Reviews in Computational Chemistry Volume 18

More documents

Recommendations

Info

110 Polarizability in Computer Simulations sites on different molecules, Uðfqg; frgÞ ¼ X X w a i 2 a 0 i qi þ 1 X X 2 i 2 a j 2 a þ X X X X a b > a i 2 a j 2 b qi qj JijðrijÞ E gp a ! qi qj JijðrijÞ ½43Š where a and b label the molecules, and i and j represent atoms (or other charge sites) in these molecules. The E gp a term represents the gas-phase energy of molecule a and defines the zero of energy as corresponding to infinitely separated molecules. The energy given by Eq. [43] replaces the Coulomb energy qiqj=rij in Eq. [1]. The charges qi are now treated as independent variables, and the polarization response is determined by variations in the charge values. These charges depend on the interactions with other molecules as well as other charge sites on the same molecule, and will change for every time step or configuration sampled during a simulation. The charge values used for each configuration are, in principle, those that minimize the energy given by Eq. [43]. This method for treating polarizability has thus been called the fluctuating charge method 126 and has been applied to a variety of systems. 10,82,104,126,142–148 The JijðrÞ interaction between different molecules is typically taken to be 1=r, although the interactions between atoms on the same molecule may be screened. Therefore, this method does not modify the intermolecular interactions. Charge conservation can be imposed in either of two ways. A charge neutrality constraint can be applied to the entire system, allowing charge to move from atomic site to atomic site until the electronegativities are equal on all the atoms of the system. Alternatively, charge can be constrained independently on each molecule (or other subgroup), so that charge flows only between atoms on the same molecule until the electronegativities are equalized within each molecule, but not between distinct molecules. 126 In most cases, charge is taken to be conserved for each molecule, so there is no charge transfer between molecules. Variations, including the atom–atom charge transfer (AACT) 149 and the bond-charge increment (BCI) 146,150 model, only allow for charge to flow between two atoms that are directly bonded to each other, guaranteeing that the total charge of each set of bonded atoms is conserved. In some situations, charge transfer is an important part of the interaction energy, so there are reasons to remove this constraint. 151–154 However, this can lead to some nonphysical charge transfer, as illustrated in the simple example of a gas-phase sodium chloride molecule. The energy for one Na atom and one Cl atom is UðqÞ ¼E 0 Na þ E0 Cl þðw Na wClÞqNa þ 1 h JNa þ JCl 2 i 2 JNaClðrNaClÞ q 2 Na ½44Š
where we have used q Cl ¼ qNa. The charge that minimizes this energy is ðwNa qNa ¼ JNa þ JCl wClÞ 2JNaClðrNaClÞ At large distances, J NaClðr NaClÞ approaches zero, and, if the w and J parameters are taken from Eqs. [33] and [34], then qNa ¼ ðw Na w ClÞ JNa þ J Cl Electronegativity Equalization Models 111 ¼ 1 2 ðIPNa þ EANa IPCl EAClÞ IPNa EANa þ IPCl EACl which gives qNa ¼ 0:391 e. Thus the model predicts a significant amount of charge transfer, even at large distances. Similar errors in the dissociation limit are seen with certain electronic structure methods. 155,156 A significant amount of charge separation, and a consequent overestimation of the dipole moment, can be found for large polymers as well. Reducing this charge transfer along the polymer can be accomplished with the AACT and BCI models. 146,149,150 In addition, when comparing fluctuating charge models with ab initio results for water trimers, agreement was found to be much better for the model without charge transfer, even after the charge-transfer model was reparameterized by fitting to the ab initio three-body energies. 145 These and associated problems with overestimated charge transfer are a general characteristic of EE-based models. Unfortunately, such errors cannot be eliminated through parameterization; the problem is a side effect of attempting to treat quantum mechanical charge-transfer effects in a purely classical way. As with all empirical potentials, the use of fractional charges is necessary for an accurate description of the electrostatic potential. Yet by allowing fractional charge transfer, the EE model has no means of enforcing the transfer of only an integral number of electrons between distant species. Indeed, the neutral dissociation products for NaCl are correctly predicted by the EE model, if the infinitely separated ions are constrained to have integer charge (see Figure 3). This constraint is difficult to apply in practice, however. As discussed recently by Morales and Martinez, 157 the EE-based models can be viewed as analytically differentiable approximations to a more rigorous statistical interpretation of UðqÞ, which is discontinuous at integer values of charge transfer and correctly predicts zero charge transfer at infinite distance. In chemically bonded systems, the assumption of partial charge transfer is not as unrealistic as in ionic compounds, as electrons are delocalized across covalent bonds. However, in these covalent cases the EE model effectively assumes that the coherence length of a delocalized electron is infinite and does not depend on the surroundings. It is for this reason that the polarizability of polymers, for example, increases too quickly with chain length under the EE model. 149 Molecular charge constraints can avoid problems at the dissociation ½45Š ½46Š
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:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Table 1 Reference List for the Most
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88: 58 The Use of Scoring Functions in
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
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: for all i ði:e:; 8 iÞ: @U @qi l
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

Create successful ePaper yourself

Delete template?

Save as template?