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# User Rev Content
1 chrisfen 2599 %\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 chrisfen 2642 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:spceMag}. The force and torque vector directionality results are displayed separately in table \ref{tab:spceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
31 chrisfen 2599 \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 chrisfen 2642 \label{tab:spceMag}
85 chrisfen 2599 \end{table}
86    
87 chrisfen 2641 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 chrisfen 2599 \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 chrisfen 2642 \label{tab:spceAng}
127 chrisfen 2599 \end{table}
128    
129 chrisfen 2642 The directionality of the force and torque vectors show a lot of parallels with the magnitude results in table \ref{tab:spceMag}.
130    
131 chrisfen 2599 \section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
132    
133     \begin{table}[htbp]
134     \centering
135     \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 $\varepsilon \approx \infty$).}
136     \begin{tabular}{@{} ccrrrrrr @{}}
137     \\
138     \toprule
139     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
140     \cmidrule(lr){3-4}
141     \cmidrule(lr){5-6}
142     \cmidrule(l){7-8}
143     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
144     \midrule
145     PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
146     SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
147     & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
148     & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
149     & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
150     SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
151     & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
152     & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
153     & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
154     GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
155     RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
156     \midrule
157     PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
158     SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
159     & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
160     & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
161     & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
162     SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
163     & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
164     & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
165     & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
166     GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
167     RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
168     \midrule
169     PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
170     SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
171     & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
172     & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
173     & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
174     SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
175     & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
176     & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
177     & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
178     GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
179     RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
180     \bottomrule
181     \end{tabular}
182 chrisfen 2641 \label{tab:iceTab}
183 chrisfen 2599 \end{table}
184    
185     \begin{table}[htbp]
186     \centering
187     \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.}
188     \begin{tabular}{@{} ccrrrrrr @{}}
189     \\
190     \toprule
191     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
192     \cmidrule(lr){3-5}
193     \cmidrule(l){6-8}
194     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
195     \midrule
196     PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
197     SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
198     & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
199     & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
200     & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
201     SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
202     & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
203     & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
204     & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
205     GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
206     RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
207     \midrule
208     GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
209     & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
210     & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
211     & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
212     GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
213     & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
214     & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
215     & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
216     \bottomrule
217     \end{tabular}
218 chrisfen 2641 \label{tab:iceTabAng}
219 chrisfen 2599 \end{table}
220    
221     \section{\label{app-melt}NaCl Melt}
222    
223     \begin{table}[htbp]
224     \centering
225     \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.}
226     \begin{tabular}{@{} ccrrrrrr @{}}
227     \\
228     \toprule
229     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
230     \cmidrule(lr){3-4}
231     \cmidrule(lr){5-6}
232     \cmidrule(l){7-8}
233     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
234     \midrule
235     PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
236     SP & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\
237     & 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\
238     & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
239     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
240     SF & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\
241     & 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\
242     & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
243     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
244     \midrule
245     PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
246     SP & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\
247     & 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\
248     & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
249     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
250     SF & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\
251     & 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\
252     & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
253     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
254     \bottomrule
255     \end{tabular}
256 chrisfen 2641 \label{tab:meltTab}
257 chrisfen 2599 \end{table}
258    
259     \begin{table}[htbp]
260     \centering
261     \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.}
262     \begin{tabular}{@{} ccrrrrrr @{}}
263     \\
264     \toprule
265     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
266     \cmidrule(lr){3-5}
267     \cmidrule(l){6-8}
268     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
269     \midrule
270     PC & & 13.294 & 8.035 & 5.366 \\
271     SP & 0.0 & 13.316 & 8.037 & 5.385 \\
272     & 0.1 & 5.705 & 1.391 & 0.360 \\
273     & 0.2 & 2.415 & 7.534 & 13.927 \\
274     & 0.3 & 23.769 & 67.306 & 57.252 \\
275     SF & 0.0 & 1.693 & 0.603 & 0.256 \\
276     & 0.1 & 1.687 & 0.653 & 0.272 \\
277     & 0.2 & 2.598 & 7.523 & 13.930 \\
278     & 0.3 & 23.734 & 67.305 & 57.252 \\
279     \bottomrule
280     \end{tabular}
281 chrisfen 2641 \label{tab:meltTabAng}
282 chrisfen 2599 \end{table}
283    
284     \section{\label{app-salt}NaCl Crystal}
285    
286     \begin{table}[htbp]
287     \centering
288     \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.}
289     \begin{tabular}{@{} ccrrrrrr @{}}
290     \\
291     \toprule
292     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
293     \cmidrule(lr){3-4}
294     \cmidrule(lr){5-6}
295     \cmidrule(l){7-8}
296     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
297     \midrule
298     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
299     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
300     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
301     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
302     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
303     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
304     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
305     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
306     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
307     \midrule
308     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
309     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
310     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
311     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
312     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
313     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
314     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
315     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
316     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
317     \bottomrule
318     \end{tabular}
319 chrisfen 2641 \label{tab:saltTab}
320 chrisfen 2599 \end{table}
321    
322     \begin{table}[htbp]
323     \centering
324     \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 $\varepsilon \approx \infty$).}
325     \begin{tabular}{@{} ccrrrrrr @{}}
326     \\
327     \toprule
328     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
329     \cmidrule(lr){3-5}
330     \cmidrule(l){6-8}
331     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
332     \midrule
333     PC & & 111.945 & 111.824 & 111.866 \\
334     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
335     & 0.1 & 52.361 & 42.574 & 2.819 \\
336     & 0.2 & 10.847 & 9.709 & 9.686 \\
337     & 0.3 & 31.128 & 31.104 & 31.029 \\
338     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
339     & 0.1 & 9.462 & 3.303 & 1.721 \\
340     & 0.2 & 11.454 & 9.813 & 9.701 \\
341     & 0.3 & 31.120 & 31.105 & 31.029 \\
342     \bottomrule
343     \end{tabular}
344 chrisfen 2641 \label{tab:saltTabAng}
345 chrisfen 2599 \end{table}
346    
347     \section{\label{app-sol1}Weak NaCl Solution}
348    
349     \begin{table}[htbp]
350     \centering
351     \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.}
352     \begin{tabular}{@{} ccrrrrrr @{}}
353     \\
354     \toprule
355     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
356     \cmidrule(lr){3-4}
357     \cmidrule(lr){5-6}
358     \cmidrule(l){7-8}
359     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
360     \midrule
361     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
362     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
363     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
364     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
365     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
366     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
367     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
368     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
369     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
370     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
371     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
372     \midrule
373     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
374     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
375     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
376     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
377     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
378     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
379     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
380     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
381     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
382     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
383     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
384     \midrule
385     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
386     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
387     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
388     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
389     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
390     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
391     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
392     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
393     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
394     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
395     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
396     \bottomrule
397     \end{tabular}
398 chrisfen 2641 \label{tab:sol1Tab}
399 chrisfen 2599 \end{table}
400    
401     \begin{table}[htbp]
402     \centering
403     \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.}
404     \begin{tabular}{@{} ccrrrrrr @{}}
405     \\
406     \toprule
407     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
408     \cmidrule(lr){3-5}
409     \cmidrule(l){6-8}
410     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
411     \midrule
412     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
413     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
414     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
415     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
416     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
417     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
418     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
419     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
420     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
421     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
422     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
423     \midrule
424     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
425     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
426     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
427     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
428     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
429     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
430     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
431     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
432     \bottomrule
433     \end{tabular}
434 chrisfen 2641 \label{tab:sol1TabAng}
435 chrisfen 2599 \end{table}
436    
437     \section{\label{app-sol10}Strong NaCl Solution}
438    
439     \begin{table}[htbp]
440     \centering
441     \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 $\varepsilon \approx \infty$).}
442     \begin{tabular}{@{} ccrrrrrr @{}}
443     \\
444     \toprule
445     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
446     \cmidrule(lr){3-4}
447     \cmidrule(lr){5-6}
448     \cmidrule(l){7-8}
449     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
450     \midrule
451     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
452     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
453     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
454     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
455     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
456     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
457     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
458     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
459     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
460     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
461     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
462     \midrule
463     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
464     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
465     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
466     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
467     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
468     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
469     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
470     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
471     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
472     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
473     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
474     \midrule
475     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
476     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
477     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
478     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
479     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
480     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
481     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
482     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
483     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
484     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
485     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
486     \bottomrule
487     \end{tabular}
488 chrisfen 2641 \label{tab:sol10Tab}
489 chrisfen 2599 \end{table}
490    
491     \begin{table}[htbp]
492     \centering
493     \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.}
494     \begin{tabular}{@{} ccrrrrrr @{}}
495     \\
496     \toprule
497     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
498     \cmidrule(lr){3-5}
499     \cmidrule(l){6-8}
500     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
501     \midrule
502     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
503     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
504     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
505     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
506     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
507     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
508     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
509     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
510     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
511     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
512     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
513     \midrule
514     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
515     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
516     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
517     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
518     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
519     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
520     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
521     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
522     \bottomrule
523     \end{tabular}
524 chrisfen 2641 \label{tab:sol10TabAng}
525 chrisfen 2599 \end{table}
526    
527     \section{\label{app-argon}Argon Sphere in Water}
528    
529     \begin{table}[htbp]
530     \centering
531     \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 $\varepsilon \approx \infty$).}
532     \begin{tabular}{@{} ccrrrrrr @{}}
533     \\
534     \toprule
535     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
536     \cmidrule(lr){3-4}
537     \cmidrule(lr){5-6}
538     \cmidrule(l){7-8}
539     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
540     \midrule
541     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
542     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
543     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
544     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
545     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
546     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
547     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
548     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
549     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
550     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
551     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
552     \midrule
553     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
554     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
555     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
556     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
557     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
558     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
559     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
560     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
561     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
562     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
563     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
564     \midrule
565     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
566     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
567     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
568     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
569     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
570     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
571     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
572     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
573     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
574     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
575     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
576     \bottomrule
577     \end{tabular}
578 chrisfen 2641 \label{tab:argonTab}
579 chrisfen 2599 \end{table}
580    
581     \begin{table}[htbp]
582     \centering
583     \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.}
584     \begin{tabular}{@{} ccrrrrrr @{}}
585     \\
586     \toprule
587     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
588     \cmidrule(lr){3-5}
589     \cmidrule(l){6-8}
590     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
591     \midrule
592     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
593     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
594     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
595     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
596     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
597     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
598     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
599     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
600     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
601     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
602     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
603     \midrule
604     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
605     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
606     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
607     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
608     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
609     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
610     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
611     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
612     \bottomrule
613     \end{tabular}
614 chrisfen 2641 \label{tab:argonTabAng}
615 chrisfen 2599 \end{table}
616    
617 chrisfen 2641 \newpage
618    
619     \bibliographystyle{jcp2}
620     \bibliography{electrostaticMethods}
621    
622 chrisfen 2599 \end{document}