Reviews in Computational Chemistry Volume 18

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192 Charge-Transfer Reactions in Condensed Phases below). The former route connects the spectral moments to ET activation parameters (Table 1). The latter is applied to extract off-diagonal matrix elements, most often the ET matrix element and the transition dipole. Band shape analysis of optical spectra has been successfully used in ET research for many years, and our present knowledge about mechanisms and energetics of ET originates largely from spectroscopic measurements. 16 The understanding of electronic and solvent effects on the ET kinetics has been recently supplemented by extensive information about the intramolecular, vibronic envelope from resonance Raman spectroscopy. 86 The fast growth of the field of ET research and, especially, the design of new bridge-coupled donor–acceptor pairs imposes new demands on the theory of optical spectra. Several major challenges are currently faced by the field. They may be summarized as follows: (1) The presently existing band-shape analysis has been created for ET transitions. 17 It has not anticipated strong electronic coupling and thus fails when applied to transitions with high magnitudes of the ET matrix element. 87 (2) The model is limited to two states only. Mixing to higher excited states, resulting in intensity borrowing, is commonly neglected. Extension to more then two states is especially important for photoinduced CT where a CT state is formed from and is strongly coupled to a locally exited state of either donor or acceptor unit. 17,88 (3) There are indications in the literature that the common assumption of complete decoupling between the intramolecular vibrational modes and solvent thermal motions may fail for some systems. 89,90 Understanding the origin of and full account for these effects should be incorporated into new models of optical bands. The challenges outlined above still await a solution. In this section, we show how some of the theoretical limitations employed in traditional formulations of the band shape analysis can be lifted. We discuss two extensions of the present-day band shape analysis. First, the two-state model of CT transitions is applied to build the Franck–Condon optical envelopes. Second, the restriction of only two electronic states is lifted within the band shape analysis of polarizable chromophores that takes higher lying excited states into account through the solute dipolar polarizability. Finally, we show how a hybrid model incorporating the electronic delocalization and chromophore’s polarizability effects can be successfully applied to the calculation of steady-state optical band shapes of the optical dye coumarin 153 (C153). We first start with a general theory and outline the connection between optical intensities and the ET matrix element and transition dipole. Optical Franck–Condon Factors Absorption of light by molecules, resulting in electronic excitations, is caused by the interaction of the bound molecular electrons with the electric field of the radiation. In the dipolar approximation, the interaction of the dipole operator of the solute ^m0 with the time-dependent electric field EðtÞ
of the radiation is the perturbation that drives the electronic excitation. The time-dependent interaction Hamiltonian is f ðnDÞ ^m0 EðtÞ ½125Š where the parameter f ðnDÞ accounts for the deviation of the local field acting on the solute dipole from the external field EðtÞ; nD is the solvent refractive index. Dielectric theories 91 predict for spherical cavities f ðnDÞ ¼ 3n2 D 2n2 ½126Š D þ 1 The perturbation given by Eq. [125] mixes the electronic states for which the off-diagonal matrix element of the dipole operator, mjk, is nonzero. The latter is called the transition dipole. 49 Mixing of electronic states by a timedependent external field leads to the dependence of the corresponding electronic state populations on time. The rate constant of the population kinetics is given by the transition probability. Quantum mechanical perturbation theory, limited to the first order in the interaction perturbation, is commonly used to calculate the one-photon transition probability and absorption intensity. 15,92 This formalism, combined with the Einstein relation between absorption intensity and the probability of spontaneous emission, 49,92 leads to experimental observables, the extinction coefficient of absorption, EðnÞ (cm 1 M 1 ), and the emission rate, IemðnÞ (number of photons per unit frequency), as functions of the light frequency n. They are given by the following relations: and EðnÞ n ¼ 8p3NA f 3000 ðln 10Þ c 2ðnDÞ G ðnÞ ½127Š IemðnÞ ¼ 64p4n3 3c3 nDf 2 ðnDÞGþðnÞ ½128Š In Eq. [127], NA is the Avogadro number, and c in Eqs. [127] and [128] is the speed of light in vacuum. The extinction coefficient and emission rate are defined through the spectral density function G ðnÞ that combines the effects of solvent-induced inhomogeneous broadening and vibrational excitations of the donor–acceptor complex. A substantial simplification of the description can be achieved if the two types of nuclear motions are not coupled to each other. The spectral density G ðnÞ is then given by the convolution 17 nD Optical Band Shape 193 G ðnÞ ¼j~m12ðhnÞj 2 ð FCWD s ðxÞFCWD v ðn xÞdx ½129Š
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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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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 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
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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
Page 219: Anharmonic higher order terms gain
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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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