| 1 |
%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
| 2 |
\documentclass[12pt]{article} |
| 3 |
\usepackage{endfloat} |
| 4 |
\usepackage{amsmath} |
| 5 |
\usepackage{amssymb} |
| 6 |
\usepackage{epsf} |
| 7 |
\usepackage{times} |
| 8 |
\usepackage{mathptm} |
| 9 |
\usepackage{setspace} |
| 10 |
\usepackage{tabularx} |
| 11 |
\usepackage{graphicx} |
| 12 |
\usepackage{booktabs} |
| 13 |
%\usepackage{berkeley} |
| 14 |
\usepackage[ref]{overcite} |
| 15 |
\pagestyle{plain} |
| 16 |
\pagenumbering{arabic} |
| 17 |
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
| 18 |
\topmargin -21pt \headsep 10pt |
| 19 |
\textheight 9.0in \textwidth 6.5in |
| 20 |
\brokenpenalty=10000 |
| 21 |
\renewcommand{\baselinestretch}{1.2} |
| 22 |
\renewcommand\citemid{\ } % no comma in optional reference note |
| 23 |
|
| 24 |
\begin{document} |
| 25 |
|
| 26 |
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. |
| 27 |
|
| 28 |
\section{\label{app-water}Liquid Water} |
| 29 |
|
| 30 |
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:spceTabTMag}. |
| 31 |
\begin{table}[htbp] |
| 32 |
\centering |
| 33 |
\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 $\varepsilon \approx \infty$).} |
| 34 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 35 |
\\ |
| 36 |
\toprule |
| 37 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 38 |
\cmidrule(lr){3-4} |
| 39 |
\cmidrule(lr){5-6} |
| 40 |
\cmidrule(l){7-8} |
| 41 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 42 |
\midrule |
| 43 |
PC & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\ |
| 44 |
SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\ |
| 45 |
& 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\ |
| 46 |
& 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\ |
| 47 |
& 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\ |
| 48 |
SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\ |
| 49 |
& 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\ |
| 50 |
& 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\ |
| 51 |
& 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\ |
| 52 |
GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\ |
| 53 |
RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
| 54 |
|
| 55 |
\midrule |
| 56 |
|
| 57 |
PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\ |
| 58 |
SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\ |
| 59 |
& 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\ |
| 60 |
& 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 61 |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
| 62 |
SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\ |
| 63 |
& 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 64 |
& 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 65 |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
| 66 |
GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 67 |
RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 68 |
|
| 69 |
\midrule |
| 70 |
|
| 71 |
PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\ |
| 72 |
SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\ |
| 73 |
& 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\ |
| 74 |
& 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
| 75 |
& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 76 |
SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\ |
| 77 |
& 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\ |
| 78 |
& 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
| 79 |
& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 80 |
GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\ |
| 81 |
RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\ |
| 82 |
\bottomrule |
| 83 |
\end{tabular} |
| 84 |
\label{tab:spceTabTMag} |
| 85 |
\end{table} |
| 86 |
|
| 87 |
Unless there is a significant change in result in any of the further systems, we are going to neglect to comment on the pure cutoff (PC) system. It is unreasonable to expect it to perform well in either energetic or dynamic studies using molecular groups, as evidenced in previous studies and in the results displayed here and in the rest of this paper.\cite{Adams79,Steinbach94} In contrast to PC, the {\sc sp} method shows variety in the results. In the weakly and undamped cases, the results are poor for both the energy gap and dynamics, and this is not surprising considering the energy oscillations observed by Wolf {\it et al.} and the discontinuity in the forces discussed in the main portion of this paper.\cite{Wolf99} Long cutoff radii, moderate damping, or a combination of the two are required for {\sc sp} to perform respectably. With a cutoff greater than 12 \AA\ and $\alpha$ of 0.2 \AA$^{-1}$, {\sc sp} provides result right in line with SPME. |
| 88 |
|
| 89 |
The {\sc sf} method displays energetic and dynamic results very similar to SPME under undamped to moderately damped conditions. The quality seems to degrade in the overdamped case ($\alpha = 0.3 \AA^{-1}$) to values identical to {\sc sp}, so it is important not to get carried away with the use of damping. A cutoff radius choice of 12 \AA\ or higher is recommended, primarily due to the energy gap results of interest in Monte Carlo (MC) calculations. |
| 90 |
|
| 91 |
The group switched cutoff (GSC) and reaction field (RF) methods seem to have very similar behavior, with the preference given to RF for the improved energy gap results. Neither mimics the energetics of SPME as well as the {\sc sp} (with moderate damping) and {\sc sf} methods, and the results seem relatively independent of cutoff radius. The dynamics for both methods, however, are quite good. Both methods utilize switching functions, which correct and discontinuities in the potential and forces, a possible reason for the improved results. It is interesting to compare the PC with the GSC cases, and recognize the significant improvement that group based cutoffs and switching functions provide. This as been recognized in previous studies,\cite{Andrea83,Steinbach94} and is a useful tactic for stably incorporating local area electrostatic effects. |
| 92 |
|
| 93 |
\begin{table}[htbp] |
| 94 |
\centering |
| 95 |
\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.} |
| 96 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 97 |
\\ |
| 98 |
\toprule |
| 99 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 100 |
\cmidrule(lr){3-5} |
| 101 |
\cmidrule(l){6-8} |
| 102 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 103 |
\midrule |
| 104 |
PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\ |
| 105 |
SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\ |
| 106 |
& 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\ |
| 107 |
& 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\ |
| 108 |
& 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\ |
| 109 |
SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\ |
| 110 |
& 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\ |
| 111 |
& 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\ |
| 112 |
& 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\ |
| 113 |
GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
| 114 |
RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\ |
| 115 |
\midrule |
| 116 |
GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
| 117 |
& 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\ |
| 118 |
& 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\ |
| 119 |
& 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\ |
| 120 |
GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\ |
| 121 |
& 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\ |
| 122 |
& 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\ |
| 123 |
& 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\ |
| 124 |
\bottomrule |
| 125 |
\end{tabular} |
| 126 |
\label{tab:spceTabAng} |
| 127 |
\end{table} |
| 128 |
|
| 129 |
\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
| 130 |
|
| 131 |
\begin{table}[htbp] |
| 132 |
\centering |
| 133 |
\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 $\varepsilon \approx \infty$).} |
| 134 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 135 |
\\ |
| 136 |
\toprule |
| 137 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 138 |
\cmidrule(lr){3-4} |
| 139 |
\cmidrule(lr){5-6} |
| 140 |
\cmidrule(l){7-8} |
| 141 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 142 |
\midrule |
| 143 |
PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\ |
| 144 |
SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\ |
| 145 |
& 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\ |
| 146 |
& 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\ |
| 147 |
& 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\ |
| 148 |
SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\ |
| 149 |
& 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\ |
| 150 |
& 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\ |
| 151 |
& 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\ |
| 152 |
GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\ |
| 153 |
RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\ |
| 154 |
\midrule |
| 155 |
PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\ |
| 156 |
SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\ |
| 157 |
& 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\ |
| 158 |
& 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\ |
| 159 |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\ |
| 160 |
SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\ |
| 161 |
& 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\ |
| 162 |
& 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\ |
| 163 |
& 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\ |
| 164 |
GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\ |
| 165 |
RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 166 |
\midrule |
| 167 |
PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\ |
| 168 |
SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\ |
| 169 |
& 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\ |
| 170 |
& 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\ |
| 171 |
& 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\ |
| 172 |
SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\ |
| 173 |
& 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\ |
| 174 |
& 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\ |
| 175 |
& 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\ |
| 176 |
GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\ |
| 177 |
RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\ |
| 178 |
\bottomrule |
| 179 |
\end{tabular} |
| 180 |
\label{tab:iceTab} |
| 181 |
\end{table} |
| 182 |
|
| 183 |
\begin{table}[htbp] |
| 184 |
\centering |
| 185 |
\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.} |
| 186 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 187 |
\\ |
| 188 |
\toprule |
| 189 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 190 |
\cmidrule(lr){3-5} |
| 191 |
\cmidrule(l){6-8} |
| 192 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 193 |
\midrule |
| 194 |
PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\ |
| 195 |
SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\ |
| 196 |
& 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\ |
| 197 |
& 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\ |
| 198 |
& 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\ |
| 199 |
SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\ |
| 200 |
& 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\ |
| 201 |
& 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\ |
| 202 |
& 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\ |
| 203 |
GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 204 |
RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\ |
| 205 |
\midrule |
| 206 |
GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 207 |
& 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\ |
| 208 |
& 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\ |
| 209 |
& 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 210 |
GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\ |
| 211 |
& 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\ |
| 212 |
& 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\ |
| 213 |
& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 214 |
\bottomrule |
| 215 |
\end{tabular} |
| 216 |
\label{tab:iceTabAng} |
| 217 |
\end{table} |
| 218 |
|
| 219 |
\section{\label{app-melt}NaCl Melt} |
| 220 |
|
| 221 |
\begin{table}[htbp] |
| 222 |
\centering |
| 223 |
\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.} |
| 224 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 225 |
\\ |
| 226 |
\toprule |
| 227 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 228 |
\cmidrule(lr){3-4} |
| 229 |
\cmidrule(lr){5-6} |
| 230 |
\cmidrule(l){7-8} |
| 231 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 232 |
\midrule |
| 233 |
PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
| 234 |
SP & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\ |
| 235 |
& 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\ |
| 236 |
& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 237 |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 238 |
SF & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\ |
| 239 |
& 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\ |
| 240 |
& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 241 |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 242 |
\midrule |
| 243 |
PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
| 244 |
SP & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\ |
| 245 |
& 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\ |
| 246 |
& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 247 |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 248 |
SF & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\ |
| 249 |
& 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\ |
| 250 |
& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 251 |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 252 |
\bottomrule |
| 253 |
\end{tabular} |
| 254 |
\label{tab:meltTab} |
| 255 |
\end{table} |
| 256 |
|
| 257 |
\begin{table}[htbp] |
| 258 |
\centering |
| 259 |
\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.} |
| 260 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 261 |
\\ |
| 262 |
\toprule |
| 263 |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 264 |
\cmidrule(lr){3-5} |
| 265 |
\cmidrule(l){6-8} |
| 266 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 267 |
\midrule |
| 268 |
PC & & 13.294 & 8.035 & 5.366 \\ |
| 269 |
SP & 0.0 & 13.316 & 8.037 & 5.385 \\ |
| 270 |
& 0.1 & 5.705 & 1.391 & 0.360 \\ |
| 271 |
& 0.2 & 2.415 & 7.534 & 13.927 \\ |
| 272 |
& 0.3 & 23.769 & 67.306 & 57.252 \\ |
| 273 |
SF & 0.0 & 1.693 & 0.603 & 0.256 \\ |
| 274 |
& 0.1 & 1.687 & 0.653 & 0.272 \\ |
| 275 |
& 0.2 & 2.598 & 7.523 & 13.930 \\ |
| 276 |
& 0.3 & 23.734 & 67.305 & 57.252 \\ |
| 277 |
\bottomrule |
| 278 |
\end{tabular} |
| 279 |
\label{tab:meltTabAng} |
| 280 |
\end{table} |
| 281 |
|
| 282 |
\section{\label{app-salt}NaCl Crystal} |
| 283 |
|
| 284 |
\begin{table}[htbp] |
| 285 |
\centering |
| 286 |
\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.} |
| 287 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 288 |
\\ |
| 289 |
\toprule |
| 290 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 291 |
\cmidrule(lr){3-4} |
| 292 |
\cmidrule(lr){5-6} |
| 293 |
\cmidrule(l){7-8} |
| 294 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 295 |
\midrule |
| 296 |
PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\ |
| 297 |
SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\ |
| 298 |
& 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\ |
| 299 |
& 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\ |
| 300 |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\ |
| 301 |
SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\ |
| 302 |
& 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\ |
| 303 |
& 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\ |
| 304 |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\ |
| 305 |
\midrule |
| 306 |
PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\ |
| 307 |
SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\ |
| 308 |
& 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\ |
| 309 |
& 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\ |
| 310 |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 311 |
SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\ |
| 312 |
& 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\ |
| 313 |
& 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\ |
| 314 |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 315 |
\bottomrule |
| 316 |
\end{tabular} |
| 317 |
\label{tab:saltTab} |
| 318 |
\end{table} |
| 319 |
|
| 320 |
\begin{table}[htbp] |
| 321 |
\centering |
| 322 |
\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 $\varepsilon \approx \infty$).} |
| 323 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 324 |
\\ |
| 325 |
\toprule |
| 326 |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 327 |
\cmidrule(lr){3-5} |
| 328 |
\cmidrule(l){6-8} |
| 329 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 330 |
\midrule |
| 331 |
PC & & 111.945 & 111.824 & 111.866 \\ |
| 332 |
SP & 0.0 & 112.414 & 152.215 & 38.087 \\ |
| 333 |
& 0.1 & 52.361 & 42.574 & 2.819 \\ |
| 334 |
& 0.2 & 10.847 & 9.709 & 9.686 \\ |
| 335 |
& 0.3 & 31.128 & 31.104 & 31.029 \\ |
| 336 |
SF & 0.0 & 10.025 & 3.555 & 1.648 \\ |
| 337 |
& 0.1 & 9.462 & 3.303 & 1.721 \\ |
| 338 |
& 0.2 & 11.454 & 9.813 & 9.701 \\ |
| 339 |
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
| 340 |
\bottomrule |
| 341 |
\end{tabular} |
| 342 |
\label{tab:saltTabAng} |
| 343 |
\end{table} |
| 344 |
|
| 345 |
\section{\label{app-sol1}Weak NaCl Solution} |
| 346 |
|
| 347 |
\begin{table}[htbp] |
| 348 |
\centering |
| 349 |
\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.} |
| 350 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 351 |
\\ |
| 352 |
\toprule |
| 353 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 354 |
\cmidrule(lr){3-4} |
| 355 |
\cmidrule(lr){5-6} |
| 356 |
\cmidrule(l){7-8} |
| 357 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 358 |
\midrule |
| 359 |
PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\ |
| 360 |
SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\ |
| 361 |
& 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\ |
| 362 |
& 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 363 |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 364 |
SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\ |
| 365 |
& 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\ |
| 366 |
& 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 367 |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 368 |
GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\ |
| 369 |
RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\ |
| 370 |
\midrule |
| 371 |
PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\ |
| 372 |
SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\ |
| 373 |
& 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\ |
| 374 |
& 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 375 |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 376 |
SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\ |
| 377 |
& 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 378 |
& 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 379 |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 380 |
GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\ |
| 381 |
RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\ |
| 382 |
\midrule |
| 383 |
PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\ |
| 384 |
SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\ |
| 385 |
& 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\ |
| 386 |
& 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 387 |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 388 |
SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\ |
| 389 |
& 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\ |
| 390 |
& 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 391 |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 392 |
GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\ |
| 393 |
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 394 |
\bottomrule |
| 395 |
\end{tabular} |
| 396 |
\label{tab:sol1Tab} |
| 397 |
\end{table} |
| 398 |
|
| 399 |
\begin{table}[htbp] |
| 400 |
\centering |
| 401 |
\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.} |
| 402 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 403 |
\\ |
| 404 |
\toprule |
| 405 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 406 |
\cmidrule(lr){3-5} |
| 407 |
\cmidrule(l){6-8} |
| 408 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 409 |
\midrule |
| 410 |
PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\ |
| 411 |
SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\ |
| 412 |
& 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\ |
| 413 |
& 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\ |
| 414 |
& 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\ |
| 415 |
SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\ |
| 416 |
& 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\ |
| 417 |
& 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\ |
| 418 |
& 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\ |
| 419 |
GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 420 |
RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\ |
| 421 |
\midrule |
| 422 |
GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 423 |
& 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\ |
| 424 |
& 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\ |
| 425 |
& 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\ |
| 426 |
GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\ |
| 427 |
& 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\ |
| 428 |
& 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\ |
| 429 |
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 430 |
\bottomrule |
| 431 |
\end{tabular} |
| 432 |
\label{tab:sol1TabAng} |
| 433 |
\end{table} |
| 434 |
|
| 435 |
\section{\label{app-sol10}Strong NaCl Solution} |
| 436 |
|
| 437 |
\begin{table}[htbp] |
| 438 |
\centering |
| 439 |
\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 $\varepsilon \approx \infty$).} |
| 440 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 441 |
\\ |
| 442 |
\toprule |
| 443 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 444 |
\cmidrule(lr){3-4} |
| 445 |
\cmidrule(lr){5-6} |
| 446 |
\cmidrule(l){7-8} |
| 447 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 448 |
\midrule |
| 449 |
PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\ |
| 450 |
SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\ |
| 451 |
& 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\ |
| 452 |
& 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 453 |
& 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 454 |
SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\ |
| 455 |
& 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\ |
| 456 |
& 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 457 |
& 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 458 |
GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\ |
| 459 |
RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\ |
| 460 |
\midrule |
| 461 |
PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\ |
| 462 |
SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\ |
| 463 |
& 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\ |
| 464 |
& 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 465 |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 466 |
SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\ |
| 467 |
& 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\ |
| 468 |
& 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 469 |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 470 |
GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\ |
| 471 |
RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\ |
| 472 |
\midrule |
| 473 |
PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\ |
| 474 |
SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\ |
| 475 |
& 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\ |
| 476 |
& 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 477 |
& 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 478 |
SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\ |
| 479 |
& 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\ |
| 480 |
& 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 481 |
& 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 482 |
GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\ |
| 483 |
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 484 |
\bottomrule |
| 485 |
\end{tabular} |
| 486 |
\label{tab:sol10Tab} |
| 487 |
\end{table} |
| 488 |
|
| 489 |
\begin{table}[htbp] |
| 490 |
\centering |
| 491 |
\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.} |
| 492 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 493 |
\\ |
| 494 |
\toprule |
| 495 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 496 |
\cmidrule(lr){3-5} |
| 497 |
\cmidrule(l){6-8} |
| 498 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 499 |
\midrule |
| 500 |
PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\ |
| 501 |
SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\ |
| 502 |
& 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\ |
| 503 |
& 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\ |
| 504 |
& 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\ |
| 505 |
SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\ |
| 506 |
& 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\ |
| 507 |
& 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\ |
| 508 |
& 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\ |
| 509 |
GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\ |
| 510 |
RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\ |
| 511 |
\midrule |
| 512 |
GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\ |
| 513 |
& 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\ |
| 514 |
& 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\ |
| 515 |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\ |
| 516 |
GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\ |
| 517 |
& 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\ |
| 518 |
& 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\ |
| 519 |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 520 |
\bottomrule |
| 521 |
\end{tabular} |
| 522 |
\label{tab:sol10TabAng} |
| 523 |
\end{table} |
| 524 |
|
| 525 |
\section{\label{app-argon}Argon Sphere in Water} |
| 526 |
|
| 527 |
\begin{table}[htbp] |
| 528 |
\centering |
| 529 |
\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 $\varepsilon \approx \infty$).} |
| 530 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 531 |
\\ |
| 532 |
\toprule |
| 533 |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 534 |
\cmidrule(lr){3-4} |
| 535 |
\cmidrule(lr){5-6} |
| 536 |
\cmidrule(l){7-8} |
| 537 |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 538 |
\midrule |
| 539 |
PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\ |
| 540 |
SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\ |
| 541 |
& 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\ |
| 542 |
& 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\ |
| 543 |
& 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\ |
| 544 |
SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\ |
| 545 |
& 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\ |
| 546 |
& 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\ |
| 547 |
& 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\ |
| 548 |
GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\ |
| 549 |
RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\ |
| 550 |
\midrule |
| 551 |
PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\ |
| 552 |
SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\ |
| 553 |
& 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\ |
| 554 |
& 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 555 |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 556 |
SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 557 |
& 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 558 |
& 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 559 |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 560 |
GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 561 |
RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 562 |
\midrule |
| 563 |
PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\ |
| 564 |
SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\ |
| 565 |
& 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\ |
| 566 |
& 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\ |
| 567 |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 568 |
SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\ |
| 569 |
& 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\ |
| 570 |
& 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\ |
| 571 |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 572 |
GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\ |
| 573 |
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 574 |
\bottomrule |
| 575 |
\end{tabular} |
| 576 |
\label{tab:argonTab} |
| 577 |
\end{table} |
| 578 |
|
| 579 |
\begin{table}[htbp] |
| 580 |
\centering |
| 581 |
\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.} |
| 582 |
\begin{tabular}{@{} ccrrrrrr @{}} |
| 583 |
\\ |
| 584 |
\toprule |
| 585 |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 586 |
\cmidrule(lr){3-5} |
| 587 |
\cmidrule(l){6-8} |
| 588 |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 589 |
\midrule |
| 590 |
PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\ |
| 591 |
SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\ |
| 592 |
& 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\ |
| 593 |
& 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\ |
| 594 |
& 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\ |
| 595 |
SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\ |
| 596 |
& 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\ |
| 597 |
& 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\ |
| 598 |
& 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\ |
| 599 |
GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 600 |
RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\ |
| 601 |
\midrule |
| 602 |
GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 603 |
& 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\ |
| 604 |
& 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\ |
| 605 |
& 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\ |
| 606 |
GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\ |
| 607 |
& 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\ |
| 608 |
& 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\ |
| 609 |
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 610 |
\bottomrule |
| 611 |
\end{tabular} |
| 612 |
\label{tab:argonTabAng} |
| 613 |
\end{table} |
| 614 |
|
| 615 |
\newpage |
| 616 |
|
| 617 |
\bibliographystyle{jcp2} |
| 618 |
\bibliography{electrostaticMethods} |
| 619 |
|
| 620 |
\end{document} |
