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root/group/trunk/electrostaticMethodsPaper/SupportingInfo.tex
Revision: 2641
Committed: Mon Mar 20 15:43:13 2006 UTC (18 years, 3 months ago) by chrisfen
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padding the supporting info.

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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 2641 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 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 2641 \label{tab:spceTabTMag}
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 2641 \label{tab:spceTabAng}
127 chrisfen 2599 \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 chrisfen 2641 \label{tab:iceTab}
181 chrisfen 2599 \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 chrisfen 2641 \label{tab:iceTabAng}
217 chrisfen 2599 \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 chrisfen 2641 \label{tab:meltTab}
255 chrisfen 2599 \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 chrisfen 2641 \label{tab:meltTabAng}
280 chrisfen 2599 \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 chrisfen 2641 \label{tab:saltTab}
318 chrisfen 2599 \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 chrisfen 2641 \label{tab:saltTabAng}
343 chrisfen 2599 \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 chrisfen 2641 \label{tab:sol1Tab}
397 chrisfen 2599 \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 chrisfen 2641 \label{tab:sol1TabAng}
433 chrisfen 2599 \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 chrisfen 2641 \label{tab:sol10Tab}
487 chrisfen 2599 \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 chrisfen 2641 \label{tab:sol10TabAng}
523 chrisfen 2599 \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 chrisfen 2641 \label{tab:argonTab}
577 chrisfen 2599 \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 chrisfen 2641 \label{tab:argonTabAng}
613 chrisfen 2599 \end{table}
614    
615 chrisfen 2641 \newpage
616    
617     \bibliographystyle{jcp2}
618     \bibliography{electrostaticMethods}
619    
620 chrisfen 2599 \end{document}