKa. Electron donating groups facilitate the removal of an electron from the phenol, therefore decreasing the ionization potential. Similarly, electron donating groups hinder the reduction of the phenol, therefore decreasing the redox potential. Finally, electron donating groups also decrease the BDE for hydrogen atom removal due to stabilization of the phenoxyl radical and destabilization of the phenol [109, 110, 111, 112]. Note that hydrogen atom removal, rather than single proton or electron removal, is most relevant to the phenoxyl/phenol self-exchange reaction. The direct connection of these physical quantities to the vibronic coupling for the phenoxyl/phenol self-exchange reaction is not straightforward because all of these quantities are determined for the isolated phenol. The phenoxyl/phenol self- exchange reaction involves the net transfer of a hydrogen atom from the phenol to the phenoxyl within a hydrogen-bonded phenoxyl/phenol complex. The most relevant quantity is the BDE for hydrogen atom removal because both an electron and a proton are being removed from the phenol, as is the case in the self-exchange reaction. A smaller BDE is expected to be associated with a lower hydrogen transfer barrier, which would lead to a larger Franck-Condon overlap between the reactant and product proton vibrational wavefunctions and hence a greater vibronic coupling. Indeed, three of the substituted species with lower BDEs (p-NH2, p-OH, and m-NH2) have signiﬁcantly greater Franck-Condon factors and vibronic couplings than the other species. In general, however, the vibronic couplings are dictated by a complex interplay among the various electronic and nuclear interactions. 65

4.4 Conclusions

The vibronic couplings for a series of substituted phenoxyl/phenol self-exchange reactions were calculated in this chapter. The vibronic couplings significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of general PCET reactions. Thus, the development and benchmarking of efficient methods for calculating these vibronic couplings is essential for the study of PCET reactions. The quantitative agreement between the three-dimensional grid-based nonadiabatic vibronic couplings and the NEO-NOCI vibronic couplings for the unsubstituted phenoxyl/phenol system provides a level of validation for both methods. The qualitative agreement in the trends for the substituted phenoxyl/phenol systems predicted by these two methods provides additional validation. We analyzed the underlying physical principles dictating the impact of substituents on the vibronic coupling for the phenoxyl/phenol self-exchange reaction. Our analysis indicates that electron donating groups enhance the vibronic coupling, and electron withdrawing groups attenuate the vibronic coupling. Thus, if all other aspects of the reaction are the same, then electron donating groups will increase the PCET rate, while electron withdrawing groups will decrease the PCET rate. Our calculations illustrate that the electronic couplings alone do not reflect these trends, but rather the Franck-Condon overlap between the reactant and product proton vibrational wavefunctions must also be considered. Furthermore, we studied the correlations between the vibronic couplings and various physical properties of the phenol. We found that negative Hammett constants correspond to higher vibronic couplings, while positive Hammett constants correspond to similar or slightly lower vibronic couplings relative to the unsubsituted phenoxyl/phenol system. In addition, lower BDEs, ionization potentials, and redox potentials, as well as higher pK a values, tend to correspond to higher vibronic couplings relative to the unsubstituted phenoxyl/phenol system. The trends observed in our calculations enable the prediction of the impact of general substituents on the vibronic coupling, and hence the rate, for the phenoxyl/phenol self-exchange reaction. Such predictions can be tested experimentally and may be extended to related systems. Moreover, the fundamental physical insights obtained from these studies are applicable to other PCET systems. Chapter 5

Conclusions

5.1 Summary and concluding remarks

The considerable impact of the vibronic coupling on the rates, kinetic isotope effects and temperature dependences of general proton-coupled electron transfer (PCET) reactions presents a need for an accurate and effective approach to evaluate this quantity. The nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) method introduced in Chapter 2 is a novel approach for evaluating proton tunneling splittings and vibronic couplings that is both reliable and tractable. Another approach is the semiclassical method of Georgievskii and Stuchebrukhov [40], which we have implemented as discussed in Chapter 3. Both approaches address the need for reliable theoretical methods to calculate vibronic couplings and additionally provide insight into the fundamental physical principles underlying general PCET reactions. The NEO-NOCI methodology with an application to calculating the proton tunneling splittings in the [He–H–He]+ model system was presented in Chapter 2. Excellent agreement between the NEO full-CI and NEO-NOCI proton tunneling splittings for a nuclear basis set small enough to be tractable for full-CI shows that the two localized configurations used in the nonorthogonal CI are capable of capturing most of the nondynamical electron-proton correlation and at consider- ably less cost than full-CI. Using a larger nuclear basis set the NEO-NOCI method quantitatively agrees with the three-dimensional grid-based methods, showing that it is a promising method for calculating proton tunneling splittings. 67

As discussed in Chapter 3, in general PCET, the proton tunneling can be either electronically adiabatic or electronically nonadiabatic. It is important to treat the proton tunneling in the correct limit of electronic adiabaticity for calculating quantitatively accurate vibronic couplings and obtaining meaningful insights. This was illustrated with two self-exchange reactions: phenoxyl/phenol and benzyl/toluene. Both systems were vibronically nonadiabatic with respect to the solvent environment, but the proton tunneling was electronically nonadiabatic in phenoxyl/phenol and electronically adiabatic in benzyl/toluene. Additionally, the dependence of the vibronic coupling on the donor-acceptor distance was found to be exponential near the equilibrium distance. More physically intuitive quantities were also presented in Chapter 3, such as the adiabaticity parameter, proton tunneling time, and electronic transition time. These quantities were used to both define electronically adiabatic and nonadiabatic proton tunneling and explain the differences between them. Distinguishing a proton transfer reaction as being electronically nonadiabatic or adiabatic through the adiabaticity parameter also offers a more physically rigorous diagnostic for classifying a general PCET system as HAT or PCET. It was also shown that treating the quantum nucleus three-dimensionally reduces the vibronic couplings by a factor of ∼2 relative to the one-dimensional treatment, indicating that a three-dimensional treatment of the proton is necessary for quantitatively accurate results. The NEO-NOCI method, similar to the semiclassical method of Georgievskii and Stuchebrukhov [40] implemented in Chapter 3, avoids the Born-Oppenheimer separation between protons and electrons, making it suitable for calculating vibronic couplings in both the nonadiabatic and adiabatic limits of proton tunneling and the intermediate regime. Moreover, the NEO-NOCI method has several advantages over the semiclassical method for calculating vibronic couplings, including its computational efficiency, the straightforward manner in which it treats the quantum proton three-dimensionally, and applicability to asymmetric systems. Chapter 4 explored the effect of chemical substituents on the vibronic couplings for a prototypical PCET system and hence the rate. The quantitative agreement between the unsubstituted phenoxyl/phenol vibronic couplings calculated with the semiclassical and NEO-NOCI methods and the qualitative agreement between the trends of the substituted phenoxyl/phenol systems for the semiclassical method 68 and the NEO-NOCI method gives credence to the validity of both methods. The trends observed for the substituted phenoxyl/phenol systems indicate that electron donating groups enhance the vibronic coupling, while electron withdrawing groups attenuate the vibronic coupling. Analysis of the correlation between the vibronic coupling and the physical properties of the phenol resulted in several useful insights on how to modulate the vibronic coupling. With the unsubstituted phenoxyl/phenol as the reference, substituents that lower the bond dissociation enthalpy, ionization potential, and redox potential, as well as raise the pKa, tend to increase the vibronic coupling. The physical insights gained from these studies are applicable to other general PCET systems. A number of future directions for the NEO-NOCI method remain to be explored. As mentioned in Chapter 2 and Chapter 4, the NEO-NOCI method performs less reliably in regimes where the classical barrier to proton transfer is fairly small (i.e. < 10 kcal mol−1). In order for the NEO-NOCI method to be capable of calculating accurate tunneling splittings or vibronic couplings in this regime, it is necessary to capture more electron-proton correlation. This can be accomplished using explicitly correlated nuclear-electronic wavefunctions [49] as basis states in- stead of NEO-HF wavefunctions. This approach may prove intractable because of the computational demands of the multi-center multi-particle integrals inherent to the explicitly correlated wavefunctions, but another more efficient approach is to use an electron-proton density functional method to generate basis states for the nonorthogonal CI. Additonally, electron-electron correlation could be included by similarly using a standard density functional method within the NEO framework to generate basis states for the nonorthogonal CI [50]. Because of the lack of electron-proton correlation, which leads to an over- localization of the nuclear density at the NEO-HF, -MP2, and -MCSCF levels of theory [44, 45, 46], it is difficult to describe the transition state along a reaction path for hydrogen transfer reactions. The NEO-NOCI approach with basis states that include electron-proton correlation could be capable of describing all points along such a reaction path. The method may also prove useful in the full multidimensional calculation of quantitatively accurate tunneling splittings comparable to experiment, because it is capable of generating a continuous NEO potential energy surface around a transition state geometry. 69

The NEO-NOCI method has already proven to be a very useful and reliable tool to calculate proton tunneling splittings, vibronic couplings, and the dependence of the vibronic coupling on the gating motion of the molecule. The method is presently capable of calculating couplings for small model proton-coupled electron transfer systems such as the intramolecular hydrogen-bonded set of phenols [113, 114], using high-end computing resources. With algorithmic improvements, including parallelization of the NEO-NOCI matrix construction, and continued gains in computing power, calculating vibronic couplings for enzyme substrates [87, 31] and larger bio-energetically relevant proton-coupled electron transfer model systems [34, 115] may soon be within the reach of the method. Appendix A

NEO-NOCI Additional Equations and Data

A.1 Expression for the tunneling splitting ∆ and

vibronic coupling VDA for asymmetric systems

In this section of Appendix A, we present a more general expression for the tunneling splitting ∆ that is applicable to both symmetric and asymmetric systems. The following expressions are also general in that they are both applicable to the conventional electronic structure methods where |Ψi = Φe and to the NEO methodology |Ψi = ΦeΦp. The 2×2 nonorthogonal CI matrix equation can be written as:

SI,II = SII,I . Let E = λ and solve for the eigenvalues λ and eigenvectors in the secular 71 equation by diagonalization:

" #" # H − λ H − S λ C I,I I,II I,II I = 0 (A.2) HI,II − SI,II λ HII,II − λ CII

The eigenvalues are:

−2H S + H + H λ− = I,II I,II I,I II,II 2 2 1 − SI,II q 2 2 2 (2HI,II SI,II − HI,I − HII,II ) − 4 HI,I HII,II − HI,II 1 − SI,II − 2 2 1 − SI,II (A.3)

−2H S + H + H λ+ = I,II I,II I,I II,II 2 2 1 − SI,II q 2 2 2 (2HI,II SI,II − HI,I − HII,II ) − 4 HI,I HII,II − HI,II 1 − SI,II + 2 2 1 − SI,II (A.4)

− + λ is the ground state energy E0 and λ is the first excited state energy E1. The tunneling splitting ∆ is expressed as the energy difference between the first excited state and the ground state energies. To make the expression for the general tunneling splitting ∆ more compact HI,I and HII,II are redefined as HI and HII , respectively.

∆ = E1 − E0 =

q 2 2 2 2 4 (HI,II + SI,II HI HII − HI,II SI,II HI − HI,II SI,II HII ) + HI + HII − 2HI HII 2 1 − SI,II (A.5)

Eq. A.5 is the expression for the more general tunneling splitting ∆, which is applicable to asymmetric systems. 72

To determine the vibronic coupling VDA for an asymmetric system in terms of the overlap matrix elements and Hamiltonian matrix elements in expression A.1, we must ﬁrst obtain a transformation matrix X that will orthogonalize the two nonorthogonal wavefunctions. This orthogonalization will lead to a matrix 0 0 H where the oﬀ-diagonal matrix element HI,II is the vibronic coupling VDA. In 0 0 0 order to ensure that the H matrix is Hermitian and thus HI,II = HII,I , symmetric orthogonalization must be employed. Applying the symmetric orthogonalization procedure to the secular equation

|H − SE| = 0 (A.6) where H and S are the Hamiltonian and overlap matrices we obtain:

|S−1/2HS−1/2 − S−1/2SS−1/2E| = 0 (A.7)

  1 √ 1 + 1 √ 1 − 1 √ 1 + 1 √ 1 2 1−SI,II 2 1+SI,II 2 1−SI,II 2 1+SI,II where: S−1/2 =   − 1 √ 1 + 1 √ 1 1 √ 1 + 1 √ 1 2 1−SI,II 2 1+SI,II 2 1−SI,II 2 1+SI,II S−1/2SS−1/2 is equal to the unit matrix 1 and S−1/2HS−1/2 is equal to H0. This allows us to rewrite Eq. A.7 as:

|H0 − E| = 0 (A.8)

0 The oﬀ-diagonal H matrix element, which is equal to VDA, is expressed as:

2H − (H + H ) S H0 = H0 = I,II I,I II,II I,II (A.9) I,II II,I 2 2 1 − SI,II

0 In the limit of small SI,II , HI,II can be approximated as:

(H + H ) S H0 = H0 ∼= H − I,I II,II I,II (A.10) I,II II,I I,II 2

For a symmetric system, Eq. A.9 reduces to the expression for the vibronic coupling in Eq. 4.4 of Chapter 4 and the tunneling splitting ∆ in Eq. A.5 is equal to twice 0 the vibronic coupling HI,II in Eq. A.9. 73

A.2 Additional data collected from the NEO- NOCI proton tunneling splitting calculations

Presented here in tabular form is additional information pertaining to the NEO- NOCI proton tunneling splitting calculations for the [He–H–He]+, phenoxyl/phenol, and benzyl/toluene systems. Tables A.1, A.2, and A.3 provide additional data for the NEO-NOCI proton tunneling splitting calculations. The additional data provided in these tables are the nuclear basis function center (nuc b.f.c.) separation, the mixed state unorthogonalized Hamiltonian matrix element HI,II , the orthogo- 0 p nalized mixed state Hamiltonian matrix element HI,II , the proton overlap S , the electronic overlap Se, the total overlap ST = Se ∗ Sp, and the associated proton tunneling splittings. For a NEO-NOCI calculation, typically two nuclear basis function centers are used to represent the quantum mechanical proton, since in proton transfer reactions the proton will often be delocalized at a symmetric geometry or electrostatically symmetric environment. Here nuclear basis function separation refers to the separation of the two nuclear basis function centers used to represent the transferring quantum proton. The nuclear basis function centers contain both electronic and nuclear basis functions and by construction are always equally spaced from the midpoint between the donor and acceptor. The nuclear basis function center separation can be minimized at either the NEO-HF level of theory, where the nuclear density is localized on one of the two centers, or at the NEO-NOCI level of theory. The proton tunneling splittings can be calculated as the energy difference between the variationally optimized NEO-NOCI ground state and first excited state energies ∆(NOCI), which may correspond to different nuclear basis function center separations for each optimized energy level. Alternatively, the proton tunneling splittings can be evaluated using NEO-NOCI at the NEO-HF minimum ∆(HF), in which case the NEO-NOCI calculation is done with the nuclear basis function centers fixed at the position corresponding to the NEO-HF minimum energy. In the deep tunneling regime, typically the basis function center separations corresponding to the minima for the ground and first excited NEO-NOCI states are the same and both are additionally the same as the localized NEO-HF optimum center separation. All values in Tables A.1, A.2, and A.3, correspond to the NEO- 74

NOCI calculation at the localized NEO-HF energy minimum with the exception of ∆(NOCI). Note that ∆(HF) and ∆(NOCI) are the same for most donor–acceptor distances in Tables A.1, A.2, and A.3, which indicates that these results correspond to the proton in the deep tunneling regime. Table A.4 provides the proton tunneling splittings for the [He–H–He]+ system calculated with the NEO-NOCI method, the NEO-FCI method, the one- dimensional grid-based method, and the three-dimensional grid-based method for fixed He–He distances of 1.86 and 2.30 A.˚ For the smaller He–He distance of 1.86 A,˚ the proton moves in a flat anharmonic single-well potential whereas for the larger He–He distance of 2.30 A,˚ the proton moves in a double-well potential. For the He–He distance of 2.30 A,˚ the NEO-FCI and NEO-NOCI methods match the 1D grid method when no d functions are present, but both NEO methods match the 3D grid method when d functions are included in the nuclear basis set because the 3D grid method includes bending modes, which cannot be described adequately with only s and p functions with either of the two NEO methods. The NEO-FCI and NEO-NOCI methods capture most of the electron-proton correlation in the double-well regime (He–He distance of 2.3 A),˚ but both methods do not adequately describe the electron-proton correlation for the single-well regime (He–He distance of 1.86 A).˚ This is evidenced by the good agreement between NEO methods and the 3D grid method for the 2.30 A˚ He–He distance and the disparity between NEO methods and the 3D grid method at the smaller He–He distance. It should be noted that the 2 × 2 NEO-NOCI method is in poor agreement with NEO-FCI for the shorter He–He distance, but the agreement can be improved, as is shown in the table, by expanding the NEO-NOCI to include all proton excitations from both localized configurations. 75 , the b p S system. + ∆(NOCI) b ∆(HF) , the proton overlap 0 I,II H T S e S p S b 0 I,II H b , and the corresponding NEO-NOCI proton tunneling splittings. ∆(HF) refers p I,II S 2 NEO-NOCI proton tunneling splitting calculations of the [He–H–He] ∗ H e × S = a,c T S , the orthogonalized mixed state Hamiltonian matrix element I,II H , the total overlap . e 1 ˚ A. nuc b.f.c. separation nuclear basis set were used for all calculations. − S 0 a Additional information for the 2 2.202.252.302.35 0.69 0.76 0.82 0.88 8419.90 5732.46 52.67 19.93 6.74E-03 947.60 0.98300 2.22E-03 318.59 7.64 6.62E-03 0.97996 2.83 2.17E-03 7.61E-04 105.35 0.97707 2.57E-04 39.86 7.44E-04 0.97434 292.24 2.50E-04 15.29 40.59 5.65 15.09 5.65 (He–He) Units are in Optimum nuclear basis function center separation for the localized NEO-HF solution with the nuclear basis Units of cm R a b c function centers equally spaced from the donor and acceptor. The additional dataHamiltonian provided matrix are element the nuclearelectronic overlap basis function center (nuc b.f.c.) separation, the mixed state unorthogonalized to the NEO-NOCI proton tunnelingtunneling splitting splitting evaluated between at the the variationallyset optimized NEO-HF and NEO-NOCI minimum ground QZSPDN and and ∆(NOCI) excited refers state to energies. the NEO-NOCI The proton 6-31G electronic basis Table A.1. 76 b , the p S ∆(NOCI) b ∆(HF) , the proton overlap 0 I,II H T S e S p S b 0 I,II H b , and the corresponding NEO-NOCI proton tunneling splittings. ∆(HF) refers p S I,II ∗ 2 NEO-NOCI proton tunneling splitting calculations of the phenoxyl/phenol sys- H e × S = T a,c S , the orthogonalized mixed state Hamiltonian matrix element I,II H , the total overlap . e 1 ˚ A. nuclear basis set were used for all calculations. S − 0 nuc b.f.c. separation a Additional information for the 2 2.252.302.352.402.452.50 0.41 0.45 0.50 0.55 0.60 249724.47 0.65 51.42 80177.32 2.02E-02 19223.68 18.37 0.09249 3893.16 7.42E-03 4.91 1.86E-03 0.08063 704.34 2.04E-03 102.86 1.10 5.98E-04 104.49 0.07017 4.76E-04 0.22 36.76 1.43E-04 107.16 0.06105 0.04 9.80E-05 2.91E-05 9.83 0.05366 1.65E-05 35.33 5.25E-06 0.04714 2.20 7.80E-07 0.43 9.83 0.07 2.20 0.43 0.07 (O–O) Units are in Optimum nuclear basis function center separation for the localized NEO-HF solution with the nuclear basis Units of cm R a b c function centers equally spaced from the donor and acceptor. electronic overlap tem. The additional dataHamiltonian provided matrix are element the nuclear basis function center (nuc b.f.c.)to separation, the the NEO-NOCI mixed proton state tunneling unorthogonalized tunneling splitting splitting evaluated between at the the variationallyset optimized NEO-HF and NEO-NOCI minimum ground QZSPDN and and ∆(NOCI) excited refers state to energies. the NEO-NOCI The proton 6-31G electronic basis Table A.2. 77 , the b p S ∆(NOCI) b ∆(HF) , the proton overlap 0 I,II H T S e S p S b 0 I,II H b , and the corresponding NEO-NOCI proton tunneling splittings. ∆(HF) refers p S I,II ∗ 2 NEO-NOCI proton tunneling splitting calculations of the benzyl/toluene system. e H × S = T a,c S , the orthogonalized mixed state Hamiltonian matrix element I,II H , the total overlap . e 1 ˚ A. nuclear basis set were used for all calculations. − S 0 nuc b.f.c. separation a Additional information for the 2 2.552.602.652.702.72 0.41 0.46 0.52 0.57 0.59 440845.98 104.67 80751.45 4.82E-03 9975.34 23.46 0.77318 3.72E-03 1433.72 9.46E-04 3.49 0.72180 209.34 619.09 0.57 6.83E-04 1.26E-04 0.66745 1.95E-05 1250.13 0.26 46.93 8.43E-05 0.62222 8.67E-06 1.21E-05 0.60375 6.97 46.75 5.23E-06 1.14 0.52 7.29 1.14 0.52 (C–C) Units are in Optimum nuclear basis function center separation for the localized NEO-HF solution with the nuclear basis Units of cm c function centers equally spaced from the donor and acceptor. a b R Table A.3. electronic overlap The additional dataHamiltonian provided matrix are element the nuclear basis function centerto the (nuc NEO-NOCI b.f.c.) proton tunnelingtunneling separation, splitting splitting evaluated between the at the the mixed variationallyset optimized NEO-HF state and NEO-NOCI minimum ground QZSPDN and unorthogonalized and ∆(NOCI) excited refers state to energies. the NEO-NOCI The proton 6-31G electronic basis 78 nuclear basis set is ) 0 0 NEO-FCI 3D-grid c ) (QZSPDN 0 NEO-NOCI b ) (QZSPDN 0 -type Gaussians. The QZSPDN p system calculated with the NEO-NOCI method, the + - and s -type Gaussians. For the grid calculations, the potential was generated d , for the [He–H–He] NEO-FCI NEO-NOCI 1 − b (QZSPNB) (QZSPNB) (QZSPDN ˚ A. For the NEO-NOCI and the NEO-FCI calculations the nuclear basis set used is indicated ˚ A. 1D-grid NEO-NOCI a 2 NEO-NOCI. Proton tunneling splittings, in cm 1.862.30 1627 38 2082 37 1927 38 2550 1826 15 1830 13 1270 14 15 × (He–He) Units are in NEO-NOCI with all proton excitations (Full-NOCI for proton active space). 2 R a b c NEO-FCI method, the one-dimensionalHe–He grid-based method, distances and of the 1.86in three-dimensional and brackets. grid-based 2.30 method The are QZSPNB presented nuclear for fixed basis set includes four each of Table A.4. the same as QZSPNBat with the the full addition CIgrid. of level two with sets For 128 the of gridbasis points NEO function for calculations centers the two contain one-dimensionalthe nuclear grid both midpoint basis and electronic between function 64 and thecorresponding grid centers nuclear NEO donor points were basis level per and used of functions dimension acceptor. theory to for and (NEO-NOCI represent the This by or three-dimensional the construction separation NEO-FCI). quantum The are is nucleus. 6-31G always minimized electronic equally for The basis spaced the set nuclear from was ground used and for excited all state calculations. energies of the 79

A.3 Memory demands and timing for the NEO- NOCI method

Since only one or a few proton(s) will typically be treated quantum mechanically with the NEO-NOCI method, the memory needs are determined primarily by the size of the electronic system. Due to the nonorthogonality of the method the atomic orbital to molecular orbital transformation for the two particle integrals is the most computationally demanding component. Presently two algorithms have been implemented to transform the two electron integrals in atomic basis (ij|kl), to integrals over the molecular orbitals (pq|rs). Here i, j, k, and l are atomic orbital indices and p, q, r, and s are the molecular orbital indices. The two- step algorithm first transforms the k and l atomic orbitals to molecular orbitals r and s. The matrix of half transformed two electron integrals (ij|rs) is then fully transformed to molecular orbital integrals (pq|rs). The memory needed for a NEO-NOCI calculation using the two-step atomic orbital to molecular orbital 1 2 2 (AO to MO) transformation is 2 (NAO + NAO)NOC . Here NAO is the number of electronic atomic orbitals and NOC is the number of occupied electronic molecular orbitals. The three-step algorithim transforms the l atomic orbitals first to molecular orbitals s. This single atomic orbital index transformed matrix is held in memory till the completion of all four indice transformations. The memory needed for a NEO-NOCI calculation using the three-step AO to MO transformation is dependent on the size of this single index transformed matrix, which is of size 1 3 2 2 (NAO + NAO)NOC . Following the first atomic orbital to molecular orbital index transformation, the k index of the single atomic orbital index transformed matrix is converted to molecular orbital index r. The third step then transforms the last two atomic orbital indices i and j together to molecular orbital indices p and q. The two step algorithm is not as memory intensive as the three-step algorithm, but it is very slow due to the looping structure of the algorithm, which is a considerable drawback to its use. Use of the two-step method is only warranted if the three-step method’s memory demands are computationally impractical. See Figure A.1 for a comparison of timing and memory demands for a set of molecular systems with increasing size. Due to the nonorthogonality of the method, symme- 80

Figure A.1. The memory footprint and timing information for the NEO-NOCI method using either the two step (black) or three-step (red) AO to MO transformation. try cannot be used to reduce the number of integrals that need to be transformed. The only reduction that can be gained by symmetry is a factor of one half due to hermicity. This is not presently incorporated into either transformation and should be the ﬁrst step at further reducing the memory demands in future improvements of the method. Parallelization of calculating the NEO-NOCI matrix elements could be implemented in the case of a multi-state NEO-NOCI calculation to reduce the time to construct the multistate NEO-NOCI matrix. Appendix B

Valence Bond Models

In this Appendix the details for the valence bond models used to ﬁt the CASSCF electronically adiabatic potential energy curves along the proton coordinate for the phenoxyl/phenol and the benzyl/toluene systems are presented. In the simplest case, the PCET system can be described in terms of two valence bond states. Here the indices I and II correspond to the electron localized on the donor and acceptor, respectively. The two diabatic state energies HI (q) and ET HII (q), as well as the electronic coupling V , are used to create a 2×2 valence bond Hamiltonian matrix: " # H (q) V ET I (B.1) ET V HII(q)

The diagonal matrix elements depend on the proton coordinate q, but the oﬀ- diagonal elements are assumed to be constant. Diagonalization of this matrix leads to the electronically adiabatic potential energy surfaces UI (q) and UII (q):

1 1q U (q) = (H (q) + H (q)) − (H (q) − H (q))2 + 4 (V ET)2 I 2 I II 2 I II 1 1q U (q) = (H (q) + H (q)) + (H (q) − H (q))2 + 4 (V ET)2 (B.2) II 2 I II 2 I II

We used this two-state model to describe the phenoxyl/phenol reaction. A Morse function with an exponential wall of the form A expB(q+q0) was used to represent the diabatic state energies HI (q) and HII (q). The minima of the Morse poten- 82 tials correspond approximately to the proton bonded to the donor and acceptor, ET respectively, for HI (q) and HII (q). The electronic coupling V was fixed to be half of the energy splitting between the CASSCF ground and excited states at q = 0. Note that the proton coordinate q has its origin at the midpoint between the proton donor and acceptor. A weighting of e−25q2 was used to fit the CASSCF data points to ensure a greater weighting for the points near the midpoint. In some cases, a four-state valence bond model is required to fit the electronically adiabatic potential energy curves. In this case, the diabatic states are defined in terms of the electron transfer state and the proton transfer state. The indices 1 and 2 denote the electron transfer state, where 1 and 2 correspond to the electron localized on the donor and acceptor, respectively. The indices a and b denote the proton transfer state, where a and b correspond to the proton bonded to its donor and acceptor, respectively. In our calculations, the four-state valence bond

Hamiltonian matrix has diagonal elements H1a(q), H1b(q), H2a(q), and H2b(q) and coupling V PT between states 1a and 1b and states 2a and 2b, as well as couplings between the other pairs of diabatic states. This 4×4 matrix can be block diagonalized for each electron transfer state to create a reduced two-state valence bond Hamiltonian matrix: " # H (q) V ET I (B.3) ET V HII (q) where

1 1q H (q) = (H (q) + H (q)) − (H (q) − H (q))2 + 4 (V PT)2 I 2 1a 1b 2 1a 1b 1 1q H (q) = (H (q) + H (q)) − (H (q) − H (q))2 + 4 (V PT)2 (B.4) II 2 2a 2b 2 2a 2b

Diagonalization of the reduced two-state valence bond Hamiltonian matrix given in Eq. B.3 leads to the electronically adiabatic potential energy surfaces

UI (q) and UII (q) given in Eq. B.2. We used this two-state model to describe the benzyl/toluene reaction. A Morse function with an exponential wall of the

B(q+q0) form A exp was used to represent the diabatic state energies H1a(q), H1b(q),

H2a(q), and H2b(q). The minima of the Morse potentials correspond approximately to the proton bonded to the donor and acceptor, respectively, for the diabatic states 83 with a and b subscripts. The electronic coupling was ﬁxed to be half of the energy splitting between the CASSCF ground and excited states at q = 0. A weighting of e−5q2 was used to ﬁt the CASSCF data points to ensure a greater weighting for the points near the midpoint. Appendix C

Substituted Phenoxyl-Phenol Optimized Geometries and Vibronic Couplings

In this Appendix, we provide the UDFT(B3LYP)/6-31G* transition state optimized geometries of the substituted phenoxyl/phenol systems and the one- dimensional semiclassical nonadiabatic vibronic couplings for the transition state structures of all systems.

C.1 Transition state optimized geometries 85

Table C.1. p-F phenoxyl/phenol transition state structure.

Energy: -812.76209825 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.401591 C 1.189910 0.000000 2.940369 C 2.374476 0.257092 2.183823 C 3.617699 0.233796 2.792521 C 3.699380 -0.027473 4.162688 C 2.564448 -0.271594 4.942744 C 1.321112 -0.259848 4.337180 H 2.278772 0.468599 1.123820 H 4.528100 0.425385 2.233715 H 2.681658 -0.468912 6.003427 H 0.416163 -0.454699 4.903212 C -0.885217 0.795156 -0.538783 C -1.594685 1.777982 0.217763 C -2.535125 2.591438 -0.390940 C -2.770448 2.451678 -1.761116 C -2.089236 1.511672 -2.541174 C -1.156431 0.689545 -1.935603 H -1.382169 1.871360 1.277772 H -3.084398 3.342327 0.167868 H -2.308263 1.443223 -3.601863 H -0.613393 -0.06012 -2.501637 H 0.005796 0.015132 1.200795 F 4.908206 -0.036869 4.753211 F -3.676010 3.252490 -2.351645 86

Table C.2. m-F phenoxyl/phenol transition state structure.

Energy: -812.76084072 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.399352 C 1.176816 0.000000 2.958305 C 2.377798 0.158459 2.204767 C 3.584410 0.131290 2.869328 C 3.686302 -0.030957 4.257060 C 2.505332 -0.176673 4.999221 C 1.269382 -0.163410 4.375350 H 2.566907 -0.302991 6.076198 H 0.345467 -0.282181 4.931067 C -0.994613 0.628996 -0.558960 C -1.924961 1.404834 0.194570 C -2.959277 2.026794 -0.470000 C -3.132104 1.944124 -1.857732 C -2.211857 1.189750 -2.599886 C -1.160178 0.540357 -1.976006 H -2.331408 1.115896 -3.676862 H -0.442787 -0.05384 -2.531718 H 0.003988 0.013776 1.199676 H 4.664742 -0.033950 4.724461 H -3.960653 2.464558 -2.325139 H 2.337807 0.296736 1.130135 H -1.817261 1.500332 1.269202 F 4.723309 0.278020 2.163816 F -3.843424 2.759539 0.235505 87

Table C.3. o-F phenoxyl/phenol transition state structure.

Energy: -812.75540742 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.391232 C 1.056769 0.000000 3.140637 C 2.384444 -0.173006 2.634583 C 3.501075 -0.175305 3.448952 C 3.340857 -0.005591 4.827180 C 2.052779 0.157886 5.370732 C 0.937881 0.155156 4.553235 H 4.478927 -0.316129 2.999672 H 1.934707 0.286445 6.442850 H -0.064276 0.280965 4.950244 C 0.701101 0.790712 -0.749401 C 1.452448 1.898920 -0.243341 C 2.191550 2.735947 -1.057700 C 2.212280 2.503455 -2.435926 C 1.480070 1.431200 -2.979484 C 0.738355 0.598807 -2.161996 H 2.734899 3.561049 -0.608416 H 1.497957 1.257553 -4.051600 H 0.167643 -0.234519 -2.559010 H 0.101297 0.045570 1.195616 H 4.212430 -0.004727 5.474287 H 2.791167 3.155022 -3.083024 F 2.532543 -0.367957 1.310350 F 1.404794 2.139082 1.080888 88

Table C.4. p-CN phenoxyl/phenol transition state structure.

Energy: -798.78276919 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.399922 C 1.164407 0.000000 2.974243 C 2.385564 0.119194 2.237091 C 3.598683 0.101608 2.891040 C 3.652894 -0.023661 4.299864 C 2.452054 -0.133759 5.041715 C 1.234902 -0.122368 4.396853 H 2.335014 0.225503 1.158176 H 4.524536 0.190836 2.331747 H 2.502932 -0.229172 6.121593 H 0.302240 -0.211640 4.943804 C -1.025410 0.551784 -0.574244 C -2.044220 1.235405 0.162991 C -3.120859 1.794801 -0.490873 C -3.228054 1.710209 -1.899692 C -2.222824 1.044227 -2.641624 C -1.145570 0.477463 -1.996847 H -1.949255 1.305045 1.241900 H -3.893841 2.312097 0.068484 H -2.312912 0.984343 -3.721497 H -0.366614 -0.043099 -2.543862 H 0.003433 0.013522 1.199960 C 4.911304 -0.032804 4.969068 N 5.940367 -0.040394 5.516164 C -4.340590 2.298482 -2.568803 N -5.250426 2.779432 -3.115819 89

Table C.5. m-CN phenoxyl/phenol transition state structure.

Energy: Energy: -798.77621530 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.397153 C 1.173690 0.000000 2.963409 C 2.375115 0.209996 2.225199 C 3.604479 0.185504 2.879762 C 3.675837 -0.030996 4.273623 C 2.496502 -0.226021 5.003196 C 1.264905 -0.213671 4.371130 H 2.317959 0.387975 1.156791 H 4.642075 -0.037903 4.766066 H 2.553511 -0.391038 6.074998 H 0.340721 -0.370165 4.917797 C -1.002146 0.610944 -0.566257 C -1.918665 1.415625 0.171953 C -2.981096 2.034637 -0.482610 C -3.154716 1.886928 -1.876472 C -2.249264 1.106528 -2.606045 C -1.191249 0.475986 -1.973978 H -1.777222 1.537836 1.240362 H -3.983324 2.383990 -2.368916 H -2.383836 0.995307 -3.677848 H -0.483600 -0.138702 -2.520645 H 0.006423 0.022877 1.198577 C 4.814587 0.394734 2.135926 N 5.798953 0.563134 1.540068 C -3.905427 2.843185 0.261225 N -4.658263 3.499366 0.857083 90

Table C.6. o-CN phenoxyl/phenol transition state structure.

Energy: -798.77986739 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.393073 C 1.152654 0.000000 2.984997 C 2.390437 0.000077 2.273583 C 3.590942 0.000073 2.957325 C 3.620437 -0.000006 4.364895 C 2.426849 -0.000083 5.088832 C 1.197693 -0.000081 4.424638 H 4.523725 0.000132 2.400679 H 2.441430 -0.000145 6.174007 C -1.152650 0.000247 -0.591932 C -2.390437 0.000586 0.119472 C -3.590937 0.000840 -0.564278 C -3.620422 0.000770 -1.971847 C -2.426830 0.000441 -2.695776 C -1.197679 0.000179 -2.031573 H -4.523725 0.001096 -0.007637 H -2.441402 0.000385 -3.780952 H -0.000003 0.000010 1.196536 H 4.570175 -0.000009 4.890093 H -4.570156 0.000971 -2.497052 H -2.359519 0.000638 1.204616 H 2.359511 0.000138 1.188440 C 0.026549 -0.000158 -2.765092 N 1.010621 -0.000426 -3.387194 C -0.026529 -0.000159 5.158165 N -1.010596 -0.000219 5.780274 91

Table C.7. p-NO2 phenoxyl/phenol transition state structure.

Energy: -1023.29636182 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.398176 C 1.160165 0.000000 2.979715 C 2.385682 0.129144 2.248926 C 3.595560 0.113722 2.910783 C 3.612538 -0.020372 4.307724 C 2.432434 -0.141940 5.057757 C 1.219052 -0.131937 4.403375 H 2.341733 0.241583 1.170442 H 4.534797 0.209018 2.379909 H 2.499491 -0.243154 6.133954 H 0.282733 -0.228888 4.942378 C -1.057370 0.477441 -0.581541 C -2.121158 1.099474 0.149246 C -3.230181 1.583320 -0.512613 C -3.300832 1.468101 -1.909555 C -2.275314 0.871660 -2.659586 C -1.165329 0.381433 -2.005202 H -2.034834 1.183860 1.227731 H -4.046983 2.056695 0.018260 H -2.378077 0.807016 -3.735784 H -0.351866 -0.092247 -2.544204 H 0.004170 0.019365 1.199088 N 4.899683 -0.029689 5.003349 O 4.885636 -0.152224 6.229467 O 5.919232 0.084739 4.320480 N -4.477764 1.989307 -2.605182 O -4.515384 1.871848 -3.831302 O -5.359889 2.513170 -1.922315 92

Table C.8. m-NO2 phenoxyl/phenol transition state structure.

Energy: -1023.29190382 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.397350 C 1.176766 0.000000 2.955563 C 2.371973 0.223638 2.208887 C 3.584741 0.190417 2.871279 C 3.693969 -0.035892 4.247777 C 2.518175 -0.241122 4.983518 C 1.281791 -0.226436 4.360965 H 2.329203 0.415167 1.143715 H 4.671113 -0.041198 4.713164 H 0.362779 -0.390392 4.913965 C -0.953307 0.689914 -0.558214 C -1.790442 1.571811 0.188460 C -2.792391 2.255919 -0.473933 C -3.013555 2.136623 -1.850433 C -2.181355 1.281021 -2.586172 C -1.171140 0.568050 -1.963617 H -1.643505 1.701894 1.253633 H -3.808259 2.705205 -2.315820 H -0.522764 -0.103570 -2.516616 H 0.007422 0.022918 1.198675 H -2.337288 1.178336 -3.655713 H 2.584299 -0.415730 6.053059 N -3.659406 3.165937 0.303460 O -3.437054 3.274758 1.506967 O -4.550420 3.756955 -0.304608 O 5.888962 0.375723 2.701951 N 4.820641 0.419318 2.093884 O 4.704309 0.637839 0.890377 93

Table C.9. o-NO2 phenoxyl/phenol transition state structure.

Energy: -1023.27646695 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.392641 C 1.155639 0.000000 2.971436 C 2.376477 -0.028981 2.221190 C 3.603046 -0.099321 2.852706 C 3.684614 -0.147761 4.254976 C 2.515119 -0.117882 5.021136 C 1.278647 -0.023747 4.402662 H 2.302609 0.001146 1.138356 H 4.511318 -0.119038 2.257539 H 4.649679 -0.202753 4.748030 H 2.549906 -0.163247 6.103352 C -1.040920 0.501995 -0.578785 C -2.153143 1.006206 0.171472 C -3.288508 1.475655 -0.460033 C -3.383035 1.467456 -1.862301 C -2.316666 0.986357 -2.628472 C -1.162048 0.534039 -2.010009 H -2.073511 1.001256 1.254305 H -4.115171 1.852437 0.135143 H -4.276186 1.837135 -2.355346 H -2.367718 0.960607 -3.710689 H -0.002345 -0.010274 1.196320 N 0.095777 0.054704 5.265564 O 0.168334 -0.502999 6.363728 O -0.873813 0.690496 4.862461 N -0.062536 0.090878 -2.872922 O -0.370159 -0.379944 -3.971084 O 1.086984 0.242378 -2.469832 94

Table C.10. p-OH phenoxyl/phenol transition state structure.

Energy: -764.73256970 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.403922 C 1.196249 0.000000 2.932086 C 2.375582 0.240700 2.166559 C 3.623659 0.215004 2.765372 C 3.744999 -0.032528 4.143208 C 2.593995 -0.259170 4.921304 C 1.347384 -0.244760 4.328929 H 2.274537 0.439810 1.104522 H 4.516607 0.398913 2.169706 H 2.713452 -0.447500 5.983754 H 0.448699 -0.427653 4.909392 C -0.857649 0.833937 -0.528164 C -1.535372 1.828650 0.237362 C -2.448092 2.680295 -0.361451 C -2.707648 2.587416 -1.739287 C -2.040435 1.622530 -2.517383 C -1.136633 0.763816 -1.925007 H -1.324122 1.900960 1.299399 H -2.960082 3.434645 0.234215 H -2.257369 1.570784 -3.579832 H -0.619822 0.006195 -2.505471 H 0.003553 0.008750 1.201961 O -3.586972 3.407848 -2.380185 H -3.984765 4.014622 -1.735677 O 4.947374 -0.057321 4.784107 H 5.655569 0.100393 4.139599 95

Table C.11. m-OH phenoxyl/phenol transition state structure.

Energy: -764.72700966 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.403468 C 1.186653 0.000000 2.945728 C 2.370659 0.240148 2.173286 C 3.609219 0.211800 2.791136 C 3.723450 -0.028174 4.165959 C 2.560384 -0.250812 4.934897 C 1.309870 -0.239158 4.339913 H 0.416994 -0.422314 4.926696 C -0.842666 0.835500 -0.542261 C -1.514367 1.839670 0.230180 C -2.413852 2.691586 -0.387671 C -2.663931 2.601603 -1.762495 C -1.994769 1.624610 -2.531432 C -1.098550 0.752424 -1.936446 H -0.593457 -0.006296 -2.523229 H 0.006757 0.016412 1.201733 H 4.700751 -0.036677 4.644065 H -3.363919 3.283663 -2.240601 H 2.270052 0.440600 1.112113 H -1.301791 1.911180 1.291353 H 4.508430 0.388943 2.207624 H -2.927677 3.450497 0.195840 O 2.624586 -0.487267 6.278299 H 3.552883 -0.481445 6.558553 O -2.206842 1.501903 -3.874834 H -2.861941 2.159637 -4.155090 96

Table C.12. o-OH phenoxyl/phenol transition state structure.

Energy: -764.72350371 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.401562 C 1.183994 0.000000 2.942866 C 2.388938 0.163272 2.204828 C 3.622021 0.139626 2.834385 C 3.706620 -0.036059 4.226217 C 2.544159 -0.189172 4.983654 C 1.294714 -0.173782 4.367853 H 4.528871 0.265705 2.249840 H 2.606130 -0.324241 6.062472 C -0.913495 0.753240 -0.541302 C -1.739278 1.645780 0.196740 C -2.705691 2.412005 -0.432816 C -2.882737 2.330276 -1.824648 C -2.083268 1.472601 -2.582087 C -1.109482 0.689598 -1.966288 H -3.325147 3.086205 0.151730 H -2.217014 1.407815 -3.660904 H 0.002966 0.008256 1.200781 H 4.674129 -0.046846 4.719144 H -3.636067 2.937468 -2.317573 H 2.307024 0.305809 1.131769 H -1.585395 1.703640 1.269797 O 0.127952 -0.321320 5.044537 H 0.327975 -0.439141 5.987221 O -0.303148 -0.166513 -2.642973 H -0.532431 -0.130167 -3.585656 97

Table C.13. p-NH2 phenoxyl/phenol transition state structure.

Energy: -725.01159529 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.407425 C 1.199644 0.000000 2.929877 C 2.381214 0.207734 2.158048 C 3.630043 0.182403 2.749424 C 3.774514 -0.037046 4.136735 C 2.609924 -0.228747 4.913947 C 1.359090 -0.212886 4.329132 H 2.278696 0.379246 1.091230 H 4.518851 0.346344 2.143256 H 2.707437 -0.391974 5.985474 H 0.463415 -0.373026 4.921335 C -0.768247 0.921159 -0.522829 C -1.365947 1.961627 0.248638 C -2.185024 2.904371 -0.343221 C -2.445442 2.874600 -1.730650 C -1.846363 1.857399 -2.507491 C -1.033269 0.907059 -1.922199 H -1.169092 1.992857 1.315551 H -2.628692 3.692019 0.262639 H -2.033710 1.827638 -3.579095 H -0.582297 0.116571 -2.514089 H 0.001059 0.002481 1.203729 N 5.028000 -0.002259 4.734210 H 5.109128 -0.457386 5.634264 H 5.818452 -0.186419 4.129912 N -3.221311 3.859437 -2.328598 H -3.622315 3.630147 -3.228846 H -3.869349 4.348550 -1.724677 98

Table C.14. m-NH2 phenoxyl/phenol transition state structure.

Energy: -725.00577696 a.u. atom x y z O 0.000000 0.000000 0.000000 O 0.000000 0.000000 2.406497 C 1.186790 0.000000 2.949924 C 2.378946 0.073295 2.186823 C 3.622670 0.050089 2.810771 C 3.697041 -0.039534 4.227401 C 2.527147 -0.108032 4.986291 C 1.282401 -0.088898 4.376196 H 2.301393 0.149071 1.105947 H 4.669718 -0.036511 4.712730 H 2.599258 -0.177866 6.068473 H 0.360570 -0.149311 4.944851 C -0.925296 0.743163 -0.543433 C -1.808887 1.546831 0.219662 C -2.793102 2.307555 -0.404292 C -2.907200 2.284250 -1.820924 C -2.037960 1.498261 -2.579807 C -1.055499 0.733723 -1.969705 H -1.700979 1.557347 1.300540 H -3.663664 2.895694 -2.306258 H -2.137905 1.488970 -3.661990 H -0.374605 0.109374 -2.538356 H -0.000414 -0.001171 1.203248 N 4.798237 0.175295 2.078545 H 5.629293 -0.215044 2.503398 H 4.727390 -0.044061 1.093225 N -3.631252 3.141310 0.327927 H -4.523623 3.357384 -0.096931 H -3.713380 2.925925 1.313247 99

Table C.15. o-NH2 phenoxyl/phenol transition state structure.

Energy: -725.02130866 a.u. atom x y z O 0.00000 0.00000 0.00000 O 0.00000 0.00000 2.40484 C 1.19892 0.00000 2.93942 C 2.39941 0.18653 2.21584 C 3.62586 0.14489 2.86171 C 3.69487 -0.07693 4.25078 C 2.53362 -0.25271 4.99388 C 1.28027 -0.21692 4.36412 H 2.33015 0.35659 1.14595 H 4.53987 0.29068 2.29280 H 4.65997 -0.10417 4.74855 H 2.58473 -0.41764 6.06783 C -0.74962 0.93568 -0.53458 C -1.35464 1.98919 0.18900 C -2.15398 2.92032 -0.45687 C -2.37024 2.83548 -1.84594 C -1.78136 1.81930 -2.58904 C -0.96977 0.86354 -1.95928 H -1.17862 2.04147 1.25889 H -2.61168 3.72479 0.11204 H -2.99493 3.57164 -2.34371 H -1.94203 1.75607 -3.66299 H 0.00273 0.00568 1.20242 N 0.09305 -0.35061 5.03613 H 0.08785 -0.79415 5.94294 H -0.71990 -0.49328 4.44982 N -0.33181 -0.14660 -2.63129 H 0.06514 -0.87025 -2.04499 H -0.67472 -0.42798 -3.53810 100

C.2 Vibronic couplings of transition state optimized substituted phenoxyl/phenol systems

Table C.16. Vibronic couplings and associated quantities for the optimized transition structures of the substituted phenoxyl/phenol systems.

a ET b c (na) d Substituent Barrier V Overlap VDA - 18.4 700 6.38×10−3 4.47

p-F 18.6 731 7.04×10−3 5.15 m-F 18.8 545 7.33×10−3 3.99 o-F 18.5 934 6.61×10−3 6.17

p-CN 17.4 459 9.66×10−3 4.43 m-CN 19.3 651 6.96×10−3 4.53 o-CN 16.8 430 1.21×10−2 5.19

−3 p-NO2 18.9 532 7.95×10 4.29 −3 m-NO2 19.8 699 6.55×10 4.57 −2 o-NO2 17.6 373 1.07×10 3.99

p-OH 16.7 662 1.04×10−2 6.87 m-OH 18.8 647 6.71×10−3 4.34 o-OH 18.0 570 8.44×10−3 4.81

−2 p-NH2 14.1 567 1.73×10 9.79 −2 m-NH2 12.5 344 3.09×10 10.6 −2 o-NH2 12.2 452 3.16×10 14.3 a barrier in kcal/mol for the CASSCF ground electronic state. b electronic coupling, which is half the splitting in cm−1 between the ground and excited CASSCF electronic states. c Franck-Condon overlap between the reactant and product proton vibrational wavefunctions for the diabatic electronic states. d one-dimensional grid-based nonadiabatic vibronic coupling in cm−1. Appendix D

Multi-Dimensional Malonaldehyde Tunneling Splitting Calculations

In this Appendix, results for an approximate treatment of the full multi-dimensional tunneling splitting calculation of malonaldehyde are presented. The full multi- dimensional tunneling splitting is approximated as a Franck-Condon overlap of the reactant and product vibrational wavefunctions corresponding to “classical” nuclei times the uncoupled proton tunneling splitting calculated at the reactant- product averaged geometry. “Classical” here refers to the nuclei not included in the electron-proton subsystem, although these nuclei are actually treated quantum mechanically as uncoupled harmonic oscillators. We will drop the quotations from “classical” throughout the rest of this Appendix. This approximate multi- dimensional tunneling splitting is expressed as:

" # Y I II 0 ∆ = ϕi (Qi) |ϕi (Qi) 2HI,II (D.1) i Q I II where ϕi (Qi) |ϕi (Qi) is the Franck-Condon overlap of the reactant and i product ground state harmonic oscillator wavefunctions for the classical nuclei I II 0 ϕi (Qi) and ϕi (Qi), repsectively. HI,II is the electronic-proton Hamiltonian matrix element between the orthogonalized proton vibrational-electronic reactant and I II product diabatic wavefunctions ψ (re, rp; Q) and ψ (re, rp; Q), respectively. To 102 arrive at the approximate expression Eq. D.1 for the multi-dimensional tunneling splitting several approximations are made. We ﬁrst begin by writing the total Hamiltonian matrix element between the reactant and product diabatic wavefunctions.

hΨR(re, rp, Q)| H |ΨP (re, rp, Q)i (D.2)

The total Hamiltonian is written as:

H = TN (Q) + Hep(re, rp, Q) (D.3)

where TN (Q) is the kinetic energy of classical nuclear motion and Hep is the Hamil- tonian for the electronic-proton subsystem, which is deﬁned as:

Hep = Tp + V (D.4)

where Tp is the kinetic energy of the tunneling proton and V is taken as the potential. At this point we can make one further assumption that the electrons in the electronic-proton subsystem are adiabatic with respect to the motion of the proton. This approximation will only aﬀect how Hep is evaluated so that the proceeding expressions are correct whether this approximation is made or not. In the NEO method this adiabatic separation is not made, but for the grid based method, that we use here to calculate the proton vibrational-electronic states, this separation is made. The form of the reactant ΨR and product ΨP wavefunctions are written as:

I Y I I ΨR = Ψ (re, rp, Q) = ϕi(Qi)ψ (re, rp; Q) (D.5) i II Y II II ΨP = Ψ (re, rp, Q) = ϕi (Qi)ψ (re, rp; Q) (D.6) i

There are several assumptions embedded in these expressions for the reactant and product diabatic wavefunctions. The classical nuclear motion is assumed to be separable from the motion of the electrons and the tunneling proton. The classical modes are assumed to be uncoupled and harmonic so that they can be written as 103 a product. Additionally it is assumed that there will be very little contribution from excited state classical nuclear wavefunctions so that only the ground state is included for each Qi normal mode. With the approximations made to the reactant and product wavefunctions we now proceed to calculate the matrix element in Eq. D.2

Y I I Y II II = ϕi(Qi)ψ (re, rp; Q) TN (Q) ϕi (Qi)ψ (re, rp; Q) i i * +

Y I Y II + ϕi(Qi) HI,II(Q) ϕi (Qi) (D.9) i i

I II where HI,II(Q) = ψ (re, rp; Q) |Hep(re, rp, Q)| ψ (re, rp; Q) is the mixed electronic-proton matrix element between reactant and product proton vibrational- electronic wavefunctions. If we neglect all nonadiabatic terms for the proton vibrational-electronic wavefunctions with respect to the Q normal mode coordinates we can further write the expression as:

* + * +

Y I Y II Y I Y II = ϕi(Qi) TN (Q)SI,II(Q) ϕi (Qi) + ϕi(Qi) HI,II(Q) ϕi (Qi) i i i i (D.10)

I II where SI,II(Q) = ψ (re, rp; Q)|ψ (re, rp; Q) is the overlap of the reactant and product proton vibrational-electronic wavefunctions. If the reactant and product proton vibrational-electronic wavefunctions are orthogonalized then SI,II = 0 and the kinetic energy of classical nuclei term in Eq. D.10 vanishes leaving the 104 expression:

* +

Y I 0 Y II = ϕi(Qi) HI,II(Q) ϕi (Qi) (D.11) i i

0 where HI,II is the mixed electronic-proton matrix element between orthogonalized reactant and product proton vibrational-electronic wavefunctions. If we make one 0 ﬁnal assumption that HI,II does not depend on the Q normal mode coordinates we can remove it from the integral over Qi normal modes. However, it is known 0 0 that HI,II depends strongly on the O–O distance, so we would expect HI,II to have dependence on the normal modes involving the O–O stretch. Therefore, it may be necessary to include the normal modes involving the O–O stretch in the integral. 0 To include this Q dependence for select normal modes we would calculate HI,II for a series of O–O distances (on a grid), and then numerically integrate over these 0 modes. Assuming here that HI,II has no Q dependence, we arrive at our ﬁnal expression for the total Hamiltonian matrix element between the reactant and product diabatic wavefunctions.

Y I II 0 hΨR(re, rp, Q)| H |ΨP (re, rp, Q)i = ϕi(Qi)|ϕi (Qi) HI,II (D.12) i

The multi-dimensional tunneling splitting is related to this matrix element through 0 0 HI,II. If the system is symmetric then 2HI,II is equal to the uncoupled proton tunneling splitting ∆ep and the full multi-dimensional tunneling splitting is expressed as:

Y I II 0 Y I II ∆ = ϕi(Qi)|ϕi (Qi) 2HI,II = ϕi(Qi)|ϕi (Qi) ∆ep (D.13) i i

We can rewrite Eq. D.13 in abbreviated form as:

∆ = (F.C. Overlap) ∗ ∆ep (D.14) where (F.C. Overlap) denotes the Franck-Condon overlap of the reactant and product harmonic oscillator wavefunctions for classical nuclei normal modes and 0 ∆ep = 2HI,II. We will calculate the quantities in Eq. D.14 to obtain the approxi- 105 mate multi-dimensional tunneling splitting.

The ﬁxed geometry of classical nuclei used in the calculation of ∆ep is the average of the reactant and product optimized structures since this geometry best describes the average classical nuclear environment that the tunneling proton feels. The average reactant-product geometry was calculated in two separate ways, which both yielded the same geometry. The ﬁrst approach took the average of the reactant and product internal coordinates (bond lengths and angles), whereas the second approach superimposed the two structures on each other to average the cartesian coordinates for each atom. In the second approach the central carbon- hydrogen bond was used to align the two structures. The reactant-product averaged geometry at the DFT(B3LYP)/6-31G** level of theory is presented in Ta- ble D.1 and the reactant-product averaged geometry at the MP2/6-31G** level of theory is presented in Table D.2. We used a Fourier 3D grid method to calculate the ∆ep tunneling splitting.

Table D.1. Malonaldehyde DFT(B3LYP)/6-31G** reactant-product averaged geometry. The geometry is the average of the reactant and product optimized structure’s internal coordinates. The tunneling hydrogen atom that is treated quantum mechanically with the 3D grid method is marked with an asterik. The position of this hydrogen atom is not used in the grid calculation, but gives some idea of where to build the hydrogen grid box.

O–O distance: 2.5525 A˚ atom x y z O 0.000000 -1.994029 -1.276292 O 0.000000 -1.994029 1.276292 H* 0.000000 -2.232450 0.000000 C 0.000000 -0.713163 -1.208213 C 0.000000 -0.713163 1.208213 C 0.000000 0.000000 0.000000 H 0.000000 -0.177494 -2.164733 H 0.000000 -0.177494 2.164733 H 0.000000 1.083179 0.000000

To calculate the Franck-Condon overlap we assume the normal mode frequencies at the transition state optimized structure are the same as those at the reactant and product structures so that we can use the normal mode frequencies caclulated 106

Table D.2. Malonaldehyde MP2/6-31G** reactant-product averaged geometry. The geometry is the average of the reactant and product optimized structure’s internal coordinates. The tunneling hydrogen atom that is treated quantum mechanically with the 3D grid method is marked with an asterik. The position of this hydrogen atom is not used in the grid calculation, but gives some idea of where to build the grid box.

O–O distance: 2.5913 A˚ atom x y z O 0.000000 -1.992396 -1.295654 O 0.000000 -1.992396 1.295654 H* 0.000000 -2.345003 0.000000 C 0.000000 -0.705999 -1.210902 C 0.000000 -0.705999 1.210902 C 0.000000 0.000000 0.000000 H 0.000000 -0.165633 -2.157936 H 0.000000 -0.165633 2.157936 H 0.000000 1.078173 0.000000

for the transition state geometry for both the reactant and product structures. We are then only left with determining the distance separating the reactant and product structures for each normal mode. Since we can write the reactant and the product internal coordinates, IR and IP respectively, in terms of the transition state internal coordinates plus a shift, the separation between reactant and product normal modes can be written in terms of the transistion state internal coordinates.

IR = ITS + ∆IR (D.15) IP = ITS + ∆IP (D.16) where ∆IR and ∆IP are the reactant and product internal coordinate shifts. The distance separating the reactant and product normal modes ∆N RP can be deﬁned in terms of these shifts as:

∆N RP = Cˆ∆IR + Cˆ∆IP (D.17) where Cˆ is an internal coordinates to normal modes transformation matrix. For 107

each normal mode Qi the reactant and product overlap is simply the overlap integral of two transition state harmonic oscillator wavefunctions for Qi separated RP by ∆N . The product of all Qi normal mode reactant-product overlaps is the Franck-Condon overlap. The normal mode corresponding to the proton vibrational motion (the imaginary mode) is excluded from the product since this mode mostly corresponds to the tunneling proton which we included in the Hep Hamiltonian and separately treated with ψ(re, rp; Q).

Table D.3. Results for the approximate multi-dimensional tunneling splitting calculation of malonaldehyde at the DFT(B3LYP)/6-31G** and the MP2/6-31G** levels of theory. ∆ep here is the 3D grid proton tunneling splitting calculated at the reactant- product averaged geometry. The full multi-dimensional tunneling splitting (Full ∆) is the product of the grid proton tunneling splitting and the Franck-Condon Overlap of reactant and product normal modes (F.C. Overlap) as expressed in Eq. D.14. 32 points per dimension were used for the 3D grid tunneling splitting calculation. ∆exp is the experimental proton tunneling splitting for malonaldehyde [15]. All splittings are in units of cm−1.

DFT(B3LYP) MP2 Experiment ∆ep F.C. Overlap Full ∆ ∆ep F.C. Overlap Full ∆ ∆exp 144.1 0.19580 28.2 36.3 0.166563 6.0 21.6

The results for the approximate full multi-dimensional tunneling splitting calculation for malonaldehyde are presented in Table D.3 along with the experimental tunneling splitting. The approximate multi-dimensional proton tunneling splitting results at the DFT(B3LYP) level of theory are in good agreement with the experimental tunneling splitting, but this is likely just fortuitous. Bibliography

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Education Pennsylvania State University State College, Pennsylvania 2002–2008 Ph.D. in Chemistry, expected May 2008 Arcadia University Glenside, PA 1998–2002 B.S. in Chemistry; minor in Mathematics Publications

1. M. K. Ludlow, J. H. Skone, and S. Hammes-Schiﬀer, “Substituent Eﬀects on the Vibronic Coupling for the Phenoxyl/Phenol Self-Exchange Reaction”, J. Phys. Chem. B 112, 336 (2008).

2. S. Hammes-Schiﬀer, E. Hatcher, H. Ishikita, J. H. Skone, and A. V. Souda- ckov, “Theoretical Studies ofProton-Coupled Electron Transfer: Models and Concepts Relevant to Photosystem II”, Coordination Chemistry Reviews 252, 384 (2008).

3. J. H. Skone, A. V. Soudackov, and S. Hammes-Schiﬀer, “Calculation of Vi- bronic Couplings for Phenoxyl/Phenol and Benzyl/Toluene Self-Exchange Reactions: Analysis of Proton-Coupled Electron Transfer Mechanisms”, J. Am. Chem. Soc. 128, 16655 (2006).

4. J. H. Skone, M. V. Pak, and S. Hammes-Schiﬀer, “Nuclear-electronic orbital nonorthogonal conﬁguration interaction approach”, J. Chem. Phys. 123, 134108 (2005).

5. J. H. Skone and E. Curotto, “A structurally driven spectral transform Lanc- zos method for mixed quantum-classical canonical simulations (MQCC): The thermodynamics of Kr10–H and the energies of Krn–H, (n: 1 → 9).” J. Mol. Struc. (THEOCHEM) 630, 151 (2003).

6. J. H. Skone and E. Curotto, Canonical parallel tempering simulations of Arn- HF clusters (n = 1-12): Thermodynamic properties and the red shift as a function of temperature, J. Chem. Phys. 117, 7137 (2002).

7. J. H. Skone and E. Curotto, Two Krylov space algorithms for repeated large scale sparse matrix diagonalization, J. Chem. Phys. 116, 3210 (2002).