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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 2652 500 liquid state configurations were generated as described in the Methods section using the SPC/E model of water.\cite{Berendsen87} The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:spce}. The force and torque vector directionality results are displayed separately in table \ref{tab:spceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
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 2652 \label{tab:spce}
85 chrisfen 2599 \end{table}
86    
87     \begin{table}[htbp]
88     \centering
89     \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.}
90     \begin{tabular}{@{} ccrrrrrr @{}}
91     \\
92     \toprule
93     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
94     \cmidrule(lr){3-5}
95     \cmidrule(l){6-8}
96     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
97     \midrule
98     PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
99     SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
100     & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
101     & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
102     & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
103     SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
104     & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
105     & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
106     & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
107     GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
108     RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\
109     \midrule
110     GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
111     & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
112     & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
113     & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
114     GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
115     & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
116     & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
117     & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
118     \bottomrule
119     \end{tabular}
120 chrisfen 2642 \label{tab:spceAng}
121 chrisfen 2599 \end{table}
122    
123 chrisfen 2652 For the most parts, the water results appear to parallel the combined results seen in the discussion in the main paper. There is good agreement with SPME in both energetic and dynamic behavior when using the {\sc sf} method with and without damping. The {\sc sp} method does well with an $\alpha$ around $0.2 \AA^{-1}$, particularly with cutoff radii greater than 12 \AA. The results for both of these methods also begin to decay as damping gets too large.
124 chrisfen 2642
125 chrisfen 2652 The pure cutoff (PC) method performs poorly, as seen in the main discussion section. In contrast to the combined values, however, the use of a switching function and group based cutoffs really improves the results for these neutral water molecules. The group switched cutoff (GSC) shows mimics the energetics of SPME more poorly than the {\sc sp} (with moderate damping) and {\sc sf} methods, but the dynamics are quite good. The switching functions corrects discontinuities in the potential and forces, leading to the improved results. Such improvements with the use of a switching function has been recognized in previous studies,\cite{Andrea83,Steinbach94} and it is a useful tactic for stably incorporating local area electrostatic effects.
126    
127     The reaction field (RF) method simply extends the results observed in the GSC case. Both methods are similar in form (i.e. neutral groups, switching function), but RF incorporates an added effect from the external dielectric. This similarity translates into the same good dynamic results and improved energetic results. These still fall short of the moderately damped {\sc sp} and {\sc sf} methods, but they display how incorporating some implicit properties of the surroundings (i.e. $\epsilon_\textrm{S}$) can improve results.
128    
129     A final note for the liquid water system, use of group cutoffs and a switching function also leads to noticeable improvements in the {\sc sp} and {\sc sf} methods, primarily in directionality of the force and torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant narrowing of the angle distribution in the cases with little to no damping and only modest improvement for the ideal conditions ($\alpha = 0.2 \AA{-1}$ and $R_\textrm{c} \geqslant 12 \AA$). The {\sc sf} method simply shows modest narrowing across all damping and cutoff ranges of interest. Group cutoffs and the switching function do nothing for cases were error is introduced by overdamping the potentials.
130    
131 chrisfen 2599 \section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
132    
133 chrisfen 2652 In addition to the disordered molecular system above, the ordered molecular system of ice I$_\textrm{c}$ was also considered. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:ice}. The force and torque vector directionality results are displayed separately in table \ref{tab:iceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
134    
135 chrisfen 2599 \begin{table}[htbp]
136     \centering
137     \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 $\varepsilon \approx \infty$).}
138     \begin{tabular}{@{} ccrrrrrr @{}}
139     \\
140     \toprule
141     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
142     \cmidrule(lr){3-4}
143     \cmidrule(lr){5-6}
144     \cmidrule(l){7-8}
145     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
146     \midrule
147     PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
148     SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
149     & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
150     & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
151     & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
152     SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
153     & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
154     & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
155     & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
156     GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
157     RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
158     \midrule
159     PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
160     SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
161     & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
162     & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
163     & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
164     SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
165     & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
166     & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
167     & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
168     GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
169     RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
170     \midrule
171     PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
172     SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
173     & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
174     & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
175     & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
176     SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
177     & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
178     & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
179     & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
180     GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
181     RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
182     \bottomrule
183     \end{tabular}
184 chrisfen 2652 \label{tab:ice}
185 chrisfen 2599 \end{table}
186    
187     \begin{table}[htbp]
188     \centering
189     \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.}
190     \begin{tabular}{@{} ccrrrrrr @{}}
191     \\
192     \toprule
193     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
194     \cmidrule(lr){3-5}
195     \cmidrule(l){6-8}
196     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
197     \midrule
198     PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
199     SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
200     & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
201     & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
202     & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
203     SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
204     & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
205     & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
206     & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
207     GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
208     RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
209     \midrule
210     GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
211     & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
212     & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
213     & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
214     GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
215     & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
216     & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
217     & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
218     \bottomrule
219     \end{tabular}
220 chrisfen 2652 \label{tab:iceAng}
221 chrisfen 2599 \end{table}
222    
223 chrisfen 2652 Highly ordered systems are a difficult test for the pairwise systems in that they lack the periodicity inherent to the Ewald summation. As expected, the energy gap agreement with SPME reduces for the {\sc sp} and {\sc sf} with parameters that were perfectly acceptable for the disordered liquid system. Moving to higher $R_\textrm{c}$ remedies this degraded performance, though at increase in computational cost. However, the dynamics of this crystalline system (both in magnitude and direction) are little affected. Both methods still reproduce the Ewald behavior with the same parameter recommendations from the previous section.
224    
225     It is also worth noting that RF exhibits a slightly improved energy gap results over the liquid water system. This can be rationalized by noting that the ice I$_\textrm{c}$ is
226    
227 chrisfen 2599 \section{\label{app-melt}NaCl Melt}
228    
229 chrisfen 2652 A high temperature NaCl melt was tested to gauge the accuracy of the pairwise summation methods in a highly charge disordered system. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:melt}. The force and torque vector directionality results are displayed separately in table \ref{tab:meltAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
230    
231 chrisfen 2599 \begin{table}[htbp]
232     \centering
233     \caption{Regression results for the molten NaCl system. Tabulated results include $\Delta E$ values (top set) and force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted Potential, and SF = Shifted Force.}
234     \begin{tabular}{@{} ccrrrrrr @{}}
235     \\
236     \toprule
237     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
238     \cmidrule(lr){3-4}
239     \cmidrule(lr){5-6}
240     \cmidrule(l){7-8}
241     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
242     \midrule
243     PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
244     SP & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\
245     & 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\
246     & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
247     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
248     SF & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\
249     & 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\
250     & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
251     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
252     \midrule
253     PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
254     SP & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\
255     & 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\
256     & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
257     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
258     SF & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\
259     & 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\
260     & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
261     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
262     \bottomrule
263     \end{tabular}
264 chrisfen 2652 \label{tab:melt}
265 chrisfen 2599 \end{table}
266    
267     \begin{table}[htbp]
268     \centering
269     \caption{Variance results from Gaussian fits to angular distributions of the force vectors in the molten NaCl system. PC = Pure Cutoff, SP = Shifted Potential, and SF = Shifted Force.}
270     \begin{tabular}{@{} ccrrrrrr @{}}
271     \\
272     \toprule
273     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
274     \cmidrule(lr){3-5}
275     \cmidrule(l){6-8}
276     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
277     \midrule
278     PC & & 13.294 & 8.035 & 5.366 \\
279     SP & 0.0 & 13.316 & 8.037 & 5.385 \\
280     & 0.1 & 5.705 & 1.391 & 0.360 \\
281     & 0.2 & 2.415 & 7.534 & 13.927 \\
282     & 0.3 & 23.769 & 67.306 & 57.252 \\
283     SF & 0.0 & 1.693 & 0.603 & 0.256 \\
284     & 0.1 & 1.687 & 0.653 & 0.272 \\
285     & 0.2 & 2.598 & 7.523 & 13.930 \\
286     & 0.3 & 23.734 & 67.305 & 57.252 \\
287     \bottomrule
288     \end{tabular}
289 chrisfen 2652 \label{tab:meltAng}
290 chrisfen 2599 \end{table}
291    
292     \section{\label{app-salt}NaCl Crystal}
293    
294 chrisfen 2652 A 1000K NaCl crystal was used to investigate the accuracy of the pairwise summation methods in an ordered system of charged particles. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:salt}. The force and torque vector directionality results are displayed separately in table \ref{tab:saltAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
295    
296 chrisfen 2599 \begin{table}[htbp]
297     \centering
298     \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.}
299     \begin{tabular}{@{} ccrrrrrr @{}}
300     \\
301     \toprule
302     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
303     \cmidrule(lr){3-4}
304     \cmidrule(lr){5-6}
305     \cmidrule(l){7-8}
306     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
307     \midrule
308     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
309     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
310     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
311     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
312     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
313     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
314     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
315     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
316     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
317     \midrule
318     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
319     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
320     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
321     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
322     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
323     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
324     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
325     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
326     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
327     \bottomrule
328     \end{tabular}
329 chrisfen 2652 \label{tab:salt}
330 chrisfen 2599 \end{table}
331    
332     \begin{table}[htbp]
333     \centering
334     \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 $\varepsilon \approx \infty$).}
335     \begin{tabular}{@{} ccrrrrrr @{}}
336     \\
337     \toprule
338     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
339     \cmidrule(lr){3-5}
340     \cmidrule(l){6-8}
341     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
342     \midrule
343     PC & & 111.945 & 111.824 & 111.866 \\
344     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
345     & 0.1 & 52.361 & 42.574 & 2.819 \\
346     & 0.2 & 10.847 & 9.709 & 9.686 \\
347     & 0.3 & 31.128 & 31.104 & 31.029 \\
348     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
349     & 0.1 & 9.462 & 3.303 & 1.721 \\
350     & 0.2 & 11.454 & 9.813 & 9.701 \\
351     & 0.3 & 31.120 & 31.105 & 31.029 \\
352     \bottomrule
353     \end{tabular}
354 chrisfen 2652 \label{tab:saltAng}
355 chrisfen 2599 \end{table}
356    
357     \section{\label{app-sol1}Weak NaCl Solution}
358    
359 chrisfen 2652 In an effort to bridge the charged atomic and neutral molecular systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into the liquid water system. This low ionic strength system consists of 4 ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}. The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
360    
361 chrisfen 2599 \begin{table}[htbp]
362     \centering
363     \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.}
364     \begin{tabular}{@{} ccrrrrrr @{}}
365     \\
366     \toprule
367     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
368     \cmidrule(lr){3-4}
369     \cmidrule(lr){5-6}
370     \cmidrule(l){7-8}
371     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
372     \midrule
373     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
374     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
375     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
376     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
377     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
378     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
379     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
380     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
381     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
382     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
383     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
384     \midrule
385     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
386     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
387     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
388     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
389     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
390     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
391     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
392     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
393     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
394     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
395     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
396     \midrule
397     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
398     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
399     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
400     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
401     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
402     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
403     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
404     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
405     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
406     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
407     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
408     \bottomrule
409     \end{tabular}
410 chrisfen 2652 \label{tab:solnWeak}
411 chrisfen 2599 \end{table}
412    
413     \begin{table}[htbp]
414     \centering
415     \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.}
416     \begin{tabular}{@{} ccrrrrrr @{}}
417     \\
418     \toprule
419     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
420     \cmidrule(lr){3-5}
421     \cmidrule(l){6-8}
422     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
423     \midrule
424     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
425     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
426     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
427     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
428     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
429     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
430     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
431     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
432     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
433     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
434     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
435     \midrule
436     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
437     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
438     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
439     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
440     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
441     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
442     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
443     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
444     \bottomrule
445     \end{tabular}
446 chrisfen 2652 \label{tab:solnWeakAng}
447 chrisfen 2599 \end{table}
448    
449     \section{\label{app-sol10}Strong NaCl Solution}
450    
451 chrisfen 2652 The bridging of the charged atomic and neutral molecular systems was furthered by considering a high ionic strength system consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}. The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
452    
453 chrisfen 2599 \begin{table}[htbp]
454     \centering
455     \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 $\varepsilon \approx \infty$).}
456     \begin{tabular}{@{} ccrrrrrr @{}}
457     \\
458     \toprule
459     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
460     \cmidrule(lr){3-4}
461     \cmidrule(lr){5-6}
462     \cmidrule(l){7-8}
463     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
464     \midrule
465     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
466     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
467     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
468     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
469     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
470     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
471     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
472     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
473     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
474     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
475     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
476     \midrule
477     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
478     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
479     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
480     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
481     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
482     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
483     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
484     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
485     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
486     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
487     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
488     \midrule
489     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
490     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
491     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
492     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
493     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
494     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
495     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
496     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
497     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
498     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
499     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
500     \bottomrule
501     \end{tabular}
502 chrisfen 2652 \label{tab:solnStr}
503 chrisfen 2599 \end{table}
504    
505     \begin{table}[htbp]
506     \centering
507     \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.}
508     \begin{tabular}{@{} ccrrrrrr @{}}
509     \\
510     \toprule
511     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
512     \cmidrule(lr){3-5}
513     \cmidrule(l){6-8}
514     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
515     \midrule
516     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
517     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
518     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
519     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
520     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
521     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
522     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
523     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
524     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
525     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
526     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
527     \midrule
528     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
529     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
530     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
531     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
532     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
533     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
534     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
535     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
536     \bottomrule
537     \end{tabular}
538 chrisfen 2652 \label{tab:solnStrAng}
539 chrisfen 2599 \end{table}
540    
541     \section{\label{app-argon}Argon Sphere in Water}
542    
543 chrisfen 2652 The final model system studied was 6 \AA\ sphere of Argon solvated by SPC/E water. The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}. The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
544    
545 chrisfen 2599 \begin{table}[htbp]
546     \centering
547     \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 $\varepsilon \approx \infty$).}
548     \begin{tabular}{@{} ccrrrrrr @{}}
549     \\
550     \toprule
551     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
552     \cmidrule(lr){3-4}
553     \cmidrule(lr){5-6}
554     \cmidrule(l){7-8}
555     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
556     \midrule
557     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
558     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
559     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
560     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
561     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
562     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
563     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
564     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
565     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
566     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
567     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
568     \midrule
569     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
570     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
571     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
572     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
573     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
574     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
575     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
576     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
577     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
578     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
579     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
580     \midrule
581     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
582     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
583     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
584     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
585     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
586     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
587     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
588     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
589     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
590     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
591     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
592     \bottomrule
593     \end{tabular}
594 chrisfen 2652 \label{tab:argon}
595 chrisfen 2599 \end{table}
596    
597     \begin{table}[htbp]
598     \centering
599     \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.}
600     \begin{tabular}{@{} ccrrrrrr @{}}
601     \\
602     \toprule
603     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
604     \cmidrule(lr){3-5}
605     \cmidrule(l){6-8}
606     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
607     \midrule
608     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
609     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
610     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
611     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
612     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
613     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
614     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
615     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
616     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
617     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
618     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
619     \midrule
620     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
621     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
622     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
623     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
624     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
625     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
626     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
627     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
628     \bottomrule
629     \end{tabular}
630 chrisfen 2652 \label{tab:argonAng}
631 chrisfen 2599 \end{table}
632    
633 chrisfen 2641 \newpage
634    
635     \bibliographystyle{jcp2}
636     \bibliography{electrostaticMethods}
637    
638 chrisfen 2599 \end{document}