Reviews in Computational Chemistry Volume 18

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154 Charge-Transfer Reactions in Condensed Phases long-distance intramolecular reactions. Many systems with intramolecular electronic transitions over a relatively short distance between the initial and final centers of electron localization have been synthesized in recent years. 21,22 They commonly incorporate the same basic design in which the donor and acceptor units are linked in one molecule through a bridge moiety. In a case of closely separated donor and acceptor units, electronic states on these two sites are strongly coupled, resulting in a substantial delocalization of the electronic density. The electronic density is only partially transferred, and the process can be classified as a CT transition. The MH formulation for the activation barrier and the related connection between activation ET parameters and optical observables generally do not apply to CT reactions. Hence the researcher is left without a procedure of calculating the activation barrier from spectroscopy. Not being able to calculate the barrier is a deficiency, and this chapter discusses some emerging approaches to develop a theory of CT processes with an explicit account for electronic delocalization effects. In application to optical transitions, this theory should lead to the development of a band shape analysis broadly applicable to Class II and III systems. The effect of electronic delocalization on the solvent component of the FCWD is emphasized here. The previously reviewed problem of delocalization effects on intramolecular vibrations 23 is not included. We also review some new approaches going beyond the two-state approximation in terms of incorporating polarizability of the donor–acceptor complex (assumption 2), and discuss some recent studies on nonlinear solvation effects (assumption 4). There are some very recent indications in the literature pointing to a possibility of an effective coupling between vibrational modes of the donor–acceptor complex and solvent fluctuations (assumption 3), but no consensus on when and why these effects are significant has yet been reached. We briefly discuss the available experimental and theoretical findings. The first part of this chapter contains an introduction to the statistical mechanical formulation of the CT free energy surfaces. Importantly, it shows how to extend the traditional MH picture of two ET parabolas to a more general case of two CT free energy surfaces of a two-state donor–acceptor complex. The notation we utilize below distinguishes between these two cases in the following fashion: we use the indices 1 and 2 to denote the two ET free energy surfaces, as in Figure 2, and refer to the lower and upper CT free energy surfaces with ‘‘ ’’ and ‘‘þ’’, respectively. The parameters entering the activation barrier of CT transitions depend on the choice of the basis set of wave functions of the initial and final states of the donor–acceptor complex. The standard MH formulation is based on the choice of a localized, diabatic basis set. When this choice is adopted, we use the superscript ‘‘d’’ to refer to diabatic wave functions. An alternative description is possible in terms of adiabatic wave functions, and this situation is distinguished by the superscript ‘‘ad’’. We also provide a basis-invariant formulation of the theory for a two-state
Paradigm of Free Energy Surfaces 155 donor–acceptor complex. A description of CT activation and spectroscopy in terms of two crossing, free energy surfaces (Figure 2) is in fact possible for any choice of the basis set as long as the off-diagonal matrix elements of the solute quantum mechanical operators can be neglected. In cases when a description in both diabatic and adiabatic representations is possible (as it is for the Q-model discussed below), we will not specify the basis by dropping the ‘‘d’’ and ‘‘ad’’ superscripts. The statistical mechanical analysis of ET and CT free energy surfaces developed in the first part of this chapter is applied to the calculation of optical absorption and emission profiles in the second part. This application of the theory, related to the band shape analysis of optical line shapes, has been a central issue in understanding CT energetics for several decades. 16 The chapter is designed to demonstrate how the extension of the basic models used to describe the thermodynamics of CT is reflected in asymmetry of the energy gap law (dependence of the CT activation barrier on the equilibrium free energy gap) and more complex and structured optical band shapes. The development of a corresponding band shape analysis incorporating these new features is in its infancy, and we will certainly see more activity in this field in the future. PARADIGM OF FREE ENERGY SURFACES The CT/ET free energy surface is the central concept in the theory of CT/ ET reactions. The surface’s main purpose is to reduce the many-body problem of a localized electron in a condensed-phase environment to a few collective reaction coordinates affecting the electronic energy levels. This idea is based on the Born–Oppenheimer (BO) separation 24 of the electronic and nuclear time scales, which in turn makes the nuclear dynamics responsible for fluctuations of electronic energy levels (Figure 1). The choice of a particular collective mode is dictated by the problem considered. One reaction coordinate stands out above all others, however, and is the energy gap between the two CT states as probed by optical spectroscopy (i.e., an experimental observable). Our discussion of the CT free energy surfaces involves a hierarchy of reaction coordinates (Figure 4). It starts from the instantaneous free energy surfaces obtained from tracing out (statistical averaging) the electronic degrees of freedom in the system density matrix (i.e., solving the electronic problem for fixed nuclear coordinates). In the case when the direction of electron transfer sets up the only preferential direction in the CT system, one can define a scalar reaction coordinate as the projection of the nuclear solvent polarization on the differential electrical field of the solute. Depending on the basis set employed, this gives the diabatic or adiabatic scalar reaction coordinates, Y d and Y ad (Figure 4). At this step, a reaction coordinate depends on the basis set of solute wave functions employed. This dependence is eliminated when a scalar reaction coordinate is projected on the energy gap between the CT surfaces.
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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
Page 131 and 132: Shell Models 103 on estimates of sh
Page 133 and 134: minimization can be replaced by mor
Page 135 and 136: The energy required to create a cha
Page 137 and 138: for all i ði:e:; 8 iÞ: @U @qi l
Page 139 and 140: where we have used q Cl ¼ qNa. The
Page 141 and 142: Electronegativity Equalization Mode
Page 143 and 144: Electronegativity Equalization Mode
Page 145 and 146: of N molecules is taken as a Hartre
Page 147 and 148: 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: Introduction 153 Equations [6]-[12]
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
Page 193 and 194: solute-solvent coupling through the
Page 195 and 196: where Z is the electrode overpotent
Page 197 and 198: energy surfaces of ET. 33,50-56 It
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Page 201 and 202: βF i (X ) 40 20 0 −20 1 2 βλ 1
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
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 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
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0.2 0.1 0 12 16 20 24 28 32 The app
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References 207 7. M. D. Newton, Adv
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References 209 59. R. Kubo and Y. T
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Reviews in Computational Chemistry Volume 18

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