Reviews in Computational Chemistry Volume 18

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90 Polarizability in Computer Simulations Carlo computer simulations, and reviews some of the systems that have been simulated with polarizable potentials. NONPOLARIZABLE MODELS Before discussing polarizable models, a useful starting point is to consider nonpolarizable models. A typical nonpolarizable potential for molecular systems is 1 U ¼ X KBðr r0Þ 2 þ X Kyðy y0Þ 2 þ X bonds þ X nonbonded pairs 4eij angles sij rij 12 sij rij 6 dihedrals X ( " # ) þ qiqj rij n Vn ð1 þ cosðnf gÞÞ 2 where U represents the potential energy of the system. There are terms for the bond length, r, with a force constant, KB, and an equilibrium bond length, r0; the bond angle, y, with a force constant Ky and an equilibrium angle, y0; and the dihedral angle, f, with barrier heights, Vn, and equilibrium angles, g. The intermolecular interactions are described with a Lennard–Jones (LJ) interaction, " # 12 6 sij sij ULJðrÞ ¼4eij ½2Š in which e and s are parameters describing the energy and distance scale of the interactions, respectively, and rij is the distance between nonbonded atoms i and j. The Coulomb interaction between charged atoms is given by qiqj=rij, where qi is the partial charge on atom i. These interactions are illustrated in Figure 1. The Lennard–Jones interaction contains a short-range repulsive part, falling off as r 12 , and a longer range attractive part, falling off as r 6 . The attractive part has the same dependence as the (dipole–dipole) London dispersion energy, which for two particles with polarizabilities a is proportional to a 2 =r 6 (Ref. 2). The Lennard–Jones parameters are not typically assigned 3 using known values of a, but this interaction is one way in which polarizability, in an average sense, is included in nonpolarizable models. Another way in which polarizability is included implicitly is in the value of the partial charges, qi, that are assigned to the atoms in the model. The charges used in potential energy models for condensed phases are often enhanced from the values that would be consistent with the gas-phase dipole moment, or those that would best reproduce the electrostatic potential (ESP) rij rij ½1Š
Vn (1 + cos(nω + γ)) 2 K θ (θ − θ 0 ) 2 from gas-phase ab initio calculations. Enhanced charge values are a means of accounting for the strong polarization of electron distributions by the electric fields of the other particles in a condensed phase environment. The enhanced charges are obtained either through explicit parameterization 4,5 or by using charges obtained via quantum chemical methods that are known to overestimate charge values. 6 Although the enhanced charge values treat polarization in an effective way, they cannot correctly reflect the dependence of charge distributions on the system’s state, nor can they respond dynamically to fluctuations in the electric field due to molecular motion. The average electric field, and therefore the charge distribution and dipole moment, will depend on the physical state and composition of the system. For example, a molecule in a solution with a high ionic strength may feel a field different from a molecule in a pure solvent; even in the bulk liquid state, the polarization of a water molecule will depend on the density, and thus on the system’s temperature and pressure. In addition, conformational changes may influence the charge distribution of a molecule. 7–13 Molecular motions in the system will result in conformational changes and fluctuations in the electric field, causing the electrostatic distribution to change on a subpicosecond time scale. Treating these effects requires a polarizable model. POLARIZABLE POINT DIPOLES C O H H O H N K B (r − r 0 ) 2 (q iq j /r ij ) + 4 ij Polarizable Point Dipoles 91 One method for treating polarizability is to add point inducible dipoles on some or all atomic sites. This polarizable point dipoles (PPD) method has been applied to a wide variety of atomic and molecular systems, ranging from noble gases to water to proteins. The dipole moment, l i, induced on a site i is ∋ σ ij r ij 12 σ 6 ij − rij Figure 1 Schematic of the interactions between an amino group and a water showing the Lennard–Jones and electrostatic nonbonded interactions along with the bond length, bond angle, and dihedral angle (torsional) interactions.
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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 117: CHAPTER 3 Potentials and Algorithms
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Page 123 and 124: Table 1 Polarizability Parameters f
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Page 167 and 168: References 139 Computational Chemis
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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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