Reviews in Computational Chemistry Volume 18

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152 Charge-Transfer Reactions in Condensed Phases where nabs and nem are the first spectral moments for absorption and emission, respectively: Ð nIabs=emðnÞdn nabs=em ¼ Ð Iabs=emðnÞdn ½7Š Here Iabs=emðnÞ is the transition intensity. The vibrational reorganization energy lv is defined in terms of force constants, ka, and displacements, Qa, of the vibrational normal coordinates Qa as lv ¼ 1 P 2 a ka Q2 a .15–17 In this chapter, we use l for the solvent component of the classical reorganization energy lcl. The subscripts 1 and 2 are used to distinguish between the reorganization energy of the initial (i ¼ 1) and final (i ¼ 2) ET states when the reorganization energies in these states are different. The mean of the first two moments gives the equilibrium free energy difference between the final and initial states of the ET reaction hnm ¼ 1 2 h n ð abs þ nemÞ ¼ F0 ¼ F02 F01 ½8Š The two parameters, l cl and F0, actually fully define the parabolic ET free energy surfaces FiðXÞ in the MH formulation (Figure 2). Calculation of these two parameters has become the main historical focus of the ET models addressing the thermodynamics of the ET activation barrier. The latter, according to MH theory, can be written in terms of F0 and l cl as F act i ¼ ðlcl F0Þ 2 ½9Š 4lcl where i ¼ 1 and ‘‘þ’’ refer to the forward transition, and i ¼ 2 and ‘‘ ’’ refer to the backward transition. The second spectral moments of absorption and emission lines s 2 abs=em ¼ are equal in the MH formulation Ð n 2 Iabs=emðnÞdn Ð Iabs=emðnÞdn s 2 abs ¼ s2 em n abs=em They are related to the classical and vibrational reorganization energies as follows 18 s 2 abs=em ¼ 2kBT l cl þ hnvlv where kB is the Boltzmann constant and T is temperature. 2 ½10Š ½11Š ½12Š
Introduction 153 Equations [6]–[12] establish a theoretical basis for calculating the activation barrier of ET from spectroscopic observables. This formalism rests on a set of fundamental assumptions of the MH picture that can be summarized as follows: (1) The electronic coupling between the donor and acceptor states is neglected in the calculation of the Franck–Condon weighted density. The latter depends only on electronic energies of localized electronic states and their coupling to the nuclear modes of the solvent and the donor–acceptor complex. (2) A two-state solute is considered. The manifold of the donor and acceptor electronic levels is limited to only two states between which the electron is transferred. (3) The intramolecular vibrations and solvent molecular motions are decoupled. (4) The linear response approximation is used for the interaction of the donor–acceptor complex with the solvent. The linear response approximation assumes that the free energy of solvation of an electric charges localized on the donor–acceptor complex is a quadratic function of this charge. The neglect of the electronic coupling in the calculation of the FCWD (assumption 1) was adopted in the original Marcus and Hush formulation. 6,12 Within this framework, the ET matrix element does not strongly affect the nuclear fluctuations, although a nonzero value of jH abj is required for electronic transitions to occur. In other words, the transferred electron is assumed to be fully localized in the calculation of the FCWD. To classify electronic delocalization, Robin and Day distinguished between three classes of symmetrical ( F0 ¼ 0) systems. 19 * In Class I systems, the coupling is very weak, and there are essentially no electronic transitions. * Class II systems remain valence-trapped (localized), and 0 < 2jH abj l cl. * In Class III systems, 2jH abj > l cl, and the electron is fully delocalized between the donor and acceptor. The MH formulation is designed to describe the case of intermediate couplings (weak-coupling limit of Class II) when jH abj can be neglected in the FCWD(0) for activated transitions and the transition moment m12 can be neglected in the FCWDðnÞ for optical transitions. In the absence of a theory incorporating jH abj and m12 into the FCWD, there is no general understanding when this approximation is applicable to particular ET systems or how the relations between optical and activation observables are affected by inclusion of electronic delocalization into the FCWD. 20 The limitations of the MH picture considerably narrow the range of systems covered by the theory. A considerable range of processes in which the donor–acceptor coupling is strong enough to change the molecular charge distribution under the influence of nuclear fluctuations cannot be treated theoretically. All such processes can be characterized as CT reactions. Weak electronic coupling characteristic of ET exists for intermolecular and
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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
Page 129 and 130: M i z i + q i d i k i and shell cha
Page 131 and 132: Shell Models 103 on estimates of sh
Page 133 and 134: 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: FCWD(∆E ) FCWD(0) ∆E = 0 ∆E
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
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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
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Page 207 and 208: λ i /eV 4 3 2 1 0 0 10 20 30 40 Bo
Page 209 and 210: the width/Stokes shift relation (Eq
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
Page 221 and 222: of the radiation is the perturbatio
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
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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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