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\begin{document} |
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This document includes system based comparisons of the studied methods with smooth particle-mesh Ewald. Each of the seven systems comprises it's own section and has it's own discussion and tabular listing of the results for the $\Delta E$, force and torque vector magnitude, and force and torque vector direction comparisons. |
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\section{\label{app-water}Liquid Water} |
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|
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500 liquid state configurations were generated as described in the Methods section using the SPC/E model of water.\cite{Berendsen87} The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:spce}. The force and torque vector directionality results are displayed separately in table \ref{tab:spceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are 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 results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).} |
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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 \\ |
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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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|
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\midrule |
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|
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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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|
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\midrule |
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|
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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 \\ |
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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 \\ |
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& 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 distributions of the force and torque vectors in the liquid water 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.} |
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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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For the most parts, the water results appear to parallel the combined results seen in the discussion in the main paper. There is good agreement with SPME in both energetic and dynamic behavior when using the {\sc sf} method with and without damping. The {\sc sp} method does well with an $\alpha$ around $0.2 \AA^{-1}$, particularly with cutoff radii greater than 12 \AA. The results for both of these methods also begin to decay as damping gets too large. |
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The pure cutoff (PC) method performs poorly, as seen in the main discussion section. In contrast to the combined values, however, the use of a switching function and group based cutoffs really improves the results for these neutral water molecules. The group switched cutoff (GSC) shows mimics the energetics of SPME more poorly than the {\sc sp} (with moderate damping) and {\sc sf} methods, but the dynamics are quite good. The switching functions corrects discontinuities in the potential and forces, leading to the improved results. Such improvements with the use of a switching function has been recognized in previous studies,\cite{Andrea83,Steinbach94} and it is a useful tactic for stably incorporating local area electrostatic effects. |
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The reaction field (RF) method simply extends the results observed in the GSC case. Both methods are similar in form (i.e. neutral groups, switching function), but RF incorporates an added effect from the external dielectric. This similarity translates into the same good dynamic results and improved energetic results. These still fall short of the moderately damped {\sc sp} and {\sc sf} methods, but they display how incorporating some implicit properties of the surroundings (i.e. $\epsilon_\textrm{S}$) can improve results. |
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A final note for the liquid water system, use of group cutoffs and a switching function also leads to noticeable improvements in the {\sc sp} and {\sc sf} methods, primarily in directionality of the force and torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant narrowing of the angle distribution in the cases with little to no damping and only modest improvement for the ideal conditions ($\alpha = 0.2 \AA{-1}$ and $R_\textrm{c} \geqslant 12 \AA$). The {\sc sf} method simply shows modest narrowing across all damping and cutoff ranges of interest. Group cutoffs and the switching function do nothing for cases were error is introduced by overdamping the potentials. |
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|
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\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
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In addition to the disordered molecular system above, the ordered molecular system of ice I$_\textrm{c}$ was also considered. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:ice}. The force and torque vector directionality results are displayed separately in table \ref{tab:iceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
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|
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\begin{table}[htbp] |
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\centering |
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\caption{Regression results for the ice I$_\textrm{c}$ system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).} |
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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 & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\ |
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SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\ |
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& 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\ |
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& 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\ |
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& 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\ |
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SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\ |
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& 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\ |
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& 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\ |
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& 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\ |
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GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\ |
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RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\ |
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\midrule |
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PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\ |
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SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\ |
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& 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\ |
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& 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\ |
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& 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\ |
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SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\ |
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& 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\ |
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& 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\ |
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& 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\ |
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GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\ |
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RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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\midrule |
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PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\ |
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SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\ |
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& 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\ |
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& 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\ |
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& 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\ |
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SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\ |
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& 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\ |
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& 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\ |
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& 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\ |
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GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\ |
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RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\ |
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\bottomrule |
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\end{tabular} |
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\label{tab:ice} |
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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 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.} |
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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 & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\ |
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SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\ |
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& 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\ |
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& 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\ |
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& 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\ |
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SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\ |
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& 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\ |
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& 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\ |
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& 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\ |
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GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
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RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\ |
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\midrule |
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GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
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& 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\ |
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& 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\ |
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& 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
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GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\ |
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& 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\ |
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& 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\ |
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& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
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\bottomrule |
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\end{tabular} |
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\label{tab:iceAng} |
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\end{table} |
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|
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Highly ordered systems are a difficult test for the pairwise systems in that they lack the periodicity inherent to the Ewald summation. As expected, the energy gap agreement with SPME reduces for the {\sc sp} and {\sc sf} with parameters that were perfectly acceptable for the disordered liquid system. Moving to higher $R_\textrm{c}$ remedies this degraded performance, though at increase in computational cost. However, the dynamics of this crystalline system (both in magnitude and direction) are little affected. Both methods still reproduce the Ewald behavior with the same parameter recommendations from the previous section. |
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|
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It is also worth noting that RF exhibits a slightly improved energy gap results over the liquid water system. One possible explanation is that the ice I$_\textrm{c}$ crystal is ordered such that the net dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = \infty$, the reaction field incorporates this structural organization by actively enforcing a zeroed dipole moment within each cutoff sphere. |
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|
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\section{\label{app-melt}NaCl Melt} |
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|
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A high temperature NaCl melt was tested to gauge the accuracy of the pairwise summation methods in a highly charge disordered system. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:melt}. The force and torque vector directionality results are displayed separately in table \ref{tab:meltAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
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|
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\begin{table}[htbp] |
| 232 |
\centering |
| 233 |
\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.} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
| 236 |
\toprule |
| 237 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
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\cmidrule(lr){3-4} |
| 239 |
\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 & $R^2$ \\ |
| 242 |
\midrule |
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PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
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SP & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\ |
| 245 |
& 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\ |
| 246 |
& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 247 |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 248 |
SF & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\ |
| 249 |
& 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\ |
| 250 |
& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 251 |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 252 |
\midrule |
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PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
| 254 |
SP & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\ |
| 255 |
& 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\ |
| 256 |
& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 257 |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 258 |
SF & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\ |
| 259 |
& 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\ |
| 260 |
& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 261 |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 262 |
\bottomrule |
| 263 |
\end{tabular} |
| 264 |
\label{tab:melt} |
| 265 |
\end{table} |
| 266 |
|
| 267 |
\begin{table}[htbp] |
| 268 |
\centering |
| 269 |
\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.} |
| 270 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 271 |
\\ |
| 272 |
\toprule |
| 273 |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 274 |
\cmidrule(lr){3-5} |
| 275 |
\cmidrule(l){6-8} |
| 276 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 277 |
\midrule |
| 278 |
PC & & 13.294 & 8.035 & 5.366 \\ |
| 279 |
SP & 0.0 & 13.316 & 8.037 & 5.385 \\ |
| 280 |
& 0.1 & 5.705 & 1.391 & 0.360 \\ |
| 281 |
& 0.2 & 2.415 & 7.534 & 13.927 \\ |
| 282 |
& 0.3 & 23.769 & 67.306 & 57.252 \\ |
| 283 |
SF & 0.0 & 1.693 & 0.603 & 0.256 \\ |
| 284 |
& 0.1 & 1.687 & 0.653 & 0.272 \\ |
| 285 |
& 0.2 & 2.598 & 7.523 & 13.930 \\ |
| 286 |
& 0.3 & 23.734 & 67.305 & 57.252 \\ |
| 287 |
\bottomrule |
| 288 |
\end{tabular} |
| 289 |
\label{tab:meltAng} |
| 290 |
\end{table} |
| 291 |
|
| 292 |
|
| 293 |
|
| 294 |
\section{\label{app-salt}NaCl Crystal} |
| 295 |
|
| 296 |
A 1000K NaCl crystal was used to investigate the accuracy of the pairwise summation methods in an ordered system of charged particles. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:salt}. The force and torque vector directionality results are displayed separately in table \ref{tab:saltAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
| 297 |
|
| 298 |
\begin{table}[htbp] |
| 299 |
\centering |
| 300 |
\caption{Regression results for the crystalline 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.} |
| 301 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 302 |
\\ |
| 303 |
\toprule |
| 304 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 305 |
\cmidrule(lr){3-4} |
| 306 |
\cmidrule(lr){5-6} |
| 307 |
\cmidrule(l){7-8} |
| 308 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 309 |
\midrule |
| 310 |
PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\ |
| 311 |
SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\ |
| 312 |
& 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\ |
| 313 |
& 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\ |
| 314 |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\ |
| 315 |
SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\ |
| 316 |
& 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\ |
| 317 |
& 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\ |
| 318 |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\ |
| 319 |
\midrule |
| 320 |
PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\ |
| 321 |
SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\ |
| 322 |
& 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\ |
| 323 |
& 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\ |
| 324 |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 325 |
SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\ |
| 326 |
& 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\ |
| 327 |
& 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\ |
| 328 |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 329 |
\bottomrule |
| 330 |
\end{tabular} |
| 331 |
\label{tab:salt} |
| 332 |
\end{table} |
| 333 |
|
| 334 |
\begin{table}[htbp] |
| 335 |
\centering |
| 336 |
\caption{Variance results from Gaussian fits to angular distributions of the force vectors in the crystalline NaCl system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).} |
| 337 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 338 |
\\ |
| 339 |
\toprule |
| 340 |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 341 |
\cmidrule(lr){3-5} |
| 342 |
\cmidrule(l){6-8} |
| 343 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 344 |
\midrule |
| 345 |
PC & & 111.945 & 111.824 & 111.866 \\ |
| 346 |
SP & 0.0 & 112.414 & 152.215 & 38.087 \\ |
| 347 |
& 0.1 & 52.361 & 42.574 & 2.819 \\ |
| 348 |
& 0.2 & 10.847 & 9.709 & 9.686 \\ |
| 349 |
& 0.3 & 31.128 & 31.104 & 31.029 \\ |
| 350 |
SF & 0.0 & 10.025 & 3.555 & 1.648 \\ |
| 351 |
& 0.1 & 9.462 & 3.303 & 1.721 \\ |
| 352 |
& 0.2 & 11.454 & 9.813 & 9.701 \\ |
| 353 |
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
| 354 |
\bottomrule |
| 355 |
\end{tabular} |
| 356 |
\label{tab:saltAng} |
| 357 |
\end{table} |
| 358 |
|
| 359 |
\section{\label{app-sol1}Weak NaCl Solution} |
| 360 |
|
| 361 |
In an effort to bridge the charged atomic and neutral molecular systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into the liquid water system. This low ionic strength system consists of 4 ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}. The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
| 362 |
|
| 363 |
\begin{table}[htbp] |
| 364 |
\centering |
| 365 |
\caption{Regression results for the weak NaCl solution system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set). 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.} |
| 366 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 367 |
\\ |
| 368 |
\toprule |
| 369 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 370 |
\cmidrule(lr){3-4} |
| 371 |
\cmidrule(lr){5-6} |
| 372 |
\cmidrule(l){7-8} |
| 373 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 374 |
\midrule |
| 375 |
PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\ |
| 376 |
SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\ |
| 377 |
& 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\ |
| 378 |
& 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 379 |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 380 |
SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\ |
| 381 |
& 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\ |
| 382 |
& 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 383 |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 384 |
GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\ |
| 385 |
RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\ |
| 386 |
\midrule |
| 387 |
PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\ |
| 388 |
SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\ |
| 389 |
& 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\ |
| 390 |
& 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 391 |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 392 |
SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\ |
| 393 |
& 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 394 |
& 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 395 |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 396 |
GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\ |
| 397 |
RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\ |
| 398 |
\midrule |
| 399 |
PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\ |
| 400 |
SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\ |
| 401 |
& 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\ |
| 402 |
& 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 403 |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 404 |
SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\ |
| 405 |
& 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\ |
| 406 |
& 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 407 |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 408 |
GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\ |
| 409 |
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 410 |
\bottomrule |
| 411 |
\end{tabular} |
| 412 |
\label{tab:solnWeak} |
| 413 |
\end{table} |
| 414 |
|
| 415 |
\begin{table}[htbp] |
| 416 |
\centering |
| 417 |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the weak 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.} |
| 418 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 419 |
\\ |
| 420 |
\toprule |
| 421 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 422 |
\cmidrule(lr){3-5} |
| 423 |
\cmidrule(l){6-8} |
| 424 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 425 |
\midrule |
| 426 |
PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\ |
| 427 |
SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\ |
| 428 |
& 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\ |
| 429 |
& 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\ |
| 430 |
& 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\ |
| 431 |
SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\ |
| 432 |
& 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\ |
| 433 |
& 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\ |
| 434 |
& 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\ |
| 435 |
GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 436 |
RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\ |
| 437 |
\midrule |
| 438 |
GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 439 |
& 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\ |
| 440 |
& 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\ |
| 441 |
& 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\ |
| 442 |
GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\ |
| 443 |
& 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\ |
| 444 |
& 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\ |
| 445 |
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 446 |
\bottomrule |
| 447 |
\end{tabular} |
| 448 |
\label{tab:solnWeakAng} |
| 449 |
\end{table} |
| 450 |
|
| 451 |
\section{\label{app-sol10}Strong NaCl Solution} |
| 452 |
|
| 453 |
The bridging of the charged atomic and neutral molecular systems was furthered by considering a high ionic strength system consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}. The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
| 454 |
|
| 455 |
\begin{table}[htbp] |
| 456 |
\centering |
| 457 |
\caption{Regression results for the strong NaCl solution system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).} |
| 458 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 459 |
\\ |
| 460 |
\toprule |
| 461 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 462 |
\cmidrule(lr){3-4} |
| 463 |
\cmidrule(lr){5-6} |
| 464 |
\cmidrule(l){7-8} |
| 465 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 466 |
\midrule |
| 467 |
PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\ |
| 468 |
SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\ |
| 469 |
& 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\ |
| 470 |
& 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 471 |
& 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 472 |
SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\ |
| 473 |
& 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\ |
| 474 |
& 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 475 |
& 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 476 |
GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\ |
| 477 |
RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\ |
| 478 |
\midrule |
| 479 |
PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\ |
| 480 |
SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\ |
| 481 |
& 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\ |
| 482 |
& 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 483 |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 484 |
SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\ |
| 485 |
& 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\ |
| 486 |
& 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 487 |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 488 |
GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\ |
| 489 |
RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\ |
| 490 |
\midrule |
| 491 |
PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\ |
| 492 |
SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\ |
| 493 |
& 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\ |
| 494 |
& 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 495 |
& 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 496 |
SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\ |
| 497 |
& 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\ |
| 498 |
& 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 499 |
& 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 500 |
GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\ |
| 501 |
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 502 |
\bottomrule |
| 503 |
\end{tabular} |
| 504 |
\label{tab:solnStr} |
| 505 |
\end{table} |
| 506 |
|
| 507 |
\begin{table}[htbp] |
| 508 |
\centering |
| 509 |
\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.} |
| 510 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 511 |
\\ |
| 512 |
\toprule |
| 513 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 514 |
\cmidrule(lr){3-5} |
| 515 |
\cmidrule(l){6-8} |
| 516 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 517 |
\midrule |
| 518 |
PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\ |
| 519 |
SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\ |
| 520 |
& 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\ |
| 521 |
& 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\ |
| 522 |
& 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\ |
| 523 |
SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\ |
| 524 |
& 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\ |
| 525 |
& 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\ |
| 526 |
& 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\ |
| 527 |
GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\ |
| 528 |
RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\ |
| 529 |
\midrule |
| 530 |
GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\ |
| 531 |
& 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\ |
| 532 |
& 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\ |
| 533 |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\ |
| 534 |
GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\ |
| 535 |
& 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\ |
| 536 |
& 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\ |
| 537 |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 538 |
\bottomrule |
| 539 |
\end{tabular} |
| 540 |
\label{tab:solnStrAng} |
| 541 |
\end{table} |
| 542 |
|
| 543 |
\section{\label{app-argon}Argon Sphere in Water} |
| 544 |
|
| 545 |
The final model system studied was 6 \AA\ sphere of Argon solvated by SPC/E water. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}. The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
| 546 |
|
| 547 |
\begin{table}[htbp] |
| 548 |
\centering |
| 549 |
\caption{Regression results for the 6 \AA\ argon sphere in liquid water system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).} |
| 550 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 551 |
\\ |
| 552 |
\toprule |
| 553 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 554 |
\cmidrule(lr){3-4} |
| 555 |
\cmidrule(lr){5-6} |
| 556 |
\cmidrule(l){7-8} |
| 557 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 558 |
\midrule |
| 559 |
PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\ |
| 560 |
SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\ |
| 561 |
& 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\ |
| 562 |
& 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\ |
| 563 |
& 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\ |
| 564 |
SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\ |
| 565 |
& 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\ |
| 566 |
& 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\ |
| 567 |
& 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\ |
| 568 |
GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\ |
| 569 |
RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\ |
| 570 |
\midrule |
| 571 |
PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\ |
| 572 |
SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\ |
| 573 |
& 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\ |
| 574 |
& 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 575 |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 576 |
SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 577 |
& 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 578 |
& 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 579 |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 580 |
GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 581 |
RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 582 |
\midrule |
| 583 |
PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\ |
| 584 |
SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\ |
| 585 |
& 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\ |
| 586 |
& 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\ |
| 587 |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 588 |
SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\ |
| 589 |
& 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\ |
| 590 |
& 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\ |
| 591 |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 592 |
GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\ |
| 593 |
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 594 |
\bottomrule |
| 595 |
\end{tabular} |
| 596 |
\label{tab:argon} |
| 597 |
\end{table} |
| 598 |
|
| 599 |
\begin{table}[htbp] |
| 600 |
\centering |
| 601 |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the 6 \AA\ sphere of argon in liquid water 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.} |
| 602 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 603 |
\\ |
| 604 |
\toprule |
| 605 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 606 |
\cmidrule(lr){3-5} |
| 607 |
\cmidrule(l){6-8} |
| 608 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 609 |
\midrule |
| 610 |
PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\ |
| 611 |
SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\ |
| 612 |
& 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\ |
| 613 |
& 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\ |
| 614 |
& 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\ |
| 615 |
SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\ |
| 616 |
& 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\ |
| 617 |
& 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\ |
| 618 |
& 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\ |
| 619 |
GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 620 |
RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\ |
| 621 |
\midrule |
| 622 |
GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 623 |
& 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\ |
| 624 |
& 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\ |
| 625 |
& 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\ |
| 626 |
GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\ |
| 627 |
& 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\ |
| 628 |
& 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\ |
| 629 |
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 630 |
\bottomrule |
| 631 |
\end{tabular} |
| 632 |
\label{tab:argonAng} |
| 633 |
\end{table} |
| 634 |
|
| 635 |
\newpage |
| 636 |
|
| 637 |
\bibliographystyle{jcp2} |
| 638 |
\bibliography{electrostaticMethods} |
| 639 |
|
| 640 |
\end{document} |
