Reviews in Computational Chemistry Volume 18

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148 Charge-Transfer Reactions in Condensed Phases reviews, but rather to highlight some very recent developments in the field that have not been reviewed. This chapter will provide the reader with a step-bystep statistical mechanical buildup of the theoretical machinery currently employed in ET research. By virtue of the ‘‘frontier’’ nature of this material, many traditional subjects of ET studies are not covered here. The reader will be referred to previous reviews whenever possible, but many excellent contributions are not directly cited. This chapter concerns the energetics of charge-transfer (CT) reactions. We will not discuss subjects dealing with nuclear dynamical effects on CT kinetics. 2–4 The more specialized topic of employing the liquid-state theories to calculate the solvation component of the reorganization parameters 5 is not considered here. We concentrate instead on the general procedure of the statistical mechanical analysis of the activation barrier to CT, as well as on its connection to optical spectroscopy. Since the very beginning of ET research, 6 steady-state optical spectroscopy has been the major source of reliable information about the activation barrier and preexponential factor for the ET rate. The main focus in this chapter is therefore on the connection between the statistical analysis of the reaction activation barrier to the steady-state optical band shape. The ET reaction is usually referred to as a process of underbarrier tunneling and subsequent localization of an electron from the potential well of the donor to the potential well of the acceptor (Figure 1). This phenomenon occurs in a broad variety of systems and reactions (see Ref. 1 for a list of E HOMO D A ∆E LUMO Figure 1 Potential energy wells for the electron localized on the donor (D) and acceptor (A) sites. The parameter h Ei indicates the average energy gap for an instantaneous (Franck–Condon) transfer of the electron from the donor HOMO to the acceptor LUMO. The dotted lines show the electronic energies on the donor and acceptor at a nonequilibrium nuclear configuration with a nonequilibrium energy gap E. The upper dashed horizontal line indicates the bottom of the conduction band of the electrons in the solvent.
Introduction 149 applications). For electron tunneling to occur, the electronic states of the donor and acceptor sites must come into resonance (degeneracy) with each other. Degeneracy occurs as a result of thermal nuclear motions of the donor–acceptor complex and the condensed-phase medium. The condition of zero energy gap, E ¼ 0, between the donor and acceptor electronic levels determines the position of the transition state for an ET reaction. The ET rate constant is proportional to the probability of such a configuration kET / FCWDð0Þ ½1Š where the Franck–Condon weighted density (FCWD), FCWD( E), determines the probability of creating a configuration with energy gap E. Electron transfer refers to the situation when essentially all the electronic density is transferred from the donor to the acceptor. The process of CT, in the present context, refers to basically the same event, but the electron density is not completely relocalized and is distributed between the two potential wells. The key factor discriminating between ET and CT reactions is the ET matrix element, 7 H ab, often called the hopping element in solid-state applications. The ET matrix element is the off-diagonal matrix element of the system Hamiltonian taken on the localized diabatic states of the donor and acceptor sites (see below). [The term diabatic refers to localized states which do not diagonalize the system Hamiltonian. These localized states are the true states of the donor and acceptor fragments when these fragments are infinitely separated. For covalently bound complexes, diabatic states become just some basis states that allow reasonable localization of the electronic density on the donor and acceptor fragments of the molecule. Adiabatic states, in contrast, are actual states of the molecule between which electronic (including optical) transitions occur.] For long-range electron transitions, the direct electronic overlap, exponentially decaying with distance between the donor and acceptor units, is weak, leading to a small magnitude of the expectation value of H ab. Such processes, especially important in biological applications, 8 can be characterized as nonadiabatic ET reactions. The small magnitude of the ET matrix element can be employed to find the transition rate using quantum mechanical perturbation theory. In this theory, the rate constant found by the Golden Rule approximation 9,10 is called the nonadiabatic ET rate constant, and the ET reaction is classified as nonadiabatic ET. 11 (The Golden Rule formula is the first-order perturbation solution for the rate of quantum mechanical transitions caused by that perturbation.) The ET rate constant is then proportional to jH abj 2 kNA /jH abj 2 FCWDð0Þ ½2Š Creation of the resonance electronic configuration of the ET transition state, E ¼ 0, is by necessity a many-body event, including complex interactions of the transferred electron with many nuclear degrees of freedom. The
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Reviews in Computational Chemistry
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Kenny B. Lipkowitz Department of Ch
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vi Preface three-dimensional struct
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viii Preface some descriptors and i
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Epilogue and Dedication My associat
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Contents 1. Clustering Methods and
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Contents xv Electron Transfer in Po
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Contributors John M. Barnard, Barna
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Contributors to Previous Volumes *
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Volume 3 (1992) Tamar Schlick, Opti
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Volume 7 (1996) Geoffrey M. Downs a
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Volume 11 (1997) Mark A. Murcko, Re
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T. Daniel Crawford* and Henry F. Sc
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Topics Covered in Volumes 1-18 * Ab
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Reviews in Computational Chemistry
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2 Clustering Methods and Their Uses
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10 Clustering Methods and Their Use
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42 The Use of Scoring Functions in
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Table 1 Reference List for the Most
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CHAPTER 3 Potentials and Algorithms
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Vn (1 + cos(nω + γ)) 2 K θ (θ
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are modified by their environment w
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Table 1 Polarizability Parameters f
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Page 139 and 140: where we have used q Cl ¼ qNa. The
Page 141 and 142: Electronegativity Equalization Mode
Page 143 and 144: Electronegativity Equalization Mode
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Page 151 and 152: Applications 123 classical and rigi
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Page 155 and 156: Comparison of the Polarization Mode
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Page 167 and 168: References 139 Computational Chemis
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Page 171 and 172: References 143 178. J. J. P. Stewar
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Page 175: CHAPTER 4 New Developments in the T
Page 179 and 180: FCWD(∆E ) FCWD(0) ∆E = 0 ∆E
Page 181 and 182: Introduction 153 Equations [6]-[12]
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In Eq. [144], the coordinates Y km
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ε/M −1 cm −1 4000 2000 0 analy
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βσ 2 /10 3 cm −1 14 12 10 8 Opt
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0.2 0.1 0 12 16 20 24 28 32 The app
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References 207 7. M. D. Newton, Adv
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References 209 59. R. Kubo and Y. T
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Reviews in Computational Chemistry Volume 18

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