| 47 |
|
|
| 48 |
|
%\preprint{AIP/123-QED} |
| 49 |
|
|
| 50 |
< |
\title{Real space alternatives to the Ewald Sum. II. Comparison of Methods} |
| 50 |
> |
\title{Real space electrostatics for multipoles. II. Comparisons with |
| 51 |
> |
the Ewald Sum} |
| 52 |
|
|
| 53 |
|
\author{Madan Lamichhane} |
| 54 |
|
\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556} |
| 325 |
|
expressed as the product of two multipole operators and a Coulombic |
| 326 |
|
kernel, |
| 327 |
|
\begin{equation} |
| 328 |
< |
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
| 328 |
> |
U_{ab}(r)= M_{a} M_{b} \frac{1}{r} \label{kernel}. |
| 329 |
|
\end{equation} |
| 330 |
< |
where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is |
| 331 |
< |
expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf |
| 332 |
< |
a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object |
| 332 |
< |
$\bf a$, etc. |
| 330 |
> |
where the multipole operator for site $a$, $M_{a}$, is |
| 331 |
> |
expressed in terms of the point charge, $C_{a}$, dipole, ${\bf D}_{a}$, and quadrupole, $\mathsf{Q}_{a}$, for object |
| 332 |
> |
$a$, etc. |
| 333 |
|
|
| 334 |
|
% Interactions between multipoles can be expressed as higher derivatives |
| 335 |
|
% of the bare Coulomb potential, so one way of ensuring that the forces |
| 384 |
|
contributions to the potential, and ensures that the forces and |
| 385 |
|
torques from each of these contributions will vanish at the cutoff |
| 386 |
|
radius. For example, the direct dipole dot product |
| 387 |
< |
($\mathbf{D}_{\bf a} |
| 388 |
< |
\cdot \mathbf{D}_{\bf b}$) is treated differently than the dipole-distance |
| 387 |
> |
($\mathbf{D}_{a} |
| 388 |
> |
\cdot \mathbf{D}_{b}$) is treated differently than the dipole-distance |
| 389 |
|
dot products: |
| 390 |
|
\begin{equation} |
| 391 |
< |
U_{D_{\bf a}D_{\bf b}}(r)= -\frac{1}{4\pi \epsilon_0} \left[ \left( |
| 392 |
< |
\mathbf{D}_{\bf a} \cdot |
| 393 |
< |
\mathbf{D}_{\bf b} \right) v_{21}(r) + |
| 394 |
< |
\left( \mathbf{D}_{\bf a} \cdot \hat{r} \right) |
| 395 |
< |
\left( \mathbf{D}_{\bf b} \cdot \hat{r} \right) v_{22}(r) \right] |
| 391 |
> |
U_{D_{a}D_{b}}(r)= -\frac{1}{4\pi \epsilon_0} \left[ \left( |
| 392 |
> |
\mathbf{D}_{a} \cdot |
| 393 |
> |
\mathbf{D}_{b} \right) v_{21}(r) + |
| 394 |
> |
\left( \mathbf{D}_{a} \cdot \hat{\mathbf{r}} \right) |
| 395 |
> |
\left( \mathbf{D}_{b} \cdot \hat{\mathbf{r}} \right) v_{22}(r) \right] |
| 396 |
|
\end{equation} |
| 397 |
|
|
| 398 |
|
For the Taylor shifted (TSF) method with the undamped kernel, |
| 417 |
|
which have been projected onto the surface of the cutoff sphere |
| 418 |
|
without changing their relative orientation, |
| 419 |
|
\begin{equation} |
| 420 |
< |
U_{D_{\bf a}D_{\bf b}}(r) = U_{D_{\bf a}D_{\bf b}}(r) - |
| 421 |
< |
U_{D_{\bf a} D_{\bf b}}(r_c) |
| 422 |
< |
- (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot |
| 423 |
< |
\nabla U_{D_{\bf a}D_{\bf b}}(r_c). |
| 420 |
> |
U_{D_{a}D_{b}}(r) = U_{D_{a}D_{b}}(r) - |
| 421 |
> |
U_{D_{a}D_{b}}(r_c) |
| 422 |
> |
- (r_{ab}-r_c) ~~~\hat{\mathbf{r}}_{ab} \cdot |
| 423 |
> |
\nabla U_{D_{a}D_{b}}(r_c). |
| 424 |
|
\end{equation} |
| 425 |
< |
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf |
| 426 |
< |
a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance |
| 425 |
> |
Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{a}$ and $\mathbf{D}_{b}$, are retained at the cutoff distance |
| 426 |
|
(although the signs are reversed for the dipole that has been |
| 427 |
|
projected onto the cutoff sphere). In many ways, this simpler |
| 428 |
|
approach is closer in spirit to the original shifted force method, in |
| 443 |
|
In general, the gradient shifted potential between a central multipole |
| 444 |
|
and any multipolar site inside the cutoff radius is given by, |
| 445 |
|
\begin{equation} |
| 446 |
< |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 447 |
< |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}} |
| 448 |
< |
\cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 446 |
> |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \mathsf{A}, \mathsf{B}) - |
| 447 |
> |
U(r_c \hat{\mathbf{r}},\mathsf{A}, \mathsf{B}) - (r-r_c) |
| 448 |
> |
\hat{\mathbf{r}} \cdot \nabla U(r_c \hat{\mathbf{r}},\mathsf{A}, \mathsf{B}) \right] |
| 449 |
|
\label{generic2} |
| 450 |
|
\end{equation} |
| 451 |
|
where the sum describes a separate force-shifting that is applied to |
| 452 |
|
each orientational contribution to the energy. In this expression, |
| 453 |
|
$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles |
| 454 |
< |
($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ |
| 454 |
> |
($a$ and $b$) in space, and $\mathsf{A}$ and $\mathsf{B}$ |
| 455 |
|
represent the orientations the multipoles. |
| 456 |
|
|
| 457 |
|
The third term converges more rapidly than the first two terms as a |
| 478 |
|
interactions with the central multipole and the image. This |
| 479 |
|
effectively shifts the total potential to zero at the cutoff radius, |
| 480 |
|
\begin{equation} |
| 481 |
< |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 482 |
< |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 481 |
> |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \mathsf{A}, \mathsf{B}) - |
| 482 |
> |
U(r_c \hat{\mathbf{r}},\mathsf{A}, \mathsf{B}) \right] |
| 483 |
|
\label{eq:SP} |
| 484 |
|
\end{equation} |
| 485 |
|
where the sum describes separate potential shifting that is done for |
| 956 |
|
in this series and provides the most comprehensive test of the new |
| 957 |
|
methods. A liquid-phase system was created with 2000 water molecules |
| 958 |
|
and 48 dissolved ions at a density of 0.98 g cm$^{-3}$ and a |
| 959 |
< |
temperature of 300K. After equilibration, this liquid-phase system |
| 960 |
< |
was run for 1 ns under the Ewald, Hard, SP, GSF, and TSF methods with |
| 961 |
< |
a cutoff radius of 12\AA. The value of the damping coefficient was |
| 962 |
< |
also varied from the undamped case ($\alpha = 0$) to a heavily damped |
| 963 |
< |
case ($\alpha = 0.3$ \AA$^{-1}$) for all of the real space methods. A |
| 964 |
< |
sample was also run using the multipolar Ewald sum with the same |
| 965 |
< |
real-space cutoff. |
| 959 |
> |
temperature of 300K. After equilibration in the canonical (NVT) |
| 960 |
> |
ensemble using a Nos\'e-Hoover thermostat, this liquid-phase system |
| 961 |
> |
was run for 1 ns in the microcanonical (NVE) ensemble under the Ewald, |
| 962 |
> |
Hard, SP, GSF, and TSF methods with a cutoff radius of 12\AA. The |
| 963 |
> |
value of the damping coefficient was also varied from the undamped |
| 964 |
> |
case ($\alpha = 0$) to a heavily damped case ($\alpha = 0.3$ |
| 965 |
> |
\AA$^{-1}$) for all of the real space methods. A sample was also run |
| 966 |
> |
using the multipolar Ewald sum with the same real-space cutoff. |
| 967 |
|
|
| 968 |
|
In figure~\ref{fig:energyDrift} we show the both the linear drift in |
| 969 |
|
energy over time, $\delta E_1$, and the standard deviation of energy |
| 987 |
|
\includegraphics[width=\textwidth]{newDrift_12.eps} |
| 988 |
|
\label{fig:energyDrift} |
| 989 |
|
\caption{Analysis of the energy conservation of the real-space |
| 990 |
< |
electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
| 991 |
< |
energy over time (in kcal / mol / particle / ns) and $\delta |
| 992 |
< |
\mathrm{E}_0$ is the standard deviation of energy fluctuations |
| 993 |
< |
around this drift (in kcal / mol / particle). All simulations were |
| 994 |
< |
of a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
| 990 |
> |
methods. $\delta \mathrm{E}_1$ is the linear drift in energy over |
| 991 |
> |
time (in kcal / mol / particle / ns) and $\delta \mathrm{E}_0$ is |
| 992 |
> |
the standard deviation of energy fluctuations around this drift (in |
| 993 |
> |
kcal / mol / particle). Points that appear below the dashed grey |
| 994 |
> |
(Ewald) lines exhibit better energy conservation than commonly-used |
| 995 |
> |
parameters for Ewald-based electrostatics. All simulations were of |
| 996 |
> |
a 2000-molecule simulation of SSDQ water with 48 ionic charges at |
| 997 |
|
300 K starting from the same initial configuration. All runs |
| 998 |
|
utilized the same real-space cutoff, $r_c = 12$\AA.} |
| 999 |
|
\end{figure} |
| 1000 |
+ |
|
| 1001 |
+ |
\subsection{Reproduction of Structural \& Dynamical Features\label{sec:structure}} |
| 1002 |
+ |
The most important test of the modified interaction potentials is the |
| 1003 |
+ |
fidelity with which they can reproduce structural features and |
| 1004 |
+ |
dynamical properties in a liquid. One commonly-utilized measure of |
| 1005 |
+ |
structural ordering is the pair distribution function, $g(r)$, which |
| 1006 |
+ |
measures local density deviations in relation to the bulk density. In |
| 1007 |
+ |
the electrostatic approaches studied here, the short-range repulsion |
| 1008 |
+ |
from the Lennard-Jones potential is identical for the various |
| 1009 |
+ |
electrostatic methods, and since short range repulsion determines much |
| 1010 |
+ |
of the local liquid ordering, one would not expect to see many |
| 1011 |
+ |
differences in $g(r)$. Indeed, the pair distributions are essentially |
| 1012 |
+ |
identical for all of the electrostatic methods studied (for each of |
| 1013 |
+ |
the different systems under investigation). An example of this |
| 1014 |
+ |
agreement for the SSDQ water/ion system is shown in |
| 1015 |
+ |
Fig. \ref{fig:gofr}. |
| 1016 |
+ |
|
| 1017 |
+ |
\begin{figure} |
| 1018 |
+ |
\centering |
| 1019 |
+ |
\includegraphics[width=\textwidth]{gofr_ssdqc.eps} |
| 1020 |
+ |
\label{fig:gofr} |
| 1021 |
+ |
\caption{The pair distribution functions, $g(r)$, for the SSDQ |
| 1022 |
+ |
water/ion system obtained using the different real-space methods are |
| 1023 |
+ |
essentially identical with the result from the Ewald |
| 1024 |
+ |
treatment.} |
| 1025 |
+ |
\end{figure} |
| 1026 |
+ |
|
| 1027 |
+ |
There is a very slight overstructuring of the first solvation shell |
| 1028 |
+ |
when using when using TSF at lower values of the damping coefficient |
| 1029 |
+ |
($\alpha \le 0.1$) or when using undamped GSF. With moderate damping, |
| 1030 |
+ |
GSF and SP produce pair distributions that are identical (within |
| 1031 |
+ |
numerical noise) to their Ewald counterparts. |
| 1032 |
+ |
|
| 1033 |
+ |
A structural property that is a more demanding test of modified |
| 1034 |
+ |
electrostatics is the mean value of the electrostatic energy $\langle |
| 1035 |
+ |
U_\mathrm{elect} \rangle / N$ which is obtained by sampling the |
| 1036 |
+ |
liquid-state configurations experienced by a liquid evolving entirely |
| 1037 |
+ |
under the influence of each of the methods. In table \ref{tab:Props} |
| 1038 |
+ |
we demonstrate how $\langle U_\mathrm{elect} \rangle / N$ varies with |
| 1039 |
+ |
the damping parameter, $\alpha$, for each of the methods. |
| 1040 |
|
|
| 1041 |
+ |
As in the crystals studied in the first paper, damping is important |
| 1042 |
+ |
for converging the mean electrostatic energy values, particularly for |
| 1043 |
+ |
the two shifted force methods (GSF and TSF). A value of $\alpha |
| 1044 |
+ |
\approx 0.2$ \AA$^{-1}$ is sufficient to converge the SP and GSF |
| 1045 |
+ |
energies with a cutoff of 12 \AA, while shorter cutoffs require more |
| 1046 |
+ |
dramatic damping ($\alpha \approx 0.3$ \AA$^{-1}$ for $r_c = 9$ \AA). |
| 1047 |
+ |
Overdamping the real-space electrostatic methods occurs with $\alpha > |
| 1048 |
+ |
0.4$, causing the estimate of the energy to drop below the Ewald |
| 1049 |
+ |
results. |
| 1050 |
|
|
| 1051 |
+ |
These ``optimal'' values of the damping coefficient are slightly |
| 1052 |
+ |
larger than what were observed for DSF electrostatics for purely |
| 1053 |
+ |
point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for |
| 1054 |
+ |
$r_c = 12$\AA appears to be an excellent compromise for mixed charge |
| 1055 |
+ |
multipole systems. |
| 1056 |
+ |
|
| 1057 |
+ |
To test the fidelity of the electrostatic methods at reproducing |
| 1058 |
+ |
dynamics in a multipolar liquid, it is also useful to look at |
| 1059 |
+ |
transport properties, particularly the diffusion constant, |
| 1060 |
+ |
\begin{equation} |
| 1061 |
+ |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle \left| |
| 1062 |
+ |
\mathbf{r}(t) -\mathbf{r}(0) \right|^2 \rangle |
| 1063 |
+ |
\label{eq:diff} |
| 1064 |
+ |
\end{equation} |
| 1065 |
+ |
which measures long-time behavior and is sensitive to the forces on |
| 1066 |
+ |
the multipoles. For the soft dipolar fluid and the SSDQ liquid |
| 1067 |
+ |
systems, the self-diffusion constants (D) were calculated from linear |
| 1068 |
+ |
fits to the long-time portion of the mean square displacement, |
| 1069 |
+ |
$\langle r^{2}(t) \rangle$.\cite{Allen87} |
| 1070 |
+ |
|
| 1071 |
+ |
In addition to translational diffusion, orientational relaxation times |
| 1072 |
+ |
were calculated for comparisons with the Ewald simulations and with |
| 1073 |
+ |
experiments. These values were determined from the same 1~ns |
| 1074 |
+ |
microcanonical trajectories used for translational diffusion by |
| 1075 |
+ |
calculating the orientational time correlation function, |
| 1076 |
+ |
\begin{equation} |
| 1077 |
+ |
C_l^\gamma(t) = \left\langle P_l\left[\hat{\mathbf{A}}_\gamma(t) |
| 1078 |
+ |
\cdot\hat{\mathbf{A}}_\gamma(0)\right]\right\rangle, |
| 1079 |
+ |
\label{eq:OrientCorr} |
| 1080 |
+ |
\end{equation} |
| 1081 |
+ |
where $P_l$ is the Legendre polynomial of order $l$ and |
| 1082 |
+ |
$\hat{\mathbf{A}}_\gamma$ is the space-frame unit vector for body axis |
| 1083 |
+ |
$\gamma$ on a molecule.. Th body-fixed reference frame used for our |
| 1084 |
+ |
models has the $z$-axis running along the dipoles, and for the SSDQ |
| 1085 |
+ |
water model, the $y$-axis connects the two implied hydrogen atom |
| 1086 |
+ |
positions. From the orientation autocorrelation functions, we can |
| 1087 |
+ |
obtain time constants for rotational relaxation either by fitting an |
| 1088 |
+ |
exponential function or by integrating the entire correlation |
| 1089 |
+ |
function. In a good water model, these decay times would be |
| 1090 |
+ |
comparable to water orientational relaxation times from nuclear |
| 1091 |
+ |
magnetic resonance (NMR). The relaxation constant obtained from |
| 1092 |
+ |
$C_2^y(t)$ is normally of experimental interest because it describes |
| 1093 |
+ |
the relaxation of the principle axis connecting the hydrogen |
| 1094 |
+ |
atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular portion |
| 1095 |
+ |
of the dipole-dipole relaxation from a proton NMR signal and should |
| 1096 |
+ |
provide an estimate of the NMR relaxation time constant.\cite{Impey82} |
| 1097 |
+ |
|
| 1098 |
+ |
Results for the diffusion constants and orientational relaxation times |
| 1099 |
+ |
are shown in figure \ref{tab:Props}. From this data, it is apparent |
| 1100 |
+ |
that the values for both $D$ and $\tau_2$ using the Ewald sum are |
| 1101 |
+ |
reproduced with reasonable fidelity by the GSF method. |
| 1102 |
+ |
|
| 1103 |
+ |
The $\tau_2$ results in \ref{tab:Props} show a much greater difference |
| 1104 |
+ |
between the real-space and the Ewald results. |
| 1105 |
+ |
|
| 1106 |
+ |
\begin{table} |
| 1107 |
+ |
\label{tab:Props} |
| 1108 |
+ |
\caption{Comparison of the structural and dynamic properties for the |
| 1109 |
+ |
soft dipolar liquid test for all of the real-space methods.} |
| 1110 |
+ |
\begin{tabular}{l|c|cccc|cccc|cccc} |
| 1111 |
+ |
& Ewald & \multicolumn{4}{c|}{SP} & \multicolumn{4}{c|}{GSF} & \multicolumn{4}{c|}{TSF} \\ |
| 1112 |
+ |
$\alpha$ (\AA$^{-1}$) & & |
| 1113 |
+ |
0.0 & 0.1 & 0.2 & 0.3 & |
| 1114 |
+ |
0.0 & 0.1 & 0.2 & 0.3 & |
| 1115 |
+ |
0.0 & 0.1 & 0.2 & 0.3 \\ \cline{2-6}\cline{6-10}\cline{10-14} |
| 1116 |
+ |
|
| 1117 |
+ |
$\langle U_\mathrm{elect} \rangle /N$ &&&&&&&&&&&&&\\ |
| 1118 |
+ |
D ($10^{-4}~\mathrm{cm}^2/\mathrm{s}$)& |
| 1119 |
+ |
470.2(6) & |
| 1120 |
+ |
416.6(5) & |
| 1121 |
+ |
379.6(5) & |
| 1122 |
+ |
438.6(5) & |
| 1123 |
+ |
476.0(6) & |
| 1124 |
+ |
412.8(5) & |
| 1125 |
+ |
421.1(5) & |
| 1126 |
+ |
400.5(5) & |
| 1127 |
+ |
437.5(6) & |
| 1128 |
+ |
434.6(5) & |
| 1129 |
+ |
411.4(5) & |
| 1130 |
+ |
545.3(7) & |
| 1131 |
+ |
459.6(6) \\ |
| 1132 |
+ |
$\tau_2$ (fs) & |
| 1133 |
+ |
1.136 & |
| 1134 |
+ |
1.041 & |
| 1135 |
+ |
1.064 & |
| 1136 |
+ |
1.109 & |
| 1137 |
+ |
1.211 & |
| 1138 |
+ |
1.119 & |
| 1139 |
+ |
1.039 & |
| 1140 |
+ |
1.058 & |
| 1141 |
+ |
1.21 & |
| 1142 |
+ |
1.15 & |
| 1143 |
+ |
1.172 & |
| 1144 |
+ |
1.153 & |
| 1145 |
+ |
1.125 \\ |
| 1146 |
+ |
\end{tabular} |
| 1147 |
+ |
\end{table} |
| 1148 |
+ |
|
| 1149 |
+ |
|
| 1150 |
|
\section{CONCLUSION} |
| 1151 |
|
In the first paper in this series, we generalized the |
| 1152 |
|
charge-neutralized electrostatic energy originally developed by Wolf |
