Reviews in Computational Chemistry Volume 18

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196 Charge-Transfer Reactions in Condensed Phases and m12 ¼ 3:092 10 8 n0 pffiffiffiffiffiffi ½ f ðnDÞŠ 1 nD ð IemðnÞn 1 dn 1=2 ½136Š where n is the wavenumber (cm 1 ) and n0 ¼ E12=hc. When the emission spectrum is not available, the radiative rate 49 k rad ¼ ð IemðnÞdn ¼ emt 1 em ½137Š can be used; em and tem are the quantum yield and emission lifetime. By defining the average frequency one gets nav ¼ ð IemðnÞdn m12 ¼ 1:786 10 3 k rad navn 2 0 nDf 2 ðnDÞ ð IemðnÞn 1 dn ½138Š 1=2 ½139Š Equation [139] is not very practical because an accurate definition of the average wavenumber, nav ¼ nav=c, demands knowledge of the emission spectrum for which Eq. [136] provides a direct route to the transition dipole. But Eq. [139] can be used in approximate calculations by assuming nav ¼ nem. Equation [139] is exact for a two-state solute, but differs from the traditionally used connection between the transition dipole and the emission intensity by the factor n0=nav. 49 The commonly used combination m12n0=nav appears as a result of neglect of the frequency dependence of the transition dipole ~m12ðnÞ entering Eq. [129]. It can be associated with the condensedphase transition dipole in the two-state approximation. 43 Exact solution for a two-state solute makes the transition dipole between the adiabatic free energy surfaces inversely proportional to the energy gap between them. This dependence, however, is eliminated when the emission intensity is integrated with the factor n 1 . 93 The transition dipole m12 in Eqs. [136] and [139] is the gas-phase adiabatic transition dipole. Therefore, emission intensities measured in different solvents should generate invariant transition dipoles when treated according to Eqs. [136] and [139]. A deviation from invariance can be used as an indication of the breakdown of the two-state approximation and the existence of intensity borrowing from other excited states of the chromophores (the Murrell mechanism 17,88,94 ). Figure 16 illustrates the difference between Eq. [139] and
m 12 /D 6 5.5 5 4.5 4 3.5 17 18 19 20 21 22 − νem /10 3 cm −1 the traditional formulation. It shows the dependence of m12 (circles) and m12n0=nav (squares, n0 ¼ 25,400 cm 1 ) on the emission frequency nem for the dye C153 measured in solvents of different polarity. 95 The two sets of transition dipoles are noticeably divergent in strongly polar solvents. Electron-Transfer Matrix Element Optical Band Shape 197 Figure 16 The transition dipole m12 according to Eq. [139] (nav ¼ nem, circles) and m12n0=nem (squares) versus nem for emission transitions in C153 in different solvents. 95 The dashed lines are regressions with the slopes 0.02 (squares) and 0.27 (circles). The transition dipole between the free energy surfaces F ðXÞ is not the only parameter that depends on the nuclear configuration of the solvent. The effective ET matrix element H eff ab ½PnŠ following from the trace of the two-state Hamiltonian over the electronic degrees of freedom also depends on the nuclear configuration of the solvent (Eq. [37]). In contrast to the case of optical transitions where the dependence on the nuclear solvent configurations is transformed into a frequency dependence of the transition dipole ~m12ðnÞ (Eq. [132]), the dependence of the ET matrix element H eff ab ½PnŠ on the nuclear field Pn should be fully included into the statistical average over Pn when the ET rate constant is calculated in the Golden Rule perturbation scheme over H eff ab ½PnŠ. 11 The Pn dependence represents a non-Condon effect of the solvent field on the rate preexponential factor. The result of the calculations 43 is the standard Golden Rule expression 9,11 for the nonadiabatic rate constant k ðiÞ NA ¼ h 1 ðpb=lÞ 1=2 H MH 2 FCWDið0Þ ½140Š
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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
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Electronegativity Equalization Mode
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Electronegativity Equalization Mode
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of N molecules is taken as a Hartre
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is treated using variable charges.
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water have been developed, includin
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Applications 123 classical and rigi
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developing polarizable models. A va
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Comparison of the Polarization Mode
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negligible errors in such propertie
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ecome significant at field strength
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noteworthy in this regard because t
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References 135 9. P. Cieplak and P.
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References 137 48. E. L. Pollock an
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References 139 Computational Chemis
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References 141 139. J. Hinze and H.
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References 143 178. J. J. P. Stewar
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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 187 and 188: In Eqs. [18] and [19], F0i is the e
Page 189 and 190: where ‘‘þ’’ and ‘‘ ’
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
Page 199 and 200: This result indicates a fundamental
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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
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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
Page 219 and 220: Anharmonic higher order terms gain
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Page 223: individual vibrational excitations
Page 227 and 228: In Eq. [144], the coordinates Y km
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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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