Reviews in Computational Chemistry Volume 18

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92 Polarizability in Computer Simulations proportional to the electric field at that site, Ei. The proportionality constant is the polarizability tensor, ai. The dipole feels an electric field both from the permanent charges of the system and from the other induced dipoles. The expression for the l i is l i ¼ ai Ei ¼ ai E 0 i " # X j 6¼ i Tijl j where E0 is the field from the permanent charges. (There also may be permanent dipoles or other multipoles present contributing to E0 .) The induced dipoles interact through the dipole field tensor, Tij, 0 1 Tij ¼ 1 I r3 ½3Š 3 r5 x2 xy xz yx y2 yz zx zy z2 B C @ A ½4Š where I is the identity matrix, r is the distance between i and j, and x, y, and z are the Cartesian components of the vector between i and j. The energy of the induced dipoles, U ind, can be split into three terms, U ind ¼ Ustat þ Umm þ U pol The energy Ustat is the interaction energy of the N induced dipoles with the permanent, or static, field Ustat ¼ XN i ¼ 1 ½5Š l i E 0 i ½6Š the energy Umm is the induced dipole–induced dipole interaction Umm ¼ 1 2 and the polarization energy, U pol, X N X li Tij lj ½7Š i ¼ 1 j 6¼ i Upol ¼ 1 2 X N i ¼ 1 l i Ei ½8Š is that required to distort the electron distribution to create the dipoles. 4,14 Any polarizable model in which dipole moments, charges, or other multipoles
are modified by their environment will have a polarization energy corresponding to U pol. Even nonpolarizable models that are parameterized to have charges enhanced from the gas-phase values should include such a term, and U pol has been called the ‘‘missing term’’ in many pair potentials. 4,5 By using Eq. [3], the electric field can be replaced by a 1 l i and U pol can be written as Upol ¼ 1 2 X N i ¼ 1 l i a 1 i l i ½9Š where a 1 i is the inverse of the polarization tensor. If the polarization matrix is isotropic ðaxx ¼ ayy ¼ azz ¼ aiÞ and diagonal, then U pol ¼ XN Combining the three energy terms gives which, using Ei ¼ E0 i static field, E0 , and Uind ¼ XN li i ¼ 1 E 0 i i ¼ 1 þ 1 2 m 2 i 2ai " # X j 6¼ i Tij l j þ 1 2 Ei ½10Š ½11Š P Tij l j (Eq. [3]), reduces to the relationship for the Uind ¼ 1 2 X N i ¼ 1 l i E 0 i ½12Š Note that the energy is the dot product of the induced dipole and the static field, not the total field. 15–19 Without a static field, there are no induced dipoles. Induced dipoles alone do not interact strongly enough to overcome the polarization energy it takes to create them (except when they are close enough to polarize catastrophically). The static field at site i due to permanent charges is E 0 i qjrij r j 6¼ i 3 ij ¼ X Polarizable Point Dipoles 93 where qj is the charge at site j and rij is the distance between i and j. A point charge at a site is generally assumed not to contribute to the field at that site. For rigid models of water and other small molecules, charges on the same molecule contribute a constant amount to the electric field at each site ½13Š
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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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Page 69 and 70: 40 Clustering Methods and Their Use
Page 71 and 72: 42 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: Vn (1 + cos(nω + γ)) 2 K θ (θ
Page 123 and 124: Table 1 Polarizability Parameters f
Page 125 and 126: The polarizable point dipole models
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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
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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
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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.
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References 143 178. J. J. P. Stewar
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References 145 216. M. W. Mahoney a
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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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Reviews in Computational Chemistry Volume 18

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