Reviews in Computational Chemistry Volume 18

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164 Charge-Transfer Reactions in Condensed Phases The diabatic solvent reorganization energy is defined by the nuclear response function w n and by the diabatic field difference The free energy gap l d ¼ 1 2 E ab w n E ab ½45Š F d 0 ¼ Fd 0b F d 0a ¼ I ab þ F d s ½46Š is composed of the gas-phase splitting I ab ¼ I b Ia and the solvation free energy F d s ¼ Eav w E ab ½47Š where w ¼ w e þ w n is the total response function of the solvent. Projection on the energy gap reaction coordinate in Eq. [18] is simple to perform for the scalar reaction coordinate Y d where FCWD s ðXÞ ¼ X k Q ¼ and Y ðkÞ are all the roots of the equation ðbQ E 0 ½Y ðkÞ ŠÞ 1 e bE ½YðkÞ Š ð e bE ðYd Þ dY d In Eq. [48], E 0 ½Y ðkÞ Š denotes the derivative ½48Š ½49Š X ¼ E½Y d Š ½50Š E 0 Y ðkÞ ¼ d EðYd Þ dY d Y d ¼Y ðkÞ where Y d ¼ Y ðkÞ indicates that the derivative is taken at the coordinate Y ðkÞ obtained as a solution of Eq. [50]. Equations [41]–[50] provide an exact solution for the CT free energy surfaces and Franck–Condon factors of a two-state system in a condensed medium with quantum electronic and classical nuclear polarization fields. The derivation does not make any specific assumptions about the off-diagonal matrix elements of the Hamiltonian. It, therefore, includes the off-diagonal ½51Š
solute–solvent coupling through the off-diagonal matrix element of the electric field of the solute. 40 This coupling represents a non-Condon dependence of the ET matrix element on the nuclear solvent polarization (this contribution is commonly neglected in MH theory 13 ). In the case of weak electronic overlap, all off-diagonal matrix elements are neglected in the free energy surfaces, and the above equations are transformed to the well-known case of two intersecting parabolas (Figure 2) representing the diabatic ET free energy surfaces FiðY d Þ¼F0i þ ðYd l d Þ 2 The reaction rate constant is then given by the Golden Rule perturbation expansion in the solvent-dependent ET matrix element Heff ab ½PnŠ. 43 Careful account for non-Condon solvent dependence of the ET matrix element generates the Mulliken-Hush matrix element in the rate preexponent (see below). In the opposite case of strong electronic overlap, the off-diagonal matrix elements cannot be neglected, and one should consider the CT free energy surfaces, instead of ET free energy surfaces, with partial transfer of the electronic density. The free energy surfaces are then substantially nonparabolic; we discuss this case in the section on Electron Delocalization Effect. Heterogeneous Discharge The diabatic two-state representation for homogeneous CT can be extended to heterogeneous CT processes between a reactant in a condensedphase solvent and a metal electrode. The system Hamiltonian is then given by the Fano–Anderson model 44,45 H ¼ HB þ½E De PnŠc þ c þ X Paradigm of Free Energy Surfaces 165 k 4l d Ek c þ k ck þ X ðHkc k þ k ½52Š c þ h:c:Þ ½53Š where k is the lattice reciprocal vector, the two summations are over the wave vectors of the electrons of a metal, ek is the kinetic energy of the conduction electrons (hence ek ¼ k 2 =2me, withme being the electron mass), and ‘‘h.c.’’ designates the corresponding Hermetian conjugate. In Eq. [53], c þ and c are the Fermionic creation and annihilation operators of the localized reactant state. c þ k and ck are the creation and annihilation operators, respectively, for a conduction electron with momentum k, andHk is the coupling of this metal state to the localized electron state on the reactant. The energy of the localized reactant state includes solvation by the solvent electronic polarization (included in E) and the interaction of the electron electric field De with the nuclear solvent polarization Pn. The transferred electron is much faster than the ions dissolved in the electrolyte. Therefore, on the time scale of charge
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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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The polarizable point dipole models
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two to three orders of magnitude sl
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M i z i + q i d i k i and shell cha
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Shell Models 103 on estimates of sh
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minimization can be replaced by mor
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The energy required to create a cha
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for all i ði:e:; 8 iÞ: @U @qi l
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where we have used q Cl ¼ qNa. The
Page 141 and 142: Electronegativity Equalization Mode
Page 143 and 144: Electronegativity Equalization Mode
Page 145 and 146: of N molecules is taken as a Hartre
Page 147 and 148: is treated using variable charges.
Page 149 and 150: water have been developed, includin
Page 151 and 152: Applications 123 classical and rigi
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Page 155 and 156: Comparison of the Polarization Mode
Page 157 and 158: negligible errors in such propertie
Page 159 and 160: ecome significant at field strength
Page 161 and 162: noteworthy in this regard because t
Page 163 and 164: References 135 9. P. Cieplak and P.
Page 165 and 166: References 137 48. E. L. Pollock an
Page 167 and 168: References 139 Computational Chemis
Page 169 and 170: References 141 139. J. Hinze and H.
Page 171 and 172: References 143 178. J. J. P. Stewar
Page 173 and 174: References 145 216. M. W. Mahoney a
Page 175 and 176: CHAPTER 4 New Developments in the T
Page 177 and 178: Introduction 149 applications). For
Page 179 and 180: FCWD(∆E ) FCWD(0) ∆E = 0 ∆E
Page 181 and 182: Introduction 153 Equations [6]-[12]
Page 183 and 184: Paradigm of Free Energy Surfaces 15
Page 185 and 186: Paradigm of Free Energy Surfaces 15
Page 187 and 188: In Eqs. [18] and [19], F0i is the e
Page 189 and 190: where ‘‘þ’’ and ‘‘ ’
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Page 197 and 198: energy surfaces of ET. 33,50-56 It
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Page 203 and 204: Table 1 Main Features of the Two-Pa
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Page 211 and 212: Table 3 Mapping of the Q Model on S
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Page 215 and 216: F ± (Y ad )/λ I 0.8 0.4 0 −0.4
Page 217 and 218: it by choosing the GMH basis set 7
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Page 223 and 224: individual vibrational excitations
Page 225 and 226: m 12 /D 6 5.5 5 4.5 4 3.5 17 18 19
Page 227 and 228: In Eq. [144], the coordinates Y km
Page 229 and 230: ε/M −1 cm −1 4000 2000 0 analy
Page 231 and 232: βσ 2 /10 3 cm −1 14 12 10 8 Opt
Page 233 and 234: 0.2 0.1 0 12 16 20 24 28 32 The app
Page 235 and 236: References 207 7. M. D. Newton, Adv
Page 237 and 238: References 209 59. R. Kubo and Y. T
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Reviews in Computational Chemistry Volume 18

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