Reviews in Computational Chemistry Volume 18

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190 Charge-Transfer Reactions in Condensed Phases Electronic transitions in the gas phase thus proceed from the lower state E1 to the upper state E2. In condensed phases, these states are of course ‘‘dressed’’ by a solvating environment, but at FI s ¼ 0 one gets a nonzero equilibrium driving force approximately equal to e E12 when E12=l I 1. The factor e in the free energy driving force appears because the free energy represents the work done to transfer the charge e ( z e at E12=l I 1, see Figure 14) over the energy barrier E12 that results in e E12 for small splittings E12. Note above that the GMH 7 and adibatic formulations are equivalent in terms of building the CT free energy surfaces. The distinctions seen in Figure 15 may seem to contradict to this statement. The problem is resolved by noting that the requirement Eab ¼ 0 imposed by the GMH formulation makes the diabatic energy gap nonzero for self-exchange transitions: I GMH ab ¼ e E12 ½120Š which is indeed the gap shown in Figure 15. The standard MH formulation 85 is then recovered when m12 ¼ 0 for symmetry reasons and thus e ¼ 1. Nonlinear Solvation versus Intramolecular Effects The origin of nonparabolic free energy surfaces of ET can be divided into two broad categories: (1) intramolecular electronic effects and (2) nonlinear solvation effects. Although these two origins can, at some instances, be treated within the same mathematical framework (Q model), there are substantial differences between them at both the quantitative and qualitative levels. From the quantitative viewpoint, nonlinear solvation produces a much weaker distortion of ET parabolas than do the polarizability change and electronic delocalization. From a qualitative viewpoint, the two categories of effects produce a nonzero nonparabolic distortion in different orders of the expansion of the system Hamiltonian in the driving solvent mode. The free energy FðPÞ invested in creation of a nonequilibrium solvent polarization P can be expressed as a series in even powers of P with the two first terms as follows: FðPÞ ¼a1P 2 þ a2P 4 ½121Š where a1; a2 > 0. The interaction energy of the solute field with the solvent polarization, U0s, is linear in P U0s ¼ bP b > 0 ½122Š For weak solute–solvent interactions, deviations from zero polarization of the solvent are small, and one can keep only the first, harmonic, term in Eq. [121].
Anharmonic higher order terms gain importance for stronger solute-solvent couplings requiring a2 6¼ 0 in Eq. [121]. The nonequilibrium solvent polarization can be considered as an ET reaction coordinate. The curvature of the corresponding free energy surface is F 00 ðP0Þ ¼2a1 þ 12a2P 2 0 ½123Š at the minimum point P0 defined by the condition F 0 ðP0Þ ¼b. Equation [123] indicates that nonlinear solvation effects, usually associated with dielectric saturation, enhance the curvature compared to the linear response result F 00 ¼ 2a1. This enhancement of curvature leads to a decrease in the solvent reorganization energy. The effect is, however, relatively small as it arises from anharmonic expansion terms. When the electron is partially delocalized, one should switch to the adiabatic representation in which the upper and lower CT surface are split by an energy gap depending on P. If this energy gap is expanded in P with truncation after the second-order term, we come to the model of a donor–acceptor complex whose dipolar polarizabilities are different in the ground and excited states. The solute–solvent interaction energy then attains the energy of solute polarization that is quadratic in P U0s ¼ bP cP 2 c > 0 ½124Š The total system energy FðPÞþU0s includes, therefore, a quadratic in P term with the coefficient ða1 cÞ. This quadratic term initiates a revision of the frequency of solvent fluctuations driving CT. The curvature of harmonic surfaces decreases producing higher reorganization energies. Since the solute polarizability contributes already to the harmonic term, its effect on the reorganization energy is stronger than that of nonlinear solvation. The revision of characteristic frequencies of nuclear modes is a general result of electronic delocalization holding for both the intramolecular vibrational modes 65 and the solvent modes. The fact that this effect shows up already in the harmonic expansion term makes it much stronger compared to nonlinear solvation in respect to nonparabolic distortion of the free energy surfaces. OPTICAL BAND SHAPE Optical Band Shape 191 Spectral measurements open a door to access the rate constant parameters of ET. The connection between optical observables and ET parameters can be divided into two broad categories: (1) analysis of the optical band profile (band shape analysis) and (2) the use of integrated spectral intensities (see
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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
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 203 and 204: Table 1 Main Features of the Two-Pa
Page 205 and 206: and Hss ¼ U rep ss 1 2 X j;k ðmj
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: it by choosing the GMH basis set 7
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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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