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\begin{document} |
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This document includes individual system-based comparisons of the |
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studied methods with smooth particle-mesh Ewald. Each of the seven |
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systems comprises its own section and has its own discussion and |
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tabular listing of the results for the $\Delta E$, force and torque |
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vector magnitude, and force and torque vector direction comparisons. |
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\section{\label{app:water}Liquid Water} |
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500 liquid state configurations were generated as described in the |
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Methods section using the SPC/E model of water.\cite{Berendsen87} The |
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results for the energy gap comparisons and the force and torque vector |
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magnitude comparisons are shown in table \ref{tab:spce}. The force |
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and torque vector directionality results are displayed separately in |
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table \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 |
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results include $\Delta E$ values (top set), force vector magnitudes |
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(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 $\varepsilon \approx |
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\infty$).} |
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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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\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 \\ |
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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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\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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For the most parts, the water results appear to parallel the combined |
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results seen in the discussion in the main paper. There is good |
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agreement with SPME in both energetic and dynamic behavior when using |
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the {\sc sf} method with and without damping. The {\sc sp} method does |
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well with an $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff |
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radii greater than 12 \AA. The results for both of these methods also |
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begin to decay as damping gets too large. |
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The pure cutoff (PC) method performs poorly, as seen in the main |
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discussion section. In contrast to the combined values, however, the |
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use of a switching function and group based cutoffs really improves |
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the results for these neutral water molecules. The group switched |
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cutoff (GSC) shows mimics the energetics of SPME more poorly than the |
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{\sc sp} (with moderate damping) and {\sc sf} methods, but the |
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dynamics are quite good. The switching functions corrects |
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discontinuities in the potential and forces, leading to the improved |
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results. Such improvements with the use of a switching function has |
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been recognized in previous studies,\cite{Andrea83,Steinbach94} and it |
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is a useful tactic for stably incorporating local area electrostatic |
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effects. |
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The reaction field (RF) method simply extends the results observed in |
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the GSC case. Both methods are similar in form (i.e. neutral groups, |
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switching function), but RF incorporates an added effect from the |
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external dielectric. This similarity translates into the same good |
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dynamic results and improved energetic results. These still fall |
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short of the moderately damped {\sc sp} and {\sc sf} methods, but they |
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display how incorporating some implicit properties of the surroundings |
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(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 |
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switching function also leads to noticeable improvements in the {\sc |
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sp} and {\sc sf} methods, primarily in directionality of the force and |
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torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant |
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narrowing of the angle distribution in the cases with little to no |
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damping and only modest improvement for the ideal conditions ($\alpha$ |
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= 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The {\sc sf} |
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method simply shows modest narrowing across all damping and cutoff |
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ranges of interest. Group cutoffs and the switching function do |
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nothing for cases were error is introduced by overdamping the |
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potentials. |
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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 |
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molecular system of ice I$_\textrm{c}$ was also considered. The |
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results for the energy gap comparisons and the force and torque vector |
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magnitude comparisons are shown in table \ref{tab:ice}. The force and |
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torque vector directionality results are displayed separately in table |
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\ref{tab:iceAng}, 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 ice I$_\textrm{c}$ |
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system. Tabulated results include $\Delta E$ values (top set), force |
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vector magnitudes (middle set) and torque vector magnitudes (bottom |
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set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
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GSC = Group Switched Cutoff, and RF = Reaction Field (where |
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$\varepsilon \approx \infty$).} |
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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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\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 \\ |
264 |
|
|
& 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\ |
265 |
|
|
& 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\ |
266 |
|
|
SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\ |
267 |
|
|
& 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\ |
268 |
|
|
& 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\ |
269 |
|
|
& 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\ |
270 |
|
|
GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
271 |
|
|
RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\ |
272 |
|
|
\midrule |
273 |
|
|
GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
274 |
|
|
& 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\ |
275 |
|
|
& 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\ |
276 |
|
|
& 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
277 |
|
|
GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\ |
278 |
|
|
& 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\ |
279 |
|
|
& 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\ |
280 |
|
|
& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
281 |
|
|
\bottomrule |
282 |
|
|
\end{tabular} |
283 |
chrisfen |
2652 |
\label{tab:iceAng} |
284 |
chrisfen |
2599 |
\end{table} |
285 |
|
|
|
286 |
gezelter |
2658 |
Highly ordered systems are a difficult test for the pairwise systems |
287 |
|
|
in that they lack the periodicity inherent to the Ewald summation. As |
288 |
|
|
expected, the energy gap agreement with SPME reduces for the {\sc sp} |
289 |
|
|
and {\sc sf} with parameters that were perfectly acceptable for the |
290 |
|
|
disordered liquid system. Moving to higher $R_\textrm{c}$ remedies |
291 |
|
|
this degraded performance, though at increase in computational cost. |
292 |
|
|
However, the dynamics of this crystalline system (both in magnitude |
293 |
|
|
and direction) are little affected. Both methods still reproduce the |
294 |
|
|
Ewald behavior with the same parameter recommendations from the |
295 |
|
|
previous section. |
296 |
chrisfen |
2652 |
|
297 |
gezelter |
2658 |
It is also worth noting that RF exhibits a slightly improved energy |
298 |
|
|
gap results over the liquid water system. One possible explanation is |
299 |
|
|
that the ice I$_\textrm{c}$ crystal is ordered such that the net |
300 |
|
|
dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = |
301 |
|
|
\infty$, the reaction field incorporates this structural organization |
302 |
|
|
by actively enforcing a zeroed dipole moment within each cutoff |
303 |
|
|
sphere. |
304 |
chrisfen |
2652 |
|
305 |
chrisfen |
2660 |
\section{\label{app:melt}NaCl Melt} |
306 |
chrisfen |
2599 |
|
307 |
gezelter |
2658 |
A high temperature NaCl melt was tested to gauge the accuracy of the |
308 |
|
|
pairwise summation methods in a highly charge disordered system. The |
309 |
|
|
results for the energy gap comparisons and the force and torque vector |
310 |
|
|
magnitude comparisons are shown in table \ref{tab:melt}. The force |
311 |
|
|
and torque vector directionality results are displayed separately in |
312 |
|
|
table \ref{tab:meltAng}, where the effect of group-based cutoffs and |
313 |
|
|
switching functions on the {\sc sp} and {\sc sf} potentials are |
314 |
|
|
investigated. |
315 |
chrisfen |
2652 |
|
316 |
chrisfen |
2599 |
\begin{table}[htbp] |
317 |
|
|
\centering |
318 |
|
|
\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.} |
319 |
|
|
\begin{tabular}{@{} ccrrrrrr @{}} |
320 |
|
|
\\ |
321 |
|
|
\toprule |
322 |
|
|
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
323 |
|
|
\cmidrule(lr){3-4} |
324 |
|
|
\cmidrule(lr){5-6} |
325 |
|
|
\cmidrule(l){7-8} |
326 |
|
|
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
327 |
|
|
\midrule |
328 |
|
|
PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
329 |
chrisfen |
2655 |
SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\ |
330 |
|
|
& 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\ |
331 |
chrisfen |
2599 |
& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
332 |
|
|
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
333 |
chrisfen |
2655 |
SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\ |
334 |
|
|
& 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\ |
335 |
chrisfen |
2599 |
& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
336 |
|
|
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
337 |
|
|
\midrule |
338 |
|
|
PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
339 |
chrisfen |
2655 |
SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\ |
340 |
|
|
& 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\ |
341 |
chrisfen |
2599 |
& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
342 |
|
|
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
343 |
chrisfen |
2655 |
SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\ |
344 |
|
|
& 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\ |
345 |
chrisfen |
2599 |
& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
346 |
|
|
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
347 |
|
|
\bottomrule |
348 |
|
|
\end{tabular} |
349 |
chrisfen |
2652 |
\label{tab:melt} |
350 |
chrisfen |
2599 |
\end{table} |
351 |
|
|
|
352 |
|
|
\begin{table}[htbp] |
353 |
|
|
\centering |
354 |
|
|
\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.} |
355 |
|
|
\begin{tabular}{@{} ccrrrrrr @{}} |
356 |
|
|
\\ |
357 |
|
|
\toprule |
358 |
|
|
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
359 |
|
|
\cmidrule(lr){3-5} |
360 |
|
|
\cmidrule(l){6-8} |
361 |
|
|
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
362 |
|
|
\midrule |
363 |
|
|
PC & & 13.294 & 8.035 & 5.366 \\ |
364 |
|
|
SP & 0.0 & 13.316 & 8.037 & 5.385 \\ |
365 |
|
|
& 0.1 & 5.705 & 1.391 & 0.360 \\ |
366 |
|
|
& 0.2 & 2.415 & 7.534 & 13.927 \\ |
367 |
|
|
& 0.3 & 23.769 & 67.306 & 57.252 \\ |
368 |
|
|
SF & 0.0 & 1.693 & 0.603 & 0.256 \\ |
369 |
|
|
& 0.1 & 1.687 & 0.653 & 0.272 \\ |
370 |
|
|
& 0.2 & 2.598 & 7.523 & 13.930 \\ |
371 |
|
|
& 0.3 & 23.734 & 67.305 & 57.252 \\ |
372 |
|
|
\bottomrule |
373 |
|
|
\end{tabular} |
374 |
chrisfen |
2652 |
\label{tab:meltAng} |
375 |
chrisfen |
2599 |
\end{table} |
376 |
|
|
|
377 |
chrisfen |
2660 |
The molten NaCl system shows more sensitivity to the electrostatic |
378 |
|
|
damping than the water systems. The most noticeable point is that the |
379 |
|
|
undamped {\sc sf} method does very well at replicating the {\sc spme} |
380 |
|
|
configurational energy differences and forces. Light damping appears |
381 |
|
|
to minimally improve the dynamics, but this comes with a deterioration |
382 |
|
|
of the energy gap results. In contrast, this light damping improves |
383 |
|
|
the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic |
384 |
|
|
damping reduce the agreement with {\sc spme} for both methods. From |
385 |
|
|
these observations, the undamped {\sc sf} method is the best choice |
386 |
|
|
for disordered systems of charges. |
387 |
chrisfen |
2654 |
|
388 |
chrisfen |
2660 |
\section{\label{app:salt}NaCl Crystal} |
389 |
chrisfen |
2599 |
|
390 |
gezelter |
2658 |
A 1000K NaCl crystal was used to investigate the accuracy of the |
391 |
|
|
pairwise summation methods in an ordered system of charged |
392 |
|
|
particles. The results for the energy gap comparisons and the force |
393 |
|
|
and torque vector magnitude comparisons are shown in table |
394 |
|
|
\ref{tab:salt}. The force and torque vector directionality results |
395 |
|
|
are displayed separately in table \ref{tab:saltAng}, where the effect |
396 |
|
|
of group-based cutoffs and switching functions on the {\sc sp} and |
397 |
|
|
{\sc sf} potentials are investigated. |
398 |
chrisfen |
2652 |
|
399 |
chrisfen |
2599 |
\begin{table}[htbp] |
400 |
|
|
\centering |
401 |
gezelter |
2658 |
\caption{Regression results for the crystalline NaCl |
402 |
|
|
system. Tabulated results include $\Delta E$ values (top set) and |
403 |
|
|
force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted |
404 |
|
|
Potential, and SF = Shifted Force.} |
405 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
406 |
|
|
\\ |
407 |
|
|
\toprule |
408 |
|
|
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
409 |
|
|
\cmidrule(lr){3-4} |
410 |
|
|
\cmidrule(lr){5-6} |
411 |
|
|
\cmidrule(l){7-8} |
412 |
|
|
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
413 |
|
|
\midrule |
414 |
|
|
PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\ |
415 |
|
|
SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\ |
416 |
|
|
& 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\ |
417 |
|
|
& 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\ |
418 |
|
|
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\ |
419 |
|
|
SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\ |
420 |
|
|
& 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\ |
421 |
|
|
& 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\ |
422 |
|
|
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\ |
423 |
|
|
\midrule |
424 |
|
|
PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\ |
425 |
|
|
SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\ |
426 |
|
|
& 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\ |
427 |
|
|
& 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\ |
428 |
|
|
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
429 |
|
|
SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\ |
430 |
|
|
& 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\ |
431 |
|
|
& 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\ |
432 |
|
|
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
433 |
|
|
\bottomrule |
434 |
|
|
\end{tabular} |
435 |
chrisfen |
2652 |
\label{tab:salt} |
436 |
chrisfen |
2599 |
\end{table} |
437 |
|
|
|
438 |
|
|
\begin{table}[htbp] |
439 |
|
|
\centering |
440 |
gezelter |
2658 |
\caption{Variance results from Gaussian fits to angular |
441 |
|
|
distributions of the force vectors in the crystalline NaCl system. PC |
442 |
|
|
= Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group |
443 |
|
|
Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
444 |
|
|
\infty$).} |
445 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
446 |
|
|
\\ |
447 |
|
|
\toprule |
448 |
|
|
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
449 |
|
|
\cmidrule(lr){3-5} |
450 |
|
|
\cmidrule(l){6-8} |
451 |
|
|
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
452 |
|
|
\midrule |
453 |
|
|
PC & & 111.945 & 111.824 & 111.866 \\ |
454 |
|
|
SP & 0.0 & 112.414 & 152.215 & 38.087 \\ |
455 |
|
|
& 0.1 & 52.361 & 42.574 & 2.819 \\ |
456 |
|
|
& 0.2 & 10.847 & 9.709 & 9.686 \\ |
457 |
|
|
& 0.3 & 31.128 & 31.104 & 31.029 \\ |
458 |
|
|
SF & 0.0 & 10.025 & 3.555 & 1.648 \\ |
459 |
|
|
& 0.1 & 9.462 & 3.303 & 1.721 \\ |
460 |
|
|
& 0.2 & 11.454 & 9.813 & 9.701 \\ |
461 |
|
|
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
462 |
|
|
\bottomrule |
463 |
|
|
\end{tabular} |
464 |
chrisfen |
2652 |
\label{tab:saltAng} |
465 |
chrisfen |
2599 |
\end{table} |
466 |
|
|
|
467 |
chrisfen |
2660 |
The crystalline NaCl system is the most challenging test case for the |
468 |
|
|
pairwise summation methods, as evidenced by the results in tables |
469 |
|
|
\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped |
470 |
|
|
{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best |
471 |
|
|
choices. These methods match well with {\sc spme} across the energy |
472 |
|
|
gap, force magnitude, and force directionality tests. The {\sc sp} |
473 |
|
|
method struggles in all cases with the exception of good dynamics |
474 |
|
|
reproduction when using weak electrostatic damping with a large cutoff |
475 |
|
|
radius. |
476 |
chrisfen |
2599 |
|
477 |
chrisfen |
2660 |
The moderate electrostatic damping case is not as good as we would |
478 |
|
|
expect given the good long-time dynamics results observed for this |
479 |
|
|
system. Since these results are a test of instantaneous dynamics, this |
480 |
|
|
indicates that good long-time dynamics comes in part at the expense of |
481 |
|
|
short-time dynamics. Further indication of this comes from the full |
482 |
|
|
power spectra shown in the main text. It appears as though a |
483 |
|
|
distortion is introduced between 200 to 300 cm$^{-1}$ with increased |
484 |
|
|
$\alpha$. |
485 |
|
|
|
486 |
|
|
\section{\label{app:solnWeak}Weak NaCl Solution} |
487 |
|
|
|
488 |
gezelter |
2658 |
In an effort to bridge the charged atomic and neutral molecular |
489 |
|
|
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
490 |
|
|
the liquid water system. This low ionic strength system consists of 4 |
491 |
|
|
ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results |
492 |
|
|
for the energy gap comparisons and the force and torque vector |
493 |
|
|
magnitude comparisons are shown in table \ref{tab:solnWeak}. The |
494 |
|
|
force and torque vector directionality results are displayed |
495 |
|
|
separately in table \ref{tab:solnWeakAng}, where the effect of |
496 |
|
|
group-based cutoffs and switching functions on the {\sc sp} and {\sc |
497 |
|
|
sf} potentials are investigated. |
498 |
chrisfen |
2652 |
|
499 |
chrisfen |
2599 |
\begin{table}[htbp] |
500 |
|
|
\centering |
501 |
gezelter |
2658 |
\caption{Regression results for the weak NaCl solution |
502 |
|
|
system. Tabulated results include $\Delta E$ values (top set), force |
503 |
|
|
vector magnitudes (middle set) and torque vector magnitudes (bottom |
504 |
|
|
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
505 |
|
|
GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon |
506 |
|
|
\approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = |
507 |
|
|
Group Switched Shifted Force.} |
508 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
509 |
|
|
\\ |
510 |
|
|
\toprule |
511 |
|
|
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
512 |
|
|
\cmidrule(lr){3-4} |
513 |
|
|
\cmidrule(lr){5-6} |
514 |
|
|
\cmidrule(l){7-8} |
515 |
|
|
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
516 |
|
|
\midrule |
517 |
|
|
PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\ |
518 |
|
|
SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\ |
519 |
|
|
& 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\ |
520 |
|
|
& 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
521 |
|
|
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
522 |
|
|
SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\ |
523 |
|
|
& 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\ |
524 |
|
|
& 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
525 |
|
|
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
526 |
|
|
GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\ |
527 |
|
|
RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\ |
528 |
|
|
\midrule |
529 |
|
|
PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\ |
530 |
|
|
SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\ |
531 |
|
|
& 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\ |
532 |
|
|
& 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
533 |
|
|
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
534 |
|
|
SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\ |
535 |
|
|
& 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
536 |
|
|
& 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
537 |
|
|
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
538 |
|
|
GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\ |
539 |
|
|
RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\ |
540 |
|
|
\midrule |
541 |
|
|
PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\ |
542 |
|
|
SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\ |
543 |
|
|
& 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\ |
544 |
|
|
& 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
545 |
|
|
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
546 |
|
|
SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\ |
547 |
|
|
& 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\ |
548 |
|
|
& 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
549 |
|
|
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
550 |
|
|
GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\ |
551 |
|
|
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
552 |
|
|
\bottomrule |
553 |
|
|
\end{tabular} |
554 |
chrisfen |
2652 |
\label{tab:solnWeak} |
555 |
chrisfen |
2599 |
\end{table} |
556 |
|
|
|
557 |
|
|
\begin{table}[htbp] |
558 |
|
|
\centering |
559 |
gezelter |
2658 |
\caption{Variance results from Gaussian fits to angular |
560 |
|
|
distributions of the force and torque vectors in the weak NaCl |
561 |
|
|
solution system. PC = Pure Cutoff, SP = Shifted Potential, SF = |
562 |
|
|
Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where |
563 |
|
|
$\varepsilon \approx \infty$), GSSP = Group Switched Shifted |
564 |
|
|
Potential, and GSSF = Group Switched Shifted Force.} |
565 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
566 |
|
|
\\ |
567 |
|
|
\toprule |
568 |
|
|
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
569 |
|
|
\cmidrule(lr){3-5} |
570 |
|
|
\cmidrule(l){6-8} |
571 |
|
|
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
572 |
|
|
\midrule |
573 |
|
|
PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\ |
574 |
|
|
SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\ |
575 |
|
|
& 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\ |
576 |
|
|
& 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\ |
577 |
|
|
& 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\ |
578 |
|
|
SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\ |
579 |
|
|
& 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\ |
580 |
|
|
& 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\ |
581 |
|
|
& 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\ |
582 |
|
|
GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
583 |
|
|
RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\ |
584 |
|
|
\midrule |
585 |
|
|
GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
586 |
|
|
& 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\ |
587 |
|
|
& 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\ |
588 |
|
|
& 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\ |
589 |
|
|
GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\ |
590 |
|
|
& 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\ |
591 |
|
|
& 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\ |
592 |
|
|
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
593 |
|
|
\bottomrule |
594 |
|
|
\end{tabular} |
595 |
chrisfen |
2652 |
\label{tab:solnWeakAng} |
596 |
chrisfen |
2599 |
\end{table} |
597 |
|
|
|
598 |
chrisfen |
2660 |
This weak ionic strength system can be considered as a perturbation of |
599 |
|
|
the pure liquid water system. The {\sc sp} and {\sc sf} methods are |
600 |
|
|
not significantly affected by the inclusion of a few ions. The aspect |
601 |
|
|
of cutoff sphere neutralization aids in the smooth incorporation of |
602 |
|
|
these ions; thus, all of the observations regarding these methods |
603 |
|
|
carry over from section \ref{app:water}. The differences between these |
604 |
|
|
systems are visible for the {\sc rf} method. Though good force |
605 |
|
|
reproduction is still maintained, the energy gaps show a significant |
606 |
|
|
increase in the data scatter. This foreshadows the breakdown of the |
607 |
|
|
method as we introduce system inhomogeneities. |
608 |
chrisfen |
2599 |
|
609 |
chrisfen |
2660 |
\section{\label{app:solnStr}Strong NaCl Solution} |
610 |
|
|
|
611 |
gezelter |
2658 |
The bridging of the charged atomic and neutral molecular systems was |
612 |
chrisfen |
2660 |
further developed by considering a high ionic strength system |
613 |
|
|
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
614 |
|
|
M). The results for the energy gap comparisons and the force and |
615 |
|
|
torque vector magnitude comparisons are shown in table |
616 |
|
|
\ref{tab:solnWeak}. The force and torque vector directionality |
617 |
|
|
results are displayed separately in table\ref{tab:solnWeakAng}, where |
618 |
|
|
the effect of group-based cutoffs and switching functions on the {\sc |
619 |
|
|
sp} and {\sc sf} potentials are investigated. |
620 |
chrisfen |
2652 |
|
621 |
chrisfen |
2599 |
\begin{table}[htbp] |
622 |
|
|
\centering |
623 |
gezelter |
2658 |
\caption{Regression results for the strong NaCl solution |
624 |
|
|
system. Tabulated results include $\Delta E$ values (top set), force |
625 |
|
|
vector magnitudes (middle set) and torque vector magnitudes (bottom |
626 |
|
|
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
627 |
|
|
GSC = Group Switched Cutoff, and RF = Reaction Field (where |
628 |
|
|
$\varepsilon \approx \infty$).} |
629 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
630 |
|
|
\\ |
631 |
|
|
\toprule |
632 |
|
|
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
633 |
|
|
\cmidrule(lr){3-4} |
634 |
|
|
\cmidrule(lr){5-6} |
635 |
|
|
\cmidrule(l){7-8} |
636 |
|
|
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
637 |
|
|
\midrule |
638 |
|
|
PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\ |
639 |
|
|
SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\ |
640 |
|
|
& 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\ |
641 |
|
|
& 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
642 |
|
|
& 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
643 |
|
|
SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\ |
644 |
|
|
& 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\ |
645 |
|
|
& 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
646 |
|
|
& 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
647 |
|
|
GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\ |
648 |
|
|
RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\ |
649 |
|
|
\midrule |
650 |
|
|
PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\ |
651 |
|
|
SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\ |
652 |
|
|
& 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\ |
653 |
|
|
& 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
654 |
|
|
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
655 |
|
|
SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\ |
656 |
|
|
& 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\ |
657 |
|
|
& 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
658 |
|
|
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
659 |
|
|
GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\ |
660 |
|
|
RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\ |
661 |
|
|
\midrule |
662 |
|
|
PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\ |
663 |
|
|
SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\ |
664 |
|
|
& 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\ |
665 |
|
|
& 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
666 |
|
|
& 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
667 |
|
|
SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\ |
668 |
|
|
& 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\ |
669 |
|
|
& 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
670 |
|
|
& 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
671 |
|
|
GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\ |
672 |
|
|
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
673 |
|
|
\bottomrule |
674 |
|
|
\end{tabular} |
675 |
chrisfen |
2652 |
\label{tab:solnStr} |
676 |
chrisfen |
2599 |
\end{table} |
677 |
|
|
|
678 |
|
|
\begin{table}[htbp] |
679 |
|
|
\centering |
680 |
|
|
\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.} |
681 |
|
|
\begin{tabular}{@{} ccrrrrrr @{}} |
682 |
|
|
\\ |
683 |
|
|
\toprule |
684 |
|
|
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
685 |
|
|
\cmidrule(lr){3-5} |
686 |
|
|
\cmidrule(l){6-8} |
687 |
|
|
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
688 |
|
|
\midrule |
689 |
|
|
PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\ |
690 |
|
|
SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\ |
691 |
|
|
& 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\ |
692 |
|
|
& 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\ |
693 |
|
|
& 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\ |
694 |
|
|
SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\ |
695 |
|
|
& 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\ |
696 |
|
|
& 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\ |
697 |
|
|
& 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\ |
698 |
|
|
GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\ |
699 |
|
|
RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\ |
700 |
|
|
\midrule |
701 |
|
|
GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\ |
702 |
|
|
& 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\ |
703 |
|
|
& 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\ |
704 |
|
|
& 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\ |
705 |
|
|
GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\ |
706 |
|
|
& 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\ |
707 |
|
|
& 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\ |
708 |
|
|
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
709 |
|
|
\bottomrule |
710 |
|
|
\end{tabular} |
711 |
chrisfen |
2652 |
\label{tab:solnStrAng} |
712 |
chrisfen |
2599 |
\end{table} |
713 |
|
|
|
714 |
chrisfen |
2660 |
The {\sc rf} method struggles with the jump in ionic strength. The |
715 |
|
|
configuration energy difference degrade to unuseable levels while the |
716 |
|
|
forces and torques degrade in a more modest fashion. The {\sc rf} |
717 |
|
|
method was designed for homogeneous systems, and this restriction is |
718 |
|
|
apparent in these results. |
719 |
chrisfen |
2599 |
|
720 |
chrisfen |
2660 |
The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
721 |
|
|
their agreement with {\sc spme}. With these results, we still |
722 |
|
|
recommend no to moderate damping for the {\sc sf} method and moderate |
723 |
|
|
damping for the {\sc sp} method, both with cutoffs greater than 12 |
724 |
|
|
\AA. |
725 |
|
|
|
726 |
|
|
\section{\label{app:argon}Argon Sphere in Water} |
727 |
|
|
|
728 |
gezelter |
2658 |
The final model system studied was 6 \AA\ sphere of Argon solvated by |
729 |
|
|
SPC/E water. The results for the energy gap comparisons and the force |
730 |
|
|
and torque vector magnitude comparisons are shown in table |
731 |
|
|
\ref{tab:solnWeak}. The force and torque vector directionality |
732 |
|
|
results are displayed separately in table \ref{tab:solnWeakAng}, where |
733 |
|
|
the effect of group-based cutoffs and switching functions on the {\sc |
734 |
|
|
sp} and {\sc sf} potentials are investigated. |
735 |
chrisfen |
2652 |
|
736 |
chrisfen |
2599 |
\begin{table}[htbp] |
737 |
|
|
\centering |
738 |
gezelter |
2658 |
\caption{Regression results for the 6 \AA\ argon sphere in liquid |
739 |
|
|
water system. Tabulated results include $\Delta E$ values (top set), |
740 |
|
|
force vector magnitudes (middle set) and torque vector magnitudes |
741 |
|
|
(bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted |
742 |
|
|
Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where |
743 |
|
|
$\varepsilon \approx \infty$).} |
744 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
745 |
|
|
\\ |
746 |
|
|
\toprule |
747 |
|
|
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
748 |
|
|
\cmidrule(lr){3-4} |
749 |
|
|
\cmidrule(lr){5-6} |
750 |
|
|
\cmidrule(l){7-8} |
751 |
|
|
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
752 |
|
|
\midrule |
753 |
|
|
PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\ |
754 |
|
|
SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\ |
755 |
|
|
& 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\ |
756 |
|
|
& 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\ |
757 |
|
|
& 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\ |
758 |
|
|
SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\ |
759 |
|
|
& 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\ |
760 |
|
|
& 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\ |
761 |
|
|
& 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\ |
762 |
|
|
GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\ |
763 |
|
|
RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\ |
764 |
|
|
\midrule |
765 |
|
|
PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\ |
766 |
|
|
SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\ |
767 |
|
|
& 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\ |
768 |
|
|
& 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
769 |
|
|
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
770 |
|
|
SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
771 |
|
|
& 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
772 |
|
|
& 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
773 |
|
|
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
774 |
|
|
GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
775 |
|
|
RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
776 |
|
|
\midrule |
777 |
|
|
PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\ |
778 |
|
|
SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\ |
779 |
|
|
& 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\ |
780 |
|
|
& 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\ |
781 |
|
|
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
782 |
|
|
SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\ |
783 |
|
|
& 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\ |
784 |
|
|
& 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\ |
785 |
|
|
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
786 |
|
|
GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\ |
787 |
|
|
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
788 |
|
|
\bottomrule |
789 |
|
|
\end{tabular} |
790 |
chrisfen |
2652 |
\label{tab:argon} |
791 |
chrisfen |
2599 |
\end{table} |
792 |
|
|
|
793 |
|
|
\begin{table}[htbp] |
794 |
|
|
\centering |
795 |
gezelter |
2658 |
\caption{Variance results from Gaussian fits to angular |
796 |
|
|
distributions of the force and torque vectors in the 6 \AA\ sphere of |
797 |
|
|
argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
798 |
|
|
Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = |
799 |
|
|
Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group |
800 |
|
|
Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
801 |
chrisfen |
2599 |
\begin{tabular}{@{} ccrrrrrr @{}} |
802 |
|
|
\\ |
803 |
|
|
\toprule |
804 |
|
|
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
805 |
|
|
\cmidrule(lr){3-5} |
806 |
|
|
\cmidrule(l){6-8} |
807 |
|
|
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
808 |
|
|
\midrule |
809 |
|
|
PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\ |
810 |
|
|
SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\ |
811 |
|
|
& 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\ |
812 |
|
|
& 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\ |
813 |
|
|
& 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\ |
814 |
|
|
SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\ |
815 |
|
|
& 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\ |
816 |
|
|
& 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\ |
817 |
|
|
& 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\ |
818 |
|
|
GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
819 |
|
|
RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\ |
820 |
|
|
\midrule |
821 |
|
|
GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
822 |
|
|
& 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\ |
823 |
|
|
& 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\ |
824 |
|
|
& 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\ |
825 |
|
|
GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\ |
826 |
|
|
& 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\ |
827 |
|
|
& 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\ |
828 |
|
|
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
829 |
|
|
\bottomrule |
830 |
|
|
\end{tabular} |
831 |
chrisfen |
2652 |
\label{tab:argonAng} |
832 |
chrisfen |
2599 |
\end{table} |
833 |
|
|
|
834 |
chrisfen |
2660 |
This system appears not to show in any significant deviation in the previously observed results. The {\sc sp} and {\sc sf} methods give result qualities similar to those observed in section \ref{app:water}. The only significant difference is the improvement for the configuration energy differences for the {\sc rf} method. This is surprising in that we are introducing an inhomogeneity to the system; however, this inhomogeneity is charge-neutral and does not result in charged cutoff spheres. The charge-neutrality, which the {\sc sp} and {\sc sf} methods explicity enforce, seems to play a greater role in the stability of the {\sc rf} method than the necessity of a homogeneous environment. |
835 |
|
|
|
836 |
chrisfen |
2641 |
\newpage |
837 |
|
|
|
838 |
|
|
\bibliographystyle{jcp2} |
839 |
|
|
\bibliography{electrostaticMethods} |
840 |
|
|
|
841 |
gezelter |
2658 |
\end{document} |