Reviews in Computational Chemistry Volume 18

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132 Polarizability in Computer Simulations with three-dimensional polarizability tensors. 126 Off-atom charge sites have been successfully used to address this limitation in some cases, as have hybrid models. 82,96,104,145,146,150,166 The Electrostatic Potential In addition to treating the polarization response in different ways, the various methods considered here also provide different levels of approximation to the external electric field. Accurate simulation of intermolecular interactions requires that the electrostatic potential be correctly represented everywhere outside the molecular surface. The correct electrostatic potentials can be reproduced, of course, by the physically correct nuclear and electronic charge distribution. At points outside the molecular surface, however, it can also be reproduced to arbitrary accuracy by a series of point monopoles, dipoles, quadrupoles, and so on. This approach is taken in most computer simulations. The simplest level of approximation is to include point charges (monopoles) at the atomic sites. The accuracy of this approximation can be improved by (1) adding more charge sites (off-atom sites); (2) increasing the number of terms in the series (dipoles, quadrupoles, etc.); and (3) by replacing the point multipoles with delocalized, diffuse charge distributions. The PPD and shell models are nearly equivalent in this sense, because they model the electrostatic potential via static point charges and polarizable dipoles (of either zero or very small extent). Accuracy can be improved by extending the expansion to include polarizable quadrupoles or higher order terms. 193 The added computational expense and difficulty in parameterizing these higher order methods has prevented them from being used widely. The accuracy of the ESP for dipole-based methods can also be improved by adding off-atom dipolar sites. 96,166 Because the EE-based methods truncate the series representation of the electrostatic potential one term earlier (i.e., by using only monopole charges), these methods would appear to sacrifice some accuracy in representing the electrostatic potential. It is becoming widely appreciated that models based solely on point charges may require the use of off-atom charge sites to successfully fit the electrostatic potential. 166,255 However, nearly all polarizable simulation methods based on charge-transfer methods have used some sort of delocalized charges, rather than point charges. 22,125,126,130,146,158,171 This approach has been shown to be successful at reproducing the electrostatic potential for most extramolecular sites, although the use of point dipoles can improve the performance for certain conformations (such as bifurcated hydrogen bonds) in which molecular symmetries prevent accurate charge distributions. 146 Indeed, it has been claimed that the better representation of intermolecular interactions due to diffuse charges is as important as the use of polarizability. 205 The chemical potential equalization (CPE) methods are
noteworthy in this regard because they use both diffuse monopoles and dipoles to represent the system’s polarization. 130,131,171 The difference between models having polarizable point dipoles and fixed point charges and those with fluctuating charges and fixed Lennard– Jones interactions reduces to considering which term is static and which is polarizable. For the PPD model, the charge–charge term is static and the induced dipole–induced dipole term is polarizable. For the EE model, the charge– charge term is polarizable and the induced dipole–induced dipole terms (included in the Lennard–Jones r 6 interaction) are static. Note that including a Lennard–Jones r 6 dispersion term is not redundant for polarizable models because this represents the interaction arising from correlated thermal fluctuations of the induced dipole. With a few exceptions, 22,57,202 most models— whether based on matrix, iterative, or extended Lagrangian algorithms—are adiabatic and do not allow for substantial fluctuations away from the minimum-energy polarization state. SUMMARY AND CONCLUSIONS Summary and Conclusions 133 There are a variety of different models used to treat polarizability in molecular simulations: polarizable point dipoles, shell models, fluctuating charge models, and semiempirical models, along with variations and combinations of these. There are advantages and disadvantages of each model, as discussed in detail in previous sections. These relative merits range from differing computational efficiencies and ease of implementation to different accuracies in representing the external electric field and transferability of parameters. Regardless of the differences in convenience and efficiency, the most important consideration when choosing a polarizable model for a particular problem should be the model’s applicability to the system in question. Despite the many differences between the various polarizable models, it is encouraging to note that the most recent models seem to be converging on the same set of necessary features. A variety of successful models based on different formalisms all share many of the same characteristics. 126,130,131,146,150,166,171,205 Regardless of the direction from which the models evolved, there is a growing consensus that accurate treatment of polarization requires (1) either diffuse charge distributions or some other type of electrostatic screening (2) a mixture of both monopoles and dipoles to represent the electrostatic charge distribution, and (3) only linear polarizability. Although much work remains to be done before there is a truly accurate, transferable model for a wide range of conditions and systems, it is fair to say that polarizable models have matured considerably since their earliest implementations. Future developments will almost certainly include continued development and parameterization of the more mainstream models, along with their incorporation into commercial and academic simulation software
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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
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
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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: ecome significant at field strength
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
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Page 203 and 204: Table 1 Main Features of the Two-Pa
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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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