Reviews in Computational Chemistry Volume 18

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98 Polarizability in Computer Simulations where t is the time step, and t is time. This method is not stable for long times, but can be combined with an iterative solution, either by providing the initial iteration of the electric field values, 52,53 or by allowing the iteration to be performed less frequently than every step. 23,54 Higher-order predictor algorithms have been used as well. 23,52,55 A different predictive procedure is to use the extended Lagrangian method, in which each dipole is treated as a dynamical variable and given a mass Mm and velocity _l. The dipoles thus have a kinetic energy, 1 P 2 i Mm _l 2 i and are propagated using the equations of motion just like the atomic coordinates. 22,56–58 The equation of motion for the dipoles is Mml i ¼ =l i U ind ¼ Ei a 1 i l i ½23Š Here l i is the second derivative with respect to time, that is, the acceleration. The dipole mass does not correspond to any physical mass of the system; it is chosen for numerical convenience, by, for example, comparing the trajectories with those from the iterative method. 57 It is desirable to keep the kinetic energy of the dipoles small so that the dipole degrees of freedom are cold and near the potential energy minimum (corresponding to the exact solution of Eq. [3]). Because this method avoids iterations, which require recalculating Ei multiple times for every sampled configuration, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. But even with methods that allow for only a single evaluation of the energy and force per time step, polarizable point dipole methods are more computationally intensive than nonpolarizable simulations. Evaluating the dipole– dipole interactions in Eqs. [7] and [20] is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. [1]. A widely used rule of thumb is that polarizable simulations based on a point dipole model take roughly four times longer than a nonpolarizable simulation of the same system. The polarizable point dipole model has also been used in Monte Carlo simulations with single particle moves. 19,21,24,59–62 When using the iterative method, a whole new set of dipoles must be computed after each molecule is moved. These updates can be made more efficient by storing the distances between all the particles, since most of them are unchanged, but this requires a lot of memory. The many-body nature of polarization makes it more amenable to molecular dynamics techniques, in which all particles move at once, compared to Monte Carlo methods where typically only one particle moves at a time. For nonpolarizable, pairwise-additive models, MC methods can be efficient because only the interactions involving the moved particle need to be recalculated [while the other ðN 1Þ ðN 1Þ interactions are unchanged]. For polarizable models, all N N interactions are, in principle, altered when one particle moves. Consequently, exact polarizable MC calculations can be
two to three orders of magnitude slower than comparable nonpolarizable calculations. 63 Various approximate methods, involving incomplete convergence or updating only a subset of the dipoles, have been suggested. 59 Unfortunately, these methods result in significant errors in computed physical properties. 19,63 Monte Carlo methods are capable of moving more than one particle at a time, with good acceptance ratios, 64,65 using, for example, the hybrid MC technique, but this method has not been applied to polarizable models, as far as we are aware. One final point concerns the long-range nature of the interactions in dipole-based models. Dipole–dipole and dipole–charge interactions are termed long range because they do not decrease faster than volume grows—that is, as r 3 . If periodic boundary conditions are used, some treatment of the long-range interactions is needed. The most complete treatment of the long-range forces is the Ewald summation technique. 64,66 All models, whether polarizable or not, face this problem if they have long-range forces, but for polarizable models this is a more significant issue. The use of cut-offs or other truncation schemes will change both the static field and the dipole field tensor. These changes to the electric field will modify the value of the induced dipole, which in turn will change the field at other sites. Accordingly, the treatment of long-range forces feeds back on itself in a way that does not occur with nonpolarizable models. It is thus crucial to treat the long-range interactions as accurately as possible in polarizable simulations. Nevertheless, a large number, if not most, of the simulations using polarizable potentials have not used Ewald sums. Recently, Nymand and Linse 67 showed that different boundary conditions (including Ewald sums, spherical cut-off, and reaction field methods) lead to more significant differences in equilibrium, dynamical, and structural properties for polarizable water models than for nonpolarizable models. Conventional methods for performing the Ewald sum scale as OðN 3=2 Þ or OðN 2 Þ, 68 and formulations specifically designed to include dipole–dipole interactions 66 are in fairly wide use. Faster scaling methods, such as the fast multipole and particle–mesh algorithms, have also been extended to the treatment of point dipoles. 50,69 SHELL MODELS Shell Models 99 A defining feature of the models discussed in the previous section, regardless of whether they are implemented via matrix inversion, iterative techniques, or predictive methods, is that they all treat the polarization response in each polarizable center using point dipoles. An alternative approach is to model the polarizable centers using dipoles of finite length, represented by a pair of point charges. A variety of different models of polarizability have used this approach, but especially noteworthy are the shell models frequently used in simulations of solid-state ionic materials.
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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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44 The Use of Scoring Functions in
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Page 79 and 80: Table 1 Reference List for the Most
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: The polarizable point dipole models
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Page 131 and 132: Shell Models 103 on estimates of sh
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Page 135 and 136: The energy required to create a cha
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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 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
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Introduction 149 applications). For
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FCWD(∆E ) FCWD(0) ∆E = 0 ∆E
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Introduction 153 Equations [6]-[12]
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Paradigm of Free Energy Surfaces 15
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Paradigm of Free Energy Surfaces 15
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In Eqs. [18] and [19], F0i is the e
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where ‘‘þ’’ and ‘‘ ’
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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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