Reviews in Computational Chemistry Volume 18

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172 Charge-Transfer Reactions in Condensed Phases and a2 ¼ 1 þ a1 An additional constraint on the magnitudes of the parameter a1 comes from the condition of the thermodynamic stability of the collective solvent mode in both states, ki > 0, resulting in two inequalities ½74Š a1 > 0 or a1 < 1 ½75Š The inequalities in Eq. [75] also define the condition for the generating function (Eq. [23]) to be analytic in the integration contour in Eq. [25]. This condition is equivalent to the linear connection between the diabatic free energy surfaces, Eq. [24]. The Q model solution thus explicitly indicates that the linear relation between the diabatic free energy surfaces is equivalent to the condition of thermodynamic stability of the collective nuclear mode driving ET. Equations [73] and [74] reduce the number of independent parameters of the Q model to three: F0, l1, and a1. Here, F0 (Eq. [21]) is the free energy gap between equilibrium configurations of the system (Figure 2). The fluctuation boundary X0 is connected to F0 by the relation X0 ¼ F0 þ l1a 2 1 =a2 ½76Š Compared to the two-parameter MH theory (l and F0), 12 the Q model introduces an additional flexibility in terms of the relative variation of the fluctuation force constant through a1. The MH theory is recovered in the limit a1 !1. Importantly, the new free energy surfaces lead to qualitatively new features for the activated ET kinetics. The standard high-temperature limit of two diabatic ET free energy surfaces FiðXÞ ¼F0i þ ðX F0 liÞ 2 is reproduced when ai 1 (the driving mode force constants ki in the two states are similar) and, additionally, jX F0 lij jaijli. Here, ‘‘ ’’ and ‘‘þ’’ correspond to i ¼ 1 and i ¼ 2, respectively. The second requirement implies that the reaction coordinate should be not too far from the free energy minimum to preserve its parabolic form. By contrast, in the limit jX X0j lijaij, the linear dependence wins over the parabolic law 4li a FiðXÞ ¼F0i þjaij X F0 þ l1 2 1 a2 ½77Š ½78Š
βF i (X ) 40 20 0 −20 1 2 βλ 1 = 40 α 1 = 1 −40 −20 0 20 40 −100 0 βX βX As a combination of these two effects, plus the existence of the fluctuation boundary, the free energy surfaces are asymmetric with a steeper branch on the side of the fluctuation boundary X0. The other branch is less steep tending to a linear dependence at large X (Figure 7). The minima of the initial and final free energy surfaces get closer to each other and to the band boundary with decreasing a1 and l1. The crossing point then moves to the inverted ET region where the free energies are nearly linear functions of the reaction coordinate. The ET activation energy follows from Eq. [66] F act i ¼ Fið0Þ F0i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼jaij j F0 l1a2 1 =a2j q ffiffiffiffiffiffiffiffiffiffi jaijli q 2 Equation [79] produces the MH quadratic energy gap law at small j F0j ja1l1j and yields a linear dependence of the activation energy on the equilibrium free energy gap at j F0 l1a 2 1 =a2j jaijli. A linear energy gap law is by no means unusual in ET kinetics. It is quite often encountered at large equilibrium energy gaps. Experimental observations of the linear energy gap law are made for intermolecular 62 as well as intramolecular 63 organic donor–acceptor complexes, in binuclear metal–metal CT complexes, 16 and in CT crystals. 64 It is commonly explained in terms of the weak coupling limit of the theory of vibronic band shapes yielding the linear-logarithmic dependence proportional to Fi ln Fi on the vertical energy gap Fi. 17 On the contrary, a strictly linear dependence proportional to Fi arises from the Q model. To complete the Q model, one needs to relate the model parameters to spectral observables. Already, the reorganization energies li are directly related to the solvent-induced inhomogeneous widths of absorption (i ¼ 1) 80 40 0 Beyond the Parabolas 173 2 1 βλ 1 = 40 α 1 = 4 Figure 7 The free energy surfaces F1ðXÞ (1) and F2ðXÞ (2) at various a1; I ¼ 0. The dashed line indicates the position of the fluctuation boundary X0. ½79Š
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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.
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 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
Page 189 and 190: where ‘‘þ’’ and ‘‘ ’
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
Page 199: This result indicates a fundamental
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
Page 219 and 220: Anharmonic higher order terms gain
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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
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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