Reviews in Computational Chemistry Volume 18

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194 Charge-Transfer Reactions in Condensed Phases of the gas-phase vibronic envelope FCWD v ðnÞ with the normalized solventinduced band shape FCWD s ðnÞ ¼hdð EðqÞ hnÞi ½130Š where the average is taken over the solvent configurations statistically weighted with the Boltzmann factor expð bF Þ with ‘‘ ’’ for absorption and ‘‘þ’’ for emission. In Eqs. [129] and [130], FCWD v ðnÞ and FCWD s ðnÞ refer to the normalized Franck–Condon weighted density of the vibrational excitations of the solute (including quantum overlap integrals of the vibrational normal modes of the solute coupled to the transferred electron 17 ) and the normalized solventinduced spectral distribution function, respectively. The gap, EðqÞ ¼ EþðqÞ E ðqÞ, in Eq. [130] is defined between the upper adiabatic surface EþðqÞ and the lower adiabatic surface E ðqÞ depending on a set of nuclear solvent modes q. Because the transitions occur between the adiabatic free energy surfaces E ðqÞ, the unperturbed basis set in the quantum mechanical perturbation theory is built on the wave functions f ~ f 1ðqÞ; ~ f 2ðqÞg diagonalizing the corresponding two-state Hamiltonian matrix (Eq. [114]). The dependence on the nuclear solvent configuration comes into the transition dipole moment (as calculated within the two-state model, TSM) j ~m12ðqÞj ¼ jh ~ f 1ðqÞj ^m0j ~ f 2ðqÞij ¼jm12j E12 EðqÞ ½131Š only through the energy gap EðqÞ, which is equal to hn according to Eq. [130]. This relationship is the reason for the dependence of the transition dipole on the light frequency in Eq. [129]. Coupling to higher lying excited states modifies Eq. [131], but if the dependence on the solvent field comes into ~m12ðqÞ only through the instantaneous energy gap, the transition dipole can still be taken out of the solvent average with, however, a more complicated dependence on the frequency of the incident light. 17,93 In the TSM, one has, according to Eq. [131] ~m12ðnÞ ¼m12 E12=hn ½132Š where m12 is the gas-phase adiabatic transition dipole moment. The vibronic envelope FCWD v ðnÞ in Eq. [129] can be an arbitrary gasphase spectral profile. In condensed-phase spectral modeling, one often simplifies the analysis by adopting the approximation of a single effective vibrational mode (Einstein model) with the frequency nv and the vibrational reorganization energy lv. The vibronic envelope is then a Poisson distribution of
individual vibrational excitations 44 FCWD v ðnÞ ¼e Sv X 1 m¼0 S m v m! dðhn mhnvÞ ½133Š where Sv the Huang–Rhys factor Sv ¼ lv=hnv (cf. to Eq. [38]). The whole inhomogeneous line shape then takes the form of a weighed sum over the solvent-induced bands, each shifted relative to the other by nv G ðnÞ ¼j~m12ðhnÞj 2 e Sv X 1 m¼0 S m v m! FCWDs ðn mhnvÞ ½134Š Equation [134], given in the form of a weighted sum of individual solvent-induced line shapes, provides an important connection between optical band shapes and CT free energy surfaces. Before turning to specific models for the Franck–Condon factor in Eq. [134], we present some useful relations, following from integrated spectral intensities, that do not depend on specific features of a particular optical line shape. Absorption Intensity and Radiative Rates Optical Band Shape 195 Extraction of activation CT parameters requires an analysis of spectral band shapes. One parameter, however, can be obtained from the integrated absorption and emission intensities. Since mixing of the electronic states in the external electric field of radiation is governed by the magnitude of the transition dipole, the transition dipole also defines the intensity of the corresponding optical line. The extinction coefficient or emission rate integrated over light frequencies then allows one to obtain the transition dipole, provided its frequency dependence is known. [Traditionally, the transition dipole is assumed to be frequency independent. 49 This leads, however, to systematic errors in estimates of transition dipoles from optical spectra, see below.] For the TSM, this procedure leads to the gas-phase transition dipole. The transition dipole is important as a parameter quantifying the extent of CT delocalization and to generate CT free energy surfaces in electronically delocalized donor–acceptor complexes. It also has an important implication due to its connection to the ET matrix element (through the Mulliken-Hush relation), 7 which enters the rate constant of nonadiabatic ET reaction rates (Eq. [2]; see below). Integration of absorption extinction coefficient (Eq. [127]) and emission rate (Eq. [128]) gives two alternative estimates for the adiabatic gas-phase transition dipole m12 (in D) within the TSM frequency-dependent ~m12ðnÞ (Eq. [132]) m12 ¼ 9:585 10 2 pffiffiffiffiffiffi ð 1=2 nD nEðnÞdn ½135Š n0f ðnDÞ
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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.
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 187 and 188: In Eqs. [18] and [19], F0i is the e
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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 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
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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
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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