Reviews in Computational Chemistry Volume 18

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150 Charge-Transfer Reactions in Condensed Phases F i (X )/λ cl 2 0 1 ∆F 0 −2 0 2 X/λcl great achievement of the Marcus–Hush (MH) model 6,12–14 of ET was to reduce the many-body problem to a one-dimensional (1D) picture of intersecting ET free energy surfaces, FiðXÞ (i ¼ 1 for the initial ET state, i ¼ 2 for the final ET state, Figure 2). Each point on the free energy surface represents the reversible work invested to create a nonequilibrium fluctuation of the nuclei resulting in a particular value of the donor–acceptor electronic energy gap X ¼ E ½3Š The electronic energy gap thus serves as a collective reaction coordinate X reflecting the strength of coupling of the nuclear modes to the electronic states of the donor and acceptor. The point of intersection of F1ðXÞ and F2ðXÞ sets up the ET transition state, X ¼ 0. The definition of the ET reaction coordinates according to Eq. [3] allows a direct connection between the activated ET kinetics and steady-state optical spectroscopy. In a spectroscopic experiment, the energy of the incident light with the frequency n (n is used for the wavenumber) is equal to the donor– acceptor energy gap hn ¼ X ½4Š and monitoring the light frequency directly probes the distribution of donor– acceptor energy gaps. The intensity of optical transitions IðnÞ is then proportional to FCWD(hn) 15 2λ cl 1/2λ cl Figure 2 Two parameters defining the Marcus–Hush model of two intersecting parabolas: the equilibrium free energy gap F0 and the classical reorganization energy l cl. The parabolas curvature is 1=ð2l clÞ. IðnÞ /jm12j 2 FCWDðhnÞ ½5Š 2
FCWD(∆E ) FCWD(0) ∆E = 0 ∆E where m12 is the adiabatic transition dipole moment. Knowledge of the spectral band shape can in principle provide the activation barrier through FCWD(0) (Figure 3), and the Mulliken-Hush relation connects jH abj to jm12j. 6 (In contrast to Marcus-Hush which refers to the theory of electron transfer activation, the Mulliken-Hush equation describes the preexponential factor of the rate constant. We spell out Mulliken–Hush each place it occurs in this chapter and use the acronym MH to refer to only Marcus–Hush.) In practice, however, FCWD(0) cannot be extracted from experimental spectra, and one needs a theoretical model to calculate FCWD(0) from experimental band shapes measured at the frequencies of the corresponding electronic transitions. This purpose is achieved by a band shape analysis of optical lines. The two main nuclear modes affecting electronic energies of the donor and acceptor are intramolecular vibrations of the molecular skeleton of the donor–acceptor complex and molecular motions of the solvent. If these two nuclear modes are uncoupled, one can arrive at a set of simple relations between the two spectral moments of absorption and/or emission transitions and the activation parameters of ET. The most transparent representation is achieved when the quantum intramolecular vibrations are represented by a single, effective vibrational mode with the frequency nv (Einstein model). 15–17 If both the forward (absorption) and backward (emission) optical transitions are available, their first spectral moments determine the reorganization energies of quantum vibrations, lv, and of the classical nuclear motions of the donor–acceptor skeleton and the solvent, l cl: σ 2 Introduction 151 Figure 3 Franck–Condon weighted density of energy gaps between the donor and acceptor electronic energy levels. The parameters h Ei and s 2 indicate the first and second spectral moments, respectively. FCWD(0) shows the probability of zero energy gap entering the ET rate (Eq. [2]). hðnabs nemÞ ¼ 2ðlcl þ lvÞ ½6Š
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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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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
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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: Introduction 149 applications). For
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
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Page 191 and 192: is, however, small for the usual co
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Page 195 and 196: where Z is the electrode overpotent
Page 197 and 198: energy surfaces of ET. 33,50-56 It
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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 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 225 and 226: m 12 /D 6 5.5 5 4.5 4 3.5 17 18 19
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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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