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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 follow- ing: 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 tunnel- ing 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 vi- bronic couplings. The NEO approach is a molecular orbital based method that avoids the Born-Oppenheimer separation of the select protons and electrons, there- by making methods developed within this scheme, such as NEO-NOCI, applica- ble to electronically nonadiabatic proton transfer. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wavefunc- tions 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 configura- tions, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between these two delocalized vi- bronic 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 calcu- lations. 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 dis- tance 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 sol- vent 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 ex- ponentially with the proton donor-acceptor distance for the range of distances studied. In addition to providing insights into the fundamental physical differ- ences 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 mecha- nism, 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 co- uplings . 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 phe- noxyl/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 de- picted 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 sys- tem 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 combina- tions 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 hy- drogen 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 repre- sent 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 adi- abatic 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 elec- tronic wavefunction is predominantly single configurational, and the lower molecular orbital is doubly occupied, while the upper molec- ular 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 lo- calized diabatic electron transfer states I and II and the correspond- (I) (II) ing proton vibrational wavefunctions 'D (blue) and 'A (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 'D j'A . (b) Electronically adiabatic
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