Reviews in Computational Chemistry Volume 18

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108 Polarizability in Computer Simulations U(q)−E 0 (eV) 15 10 5 0 −5 −1 0 q (e) Figure 3 Energy versus charge for chlorine (solid line) and sodium (dashed line). The lines are a quadratic fit through the energies of the ions relative to the neutral atom. from the Coulomb interaction between delocalized charge distributions rðrÞ, rather than point charges, ð riðriÞrjðrjÞ JijðrÞ ¼ jri rj rj dri drj ½37Š The charge distributions are frequently assumed to be spherical, for simplicity. 125–128 Directional interactions can be incorporated with nonspherical charge distributions, at some added computational expense. 129–131 The partial charges on each atom of the molecule are found by minimizing the energy, subject to a constraint that the total charge is conserved. X ½38Š i qi ¼ qtot The charge conservation constraint can be enforced using an undetermined multiplier, UðqÞ ¼UðqÞ l X ! ½39Š i qi qtot Minimizing this expression for the energy with respect to each of the qi under the assumption that the molecule in question is neutral ðqtot ¼ 0Þ gives 1
for all i ði:e:; 8 iÞ: @U @qi l ¼ 0 8 i ½40Š Because ð@U=@qiÞ for each atom is equal to the same undetermined multiplier l, this quantity must be identical for all atoms in the molecule, 132 @U @qi ¼ @U @qj 8 i; j ½41Š Through Mulliken’s identification of @U=@q as the electronegativity, we see that minimizing the energy with respect to the charges is equivalent to equalizing the electronegativities, w i Electronegativity Equalization Models 109 @U @qi ¼ w 0 i þ Jii qi þ X JijðrijÞqj for all atoms. Notice that the electronegativity of atom i in a molecule, wi, differs from the electronegativity of the isolated atom, w0 i , and now depends on its charge, the charge of the other atoms, its hardness, and the interactions with other atoms through JijðrijÞ. In addition, Parr et al. 132 identified the chemical potential of an electron as the negative of the electronegativity, m ¼ @U=@q. So electronegativity equalization is equivalent to chemical potential equalization. Thus, this model effectively moves charge around a molecule to minimize the energy or to equalize the electronegativity or chemical potential. These interpretations are all equivalent (for a dissenting opinion, see Ref. 133). Electronegativity equalization (EE) was first proposed by Sanderson. 134 The EE model, with appropriate parameterization, has been successful in predicting the charges of a variety of molecules. 125,135–138 The parameters w0 and J are not typically assigned from Eqs. [33] and [34], but instead are taken as parameters to be optimized and can be viewed as depending on the valence state of the atom, as indicated by electronic structure calculations. 139,140 Some models 136 set JabðrijÞ ¼1=rij, and others use some type of screening. 125,135,137 In addition, some models have an expression for the energy that is not quadratic. 125,135,141 Going beyond the quadratic term in the Taylor expansion of Eq. [32] can possibly extend the validity of the model, but it introduces complications in the methods available for treating the charge dynamics, as will be discussed below. For a collection of molecules, the overall energy is comprised of the energy given by Eq. [36] for each molecule and an interaction between charge j 6¼ i ½42Š
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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
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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: The energy required to create a cha
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
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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
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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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