Reviews in Computational Chemistry Volume 18

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158 Charge-Transfer Reactions in Condensed Phases In such cases, the instantaneous eigenvalues EiðqÞ of the solute electronic Hamiltonian that form the free energy F elðqÞ X i e bEiðqÞ ¼ e bF elðqÞ should be considered as the basis for building the ET free energy surfaces. The energies EiðqÞ can be used for nonequilibrium dynamics, since the population of each surface is not limited by the condition of equilibrium as it is the case in Eq. [16]. An electron is transferred between its centers of localization as a result of underbarrier tunneling when the instantaneous electronic energies EiðqÞ come into resonance due to thermal fluctuation or radiation of the medium (Figure 1). The difference between the energies EiðqÞ thus makes a natural choice for the ET reaction coordinate (cf. to Eq. [3]) ½16Š X ¼ EðqÞ ¼E2ðqÞ E1ðqÞ ½17Š as first suggested by Lax 15 and then utilized in many ET studies. 5,17,32,33 The reversible work necessary to achieve a particular magnitude of the energy gap X defines the free energy profile of CT in terms of a Dirac delta function e bFiðXÞþbF0i 1 ¼ b Trn½dðXEðqÞÞe bEiðqÞ Š=Trn½e bEiðqÞ Š ½18Š The partial trace in nuclear degrees of freedom in Eq. [13] is replaced in Eq. [18] by the constraint imposed on the collective reaction coordinate X representing the energy gap between the two levels involved in the transition. This reduces the many-body problem of calculating the activation dynamics in the coordinate space q to the dynamics over just one coordinate X. As we show in the discussion of optical transition below, the same Boltzmann factor as in Eq. [18] comes into expressions for optical profiles of CT bands. The solvent component of the FCWD then becomes FCWD s i ðXÞ ¼be bFiðXÞþbF0i ½19Š A more general definition of the FCWD includes overlap integrals of quantum nuclear modes. 15,17 The definition given by Eq. [19] includes only classical solvent modes (superscript ‘‘s’’) for which these overlap integrals are identically equal to unity. An extension of Eq. [19] to the case of quantum intramolecular excitations of the donor–acceptor complex is given below in the section discussing optical Franck–Condon factors.
In Eqs. [18] and [19], F0i is the equilibrium free energy of the system in each CT state e bF0i ¼ Trn½e bEiðqÞ Š ½20Š Although the free energy profile FiðXÞ and the free energy F0i are combined in one equation (Eq. [18]), they have a somewhat different physical meaning. The free energy F0i is the total, equilibrium free energy of the system calculated for all its configurations. The difference of F02 and F01 makes the free energy gap F0 entering the MH theory of ET (Figure 2). Thus F0 ¼ F02 F01 ½21Š On the other hand, FiðXÞ is the constrained, incomplete free energy implying that some of the configurations of the system separated by the d-function in Eq. [18] are not included in the calculation of FiðXÞ. 33 The phase space of the system is not completely sampled in defining FiðXÞ, in contrast to the complete sampling for F0i. Using molecular dynamics simulations and explicit atomistic models, the free energy in Eq. [18] can be explicitly mapped out. This kind of calculation has become fairly routine (see, e.g., Refs. 32 and 33). It should be noted, however, that such simulations usually neglect the electronic polarizability of both the CT complex and the solvent. These effects may be large (cf. Ref. 32 and the later discussion in this chapter). When the number of electronic states can be limited to two (two-state model), the analytic properties of the generating function for the two CT free energy surfaces can be used to establish a linear relation between them. 32 The d-function in Eq. [18] can be represented as a Fourier integral that allows one to rewrite the CT free energy in the integral form e bFiðXÞþbF0i ¼ ð 1 1 dx 2p Giðx; XÞ ½22Š The integral is taken over one of the variables of the generating function Giðx; XÞ ¼e ixbX Trn e ixb E bEi Trn e bEi ½23Š Analytic properties of Giðx; XÞ in the complex x-plane then allow one to obtain a linear connection between the free energy surfaces F2ðXÞ ¼F1ðXÞþX ½24Š as first established by Warshel. 32,34 This relation is based on the transforma- tion of the integral ð 1 1 Paradigm of Free Energy Surfaces 159 dx 2p G2ðx; XÞ ¼e bð F0 XÞ ð iþ1 i 1 dx 2p G1ðx; XÞ ½25Š
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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
Page 135 and 136: The energy required to create a cha
Page 137 and 138: for all i ði:e:; 8 iÞ: @U @qi l
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
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
Page 153 and 154: developing polarizable models. A va
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
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Page 191 and 192: is, however, small for the usual co
Page 193 and 194: solute-solvent coupling through the
Page 195 and 196: where Z is the electrode overpotent
Page 197 and 198: energy surfaces of ET. 33,50-56 It
Page 199 and 200: This result indicates a fundamental
Page 201 and 202: βF i (X ) 40 20 0 −20 1 2 βλ 1
Page 203 and 204: Table 1 Main Features of the Two-Pa
Page 205 and 206: and Hss ¼ U rep ss 1 2 X j;k ðmj
Page 207 and 208: λ i /eV 4 3 2 1 0 0 10 20 30 40 Bo
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Page 211 and 212: Table 3 Mapping of the Q Model on S
Page 213 and 214: This situation is of course not sat
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
Page 219 and 220: Anharmonic higher order terms gain
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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
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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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