Reviews in Computational Chemistry Volume 18

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174 Charge-Transfer Reactions in Condensed Phases and emission (i ¼ 2) li ¼ 1 2 bh2 hdn 2 i i ¼ 1 2 bs2 i where the Gaussian spectral width si is experimentally defined through the half-intensity width i as ½80Š s 2 i ¼ 2 i =ð8ln2Þ ½81Š As is easy to see from Eq. [80] and Figure 7, the Q model predicts the breaking of the symmetry between the absorption and emission widths (Eq. [11]) generated by a statistical distribution of solvent configurations around a donor– acceptor complex (inhomogeneous broadening). This fact may have a significant application to the band shape analysis of optical transitions since unequal absorption and emission width are often observed experimently. 65,66 The parameter a1 is defined through the Stokes shift and two reorganization energies from optical widths a1 ¼ l 1 ðhnst þ l2Þ l ¼ l2 l1 ½82Š Similarly, the equilibrium energy gap is (cf. to Eq. [8]) which is equivalent to F0 ¼ hnm F0 ¼ hnm þ l1 l 2 l1 a1 2 a2 2 h nst þ l2 ½83Š ðh nst þ l1Þ 2 ½84Š The Stokes shift and two second spectral moments fully define the parameters of the model. In addition, they should satisfy Eqs. [73] and [74]. The latter feature establishes the condition of model consistency that is important for mapping the model onto condensed-phase simulations that we discuss below. The connection of the model parameters to the first and second spectral cumulants enables one to build global, nonequilibrium free energy surfaces of ET based on two cumulants obtained at equilibrium configuration of the system. This allows one to apply the model to equilibrium computer simulations data or to spectral modeling. Compared to the MH picture of intersecting parabolas, the Q model predicts a more diverse pattern of possible system regimes including (1) an existence of a one-sided band restricting the range of permissible reaction coordinates, (2) singular free energies outside the fluctuation band, and (3) a linear energy gap law at large activation barriers. The main features of the Q and L models are compared in Table 1.
Table 1 Main Features of the Two-Parameter L Model (MH) and the Three-Parameter Q Model L Model Q Model Parameters Reaction coordinate Spectral moments F0; l 1 < X < 1 F0 ¼ hnm F0; l1; a1 X > X0 at a1 > 0 X < X0 at a1 < 0 F0 ¼ hnm ½l1a1=2ð1 þ a1Þ 2 Š Energy gap law F0 þ l1 l1 F act Electron Transfer in Polarizable Donor–Acceptor Complexes Beyond the Parabolas 175 l ¼ 1 2 h nst l1 ¼ 1 2 bh2 hðdnÞ 2 i 1 a1 ¼ðh nst þ l2Þ=ðl1 l2Þ 1 /ð F0 þ lÞ 2 Fact 1 /ð F0 þ l1Þ 2 j F0j l1 Fact 1 / F2 0 Fact 1 /j F0j The mathematical model incorporating the bilinear solute–solvent coupling considered above can be realized in various situations involving nonlinear interactions of the CT electronic subsystem with the condensed-phase environment. The most obvious reason for such effects is the coupling of the two states participating in the transition to other excited states of the donor– acceptor complex. These effects bring about polarizability and electronic delocalization in CT systems. The instantaneous energies obtained for a two-state donor–acceptor complex contain a highly nonlinear dependence on the solvent field through the instantaneous adiabatic energy gap. Expansion of the energy gap in the solvent field truncated after the second term generates a statedependent bilinear solute–solvent coupling characteristic of the Q model. The second derivative of the energy in the external field is the system’s polarizability. It is therefore hardly surprising that models incorporating the polarizability of the solute 67 turn out to be isomorphic to the Q model. 61 Here, we focus on some specific features of polarizable CT systems. The common starting point to build a theoretical description of the thermodynamics and dynamics of the condensed environment response to an electronic transition is to assume that the transition alters the long-range solute– solvent electrostatic forces. This change comes about due to the variation of the electronic density distribution caused by the transition. The combined electron and nuclear charge distributions are represented by a set of partial charges that are assumed to change when the transition occurs. Actually, a change in the electronic state of a molecule changes not only the electronic charge distribution, but also the ability of the electron cloud to polarize in the external field. In other words, the set of transition dipoles to other electronic states is individual for each state of the molecule, and the dipolar (and higher order) polarizability changes with the transition.
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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
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
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 197 and 198: energy surfaces of ET. 33,50-56 It
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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
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
Page 237 and 238: References 209 59. R. Kubo and Y. T
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Reviews in Computational Chemistry Volume 18

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