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\begin{document} |
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|
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This document includes individual system-based comparisons of the |
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studied methods with smooth particle mesh Ewald {\sc spme}. Each of |
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the seven systems comprises its own section and has its own discussion |
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and tabular listing of the results for the $\Delta E$, force and |
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torque vector magnitude, and force and torque vector direction |
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comparisons. |
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|
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\section{\label{app:water}Liquid Water} |
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|
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The first system considered was liquid water at 300K using the SPC/E |
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model of water.\cite{Berendsen87} The results for the energy gap |
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comparisons and the force and torque vector magnitude comparisons are |
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shown in table \ref{tab:spce}. The force and torque vector |
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directionality results are displayed separately in table |
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\ref{tab:spceAng}, where the effect of group-based cutoffs and |
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switching functions on the {\sc sp} and {\sc sf} potentials are |
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investigated. |
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\begin{table}[htbp] |
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\centering |
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\caption{Regression results for the liquid water system. Tabulated |
| 46 |
results include $\Delta E$ values (top set), force vector magnitudes |
| 47 |
(middle set) and torque vector magnitudes (bottom set). PC = Pure |
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Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group |
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Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
| 50 |
\infty$).} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
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\cmidrule(lr){3-4} |
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\cmidrule(lr){5-6} |
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\cmidrule(l){7-8} |
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Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
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\midrule |
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PC & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\ |
| 61 |
SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\ |
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& 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\ |
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& 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\ |
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& 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\ |
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SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\ |
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& 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\ |
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& 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\ |
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& 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\ |
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GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\ |
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RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
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\midrule |
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PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\ |
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SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\ |
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& 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\ |
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& 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
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& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
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SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\ |
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& 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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& 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
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& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
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GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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\midrule |
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PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\ |
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SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\ |
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& 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\ |
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& 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
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& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 89 |
SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\ |
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& 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\ |
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& 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
| 92 |
& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
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GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\ |
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RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\ |
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\bottomrule |
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\end{tabular} |
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\label{tab:spce} |
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\end{table} |
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|
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\begin{table}[htbp] |
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\centering |
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\caption{Variance results from Gaussian fits to angular |
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distributions of the force and torque vectors in the liquid water |
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system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
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GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon |
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\approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = |
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Group Switched Shifted Force.} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
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\cmidrule(lr){3-5} |
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\cmidrule(l){6-8} |
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Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
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\midrule |
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PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\ |
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SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\ |
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& 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\ |
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& 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\ |
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& 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\ |
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SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\ |
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& 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\ |
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& 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\ |
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& 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\ |
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GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
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RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\ |
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\midrule |
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GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
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& 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\ |
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& 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\ |
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& 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\ |
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GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\ |
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& 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\ |
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& 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\ |
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& 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\ |
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\bottomrule |
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\end{tabular} |
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\label{tab:spceAng} |
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\end{table} |
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|
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The water results appear to parallel the combined results seen in the |
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discussion section of the main paper. There is good agreement with |
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{\sc spme} in both energetic and dynamic behavior when using the {\sc sf} |
| 144 |
method with and without damping. The {\sc sp} method does well with an |
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$\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater |
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than 12 \AA. Overdamping the electrostatics reduces the agreement between both these methods and {\sc spme}. |
| 147 |
|
| 148 |
The pure cutoff ({\sc pc}) method performs poorly, again mirroring the |
| 149 |
observations in the main portion of this paper. In contrast to the |
| 150 |
combined values, however, the use of a switching function and group |
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based cutoffs greatly improves the results for these neutral water |
| 152 |
molecules. The group switched cutoff ({\sc gsc}) does not mimic the |
| 153 |
energetics of {\sc spme} as well as the {\sc sp} (with moderate |
| 154 |
damping) and {\sc sf} methods, but the dynamics are quite good. The |
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switching functions correct discontinuities in the potential and |
| 156 |
forces, leading to these improved results. Such improvements with the |
| 157 |
use of a switching function have been recognized in previous |
| 158 |
studies,\cite{Andrea83,Steinbach94} and this proves to be a useful |
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tactic for stably incorporating local area electrostatic effects. |
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|
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The reaction field ({\sc rf}) method simply extends upon the results |
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observed in the {\sc gsc} case. Both methods are similar in form |
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(i.e. neutral groups, switching function), but {\sc rf} incorporates |
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an added effect from the external dielectric. This similarity |
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translates into the same good dynamic results and improved energetic |
| 166 |
agreement with {\sc spme}. Though this agreement is not to the level |
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of the moderately damped {\sc sp} and {\sc sf} methods, these results |
| 168 |
show how incorporating some implicit properties of the surroundings |
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(i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction. |
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|
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As a final note for the liquid water system, use of group cutoffs and a |
| 172 |
switching function leads to noticeable improvements in the {\sc sp} |
| 173 |
and {\sc sf} methods, primarily in directionality of the force and |
| 174 |
torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows |
| 175 |
significant narrowing of the angle distribution when using little to |
| 176 |
no damping and only modest improvement for the recommended conditions |
| 177 |
($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The |
| 178 |
{\sc sf} method shows modest narrowing across all damping and cutoff |
| 179 |
ranges of interest. When overdamping these methods, group cutoffs and |
| 180 |
the switching function do not improve the force and torque |
| 181 |
directionalities. |
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|
| 183 |
\section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$} |
| 184 |
|
| 185 |
In addition to the disordered molecular system above, the ordered |
| 186 |
molecular system of ice I$_\textrm{c}$ was also considered. The |
| 187 |
results for the energy gap comparisons and the force and torque vector |
| 188 |
magnitude comparisons are shown in table \ref{tab:ice}. The force and |
| 189 |
torque vector directionality results are displayed separately in table |
| 190 |
\ref{tab:iceAng}, where the effect of group-based cutoffs and |
| 191 |
switching functions on the {\sc sp} and {\sc sf} potentials are |
| 192 |
investigated. |
| 193 |
|
| 194 |
\begin{table}[htbp] |
| 195 |
\centering |
| 196 |
\caption{Regression results for the ice I$_\textrm{c}$ |
| 197 |
system. Tabulated results include $\Delta E$ values (top set), force |
| 198 |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 199 |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 200 |
GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 201 |
$\varepsilon \approx \infty$).} |
| 202 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 203 |
\\ |
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\toprule |
| 205 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 206 |
\cmidrule(lr){3-4} |
| 207 |
\cmidrule(lr){5-6} |
| 208 |
\cmidrule(l){7-8} |
| 209 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 210 |
\midrule |
| 211 |
PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\ |
| 212 |
SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\ |
| 213 |
& 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\ |
| 214 |
& 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\ |
| 215 |
& 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\ |
| 216 |
SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\ |
| 217 |
& 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\ |
| 218 |
& 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\ |
| 219 |
& 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\ |
| 220 |
GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\ |
| 221 |
RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\ |
| 222 |
\midrule |
| 223 |
PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\ |
| 224 |
SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\ |
| 225 |
& 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\ |
| 226 |
& 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\ |
| 227 |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\ |
| 228 |
SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\ |
| 229 |
& 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\ |
| 230 |
& 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\ |
| 231 |
& 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\ |
| 232 |
GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\ |
| 233 |
RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 234 |
\midrule |
| 235 |
PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\ |
| 236 |
SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\ |
| 237 |
& 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\ |
| 238 |
& 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\ |
| 239 |
& 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\ |
| 240 |
SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\ |
| 241 |
& 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\ |
| 242 |
& 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\ |
| 243 |
& 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\ |
| 244 |
GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\ |
| 245 |
RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\ |
| 246 |
\bottomrule |
| 247 |
\end{tabular} |
| 248 |
\label{tab:ice} |
| 249 |
\end{table} |
| 250 |
|
| 251 |
\begin{table}[htbp] |
| 252 |
\centering |
| 253 |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the ice I$_\textrm{c}$ system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
| 254 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 255 |
\\ |
| 256 |
\toprule |
| 257 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 258 |
\cmidrule(lr){3-5} |
| 259 |
\cmidrule(l){6-8} |
| 260 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 261 |
\midrule |
| 262 |
PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\ |
| 263 |
SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\ |
| 264 |
& 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\ |
| 265 |
& 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\ |
| 266 |
& 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\ |
| 267 |
SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\ |
| 268 |
& 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\ |
| 269 |
& 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\ |
| 270 |
& 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\ |
| 271 |
GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 272 |
RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\ |
| 273 |
\midrule |
| 274 |
GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 275 |
& 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\ |
| 276 |
& 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\ |
| 277 |
& 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 278 |
GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\ |
| 279 |
& 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\ |
| 280 |
& 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\ |
| 281 |
& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 282 |
\bottomrule |
| 283 |
\end{tabular} |
| 284 |
\label{tab:iceAng} |
| 285 |
\end{table} |
| 286 |
|
| 287 |
Highly ordered systems are a difficult test for the pairwise methods |
| 288 |
in that they lack the periodicity term of the Ewald summation. As |
| 289 |
expected, the energy gap agreement with {\sc spme} is reduced for the |
| 290 |
{\sc sp} and {\sc sf} methods with parameters that were acceptable for |
| 291 |
the disordered liquid system. Moving to higher $R_\textrm{c}$ helps |
| 292 |
improve the agreement, though at an increase in computational cost. |
| 293 |
The dynamics of this crystalline system (both in magnitude and |
| 294 |
direction) are little affected. Both methods still reproduce the Ewald |
| 295 |
behavior with the same parameter recommendations from the previous |
| 296 |
section. |
| 297 |
|
| 298 |
It is also worth noting that {\sc rf} exhibits improved energy gap |
| 299 |
results over the liquid water system. One possible explanation is |
| 300 |
that the ice I$_\textrm{c}$ crystal is ordered such that the net |
| 301 |
dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = |
| 302 |
\infty$, the reaction field incorporates this structural organization |
| 303 |
by actively enforcing a zeroed dipole moment within each cutoff |
| 304 |
sphere. |
| 305 |
|
| 306 |
\section{\label{app:melt}NaCl Melt} |
| 307 |
|
| 308 |
A high temperature NaCl melt was tested to gauge the accuracy of the |
| 309 |
pairwise summation methods in a charged disordered system. The results |
| 310 |
for the energy gap comparisons and the force vector magnitude |
| 311 |
comparisons are shown in table \ref{tab:melt}. The force vector |
| 312 |
directionality results are displayed separately in table |
| 313 |
\ref{tab:meltAng}. |
| 314 |
|
| 315 |
\begin{table}[htbp] |
| 316 |
\centering |
| 317 |
\caption{Regression results for the molten NaCl system. Tabulated results include $\Delta E$ values (top set) and force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted Potential, and SF = Shifted Force.} |
| 318 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 319 |
\\ |
| 320 |
\toprule |
| 321 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 322 |
\cmidrule(lr){3-4} |
| 323 |
\cmidrule(lr){5-6} |
| 324 |
\cmidrule(l){7-8} |
| 325 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 326 |
\midrule |
| 327 |
PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
| 328 |
SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\ |
| 329 |
& 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\ |
| 330 |
& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 331 |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 332 |
SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\ |
| 333 |
& 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\ |
| 334 |
& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 335 |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 336 |
\midrule |
| 337 |
PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
| 338 |
SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\ |
| 339 |
& 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\ |
| 340 |
& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 341 |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 342 |
SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\ |
| 343 |
& 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\ |
| 344 |
& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 345 |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 346 |
\bottomrule |
| 347 |
\end{tabular} |
| 348 |
\label{tab:melt} |
| 349 |
\end{table} |
| 350 |
|
| 351 |
\begin{table}[htbp] |
| 352 |
\centering |
| 353 |
\caption{Variance results from Gaussian fits to angular distributions of the force vectors in the molten NaCl system. PC = Pure Cutoff, SP = Shifted Potential, and SF = Shifted Force.} |
| 354 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 355 |
\\ |
| 356 |
\toprule |
| 357 |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 358 |
\cmidrule(lr){3-5} |
| 359 |
\cmidrule(l){6-8} |
| 360 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 361 |
\midrule |
| 362 |
PC & & 13.294 & 8.035 & 5.366 \\ |
| 363 |
SP & 0.0 & 13.316 & 8.037 & 5.385 \\ |
| 364 |
& 0.1 & 5.705 & 1.391 & 0.360 \\ |
| 365 |
& 0.2 & 2.415 & 7.534 & 13.927 \\ |
| 366 |
& 0.3 & 23.769 & 67.306 & 57.252 \\ |
| 367 |
SF & 0.0 & 1.693 & 0.603 & 0.256 \\ |
| 368 |
& 0.1 & 1.687 & 0.653 & 0.272 \\ |
| 369 |
& 0.2 & 2.598 & 7.523 & 13.930 \\ |
| 370 |
& 0.3 & 23.734 & 67.305 & 57.252 \\ |
| 371 |
\bottomrule |
| 372 |
\end{tabular} |
| 373 |
\label{tab:meltAng} |
| 374 |
\end{table} |
| 375 |
|
| 376 |
The molten NaCl system shows more sensitivity to the electrostatic |
| 377 |
damping than the water systems. The most noticeable point is that the |
| 378 |
undamped {\sc sf} method does very well at replicating the {\sc spme} |
| 379 |
configurational energy differences and forces. Light damping appears |
| 380 |
to minimally improve the dynamics, but this comes with a deterioration |
| 381 |
of the energy gap results. In contrast, this light damping improves |
| 382 |
the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic |
| 383 |
damping reduce the agreement with {\sc spme} for both methods. From |
| 384 |
these observations, the undamped {\sc sf} method is the best choice |
| 385 |
for disordered systems of charges. |
| 386 |
|
| 387 |
\section{\label{app:salt}NaCl Crystal} |
| 388 |
|
| 389 |
A 1000K NaCl crystal was used to investigate the accuracy of the |
| 390 |
pairwise summation methods in an ordered system of charged |
| 391 |
particles. The results for the energy gap comparisons and the force |
| 392 |
vector magnitude comparisons are shown in table \ref{tab:salt}. The |
| 393 |
force vector directionality results are displayed separately in table |
| 394 |
\ref{tab:saltAng}. |
| 395 |
|
| 396 |
\begin{table}[htbp] |
| 397 |
\centering |
| 398 |
\caption{Regression results for the crystalline NaCl |
| 399 |
system. Tabulated results include $\Delta E$ values (top set) and |
| 400 |
force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted |
| 401 |
Potential, and SF = Shifted Force.} |
| 402 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 403 |
\\ |
| 404 |
\toprule |
| 405 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 406 |
\cmidrule(lr){3-4} |
| 407 |
\cmidrule(lr){5-6} |
| 408 |
\cmidrule(l){7-8} |
| 409 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 410 |
\midrule |
| 411 |
PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\ |
| 412 |
SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\ |
| 413 |
& 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\ |
| 414 |
& 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\ |
| 415 |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\ |
| 416 |
SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\ |
| 417 |
& 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\ |
| 418 |
& 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\ |
| 419 |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\ |
| 420 |
\midrule |
| 421 |
PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\ |
| 422 |
SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\ |
| 423 |
& 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\ |
| 424 |
& 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\ |
| 425 |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 426 |
SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\ |
| 427 |
& 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\ |
| 428 |
& 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\ |
| 429 |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 430 |
\bottomrule |
| 431 |
\end{tabular} |
| 432 |
\label{tab:salt} |
| 433 |
\end{table} |
| 434 |
|
| 435 |
\begin{table}[htbp] |
| 436 |
\centering |
| 437 |
\caption{Variance results from Gaussian fits to angular |
| 438 |
distributions of the force vectors in the crystalline NaCl system. PC |
| 439 |
= Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group |
| 440 |
Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
| 441 |
\infty$).} |
| 442 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 443 |
\\ |
| 444 |
\toprule |
| 445 |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 446 |
\cmidrule(lr){3-5} |
| 447 |
\cmidrule(l){6-8} |
| 448 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 449 |
\midrule |
| 450 |
PC & & 111.945 & 111.824 & 111.866 \\ |
| 451 |
SP & 0.0 & 112.414 & 152.215 & 38.087 \\ |
| 452 |
& 0.1 & 52.361 & 42.574 & 2.819 \\ |
| 453 |
& 0.2 & 10.847 & 9.709 & 9.686 \\ |
| 454 |
& 0.3 & 31.128 & 31.104 & 31.029 \\ |
| 455 |
SF & 0.0 & 10.025 & 3.555 & 1.648 \\ |
| 456 |
& 0.1 & 9.462 & 3.303 & 1.721 \\ |
| 457 |
& 0.2 & 11.454 & 9.813 & 9.701 \\ |
| 458 |
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
| 459 |
\bottomrule |
| 460 |
\end{tabular} |
| 461 |
\label{tab:saltAng} |
| 462 |
\end{table} |
| 463 |
|
| 464 |
The crystalline NaCl system is the most challenging test case for the |
| 465 |
pairwise summation methods, as evidenced by the results in tables |
| 466 |
\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped |
| 467 |
{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best |
| 468 |
choices. These methods match well with {\sc spme} across the energy |
| 469 |
gap, force magnitude, and force directionality tests. The {\sc sp} |
| 470 |
method struggles in all cases, with the exception of good dynamics |
| 471 |
reproduction when using weak electrostatic damping with a large cutoff |
| 472 |
radius. |
| 473 |
|
| 474 |
The moderate electrostatic damping case is not as good as we would |
| 475 |
expect given the long-time dynamics results observed for this |
| 476 |
system. Since the data tabulated in tables \ref{tab:salt} and |
| 477 |
\ref{tab:saltAng} are a test of instantaneous dynamics, this indicates |
| 478 |
that good long-time dynamics comes in part at the expense of |
| 479 |
short-time dynamics. |
| 480 |
|
| 481 |
\section{\label{app:solnWeak}Weak NaCl Solution} |
| 482 |
|
| 483 |
In an effort to bridge the charged atomic and neutral molecular |
| 484 |
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
| 485 |
the liquid water system. This low ionic strength system consists of 4 |
| 486 |
ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results |
| 487 |
for the energy gap comparisons and the force and torque vector |
| 488 |
magnitude comparisons are shown in table \ref{tab:solnWeak}. The |
| 489 |
force and torque vector directionality results are displayed |
| 490 |
separately in table \ref{tab:solnWeakAng}, where the effect of |
| 491 |
group-based cutoffs and switching functions on the {\sc sp} and {\sc |
| 492 |
sf} potentials are investigated. |
| 493 |
|
| 494 |
\begin{table}[htbp] |
| 495 |
\centering |
| 496 |
\caption{Regression results for the weak NaCl solution |
| 497 |
system. Tabulated results include $\Delta E$ values (top set), force |
| 498 |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 499 |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 500 |
GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon |
| 501 |
\approx \infty$).} |
| 502 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 503 |
\\ |
| 504 |
\toprule |
| 505 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 506 |
\cmidrule(lr){3-4} |
| 507 |
\cmidrule(lr){5-6} |
| 508 |
\cmidrule(l){7-8} |
| 509 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 510 |
\midrule |
| 511 |
PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\ |
| 512 |
SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\ |
| 513 |
& 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\ |
| 514 |
& 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 515 |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 516 |
SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\ |
| 517 |
& 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\ |
| 518 |
& 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 519 |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 520 |
GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\ |
| 521 |
RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\ |
| 522 |
\midrule |
| 523 |
PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\ |
| 524 |
SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\ |
| 525 |
& 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\ |
| 526 |
& 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 527 |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 528 |
SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\ |
| 529 |
& 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 530 |
& 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 531 |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 532 |
GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\ |
| 533 |
RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\ |
| 534 |
\midrule |
| 535 |
PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\ |
| 536 |
SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\ |
| 537 |
& 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\ |
| 538 |
& 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 539 |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 540 |
SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\ |
| 541 |
& 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\ |
| 542 |
& 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 543 |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 544 |
GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\ |
| 545 |
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 546 |
\bottomrule |
| 547 |
\end{tabular} |
| 548 |
\label{tab:solnWeak} |
| 549 |
\end{table} |
| 550 |
|
| 551 |
\begin{table}[htbp] |
| 552 |
\centering |
| 553 |
\caption{Variance results from Gaussian fits to angular |
| 554 |
distributions of the force and torque vectors in the weak NaCl |
| 555 |
solution system. PC = Pure Cutoff, SP = Shifted Potential, SF = |
| 556 |
Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where |
| 557 |
$\varepsilon \approx \infty$), GSSP = Group Switched Shifted |
| 558 |
Potential, and GSSF = Group Switched Shifted Force.} |
| 559 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 560 |
\\ |
| 561 |
\toprule |
| 562 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 563 |
\cmidrule(lr){3-5} |
| 564 |
\cmidrule(l){6-8} |
| 565 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 566 |
\midrule |
| 567 |
PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\ |
| 568 |
SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\ |
| 569 |
& 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\ |
| 570 |
& 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\ |
| 571 |
& 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\ |
| 572 |
SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\ |
| 573 |
& 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\ |
| 574 |
& 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\ |
| 575 |
& 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\ |
| 576 |
GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 577 |
RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\ |
| 578 |
\midrule |
| 579 |
GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 580 |
& 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\ |
| 581 |
& 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\ |
| 582 |
& 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\ |
| 583 |
GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\ |
| 584 |
& 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\ |
| 585 |
& 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\ |
| 586 |
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 587 |
\bottomrule |
| 588 |
\end{tabular} |
| 589 |
\label{tab:solnWeakAng} |
| 590 |
\end{table} |
| 591 |
|
| 592 |
Because this system is a perturbation of the pure liquid water system, |
| 593 |
comparisons are best drawn between these two sets. The {\sc sp} and |
| 594 |
{\sc sf} methods are not significantly affected by the inclusion of a |
| 595 |
few ions. The aspect of cutoff sphere neutralization aids in the |
| 596 |
smooth incorporation of these ions; thus, all of the observations |
| 597 |
regarding these methods carry over from section \ref{app:water}. The |
| 598 |
differences between these systems are more visible for the {\sc rf} |
| 599 |
method. Though good force agreement is still maintained, the energy |
| 600 |
gaps show a significant increase in the data scatter. This foreshadows |
| 601 |
the breakdown of the method as we introduce charged inhomogeneities. |
| 602 |
|
| 603 |
\section{\label{app:solnStr}Strong NaCl Solution} |
| 604 |
|
| 605 |
The bridging of the charged atomic and neutral molecular systems was |
| 606 |
further developed by considering a high ionic strength system |
| 607 |
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
| 608 |
M). The results for the energy gap comparisons and the force and |
| 609 |
torque vector magnitude comparisons are shown in table |
| 610 |
\ref{tab:solnStr}. The force and torque vector directionality |
| 611 |
results are displayed separately in table \ref{tab:solnStrAng}, where |
| 612 |
the effect of group-based cutoffs and switching functions on the {\sc |
| 613 |
sp} and {\sc sf} potentials are investigated. |
| 614 |
|
| 615 |
\begin{table}[htbp] |
| 616 |
\centering |
| 617 |
\caption{Regression results for the strong NaCl solution |
| 618 |
system. Tabulated results include $\Delta E$ values (top set), force |
| 619 |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 620 |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 621 |
GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 622 |
$\varepsilon \approx \infty$).} |
| 623 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 624 |
\\ |
| 625 |
\toprule |
| 626 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 627 |
\cmidrule(lr){3-4} |
| 628 |
\cmidrule(lr){5-6} |
| 629 |
\cmidrule(l){7-8} |
| 630 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 631 |
\midrule |
| 632 |
PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\ |
| 633 |
SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\ |
| 634 |
& 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\ |
| 635 |
& 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 636 |
& 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 637 |
SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\ |
| 638 |
& 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\ |
| 639 |
& 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 640 |
& 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 641 |
GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\ |
| 642 |
RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\ |
| 643 |
\midrule |
| 644 |
PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\ |
| 645 |
SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\ |
| 646 |
& 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\ |
| 647 |
& 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 648 |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 649 |
SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\ |
| 650 |
& 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\ |
| 651 |
& 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 652 |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 653 |
GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\ |
| 654 |
RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\ |
| 655 |
\midrule |
| 656 |
PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\ |
| 657 |
SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\ |
| 658 |
& 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\ |
| 659 |
& 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 660 |
& 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 661 |
SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\ |
| 662 |
& 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\ |
| 663 |
& 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 664 |
& 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 665 |
GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\ |
| 666 |
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 667 |
\bottomrule |
| 668 |
\end{tabular} |
| 669 |
\label{tab:solnStr} |
| 670 |
\end{table} |
| 671 |
|
| 672 |
\begin{table}[htbp] |
| 673 |
\centering |
| 674 |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the strong NaCl solution system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
| 675 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 676 |
\\ |
| 677 |
\toprule |
| 678 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 679 |
\cmidrule(lr){3-5} |
| 680 |
\cmidrule(l){6-8} |
| 681 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 682 |
\midrule |
| 683 |
PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\ |
| 684 |
SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\ |
| 685 |
& 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\ |
| 686 |
& 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\ |
| 687 |
& 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\ |
| 688 |
SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\ |
| 689 |
& 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\ |
| 690 |
& 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\ |
| 691 |
& 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\ |
| 692 |
GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\ |
| 693 |
RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\ |
| 694 |
\midrule |
| 695 |
GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\ |
| 696 |
& 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\ |
| 697 |
& 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\ |
| 698 |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\ |
| 699 |
GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\ |
| 700 |
& 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\ |
| 701 |
& 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\ |
| 702 |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 703 |
\bottomrule |
| 704 |
\end{tabular} |
| 705 |
\label{tab:solnStrAng} |
| 706 |
\end{table} |
| 707 |
|
| 708 |
The {\sc rf} method struggles with the jump in ionic strength. The |
| 709 |
configuration energy differences degrade to unusable levels while the |
| 710 |
forces and torques show a more modest reduction in the agreement with |
| 711 |
{\sc spme}. The {\sc rf} method was designed for homogeneous systems, |
| 712 |
and this attribute is apparent in these results. |
| 713 |
|
| 714 |
The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
| 715 |
their agreement with {\sc spme}. With these results, we still |
| 716 |
recommend no to moderate damping for the {\sc sf} method and moderate |
| 717 |
damping for the {\sc sp} method, both with cutoffs greater than 12 |
| 718 |
\AA. |
| 719 |
|
| 720 |
\section{\label{app:argon}Argon Sphere in Water} |
| 721 |
|
| 722 |
The final model system studied was a 6 \AA\ sphere of Argon solvated |
| 723 |
by SPC/E water. The results for the energy gap comparisons and the |
| 724 |
force and torque vector magnitude comparisons are shown in table |
| 725 |
\ref{tab:argon}. The force and torque vector directionality |
| 726 |
results are displayed separately in table \ref{tab:argonAng}, where |
| 727 |
the effect of group-based cutoffs and switching functions on the {\sc |
| 728 |
sp} and {\sc sf} potentials are investigated. |
| 729 |
|
| 730 |
\begin{table}[htbp] |
| 731 |
\centering |
| 732 |
\caption{Regression results for the 6 \AA\ Argon sphere in liquid |
| 733 |
water system. Tabulated results include $\Delta E$ values (top set), |
| 734 |
force vector magnitudes (middle set) and torque vector magnitudes |
| 735 |
(bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted |
| 736 |
Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 737 |
$\varepsilon \approx \infty$).} |
| 738 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 739 |
\\ |
| 740 |
\toprule |
| 741 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 742 |
\cmidrule(lr){3-4} |
| 743 |
\cmidrule(lr){5-6} |
| 744 |
\cmidrule(l){7-8} |
| 745 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 746 |
\midrule |
| 747 |
PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\ |
| 748 |
SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\ |
| 749 |
& 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\ |
| 750 |
& 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\ |
| 751 |
& 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\ |
| 752 |
SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\ |
| 753 |
& 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\ |
| 754 |
& 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\ |
| 755 |
& 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\ |
| 756 |
GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\ |
| 757 |
RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\ |
| 758 |
\midrule |
| 759 |
PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\ |
| 760 |
SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\ |
| 761 |
& 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\ |
| 762 |
& 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 763 |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 764 |
SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 765 |
& 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 766 |
& 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 767 |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 768 |
GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 769 |
RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 770 |
\midrule |
| 771 |
PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\ |
| 772 |
SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\ |
| 773 |
& 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\ |
| 774 |
& 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\ |
| 775 |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 776 |
SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\ |
| 777 |
& 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\ |
| 778 |
& 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\ |
| 779 |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 780 |
GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\ |
| 781 |
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 782 |
\bottomrule |
| 783 |
\end{tabular} |
| 784 |
\label{tab:argon} |
| 785 |
\end{table} |
| 786 |
|
| 787 |
\begin{table}[htbp] |
| 788 |
\centering |
| 789 |
\caption{Variance results from Gaussian fits to angular |
| 790 |
distributions of the force and torque vectors in the 6 \AA\ sphere of |
| 791 |
Argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
| 792 |
Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = |
| 793 |
Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group |
| 794 |
Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
| 795 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 796 |
\\ |
| 797 |
\toprule |
| 798 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 799 |
\cmidrule(lr){3-5} |
| 800 |
\cmidrule(l){6-8} |
| 801 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 802 |
\midrule |
| 803 |
PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\ |
| 804 |
SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\ |
| 805 |
& 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\ |
| 806 |
& 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\ |
| 807 |
& 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\ |
| 808 |
SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\ |
| 809 |
& 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\ |
| 810 |
& 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\ |
| 811 |
& 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\ |
| 812 |
GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 813 |
RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\ |
| 814 |
\midrule |
| 815 |
GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 816 |
& 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\ |
| 817 |
& 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\ |
| 818 |
& 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\ |
| 819 |
GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\ |
| 820 |
& 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\ |
| 821 |
& 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\ |
| 822 |
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 823 |
\bottomrule |
| 824 |
\end{tabular} |
| 825 |
\label{tab:argonAng} |
| 826 |
\end{table} |
| 827 |
|
| 828 |
This system does not appear to show any significant deviations from |
| 829 |
the previously observed results. The {\sc sp} and {\sc sf} methods |
| 830 |
have aggrements similar to those observed in section |
| 831 |
\ref{app:water}. The only significant difference is the improvement |
| 832 |
in the configuration energy differences for the {\sc rf} method. This |
| 833 |
is surprising in that we are introducing an inhomogeneity to the |
| 834 |
system; however, this inhomogeneity is charge-neutral and does not |
| 835 |
result in charged cutoff spheres. The charge-neutrality of the cutoff |
| 836 |
spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce, |
| 837 |
seems to play a greater role in the stability of the {\sc rf} method |
| 838 |
than the required homogeneity of the environment. |
| 839 |
|
| 840 |
\newpage |
| 841 |
|
| 842 |
\bibliographystyle{jcp2} |
| 843 |
\bibliography{electrostaticMethods} |
| 844 |
|
| 845 |
\end{document} |
