Reviews in Computational Chemistry Volume 18

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112 Polarizability in Computer Simulations limit, and methods constraining the charge based on the bonding network are an extension of EE models that appear to be successful at controlling the coherence lengths. 149 There now exist classical models that can describe the charge transfer reasonably across the full range of a dissociating bond, but these are currently less well developed. 157 The polarization energy in the EE models can be compared directly to that in the polarizable point dipole and shell models. Consider the first term in Eq. [43], X X X w i 2 a 0 i qi þ 1 2 i 2 a j 2 a qi qj JijðrijÞ E gp a This term represents the energy required to induce charges qi on the atoms of molecule a in the electric field of its neighbors, relative to the energy of the isolated molecule. This quantity is simply the polarization energy of the molecule. The polarization energy of the full system can thus be written Upol ¼ X X w 0 i qi þ 1 X X qi qj JijðrijÞ E 2 gp " # a a i 2 a i 2 a j 2 a which can be compared to Eqs. [9] and [26]. 10 There exist other models that treat polarizability using variable charges in a way similar to the fluctuating charge model. 22,53,127,143,158 In the Sprik and Klein 22 model for water, four charge sites are located near the oxygen atom in a tetrahedral geometry, in addition to the three atom-centered permanent charges. The tetrahedron of charges is used to represent an induced dipole moment on the oxygen center. This approach differs from a polarizable point dipole model in using a dipole of finite extent. It also differs from a shell model in that the point charges are fixed in the molecular frame. Consequently, the Sprik–Klein model should perhaps best be considered an entirely different type of model. The model of Zhu, Singh, and Robinson 158 is similar to the Sprik– Klein model, but it has no permanent charges. The four charge sites, two on hydrogen atoms and two on lone-pair positions 1 A˚ from the oxygen atom, are all variables coupled to the electric field. For both these models, the coupling is described by the polarizability, a, just as with other dipole polarizable models. Wilson and Madden 159 described a model for ions in which charge is transferred between ends of a rigid, rotating rod. In the model of Perng et al., 143 the charge, qi, on an atom is equal to a permanent value, q0 i , plus an induced part, dqi. The induced charge is dependent on the electrostatic potential at that site and all the induced charges are coupled through Coulombic interactions, similar to the fluctuating charge models. In the polarizable point charge (PPC) model of Svishchev et al., 53 charges are coupled directly to the electric field at that site, so this model is slightly different from the fluctuating charge model. ½47Š ½48Š
Electronegativity Equalization Models 113 Although a valence-type force field of the type illustrated by Eq. [1] is most suitable for modeling molecular systems, the electronegativity equalization approach to treating polarization can be coupled equally well to other types of potentials. Streitz and Mintmire 127 used an EE-based model in conjunction with an embedded atom method (EAM) potential to treat polarization effects in bulk metals and oxides. The resulting ES þ EAM model has been parameterized for aluminum and titanium oxides, and has been used to study both charge-transfer effects and reactivity at interfaces. 127,128,160,161 In most electronegativity equalization models, if the energy is quadratic in the charges (as in Eq. [36]), the minimization condition (Eq. [41]) leads to a coupled set of linear equations for the charges. As with the polarizable point dipole and shell models, solving for the charges can be done by matrix inversion, iteration, or extended Lagrangian methods. As with other polarizable models, the matrix methods tend to be avoided by most researchers because of their computational expense. And when they are used, the matrix inversion is typically not performed at every step. 160,162 Some EE applications have relied on iterative methods to determine the charges. 53,127 For very large-scale systems, multilevel methods are available. 161,163 As with the dipole polarizable models, the proper treatment of long-range electrostatic interactions is especially important for fluctuating charge models. 164 Monte Carlo methods have also been developed for use with fluctuating charge models. 162,165 Despite this variety of available techniques, the most common approach is to use a matrix inversion or iterative method only to obtain the initial energy-minimizing charge distribution; an extended Lagrangian method is then used to propagate the charges dynamically in order to take advantage of its computational efficiency. In the extended Lagrangian method, as applied to a fluctuating charge system, 126 the charges are given a fictitious mass, Mq, and evolved in time according to Newton’s equation of motion, analogous to Eq. [23], Mq qi ¼ @U where la is the average of the negative of the electronegativity of the molecule a containing atom i, la ¼ 1 Na @qi la ½49Š X wi wa ½50Š i 2 a Here, Na is the number of atoms in molecule a. Combining Eq. [49], [50], and [42], we have Mq qi ¼ w a w i ½51Š
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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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Page 117 and 118: CHAPTER 3 Potentials and Algorithms
Page 119 and 120: Vn (1 + cos(nω + γ)) 2 K θ (θ
Page 121 and 122: are modified by their environment w
Page 123 and 124: Table 1 Polarizability Parameters f
Page 125 and 126: The polarizable point dipole models
Page 127 and 128: two to three orders of magnitude sl
Page 129 and 130: 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
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Page 139: where we have used q Cl ¼ qNa. The
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 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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is, however, small for the usual co
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solute-solvent coupling through the
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where Z is the electrode overpotent
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energy surfaces of ET. 33,50-56 It
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This result indicates a fundamental
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βF i (X ) 40 20 0 −20 1 2 βλ 1
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Table 1 Main Features of the Two-Pa
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and Hss ¼ U rep ss 1 2 X j;k ðmj
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λ i /eV 4 3 2 1 0 0 10 20 30 40 Bo
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the width/Stokes shift relation (Eq
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Table 3 Mapping of the Q Model on S
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This situation is of course not sat
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F ± (Y ad )/λ I 0.8 0.4 0 −0.4
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it by choosing the GMH basis set 7
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Anharmonic higher order terms gain
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of the radiation is the perturbatio
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individual vibrational excitations
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m 12 /D 6 5.5 5 4.5 4 3.5 17 18 19
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In Eq. [144], the coordinates Y km
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ε/M −1 cm −1 4000 2000 0 analy
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βσ 2 /10 3 cm −1 14 12 10 8 Opt
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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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