Reviews in Computational Chemistry Volume 18

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106 Polarizability in Computer Simulations comparable energy conservation, 90,99,101,105 the dynamic methods can have a computational advantage, in some cases by as much as a factor of two to four. 90 Because the shell model represents each polarizable site with two point charges, it replaces the dipole–dipole interactions in the polarizable point dipole models with four charge–charge interactions. The greater number of pair distances largely offsets the computational advantage of the simpler interaction, and energy and force evaluations in the two methods are comparable in speed. Between the reduced time step and the greater number of interactions, shell models typically require 10 times more CPU time than corresponding nonpolarizable simulations. 105,106 In the shell model, as mentioned above, the short-range repulsion and van der Waals interactions are taken to act between the shell particles. This finding has the effect of coupling the electrostatic and steric interactions in the system: in a solid-state system where the nuclei are fixed at the lattice positions, polarization can occur not only from the electric field generated by neighboring atoms, but also from the short-range interactions with close neighbors (as, e.g., in the case of defects, substitutions, or surfaces). This ability to model both electrical and mechanical polarizability is one reason for the success of shell models in solid-state ionic materials. 73,107 There exist a variety of extensions of the basic shell model. One variation for molecular systems uses an anisotropic oscillator to couple the core and shell charges, 99,108 thus allowing for anisotropic polarizability in nonspherical systems. Other modifications of the basic shell model that account for explicit environment dependence include a deformable or ‘‘breathing’’ shell 75,76,109 and shell models allowing for charge transfer between neighboring sites. 75,76,110 Shell models have been used successfully in a wide variety of systems. The greatest number of applications have been in the simulation of ionic materials, 86–88,90,111 especially systems including alkali halides, 83 ox- ides, 85,89,91,92,112–115 and zeolites. 93,94 The shell model is also commonly used for the simulation of molten salts, 77,84,90,101,116–120 and shell-type models have been developed for various molecular 95–99 and polymeric species. 100,121 ELECTRONEGATIVITY EQUALIZATION MODELS Polarizability can also be introduced into standard potentials (Eq. [1]) by allowing the values of the partial charges to respond to the electric field of their environment. A practical advantage of this approach is that it introduces polarizability without introducing new interactions. And unlike the shell model, this can be accomplished using the same number of charge–charge interactions as would be present in a nonpolarizable simulation. Another more conceptual advantage is that this treats the polarizable and permanent electrostatic interactions with the same multipoles. One way to couple the charges to their environment is by using electronegativity equalization.
The energy required to create a charge, q, on an atom can be expressed as a Taylor series expansion, UðqÞ ¼E 0 þ w 0 q þ 1 2 Jq2 which has been truncated after the second-order terms. If the Taylor series is valid for charges of up to 1 e, then, because the ionization potential, IP, is equal to Uð1Þ Uð0Þ and the electron affinity, EA, is Uð 1Þ Uð0Þ, the Taylor series coefficients are ½32Š w 0 ¼ðIP þ EAÞ=2 ½33Š J ¼ IP EA ½34Š Equation [33] is Mulliken’s definition of electronegativity, 122 so the linear coefficient in the Taylor series is the electronegativity of the atom. Mulliken’s definition is consistent with other electronegativity scales. The second-order coefficient, 1 2 J, is the ‘‘hardness’’ of the atom, Z.123 For semiconductors, the hardness is half the band gap, and Z is an important property in inorganic and acid–base chemistry. 124 Physically, IP EA is the energy required to transfer an electron from one atom to another atom of the same type, 2AðgÞ !A þ ðgÞþA ðgÞ E ¼ IP EA ½35Š This energy is always positive (in fact, it is positive even if the two atoms are not the same element), so J 0. Figure 3 shows UðqÞ for chlorine and sodium, as calculated from the experimental IP and EA. The energies of the ions, w0 , and J are all calculated using the experimental IP and EA. Chlorine is more electronegative than sodium ðw0 Na ¼ 2:84 eV; w0 Cl ¼ 8:29 eVÞ and also harder ðJNa ¼ 4:59 eV, JCl ¼ 9:35 eVÞ. This means that both the slope and the second derivative of UðqÞ are larger for Cl than for Na. When atoms are brought together to form molecules, the energy of the charges is described in the EE model as UðqÞ ¼ X i Electronegativity Equalization Models 107 E 0 i þ w0 i qi þ 1 2 Jiiq 2 i X X þ JijðrijÞqiqj The vector q represents the set of qi. The second-order coefficient, JijðrijÞ, depends on the distance between the two atoms i and j, and at large distances should equal 1=rij. At shorter distances, there may be screening of the interactions, just as for the dipole–dipole interactions in the earlier section on Polarizable Point Dipoles. This screened interaction is typically assumed to arise i j > i ½36Š
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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 131 and 132: Shell Models 103 on estimates of sh
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Page 139 and 140: where we have used q Cl ¼ qNa. The
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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
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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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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