Reviews in Computational Chemistry Volume 18

More documents

Recommendations

Info

96 Polarizability in Computer Simulations molecular polarizabilities are not that far off, with the Applequist method tending to overestimate the polarization anisotropies. Various computer simulation models have used either the Applequist parameters and no screening 15,16,24,27,28,32,34 or the Thole parameters and screening of Tij. 19,31,38 Different screening functions have been used as well. A large number of polarizable models have been developed for water, many of them with one polarizable site (with a ¼ 1:44 A ˚ 3 ) on or near the oxygen position. 20–23,26,29,30,33,35–37 For these models, the polarizable sites do not typically get close enough for polarization catastrophes {ð4aaÞ 1=6 ¼ 1:4 A ˚ , see comments after Eq. [16]}, so screening is not as necessary as it would be if polarization sites were on all atoms. However, some water models with a single polarizable site do screen the dipole field tensor. 20,22,37 Another model for water places polarizable sites on bonds. 25 Other polarizable models have been used for monatomic ions and used no screening of T or E 0 . 15,16,27,34 Polarizable models have been developed for proteins as well, by Warshel and co-workers (with screening of T but not E 0 ), 44,45 and by Wodak and co-workers (with no screening). 46 An attractive feature of the dipole polarizable model is that the assignment of electrostatic potential parameters is more straightforward than for nonpolarizable models. Charges can be assigned on the basis of experimental dipole moments or ab initio electrostatic potential charges for the isolated molecule. The polarizabilities can be assigned from the literature (as in Table 1) or calculated. Contrarily, with nonpolarizable models, charges may have some permanent polarization to reflect their enhanced values in the condensed phase. 6,47 The degree of enhancement is part of the art of constructing potentials and limits the transferability of these potentials. By explicitly including polarizability, the polarizable models are a more systematic approach for potential parameterization and are therefore more transferrable. Using Eqs. [9] and [11], the energy can be rewritten as U ind ¼ XN i ¼ 1 l i E 0 i 1 þ 2 X N X li Tij lj þ 1 2 i ¼ 1 j 6¼ i and the derivative of U ind with respect to the induced dipoles is =l i U ind ¼ E 0 i þ X j 6¼ i X N i ¼ 1 l ia 1 i l i ½18Š Tij l j þ a 1 i l i ¼ 0 ½19Š The derivative in Eq. [19] is zero because a 1 i li ¼ E0 P i Tij lj, according to Eq. [3]. The values of the induced dipoles are therefore those that minimize the energy. Other polarizable models also have auxiliary variables, analogous to l, which likewise adjust to minimize the energy.
The polarizable point dipole models have been used in molecular dynamics (MD) simulations since the 1970s. 48 For these simulations, the forces, or spatial derivatives of the potential, are needed. From Eq. [18], the force 23 on atomic site k is F k ¼ r kU ind ¼ XN i ¼ 1 l ir kE 0 i X þ lk ðrkTkiÞ li ½20Š All contributions to the forces from terms involving derivatives with respect to the dipoles are zero from the extremum condition of Eq. [19]. 23,49 Finding the inducible dipoles requires a self-consistent method, because the field that each dipole feels depends on all of the other induced dipoles. There exist three methods for determining the dipoles: matrix inversion, iterative methods, and predictive methods. We describe each of these in turn. The dipoles are coupled through the matrix equation, A l ¼ E 0 Polarizable Point Dipoles 97 where the diagonal elements of the matrix, Aii, are a 1 i and the off-diagonal elements Aij are Tij. For a system with N dipoles, solving for each of them involves inverting the N N matrix, A—an OðN3Þ operation that is typically too computationally expensive to perform at each step of an OðNÞ or OðN2Þ simulation. Consequently, this method has been used only rarely. 31 Note that since Eq. [21] for l is linear, there is only one solution for the dipoles. In the iterative method, an initial guess for the field is made by, for example, just using the static field, E0 , or by using the dipoles from the previous time step of the MD simulation. 48,49 The dipole moments resulting from this field are evaluated using Eq. [3], which can be iterated to self-consistency. Typical convergence limits on the dipoles range from 1 10 2 D to 1 10 6 D. 21,27,34–36,50 Long simulations require very strict convergence limits or mild thermostatting 50 to prevent problems due to poor energy conservation. Alternatively, the energy Upol can be monitored for convergence. 19,51 The level of convergence, and therefore the number of iterations required, varies considerably. Between 2 and 10 iterations are typically required. For some calculations, including free energy calculations, a high level of convergence may be necessary. 38 The iterative method is the most common method for finding the dipoles. The predictive methods determine l for the next time step based on information from previous time steps. Ahlström et al. 23 used a first-order predictor algorithm, which uses the l values from the two previous times steps to predict l at the next time step, i 6¼ k ½21Š l iðtÞ ¼2l iðt tÞ l iðt 2 tÞ ½22Š
Page 1 and 2:
Reviews in Computational Chemistry
Page 3 and 4:
Kenny B. Lipkowitz Department of Ch
Page 5 and 6:
vi Preface three-dimensional struct
Page 7 and 8:
viii Preface some descriptors and i
Page 9 and 10:
Epilogue and Dedication My associat
Page 11 and 12:
Contents 1. Clustering Methods and
Page 13 and 14:
Contents xv Electron Transfer in Po
Page 15 and 16:
Contributors John M. Barnard, Barna
Page 17 and 18:
Contributors to Previous Volumes *
Page 19 and 20:
Volume 3 (1992) Tamar Schlick, Opti
Page 21 and 22:
Volume 7 (1996) Geoffrey M. Downs a
Page 23 and 24:
Volume 11 (1997) Mark A. Murcko, Re
Page 25 and 26:
T. Daniel Crawford* and Henry F. Sc
Page 27 and 28:
Topics Covered in Volumes 1-18 * Ab
Page 29 and 30:
Reviews in Computational Chemistry
Page 31 and 32:
2 Clustering Methods and Their Uses
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
10 Clustering Methods and Their Use
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
42 The Use of Scoring Functions in
Page 73 and 74: 44 The Use of Scoring Functions in
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: Table 1 Polarizability Parameters f
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
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
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
Page 199 and 200:
This result indicates a fundamental
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
Page 213 and 214:
This situation is of course not sat
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
Page 221 and 222:
of the radiation is the perturbatio
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
Page 229 and 230:
ε/M −1 cm −1 4000 2000 0 analy
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
show all

Reviews in Computational Chemistry Volume 18

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?