| 556 |
|
differences. |
| 557 |
|
|
| 558 |
|
Results and discussion for the individual analysis of each of the |
| 559 |
< |
system types appear in sections \ref{sec:SystemResults}, while the |
| 559 |
> |
system types appear in sections \ref{sec:IndividualResults}, while the |
| 560 |
|
cumulative results over all the investigated systems appear below in |
| 561 |
|
sections \ref{sec:EnergyResults}. |
| 562 |
|
|
| 721 |
|
0.3119, and 0.2476\AA$^{-1}$ for cutoff radii of 9, 12, and 15\AA\ |
| 722 |
|
respectively. |
| 723 |
|
|
| 724 |
< |
|
| 725 |
< |
\section{Discussion on the Pairwise Technique Evaluation} |
| 726 |
< |
|
| 727 |
< |
\subsection{Configuration Energy Differences}\label{sec:EnergyResults} |
| 724 |
> |
\section{Configuration Energy Difference Results}\label{sec:EnergyResults} |
| 725 |
|
In order to evaluate the performance of the pairwise electrostatic |
| 726 |
< |
summation methods for Monte Carlo simulations, the energy differences |
| 727 |
< |
between configurations were compared to the values obtained when using |
| 728 |
< |
{\sc spme}. The results for the combined regression analysis of all |
| 729 |
< |
of the systems are shown in figure \ref{fig:delE}. |
| 726 |
> |
summation methods for Monte Carlo (MC) simulations, the energy |
| 727 |
> |
differences between configurations were compared to the values |
| 728 |
> |
obtained when using {\sc spme}. The results for the combined |
| 729 |
> |
regression analysis of all of the systems are shown in figure |
| 730 |
> |
\ref{fig:delE}. |
| 731 |
|
|
| 732 |
|
\begin{figure} |
| 733 |
|
\centering |
| 757 |
|
function.\cite{Adams79,Steinbach94,Leach01} However, we do not see |
| 758 |
|
significant improvement using the group-switched cutoff because the |
| 759 |
|
salt and salt solution systems contain non-neutral groups. Section |
| 760 |
< |
\ref{sec:SystemResults} includes results for systems comprised entirely |
| 760 |
> |
\ref{sec:IndividualResults} includes results for systems comprised entirely |
| 761 |
|
of neutral groups. |
| 762 |
|
|
| 763 |
|
For the {\sc sp} method, inclusion of electrostatic damping improves |
| 780 |
|
systems; although it does provide results that are an improvement over |
| 781 |
|
those from an unmodified cutoff. |
| 782 |
|
|
| 783 |
< |
\sub |
| 783 |
> |
\section{Magnitude of the Force and Torque Vector Results}\label{sec:FTMagResults} |
| 784 |
|
|
| 787 |
– |
\subsection{Magnitudes of the Force and Torque Vectors} |
| 788 |
– |
|
| 785 |
|
Evaluation of pairwise methods for use in Molecular Dynamics |
| 786 |
|
simulations requires consideration of effects on the forces and |
| 787 |
|
torques. Figures \ref{fig:frcMag} and \ref{fig:trqMag} show the |
| 856 |
|
molecular bodies. Therefore it is not surprising that reaction field |
| 857 |
|
performs best of all of the methods on molecular torques. |
| 858 |
|
|
| 859 |
< |
\subsection{Directionality of the Force and Torque Vectors} |
| 859 |
> |
\section{Directionality of the Force and Torque Vector Results}\label{sec:FTDirResults} |
| 860 |
|
|
| 861 |
|
It is clearly important that a new electrostatic method can reproduce |
| 862 |
|
the magnitudes of the force and torque vectors obtained via the Ewald |
| 906 |
|
particles in all seven systems, while torque vectors are only |
| 907 |
|
available for neutral molecular groups. Damping is more beneficial to |
| 908 |
|
charged bodies, and this observation is investigated further in |
| 909 |
< |
section \ref{SystemResults}. |
| 909 |
> |
section \ref{IndividualResults}. |
| 910 |
|
|
| 911 |
|
Although not discussed previously, group based cutoffs can be applied |
| 912 |
|
to both the {\sc sp} and {\sc sf} methods. The group-based cutoffs |
| 984 |
|
observations, empirical damping up to 0.2\AA$^{-1}$ is beneficial, |
| 985 |
|
but damping may be unnecessary when using the {\sc sf} method. |
| 986 |
|
|
| 987 |
< |
\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals} |
| 987 |
> |
\section{Individual System Analysis Results}\label{sec:IndividualResults} |
| 988 |
> |
|
| 989 |
> |
The combined results of the previous sections show how the pairwise |
| 990 |
> |
methods compare to the Ewald summation in the general sense over all |
| 991 |
> |
of the system types. It is also useful to consider each of the |
| 992 |
> |
studied systems in an individual fashion, so that we can identify |
| 993 |
> |
conditions that are particularly difficult for a selected pairwise |
| 994 |
> |
method to address. This allows us to further establish the limitations |
| 995 |
> |
of these pairwise techniques. Below, the energy difference, force |
| 996 |
> |
vector, and torque vector analyses are presented on an individual |
| 997 |
> |
system basis. |
| 998 |
> |
|
| 999 |
> |
\subsection{SPC/E Water Results}\label{sec:WaterResults} |
| 1000 |
> |
|
| 1001 |
> |
The first system considered was liquid water at 300K using the SPC/E |
| 1002 |
> |
model of water.\cite{Berendsen87} The results for the energy gap |
| 1003 |
> |
comparisons and the force and torque vector magnitude comparisons are |
| 1004 |
> |
shown in table \ref{tab:spce}. The force and torque vector |
| 1005 |
> |
directionality results are displayed separately in table |
| 1006 |
> |
\ref{tab:spceAng}, where the effect of group-based cutoffs and |
| 1007 |
> |
switching functions on the {\sc sp} and {\sc sf} potentials are also |
| 1008 |
> |
investigated. In all of the individual results table, the method |
| 1009 |
> |
abbreviations are as follows: |
| 1010 |
> |
|
| 1011 |
> |
\begin{itemize} |
| 1012 |
> |
\item PC = Pure Cutoff, |
| 1013 |
> |
\item SP = Shifted Potential, |
| 1014 |
> |
\item SF = Shifted Force, |
| 1015 |
> |
\item GSC = Group Switched Cutoff, |
| 1016 |
> |
\item RF = Reaction Field (where $\varepsilon \approx\infty$), |
| 1017 |
> |
\item GSSP = Group Switched Shifted Potential, and |
| 1018 |
> |
\item GSSF = Group Switched Shifted Force. |
| 1019 |
> |
\end{itemize} |
| 1020 |
> |
|
| 1021 |
> |
\begin{table}[htbp] |
| 1022 |
> |
\centering |
| 1023 |
> |
\caption{REGRESSION RESULTS OF THE LIQUID WATER SYSTEM FOR THE |
| 1024 |
> |
$\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it middle}) |
| 1025 |
> |
AND TORQUE VECTOR MAGNITUDES ({\it lower})} |
| 1026 |
> |
|
| 1027 |
> |
\footnotesize |
| 1028 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1029 |
> |
\\ |
| 1030 |
> |
\toprule |
| 1031 |
> |
\toprule |
| 1032 |
> |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1033 |
> |
\cmidrule(lr){3-4} |
| 1034 |
> |
\cmidrule(lr){5-6} |
| 1035 |
> |
\cmidrule(l){7-8} |
| 1036 |
> |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1037 |
> |
\midrule |
| 1038 |
> |
PC & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\ |
| 1039 |
> |
SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\ |
| 1040 |
> |
& 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\ |
| 1041 |
> |
& 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\ |
| 1042 |
> |
& 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\ |
| 1043 |
> |
SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\ |
| 1044 |
> |
& 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\ |
| 1045 |
> |
& 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\ |
| 1046 |
> |
& 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\ |
| 1047 |
> |
GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\ |
| 1048 |
> |
RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
| 1049 |
> |
\midrule |
| 1050 |
> |
PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\ |
| 1051 |
> |
SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\ |
| 1052 |
> |
& 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\ |
| 1053 |
> |
& 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1054 |
> |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
| 1055 |
> |
SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\ |
| 1056 |
> |
& 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 1057 |
> |
& 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1058 |
> |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
| 1059 |
> |
GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 1060 |
> |
RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 1061 |
> |
\midrule |
| 1062 |
> |
PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\ |
| 1063 |
> |
SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\ |
| 1064 |
> |
& 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\ |
| 1065 |
> |
& 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
| 1066 |
> |
& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 1067 |
> |
SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\ |
| 1068 |
> |
& 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\ |
| 1069 |
> |
& 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
| 1070 |
> |
& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 1071 |
> |
GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\ |
| 1072 |
> |
RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\ |
| 1073 |
> |
\bottomrule |
| 1074 |
> |
\end{tabular} |
| 1075 |
> |
\label{tab:spce} |
| 1076 |
> |
\end{table} |
| 1077 |
> |
|
| 1078 |
> |
\begin{table}[htbp] |
| 1079 |
> |
\centering |
| 1080 |
> |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR |
| 1081 |
> |
DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE LIQUID WATER |
| 1082 |
> |
SYSTEM} |
| 1083 |
> |
|
| 1084 |
> |
\footnotesize |
| 1085 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1086 |
> |
\\ |
| 1087 |
> |
\toprule |
| 1088 |
> |
\toprule |
| 1089 |
> |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 1090 |
> |
\cmidrule(lr){3-5} |
| 1091 |
> |
\cmidrule(l){6-8} |
| 1092 |
> |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA & 9\AA & 12\AA & 15\AA \\ |
| 1093 |
> |
\midrule |
| 1094 |
> |
PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\ |
| 1095 |
> |
SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\ |
| 1096 |
> |
& 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\ |
| 1097 |
> |
& 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\ |
| 1098 |
> |
& 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\ |
| 1099 |
> |
SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\ |
| 1100 |
> |
& 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\ |
| 1101 |
> |
& 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\ |
| 1102 |
> |
& 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\ |
| 1103 |
> |
GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
| 1104 |
> |
RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\ |
| 1105 |
> |
\midrule |
| 1106 |
> |
GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
| 1107 |
> |
& 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\ |
| 1108 |
> |
& 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\ |
| 1109 |
> |
& 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\ |
| 1110 |
> |
GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\ |
| 1111 |
> |
& 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\ |
| 1112 |
> |
& 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\ |
| 1113 |
> |
& 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\ |
| 1114 |
> |
\bottomrule |
| 1115 |
> |
\end{tabular} |
| 1116 |
> |
\label{tab:spceAng} |
| 1117 |
> |
\end{table} |
| 1118 |
> |
|
| 1119 |
> |
The water results parallel the combined results seen in sections |
| 1120 |
> |
\ref{sec:EnergyResults} through \ref{sec:FTDirResults}. There is good |
| 1121 |
> |
agreement with {\sc spme} in both energetic and dynamic behavior when |
| 1122 |
> |
using the {\sc sf} method with and without damping. The {\sc sp} |
| 1123 |
> |
method does well with an $\alpha$ around 0.2\AA$^{-1}$, particularly |
| 1124 |
> |
with cutoff radii greater than 12\AA. Overdamping the electrostatics |
| 1125 |
> |
reduces the agreement between both these methods and {\sc spme}. |
| 1126 |
> |
|
| 1127 |
> |
The pure cutoff ({\sc pc}) method performs poorly, again mirroring the |
| 1128 |
> |
observations from the combined results. In contrast to these results, however, the use of a switching function and group |
| 1129 |
> |
based cutoffs greatly improves the results for these neutral water |
| 1130 |
> |
molecules. The group switched cutoff ({\sc gsc}) does not mimic the |
| 1131 |
> |
energetics of {\sc spme} as well as the {\sc sp} (with moderate |
| 1132 |
> |
damping) and {\sc sf} methods, but the dynamics are quite good. The |
| 1133 |
> |
switching functions correct discontinuities in the potential and |
| 1134 |
> |
forces, leading to these improved results. Such improvements with the |
| 1135 |
> |
use of a switching function have been recognized in previous |
| 1136 |
> |
studies,\cite{Andrea83,Steinbach94} and this proves to be a useful |
| 1137 |
> |
tactic for stably incorporating local area electrostatic effects. |
| 1138 |
> |
|
| 1139 |
> |
The reaction field ({\sc rf}) method simply extends upon the results |
| 1140 |
> |
observed in the {\sc gsc} case. Both methods are similar in form |
| 1141 |
> |
(i.e. neutral groups, switching function), but {\sc rf} incorporates |
| 1142 |
> |
an added effect from the external dielectric. This similarity |
| 1143 |
> |
translates into the same good dynamic results and improved energetic |
| 1144 |
> |
agreement with {\sc spme}. Though this agreement is not to the level |
| 1145 |
> |
of the moderately damped {\sc sp} and {\sc sf} methods, these results |
| 1146 |
> |
show how incorporating some implicit properties of the surroundings |
| 1147 |
> |
(i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction. |
| 1148 |
> |
|
| 1149 |
> |
As a final note for the liquid water system, use of group cutoffs and a |
| 1150 |
> |
switching function leads to noticeable improvements in the {\sc sp} |
| 1151 |
> |
and {\sc sf} methods, primarily in directionality of the force and |
| 1152 |
> |
torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows |
| 1153 |
> |
significant narrowing of the angle distribution when using little to |
| 1154 |
> |
no damping and only modest improvement for the recommended conditions |
| 1155 |
> |
($\alpha = 0.2$\AA$^{-1}$ and $R_\textrm{c}~\geqslant~12$\AA). The |
| 1156 |
> |
{\sc sf} method shows modest narrowing across all damping and cutoff |
| 1157 |
> |
ranges of interest. When overdamping these methods, group cutoffs and |
| 1158 |
> |
the switching function do not improve the force and torque |
| 1159 |
> |
directionalities. |
| 1160 |
> |
|
| 1161 |
> |
\subsection{SPC/E Ice I$_\textrm{c}$ Results}\label{sec:IceResults} |
| 1162 |
> |
|
| 1163 |
> |
In addition to the disordered molecular system above, the ordered |
| 1164 |
> |
molecular system of ice I$_\textrm{c}$ was also considered. Ice |
| 1165 |
> |
polymorph could have been used to fit this role; however, ice |
| 1166 |
> |
I$_\textrm{c}$ was chosen because it can form an ideal periodic |
| 1167 |
> |
lattice with the same number of water molecules used in the disordered |
| 1168 |
> |
liquid state case. The results for the energy gap comparisons and the |
| 1169 |
> |
force and torque vector magnitude comparisons are shown in table |
| 1170 |
> |
\ref{tab:ice}. The force and torque vector directionality results are |
| 1171 |
> |
displayed separately in table \ref{tab:iceAng}, where the effect of |
| 1172 |
> |
group-based cutoffs and switching functions on the {\sc sp} and {\sc |
| 1173 |
> |
sf} potentials are also displayed. |
| 1174 |
> |
|
| 1175 |
> |
\begin{table}[htbp] |
| 1176 |
> |
\centering |
| 1177 |
> |
\caption{REGRESSION RESULTS OF THE ICE I$_\textrm{c}$ SYSTEM FOR |
| 1178 |
> |
$\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it |
| 1179 |
> |
middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})} |
| 1180 |
> |
|
| 1181 |
> |
\footnotesize |
| 1182 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1183 |
> |
\\ |
| 1184 |
> |
\toprule |
| 1185 |
> |
\toprule |
| 1186 |
> |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1187 |
> |
\cmidrule(lr){3-4} |
| 1188 |
> |
\cmidrule(lr){5-6} |
| 1189 |
> |
\cmidrule(l){7-8} |
| 1190 |
> |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1191 |
> |
\midrule |
| 1192 |
> |
PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\ |
| 1193 |
> |
SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\ |
| 1194 |
> |
& 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\ |
| 1195 |
> |
& 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\ |
| 1196 |
> |
& 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\ |
| 1197 |
> |
SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\ |
| 1198 |
> |
& 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\ |
| 1199 |
> |
& 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\ |
| 1200 |
> |
& 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\ |
| 1201 |
> |
GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\ |
| 1202 |
> |
RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\ |
| 1203 |
> |
\midrule |
| 1204 |
> |
PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\ |
| 1205 |
> |
SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\ |
| 1206 |
> |
& 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\ |
| 1207 |
> |
& 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\ |
| 1208 |
> |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\ |
| 1209 |
> |
SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\ |
| 1210 |
> |
& 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\ |
| 1211 |
> |
& 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\ |
| 1212 |
> |
& 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\ |
| 1213 |
> |
GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\ |
| 1214 |
> |
RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 1215 |
> |
\midrule |
| 1216 |
> |
PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\ |
| 1217 |
> |
SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\ |
| 1218 |
> |
& 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\ |
| 1219 |
> |
& 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\ |
| 1220 |
> |
& 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\ |
| 1221 |
> |
SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\ |
| 1222 |
> |
& 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\ |
| 1223 |
> |
& 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\ |
| 1224 |
> |
& 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\ |
| 1225 |
> |
GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\ |
| 1226 |
> |
RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\ |
| 1227 |
> |
\bottomrule |
| 1228 |
> |
\end{tabular} |
| 1229 |
> |
\label{tab:ice} |
| 1230 |
> |
\end{table} |
| 1231 |
> |
|
| 1232 |
> |
\begin{table}[htbp] |
| 1233 |
> |
\centering |
| 1234 |
> |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS |
| 1235 |
> |
OF THE FORCE AND TORQUE VECTORS IN THE ICE I$_\textrm{c}$ SYSTEM} |
| 1236 |
> |
|
| 1237 |
> |
\footnotesize |
| 1238 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1239 |
> |
\\ |
| 1240 |
> |
\toprule |
| 1241 |
> |
\toprule |
| 1242 |
> |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque |
| 1243 |
> |
$\sigma^2$} \\ |
| 1244 |
> |
\cmidrule(lr){3-5} |
| 1245 |
> |
\cmidrule(l){6-8} |
| 1246 |
> |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA & 9\AA & 12\AA & 15\AA \\ |
| 1247 |
> |
\midrule |
| 1248 |
> |
PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\ |
| 1249 |
> |
SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\ |
| 1250 |
> |
& 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\ |
| 1251 |
> |
& 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\ |
| 1252 |
> |
& 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\ |
| 1253 |
> |
SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\ |
| 1254 |
> |
& 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\ |
| 1255 |
> |
& 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\ |
| 1256 |
> |
& 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\ |
| 1257 |
> |
GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 1258 |
> |
RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\ |
| 1259 |
> |
\midrule |
| 1260 |
> |
GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 1261 |
> |
& 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\ |
| 1262 |
> |
& 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\ |
| 1263 |
> |
& 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 1264 |
> |
GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\ |
| 1265 |
> |
& 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\ |
| 1266 |
> |
& 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\ |
| 1267 |
> |
& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 1268 |
> |
\bottomrule |
| 1269 |
> |
\end{tabular} |
| 1270 |
> |
\label{tab:iceAng} |
| 1271 |
> |
\end{table} |
| 1272 |
> |
|
| 1273 |
> |
Highly ordered systems are a difficult test for the pairwise methods |
| 1274 |
> |
in that they lack the implicit periodicity of the Ewald summation. As |
| 1275 |
> |
expected, the energy gap agreement with {\sc spme} is reduced for the |
| 1276 |
> |
{\sc sp} and {\sc sf} methods with parameters that were ideal for the |
| 1277 |
> |
disordered liquid system. Moving to higher $R_\textrm{c}$ helps |
| 1278 |
> |
improve the agreement, though at an increase in computational cost. |
| 1279 |
> |
The dynamics of this crystalline system (both in magnitude and |
| 1280 |
> |
direction) are little affected. Both methods still reproduce the Ewald |
| 1281 |
> |
behavior with the same parameter recommendations from the previous |
| 1282 |
> |
section. |
| 1283 |
> |
|
| 1284 |
> |
It is also worth noting that {\sc rf} exhibits improved energy gap |
| 1285 |
> |
results over the liquid water system. One possible explanation is |
| 1286 |
> |
that the ice I$_\textrm{c}$ crystal is ordered such that the net |
| 1287 |
> |
dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = |
| 1288 |
> |
\infty$, the reaction field incorporates this structural organization |
| 1289 |
> |
by actively enforcing a zeroed dipole moment within each cutoff |
| 1290 |
> |
sphere. |
| 1291 |
> |
|
| 1292 |
> |
\subsection{NaCl Melt Results}\label{sec:SaltMeltResults} |
| 1293 |
> |
|
| 1294 |
> |
A high temperature NaCl melt was tested to gauge the accuracy of the |
| 1295 |
> |
pairwise summation methods in a disordered system of charges. The |
| 1296 |
> |
results for the energy gap comparisons and the force vector magnitude |
| 1297 |
> |
comparisons are shown in table \ref{tab:melt}. The force vector |
| 1298 |
> |
directionality results are displayed separately in table |
| 1299 |
> |
\ref{tab:meltAng}. |
| 1300 |
> |
|
| 1301 |
> |
\begin{table}[htbp] |
| 1302 |
> |
\centering |
| 1303 |
> |
\caption{REGRESSION RESULTS OF THE MOLTEN SODIUM CHLORIDE SYSTEM FOR |
| 1304 |
> |
$\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES ({\it |
| 1305 |
> |
lower})} |
| 1306 |
> |
|
| 1307 |
> |
\footnotesize |
| 1308 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1309 |
> |
\\ |
| 1310 |
> |
\toprule |
| 1311 |
> |
\toprule |
| 1312 |
> |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1313 |
> |
\cmidrule(lr){3-4} |
| 1314 |
> |
\cmidrule(lr){5-6} |
| 1315 |
> |
\cmidrule(l){7-8} |
| 1316 |
> |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1317 |
> |
\midrule |
| 1318 |
> |
PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
| 1319 |
> |
SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\ |
| 1320 |
> |
& 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\ |
| 1321 |
> |
& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 1322 |
> |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 1323 |
> |
SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\ |
| 1324 |
> |
& 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\ |
| 1325 |
> |
& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 1326 |
> |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 1327 |
> |
\midrule |
| 1328 |
> |
PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
| 1329 |
> |
SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\ |
| 1330 |
> |
& 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\ |
| 1331 |
> |
& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 1332 |
> |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 1333 |
> |
SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\ |
| 1334 |
> |
& 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\ |
| 1335 |
> |
& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 1336 |
> |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 1337 |
> |
\bottomrule |
| 1338 |
> |
\end{tabular} |
| 1339 |
> |
\label{tab:melt} |
| 1340 |
> |
\end{table} |
| 1341 |
> |
|
| 1342 |
> |
\begin{table}[htbp] |
| 1343 |
> |
\centering |
| 1344 |
> |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS |
| 1345 |
> |
OF THE FORCE VECTORS IN THE MOLTEN SODIUM CHLORIDE SYSTEM} |
| 1346 |
> |
|
| 1347 |
> |
\footnotesize |
| 1348 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1349 |
> |
\\ |
| 1350 |
> |
\toprule |
| 1351 |
> |
\toprule |
| 1352 |
> |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 1353 |
> |
\cmidrule(lr){3-5} |
| 1354 |
> |
\cmidrule(l){6-8} |
| 1355 |
> |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA \\ |
| 1356 |
> |
\midrule |
| 1357 |
> |
PC & & 13.294 & 8.035 & 5.366 \\ |
| 1358 |
> |
SP & 0.0 & 13.316 & 8.037 & 5.385 \\ |
| 1359 |
> |
& 0.1 & 5.705 & 1.391 & 0.360 \\ |
| 1360 |
> |
& 0.2 & 2.415 & 7.534 & 13.927 \\ |
| 1361 |
> |
& 0.3 & 23.769 & 67.306 & 57.252 \\ |
| 1362 |
> |
SF & 0.0 & 1.693 & 0.603 & 0.256 \\ |
| 1363 |
> |
& 0.1 & 1.687 & 0.653 & 0.272 \\ |
| 1364 |
> |
& 0.2 & 2.598 & 7.523 & 13.930 \\ |
| 1365 |
> |
& 0.3 & 23.734 & 67.305 & 57.252 \\ |
| 1366 |
> |
\bottomrule |
| 1367 |
> |
\end{tabular} |
| 1368 |
> |
\label{tab:meltAng} |
| 1369 |
> |
\end{table} |
| 1370 |
> |
|
| 1371 |
> |
The molten NaCl system shows more sensitivity to the electrostatic |
| 1372 |
> |
damping than the water systems. The most noticeable point is that the |
| 1373 |
> |
undamped {\sc sf} method does very well at replicating the {\sc spme} |
| 1374 |
> |
configurational energy differences and forces. Light damping appears |
| 1375 |
> |
to minimally improve the dynamics, but this comes with a deterioration |
| 1376 |
> |
of the energy gap results. In contrast, this light damping improves |
| 1377 |
> |
the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic |
| 1378 |
> |
damping reduce the agreement with {\sc spme} for both methods. From |
| 1379 |
> |
these observations, the undamped {\sc sf} method is the best choice |
| 1380 |
> |
for disordered systems of charges. |
| 1381 |
> |
|
| 1382 |
> |
\subsection{NaCl Crystal Results}\label{sec:SaltCrystalResults} |
| 1383 |
> |
|
| 1384 |
> |
Similar to the use of ice I$_\textrm{c}$ to investigate the role of |
| 1385 |
> |
order in molecular systems on the effectiveness of the pairwise |
| 1386 |
> |
methods, the 1000K NaCl crystal system was used to investigate the |
| 1387 |
> |
accuracy of the pairwise summation methods in an ordered system of |
| 1388 |
> |
charged particles. The results for the energy gap comparisons and the |
| 1389 |
> |
force vector magnitude comparisons are shown in table \ref{tab:salt}. |
| 1390 |
> |
The force vector directionality results are displayed separately in |
| 1391 |
> |
table \ref{tab:saltAng}. |
| 1392 |
> |
|
| 1393 |
> |
\begin{table}[htbp] |
| 1394 |
> |
\centering |
| 1395 |
> |
\caption{REGRESSION RESULTS OF THE CRYSTALLINE SODIUM CHLORIDE |
| 1396 |
> |
SYSTEM FOR $\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES |
| 1397 |
> |
({\it lower})} |
| 1398 |
> |
|
| 1399 |
> |
\footnotesize |
| 1400 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1401 |
> |
\\ |
| 1402 |
> |
\toprule |
| 1403 |
> |
\toprule |
| 1404 |
> |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1405 |
> |
\cmidrule(lr){3-4} |
| 1406 |
> |
\cmidrule(lr){5-6} |
| 1407 |
> |
\cmidrule(l){7-8} |
| 1408 |
> |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1409 |
> |
\midrule |
| 1410 |
> |
PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\ |
| 1411 |
> |
SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\ |
| 1412 |
> |
& 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\ |
| 1413 |
> |
& 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\ |
| 1414 |
> |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\ |
| 1415 |
> |
SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\ |
| 1416 |
> |
& 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\ |
| 1417 |
> |
& 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\ |
| 1418 |
> |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\ |
| 1419 |
> |
\midrule |
| 1420 |
> |
PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\ |
| 1421 |
> |
SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\ |
| 1422 |
> |
& 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\ |
| 1423 |
> |
& 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\ |
| 1424 |
> |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 1425 |
> |
SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\ |
| 1426 |
> |
& 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\ |
| 1427 |
> |
& 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\ |
| 1428 |
> |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 1429 |
> |
\bottomrule |
| 1430 |
> |
\end{tabular} |
| 1431 |
> |
\label{tab:salt} |
| 1432 |
> |
\end{table} |
| 1433 |
> |
|
| 1434 |
> |
\begin{table}[htbp] |
| 1435 |
> |
\centering |
| 1436 |
> |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR |
| 1437 |
> |
DISTRIBUTIONS OF THE FORCE VECTORS IN THE CRYSTALLINE SODIUM CHLORIDE |
| 1438 |
> |
SYSTEM} |
| 1439 |
> |
|
| 1440 |
> |
\footnotesize |
| 1441 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1442 |
> |
\\ |
| 1443 |
> |
\toprule |
| 1444 |
> |
\toprule |
| 1445 |
> |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 1446 |
> |
\cmidrule(lr){3-5} |
| 1447 |
> |
\cmidrule(l){6-8} |
| 1448 |
> |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA \\ |
| 1449 |
> |
\midrule |
| 1450 |
> |
PC & & 111.945 & 111.824 & 111.866 \\ |
| 1451 |
> |
SP & 0.0 & 112.414 & 152.215 & 38.087 \\ |
| 1452 |
> |
& 0.1 & 52.361 & 42.574 & 2.819 \\ |
| 1453 |
> |
& 0.2 & 10.847 & 9.709 & 9.686 \\ |
| 1454 |
> |
& 0.3 & 31.128 & 31.104 & 31.029 \\ |
| 1455 |
> |
SF & 0.0 & 10.025 & 3.555 & 1.648 \\ |
| 1456 |
> |
& 0.1 & 9.462 & 3.303 & 1.721 \\ |
| 1457 |
> |
& 0.2 & 11.454 & 9.813 & 9.701 \\ |
| 1458 |
> |
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
| 1459 |
> |
\bottomrule |
| 1460 |
> |
\end{tabular} |
| 1461 |
> |
\label{tab:saltAng} |
| 1462 |
> |
\end{table} |
| 1463 |
> |
|
| 1464 |
> |
The crystalline NaCl system is the most challenging test case for the |
| 1465 |
> |
pairwise summation methods, as evidenced by the results in tables |
| 1466 |
> |
\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped |
| 1467 |
> |
{\sc sf} methods seem to be the best choices. These methods match well |
| 1468 |
> |
with {\sc spme} across the energy gap, force magnitude, and force |
| 1469 |
> |
directionality tests. The {\sc sp} method struggles in all cases, |
| 1470 |
> |
with the exception of good dynamics reproduction when using weak |
| 1471 |
> |
electrostatic damping with a large cutoff radius. |
| 1472 |
> |
|
| 1473 |
> |
The moderate electrostatic damping case is not as good as we would |
| 1474 |
> |
expect given the long-time dynamics results observed for this system |
| 1475 |
> |
(see section \ref{sec:LongTimeDynamics}). Since the data tabulated in |
| 1476 |
> |
tables \ref{tab:salt} and \ref{tab:saltAng} are a test of |
| 1477 |
> |
instantaneous dynamics, this indicates that good long-time dynamics |
| 1478 |
> |
comes in part at the expense of short-time dynamics. |
| 1479 |
> |
|
| 1480 |
> |
\subsection{0.11M NaCl Solution Results} |
| 1481 |
> |
|
| 1482 |
> |
In an effort to bridge the charged atomic and neutral molecular |
| 1483 |
> |
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
| 1484 |
> |
the liquid water system. This low ionic strength system consists of 4 |
| 1485 |
> |
ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results |
| 1486 |
> |
for the energy gap comparisons and the force and torque vector |
| 1487 |
> |
magnitude comparisons are shown in table \ref{tab:solnWeak}. The |
| 1488 |
> |
force and torque vector directionality results are displayed |
| 1489 |
> |
separately in table \ref{tab:solnWeakAng}, where the effect of |
| 1490 |
> |
group-based cutoffs and switching functions on the {\sc sp} and {\sc |
| 1491 |
> |
sf} potentials are investigated. |
| 1492 |
> |
|
| 1493 |
> |
\begin{table}[htbp] |
| 1494 |
> |
\centering |
| 1495 |
> |
\caption{REGRESSION RESULTS OF THE WEAK SODIUM CHLORIDE SOLUTION |
| 1496 |
> |
SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES |
| 1497 |
> |
({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})} |
| 1498 |
> |
|
| 1499 |
> |
\footnotesize |
| 1500 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1501 |
> |
\\ |
| 1502 |
> |
\toprule |
| 1503 |
> |
\toprule |
| 1504 |
> |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1505 |
> |
\cmidrule(lr){3-4} |
| 1506 |
> |
\cmidrule(lr){5-6} |
| 1507 |
> |
\cmidrule(l){7-8} |
| 1508 |
> |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1509 |
> |
\midrule |
| 1510 |
> |
PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\ |
| 1511 |
> |
SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\ |
| 1512 |
> |
& 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\ |
| 1513 |
> |
& 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 1514 |
> |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 1515 |
> |
SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\ |
| 1516 |
> |
& 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\ |
| 1517 |
> |
& 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 1518 |
> |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 1519 |
> |
GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\ |
| 1520 |
> |
RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\ |
| 1521 |
> |
\midrule |
| 1522 |
> |
PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\ |
| 1523 |
> |
SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\ |
| 1524 |
> |
& 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\ |
| 1525 |
> |
& 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 1526 |
> |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 1527 |
> |
SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\ |
| 1528 |
> |
& 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 1529 |
> |
& 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 1530 |
> |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 1531 |
> |
GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\ |
| 1532 |
> |
RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\ |
| 1533 |
> |
\midrule |
| 1534 |
> |
PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\ |
| 1535 |
> |
SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\ |
| 1536 |
> |
& 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\ |
| 1537 |
> |
& 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 1538 |
> |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 1539 |
> |
SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\ |
| 1540 |
> |
& 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\ |
| 1541 |
> |
& 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 1542 |
> |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 1543 |
> |
GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\ |
| 1544 |
> |
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 1545 |
> |
\bottomrule |
| 1546 |
> |
\end{tabular} |
| 1547 |
> |
\label{tab:solnWeak} |
| 1548 |
> |
\end{table} |
| 1549 |
> |
|
| 1550 |
> |
\begin{table}[htbp] |
| 1551 |
> |
\centering |
| 1552 |
> |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR |
| 1553 |
> |
DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE WEAK SODIUM |
| 1554 |
> |
CHLORIDE SOLUTION SYSTEM} |
| 1555 |
> |
|
| 1556 |
> |
\footnotesize |
| 1557 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1558 |
> |
\\ |
| 1559 |
> |
\toprule |
| 1560 |
> |
\toprule |
| 1561 |
> |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 1562 |
> |
\cmidrule(lr){3-5} |
| 1563 |
> |
\cmidrule(l){6-8} |
| 1564 |
> |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA & 9\AA & 12\AA & 15\AA \\ |
| 1565 |
> |
\midrule |
| 1566 |
> |
PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\ |
| 1567 |
> |
SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\ |
| 1568 |
> |
& 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\ |
| 1569 |
> |
& 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\ |
| 1570 |
> |
& 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\ |
| 1571 |
> |
SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\ |
| 1572 |
> |
& 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\ |
| 1573 |
> |
& 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\ |
| 1574 |
> |
& 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\ |
| 1575 |
> |
GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 1576 |
> |
RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\ |
| 1577 |
> |
\midrule |
| 1578 |
> |
GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 1579 |
> |
& 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\ |
| 1580 |
> |
& 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\ |
| 1581 |
> |
& 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\ |
| 1582 |
> |
GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\ |
| 1583 |
> |
& 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\ |
| 1584 |
> |
& 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\ |
| 1585 |
> |
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 1586 |
> |
\bottomrule |
| 1587 |
> |
\end{tabular} |
| 1588 |
> |
\label{tab:solnWeakAng} |
| 1589 |
> |
\end{table} |
| 1590 |
> |
|
| 1591 |
> |
Because this system is a perturbation of the pure liquid water system, |
| 1592 |
> |
comparisons are best drawn between these two sets. The {\sc sp} and |
| 1593 |
> |
{\sc sf} methods are not significantly affected by the inclusion of a |
| 1594 |
> |
few ions. The aspect of cutoff sphere neutralization aids in the |
| 1595 |
> |
smooth incorporation of these ions; thus, all of the observations |
| 1596 |
> |
regarding these methods carry over from section |
| 1597 |
> |
\ref{sec:WaterResults}. The differences between these systems are more |
| 1598 |
> |
visible for the {\sc rf} method. Though good force agreement is still |
| 1599 |
> |
maintained, the energy gaps show a significant increase in the scatter |
| 1600 |
> |
of the data. |
| 1601 |
> |
|
| 1602 |
> |
\subsection{1.1M NaCl Solution Results} |
| 1603 |
> |
|
| 1604 |
> |
The bridging of the charged atomic and neutral molecular systems was |
| 1605 |
> |
further developed by considering a high ionic strength system |
| 1606 |
> |
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
| 1607 |
> |
M). The results for the energy gap comparisons and the force and |
| 1608 |
> |
torque vector magnitude comparisons are shown in table |
| 1609 |
> |
\ref{tab:solnStr}. The force and torque vector directionality |
| 1610 |
> |
results are displayed separately in table \ref{tab:solnStrAng}, where |
| 1611 |
> |
the effect of group-based cutoffs and switching functions on the {\sc |
| 1612 |
> |
sp} and {\sc sf} potentials are investigated. |
| 1613 |
> |
|
| 1614 |
> |
\begin{table}[htbp] |
| 1615 |
> |
\centering |
| 1616 |
> |
\caption{REGRESSION RESULTS OF THE STRONG SODIUM CHLORIDE SOLUTION |
| 1617 |
> |
SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES |
| 1618 |
> |
({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})} |
| 1619 |
> |
|
| 1620 |
> |
\footnotesize |
| 1621 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1622 |
> |
\\ |
| 1623 |
> |
\toprule |
| 1624 |
> |
\toprule |
| 1625 |
> |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1626 |
> |
\cmidrule(lr){3-4} |
| 1627 |
> |
\cmidrule(lr){5-6} |
| 1628 |
> |
\cmidrule(l){7-8} |
| 1629 |
> |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1630 |
> |
\midrule |
| 1631 |
> |
PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\ |
| 1632 |
> |
SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\ |
| 1633 |
> |
& 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\ |
| 1634 |
> |
& 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 1635 |
> |
& 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 1636 |
> |
SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\ |
| 1637 |
> |
& 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\ |
| 1638 |
> |
& 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 1639 |
> |
& 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 1640 |
> |
GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\ |
| 1641 |
> |
RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\ |
| 1642 |
> |
\midrule |
| 1643 |
> |
PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\ |
| 1644 |
> |
SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\ |
| 1645 |
> |
& 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\ |
| 1646 |
> |
& 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 1647 |
> |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 1648 |
> |
SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\ |
| 1649 |
> |
& 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\ |
| 1650 |
> |
& 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 1651 |
> |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 1652 |
> |
GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\ |
| 1653 |
> |
RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\ |
| 1654 |
> |
\midrule |
| 1655 |
> |
PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\ |
| 1656 |
> |
SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\ |
| 1657 |
> |
& 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\ |
| 1658 |
> |
& 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 1659 |
> |
& 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 1660 |
> |
SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\ |
| 1661 |
> |
& 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\ |
| 1662 |
> |
& 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 1663 |
> |
& 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 1664 |
> |
GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\ |
| 1665 |
> |
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 1666 |
> |
\bottomrule |
| 1667 |
> |
\end{tabular} |
| 1668 |
> |
\label{tab:solnStr} |
| 1669 |
> |
\end{table} |
| 1670 |
> |
|
| 1671 |
> |
\begin{table}[htbp] |
| 1672 |
> |
\centering |
| 1673 |
> |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS |
| 1674 |
> |
OF THE FORCE AND TORQUE VECTORS IN THE STRONG SODIUM CHLORIDE SOLUTION |
| 1675 |
> |
SYSTEM} |
| 1676 |
> |
|
| 1677 |
> |
\footnotesize |
| 1678 |
> |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1679 |
> |
\\ |
| 1680 |
> |
\toprule |
| 1681 |
> |
\toprule |
| 1682 |
> |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 1683 |
> |
\cmidrule(lr){3-5} |
| 1684 |
> |
\cmidrule(l){6-8} |
| 1685 |
> |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA & 9\AA & 12\AA & 15\AA \\ |
| 1686 |
> |
\midrule |
| 1687 |
> |
PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\ |
| 1688 |
> |
SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\ |
| 1689 |
> |
& 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\ |
| 1690 |
> |
& 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\ |
| 1691 |
> |
& 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\ |
| 1692 |
> |
SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\ |
| 1693 |
> |
& 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\ |
| 1694 |
> |
& 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\ |
| 1695 |
> |
& 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\ |
| 1696 |
> |
GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\ |
| 1697 |
> |
RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\ |
| 1698 |
> |
\midrule |
| 1699 |
> |
GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\ |
| 1700 |
> |
& 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\ |
| 1701 |
> |
& 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\ |
| 1702 |
> |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\ |
| 1703 |
> |
GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\ |
| 1704 |
> |
& 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\ |
| 1705 |
> |
& 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\ |
| 1706 |
> |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 1707 |
> |
\bottomrule |
| 1708 |
> |
\end{tabular} |
| 1709 |
> |
\label{tab:solnStrAng} |
| 1710 |
> |
\end{table} |
| 1711 |
|
|
| 1712 |
+ |
The {\sc rf} method struggles with the jump in ionic strength. The |
| 1713 |
+ |
configuration energy differences degrade to unusable levels while the |
| 1714 |
+ |
forces and torques show a more modest reduction in the agreement with |
| 1715 |
+ |
{\sc spme}. The {\sc rf} method was designed for homogeneous systems, |
| 1716 |
+ |
and this attribute is apparent in these results. |
| 1717 |
+ |
|
| 1718 |
+ |
The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
| 1719 |
+ |
their agreement with {\sc spme}. With these results, we still |
| 1720 |
+ |
recommend undamped to moderate damping for the {\sc sf} method and |
| 1721 |
+ |
moderate damping for the {\sc sp} method, both with cutoffs greater |
| 1722 |
+ |
than 12\AA. |
| 1723 |
+ |
|
| 1724 |
+ |
\subsection{6\AA\ Argon Sphere in SPC/E Water Results} |
| 1725 |
+ |
|
| 1726 |
+ |
The final model system studied was a 6\AA\ sphere of Argon solvated |
| 1727 |
+ |
by SPC/E water. This serves as a test case of a specifically sized |
| 1728 |
+ |
electrostatic defect in a disordered molecular system. The results for |
| 1729 |
+ |
the energy gap comparisons and the force and torque vector magnitude |
| 1730 |
+ |
comparisons are shown in table \ref{tab:argon}. The force and torque |
| 1731 |
+ |
vector directionality results are displayed separately in table |
| 1732 |
+ |
\ref{tab:argonAng}, where the effect of group-based cutoffs and |
| 1733 |
+ |
switching functions on the {\sc sp} and {\sc sf} potentials are |
| 1734 |
+ |
investigated. |
| 1735 |
+ |
|
| 1736 |
+ |
\begin{table}[htbp] |
| 1737 |
+ |
\centering |
| 1738 |
+ |
\caption{REGRESSION RESULTS OF THE 6\AA\ ARGON SPHERE IN LIQUID |
| 1739 |
+ |
WATER SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR |
| 1740 |
+ |
MAGNITUDES ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})} |
| 1741 |
+ |
|
| 1742 |
+ |
\footnotesize |
| 1743 |
+ |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1744 |
+ |
\\ |
| 1745 |
+ |
\toprule |
| 1746 |
+ |
\toprule |
| 1747 |
+ |
& & \multicolumn{2}{c}{9\AA} & \multicolumn{2}{c}{12\AA} & \multicolumn{2}{c}{15\AA}\\ |
| 1748 |
+ |
\cmidrule(lr){3-4} |
| 1749 |
+ |
\cmidrule(lr){5-6} |
| 1750 |
+ |
\cmidrule(l){7-8} |
| 1751 |
+ |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 1752 |
+ |
\midrule |
| 1753 |
+ |
PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\ |
| 1754 |
+ |
SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\ |
| 1755 |
+ |
& 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\ |
| 1756 |
+ |
& 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\ |
| 1757 |
+ |
& 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\ |
| 1758 |
+ |
SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\ |
| 1759 |
+ |
& 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\ |
| 1760 |
+ |
& 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\ |
| 1761 |
+ |
& 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\ |
| 1762 |
+ |
GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\ |
| 1763 |
+ |
RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\ |
| 1764 |
+ |
\midrule |
| 1765 |
+ |
PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\ |
| 1766 |
+ |
SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\ |
| 1767 |
+ |
& 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\ |
| 1768 |
+ |
& 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1769 |
+ |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 1770 |
+ |
SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1771 |
+ |
& 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1772 |
+ |
& 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1773 |
+ |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 1774 |
+ |
GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1775 |
+ |
RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 1776 |
+ |
\midrule |
| 1777 |
+ |
PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\ |
| 1778 |
+ |
SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\ |
| 1779 |
+ |
& 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\ |
| 1780 |
+ |
& 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\ |
| 1781 |
+ |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 1782 |
+ |
SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\ |
| 1783 |
+ |
& 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\ |
| 1784 |
+ |
& 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\ |
| 1785 |
+ |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 1786 |
+ |
GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\ |
| 1787 |
+ |
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 1788 |
+ |
\bottomrule |
| 1789 |
+ |
\end{tabular} |
| 1790 |
+ |
\label{tab:argon} |
| 1791 |
+ |
\end{table} |
| 1792 |
+ |
|
| 1793 |
+ |
\begin{table}[htbp] |
| 1794 |
+ |
\centering |
| 1795 |
+ |
\caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR |
| 1796 |
+ |
DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE 6\AA\ SPHERE OF |
| 1797 |
+ |
ARGON IN LIQUID WATER SYSTEM} |
| 1798 |
+ |
|
| 1799 |
+ |
\footnotesize |
| 1800 |
+ |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 1801 |
+ |
\\ |
| 1802 |
+ |
\toprule |
| 1803 |
+ |
\toprule |
| 1804 |
+ |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 1805 |
+ |
\cmidrule(lr){3-5} |
| 1806 |
+ |
\cmidrule(l){6-8} |
| 1807 |
+ |
Method & $\alpha$ & 9\AA & 12\AA & 15\AA & 9\AA & 12\AA & 15\AA \\ |
| 1808 |
+ |
\midrule |
| 1809 |
+ |
PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\ |
| 1810 |
+ |
SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\ |
| 1811 |
+ |
& 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\ |
| 1812 |
+ |
& 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\ |
| 1813 |
+ |
& 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\ |
| 1814 |
+ |
SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\ |
| 1815 |
+ |
& 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\ |
| 1816 |
+ |
& 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\ |
| 1817 |
+ |
& 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\ |
| 1818 |
+ |
GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 1819 |
+ |
RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\ |
| 1820 |
+ |
\midrule |
| 1821 |
+ |
GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 1822 |
+ |
& 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\ |
| 1823 |
+ |
& 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\ |
| 1824 |
+ |
& 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\ |
| 1825 |
+ |
GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\ |
| 1826 |
+ |
& 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\ |
| 1827 |
+ |
& 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\ |
| 1828 |
+ |
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 1829 |
+ |
\bottomrule |
| 1830 |
+ |
\end{tabular} |
| 1831 |
+ |
\label{tab:argonAng} |
| 1832 |
+ |
\end{table} |
| 1833 |
+ |
|
| 1834 |
+ |
This system does not appear to show any significant deviations from |
| 1835 |
+ |
the previously observed results. The {\sc sp} and {\sc sf} methods |
| 1836 |
+ |
have aggrements similar to those observed in section |
| 1837 |
+ |
\ref{sec:WaterResults}. The only significant difference is the |
| 1838 |
+ |
improvement in the configuration energy differences for the {\sc rf} |
| 1839 |
+ |
method. This is surprising in that we are introducing an inhomogeneity |
| 1840 |
+ |
to the system; however, this inhomogeneity is charge-neutral and does |
| 1841 |
+ |
not result in charged cutoff spheres. The charge-neutrality of the |
| 1842 |
+ |
cutoff spheres, which the {\sc sp} and {\sc sf} methods explicitly |
| 1843 |
+ |
enforce, seems to play a greater role in the stability of the {\sc rf} |
| 1844 |
+ |
method than the required homogeneity of the environment. |
| 1845 |
+ |
|
| 1846 |
+ |
|
| 1847 |
+ |
\section{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals}\label{sec:ShortTimeDynamics} |
| 1848 |
+ |
|
| 1849 |
|
Zahn {\it et al.} investigated the structure and dynamics of water |
| 1850 |
|
using eqs. (\ref{eq:ZahnPot}) and |
| 1851 |
|
(\ref{eq:WolfForces}).\cite{Zahn02,Kast03} Their results indicated |
| 1871 |
|
\centering |
| 1872 |
|
\includegraphics[width = \linewidth]{./figures/vCorrPlot.pdf} |
| 1873 |
|
\caption{Velocity autocorrelation functions of NaCl crystals at |
| 1874 |
< |
1000K using {\sc spme}, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc |
| 1875 |
< |
sp} ($\alpha$ = 0.2). The inset is a magnification of the area around |
| 1876 |
< |
the first minimum. The times to first collision are nearly identical, |
| 1877 |
< |
but differences can be seen in the peaks and troughs, where the |
| 1878 |
< |
undamped and weakly damped methods are stiffer than the moderately |
| 1879 |
< |
damped and {\sc spme} methods.} |
| 1874 |
> |
1000K using {\sc spme}, {\sc sf} ($\alpha$ = 0.0, 0.1, \& |
| 1875 |
> |
0.2\AA$^{-1}$), and {\sc sp} ($\alpha$ = 0.2\AA$^{-1}$). The inset is |
| 1876 |
> |
a magnification of the area around the first minimum. The times to |
| 1877 |
> |
first collision are nearly identical, but differences can be seen in |
| 1878 |
> |
the peaks and troughs, where the undamped and weakly damped methods |
| 1879 |
> |
are stiffer than the moderately damped and {\sc spme} methods.} |
| 1880 |
|
\label{fig:vCorrPlot} |
| 1881 |
|
\end{figure} |
| 1882 |
|
|
| 1896 |
|
constructed out of the damped electrostatic interaction are less |
| 1897 |
|
important. |
| 1898 |
|
|
| 1899 |
< |
\subsection{Collective Motion: Power Spectra of NaCl Crystals} |
| 1899 |
> |
\section{Collective Motion: Power Spectra of NaCl Crystals}\label{sec:LongTimeDynamics} |
| 1900 |
|
|
| 1901 |
|
To evaluate how the differences between the methods affect the |
| 1902 |
|
collective long-time motion, we computed power spectra from long-time |
| 1912 |
|
\includegraphics[width = \linewidth]{./figures/spectraSquare.pdf} |
| 1913 |
|
\caption{Power spectra obtained from the velocity auto-correlation |
| 1914 |
|
functions of NaCl crystals at 1000K while using {\sc spme}, {\sc sf} |
| 1915 |
< |
($\alpha$ = 0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). The inset |
| 1916 |
< |
shows the frequency region below 100 cm$^{-1}$ to highlight where the |
| 1917 |
< |
spectra differ.} |
| 1915 |
> |
($\alpha$ = 0, 0.1, \& 0.2\AA$^{-1}$), and {\sc sp} ($\alpha$ = |
| 1916 |
> |
0.2\AA$^{-1}$). The inset shows the frequency region below 100 |
| 1917 |
> |
cm$^{-1}$ to highlight where the spectra differ.} |
| 1918 |
|
\label{fig:methodPS} |
| 1919 |
|
\end{figure} |
| 1920 |
|
|
| 1959 |
|
\label{fig:dampInc} |
| 1960 |
|
\end{figure} |
| 1961 |
|
|
| 1962 |
+ |
\section{Synopsis of the Pairwise Method Evaluation}\label{sec:PairwiseSynopsis} |
| 1963 |
|
|
| 1964 |
+ |
The above investigation of pairwise electrostatic summation techniques |
| 1965 |
+ |
shows that there are viable and computationally efficient alternatives |
| 1966 |
+ |
to the Ewald summation. These methods are derived from the damped and |
| 1967 |
+ |
cutoff-neutralized Coulombic sum originally proposed by Wolf |
| 1968 |
+ |
\textit{et al.}\cite{Wolf99} In particular, the {\sc sf} |
| 1969 |
+ |
method, reformulated above as eqs. (\ref{eq:DSFPot}) and |
| 1970 |
+ |
(\ref{eq:DSFForces}), shows a remarkable ability to reproduce the |
| 1971 |
+ |
energetic and dynamic characteristics exhibited by simulations |
| 1972 |
+ |
employing lattice summation techniques. The cumulative energy |
| 1973 |
+ |
difference results showed the undamped {\sc sf} and moderately damped |
| 1974 |
+ |
{\sc sp} methods produced results nearly identical to {\sc spme}. |
| 1975 |
+ |
Similarly for the dynamic features, the undamped or moderately damped |
| 1976 |
+ |
{\sc sf} and moderately damped {\sc sp} methods produce force and |
| 1977 |
+ |
torque vector magnitude and directions very similar to the expected |
| 1978 |
+ |
values. These results translate into long-time dynamic behavior |
| 1979 |
+ |
equivalent to that produced in simulations using {\sc spme}. |
| 1980 |
+ |
|
| 1981 |
+ |
As in all purely-pairwise cutoff methods, these methods are expected |
| 1982 |
+ |
to scale approximately {\it linearly} with system size, and they are |
| 1983 |
+ |
easily parallelizable. This should result in substantial reductions |
| 1984 |
+ |
in the computational cost of performing large simulations. |
| 1985 |
+ |
|
| 1986 |
+ |
Aside from the computational cost benefit, these techniques have |
| 1987 |
+ |
applicability in situations where the use of the Ewald sum can prove |
| 1988 |
+ |
problematic. Of greatest interest is their potential use in |
| 1989 |
+ |
interfacial systems, where the unmodified lattice sum techniques |
| 1990 |
+ |
artificially accentuate the periodicity of the system in an |
| 1991 |
+ |
undesirable manner. There have been alterations to the standard Ewald |
| 1992 |
+ |
techniques, via corrections and reformulations, to compensate for |
| 1993 |
+ |
these systems; but the pairwise techniques discussed here require no |
| 1994 |
+ |
modifications, making them natural tools to tackle these problems. |
| 1995 |
+ |
Additionally, this transferability gives them benefits over other |
| 1996 |
+ |
pairwise methods, like reaction field, because estimations of physical |
| 1997 |
+ |
properties (e.g. the dielectric constant) are unnecessary. |
| 1998 |
+ |
|
| 1999 |
+ |
If a researcher is using Monte Carlo simulations of large chemical |
| 2000 |
+ |
systems containing point charges, most structural features will be |
| 2001 |
+ |
accurately captured using the undamped {\sc sf} method or the {\sc sp} |
| 2002 |
+ |
method with an electrostatic damping of 0.2\AA$^{-1}$. These methods |
| 2003 |
+ |
would also be appropriate for molecular dynamics simulations where the |
| 2004 |
+ |
data of interest is either structural or short-time dynamical |
| 2005 |
+ |
quantities. For long-time dynamics and collective motions, the safest |
| 2006 |
+ |
pairwise method we have evaluated is the {\sc sf} method with an |
| 2007 |
+ |
electrostatic damping between 0.2 and 0.25\AA$^{-1}$. |
| 2008 |
+ |
|
| 2009 |
+ |
We are not suggesting that there is any flaw with the Ewald sum; in |
| 2010 |
+ |
fact, it is the standard by which these simple pairwise sums have been |
| 2011 |
+ |
judged. However, these results do suggest that in the typical |
| 2012 |
+ |
simulations performed today, the Ewald summation may no longer be |
| 2013 |
+ |
required to obtain the level of accuracy most researchers have come to |
| 2014 |
+ |
expect. |
| 2015 |
+ |
|
| 2016 |
+ |
\section{An Application: TIP5P-E Water} |
| 2017 |
+ |
|
| 2018 |
+ |
|
| 2019 |
|
\chapter{\label{chap:water}SIMPLE MODELS FOR WATER} |
| 2020 |
|
|
| 2021 |
|
\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER SIMULATIONS} |
| 2028 |
|
\backmatter |
| 2029 |
|
|
| 2030 |
|
\bibliographystyle{ndthesis} |
| 2031 |
< |
\bibliography{dissertation} |
| 2031 |
> |
\bibliography{dissertation} |
| 2032 |
|
|
| 2033 |
|
\end{document} |
| 2034 |
|
|
