| 60 |
|
\section{Introduction} |
| 61 |
|
|
| 62 |
|
Over the past several years, there has been increasing interest in |
| 63 |
< |
pairwise methods for correcting electrostatic interactions in computer |
| 63 |
> |
pairwise methods for computing electrostatic interactions in |
| 64 |
|
simulations of condensed |
| 65 |
|
systems.\cite{Wolf99,Zahn02,Kast03,Beckd.A.C._Bi0486381,Ma05,Fennell06} |
| 66 |
|
These techniques were initially developed from the observations and |
| 73 |
|
using shifted force approximations at the cutoff distance in order to |
| 74 |
|
conserve total energy in molecular dynamics simulations.\cite{Zahn02, |
| 75 |
|
Fennell06} In the previous two papers in this series we developed |
| 76 |
< |
three generalized real space methods charges: shifted potential (SP), |
| 77 |
< |
gradient shifted force (GSF), and Taylor shifted force |
| 78 |
< |
(TSF).\cite{PaperI, PaperII} These methods provide electrostatic |
| 79 |
< |
interactions for higher order multipoles (dipoles and quadrupoles) |
| 80 |
< |
using finite cutoff sphere with the neutralization of the |
| 81 |
< |
electrostatic moments within the cutoff sphere. The multipolar |
| 82 |
< |
generalizations of shifted force terms provides additional terms in |
| 83 |
< |
the potential energy so that force and torque vanish smoothly at the |
| 84 |
< |
cutoff radius. This ensures that the total energy is conserved in |
| 85 |
< |
molecular dynamics simulations. |
| 76 |
> |
three generalized real space methods: shifted potential (SP), gradient |
| 77 |
> |
shifted force (GSF), and Taylor shifted force (TSF).\cite{PaperI, |
| 78 |
> |
PaperII} These methods provide real-space electrostatic interactions |
| 79 |
> |
for higher order multipoles (e.g. dipoles and quadrupoles) using a |
| 80 |
> |
finite cutoff sphere with neutralization of the electrostatic moments |
| 81 |
> |
within the cutoff region. The multipolar generalizations of the |
| 82 |
> |
shifted force approach provide additional terms in the potential |
| 83 |
> |
energy so that force and torque vanish smoothly at the cutoff |
| 84 |
> |
radius. This ensures that the total energy is conserved in molecular |
| 85 |
> |
dynamics simulations. |
| 86 |
|
|
| 87 |
|
One of the most difficult tests of any new electrostatic method is the |
| 88 |
|
fidelity with which that method can reproduce the bulk-phase |
| 89 |
|
polarizability or equivalently, the dielectric properties of a |
| 90 |
|
fluid. Since dielectric properties are macroscopic properties, all |
| 91 |
< |
interactions between molecules in the system contribute. Well before |
| 92 |
< |
the advent of computer simulations, Kirkwood and Onsager developed |
| 91 |
> |
interactions between molecules in the system contribute. Before the |
| 92 |
> |
advent of computer simulations, Kirkwood and Onsager developed |
| 93 |
|
fluctuation formulae for the dielectric properties of dipolar |
| 94 |
|
fluids.\cite{Kirkwood39,Onsagar36} Similar developments were made by |
| 95 |
|
Logan \textit{et al.} for the bulk polarizability of quadrupolar |
| 126 |
|
% cutoff method (See equation |
| 127 |
|
% \ref{dipole-diopleTensor}).\cite{Neumann83} |
| 128 |
|
|
| 129 |
< |
Zahn \textit{et al.}\cite{Zahn02} evaluated the correction factor for |
| 130 |
< |
using damped shifted charge-charge kernel (see equation |
| 131 |
< |
\ref{dipole-chargeTensor}). This was later generalized by Izvekove |
| 132 |
< |
\textit{et al.},\cite{Izvekov:2008wo} who showed that the expression |
| 133 |
< |
for the dielectric constant reduces to widely-used \textit{conducting |
| 129 |
> |
Zahn \textit{et al.}\cite{Zahn02} utilized this approach and evaluated |
| 130 |
> |
the correction factor for using damped shifted charge-charge |
| 131 |
> |
kernel. This was later generalized by Izvekov \textit{et |
| 132 |
> |
al.},\cite{Izvekov:2008wo} who showed that the expression for the |
| 133 |
> |
dielectric constant reduces to widely-used \textit{conducting |
| 134 |
|
boundary} formula for real-space cutoff methods that have first |
| 135 |
< |
derivatives (forces) that vanish at the cutoff sphere. |
| 136 |
< |
|
| 137 |
< |
The relationship between quadrupolar susceptibility and dielectric |
| 138 |
< |
constant is not as straight forward for quadrupolar fluids as in the |
| 139 |
< |
dipolar case. The dielectric constant depends on the geometry of the |
| 140 |
< |
external field perturbation.\cite{Ernst92} Many studies have also been |
| 141 |
< |
conducted to understand solvation theory using dielectric properties |
| 142 |
< |
of these fluids,\cite{JeonI03,JeonII03,Chitanvis96} although a |
| 143 |
< |
correction formula for different cutoff methods has not yet been |
| 144 |
< |
developed. |
| 135 |
> |
derivatives that vanish at the cutoff sphere. |
| 136 |
|
|
| 137 |
< |
In this paper we derive general formulas for calculating the |
| 137 |
> |
In quadrupolar fluids, the relationship between quadrupolar |
| 138 |
> |
susceptibility and the dielectric constant is not as straightforward |
| 139 |
> |
as in the dipolar case. The dielectric constant depends on the |
| 140 |
> |
geometry of the external field perturbation.\cite{Ernst92} Many |
| 141 |
> |
studies have also been conducted to understand solvation theory using |
| 142 |
> |
dielectric properties of these |
| 143 |
> |
fluids,\cite{JeonI03,JeonII03,Chitanvis96} although a correction |
| 144 |
> |
formula for different cutoff methods has not yet been developed. |
| 145 |
> |
|
| 146 |
> |
In this paper we derive general formulae for calculating the |
| 147 |
|
dielectric properties of quadrupolar fluids. We also evaluate the |
| 148 |
|
correction factor for SP, GSF, and TSF methods for both dipolar and |
| 149 |
|
quadrupolar fluids interacting via charge-charge, dipole-dipole or |
| 150 |
< |
quadrupole-quadrupole interactions. |
| 150 |
> |
quadrupole-quadrupole interactions. |
| 151 |
|
|
| 152 |
|
We have also calculated the geometrical factor for two ions immersed |
| 153 |
|
quadrupolar system to evaluate dielectric constant from the |
