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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 chrisfen 2654 It is also worth noting that RF exhibits a slightly improved energy gap results over the liquid water system. One possible explanation is that the ice I$_\textrm{c}$ crystal is ordered such that the net dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = \infty$, the reaction field incorporates this structural organization by actively enforcing a zeroed dipole moment within each cutoff sphere.
226 chrisfen 2652
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 chrisfen 2654
293    
294 chrisfen 2599 \section{\label{app-salt}NaCl Crystal}
295    
296 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.
297    
298 chrisfen 2599 \begin{table}[htbp]
299     \centering
300     \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.}
301     \begin{tabular}{@{} ccrrrrrr @{}}
302     \\
303     \toprule
304     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
305     \cmidrule(lr){3-4}
306     \cmidrule(lr){5-6}
307     \cmidrule(l){7-8}
308     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
309     \midrule
310     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
311     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
312     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
313     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
314     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
315     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
316     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
317     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
318     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
319     \midrule
320     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
321     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
322     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
323     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
324     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
325     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
326     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
327     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
328     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
329     \bottomrule
330     \end{tabular}
331 chrisfen 2652 \label{tab:salt}
332 chrisfen 2599 \end{table}
333    
334     \begin{table}[htbp]
335     \centering
336     \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 $\varepsilon \approx \infty$).}
337     \begin{tabular}{@{} ccrrrrrr @{}}
338     \\
339     \toprule
340     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
341     \cmidrule(lr){3-5}
342     \cmidrule(l){6-8}
343     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
344     \midrule
345     PC & & 111.945 & 111.824 & 111.866 \\
346     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
347     & 0.1 & 52.361 & 42.574 & 2.819 \\
348     & 0.2 & 10.847 & 9.709 & 9.686 \\
349     & 0.3 & 31.128 & 31.104 & 31.029 \\
350     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
351     & 0.1 & 9.462 & 3.303 & 1.721 \\
352     & 0.2 & 11.454 & 9.813 & 9.701 \\
353     & 0.3 & 31.120 & 31.105 & 31.029 \\
354     \bottomrule
355     \end{tabular}
356 chrisfen 2652 \label{tab:saltAng}
357 chrisfen 2599 \end{table}
358    
359     \section{\label{app-sol1}Weak NaCl Solution}
360    
361 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.
362    
363 chrisfen 2599 \begin{table}[htbp]
364     \centering
365     \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.}
366     \begin{tabular}{@{} ccrrrrrr @{}}
367     \\
368     \toprule
369     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
370     \cmidrule(lr){3-4}
371     \cmidrule(lr){5-6}
372     \cmidrule(l){7-8}
373     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
374     \midrule
375     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
376     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
377     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
378     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
379     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
380     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
381     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
382     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
383     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
384     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
385     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
386     \midrule
387     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
388     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
389     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
390     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
391     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
392     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
393     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
394     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
395     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
396     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
397     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
398     \midrule
399     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
400     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
401     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
402     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
403     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
404     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
405     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
406     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
407     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
408     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
409     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
410     \bottomrule
411     \end{tabular}
412 chrisfen 2652 \label{tab:solnWeak}
413 chrisfen 2599 \end{table}
414    
415     \begin{table}[htbp]
416     \centering
417     \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.}
418     \begin{tabular}{@{} ccrrrrrr @{}}
419     \\
420     \toprule
421     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
422     \cmidrule(lr){3-5}
423     \cmidrule(l){6-8}
424     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
425     \midrule
426     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
427     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
428     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
429     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
430     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
431     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
432     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
433     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
434     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
435     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
436     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
437     \midrule
438     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
439     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
440     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
441     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
442     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
443     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
444     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
445     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
446     \bottomrule
447     \end{tabular}
448 chrisfen 2652 \label{tab:solnWeakAng}
449 chrisfen 2599 \end{table}
450    
451     \section{\label{app-sol10}Strong NaCl Solution}
452    
453 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.
454    
455 chrisfen 2599 \begin{table}[htbp]
456     \centering
457     \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 $\varepsilon \approx \infty$).}
458     \begin{tabular}{@{} ccrrrrrr @{}}
459     \\
460     \toprule
461     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
462     \cmidrule(lr){3-4}
463     \cmidrule(lr){5-6}
464     \cmidrule(l){7-8}
465     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
466     \midrule
467     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
468     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
469     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
470     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
471     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
472     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
473     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
474     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
475     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
476     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
477     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
478     \midrule
479     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
480     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
481     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
482     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
483     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
484     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
485     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
486     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
487     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
488     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
489     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
490     \midrule
491     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
492     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
493     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
494     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
495     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
496     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
497     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
498     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
499     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
500     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
501     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
502     \bottomrule
503     \end{tabular}
504 chrisfen 2652 \label{tab:solnStr}
505 chrisfen 2599 \end{table}
506    
507     \begin{table}[htbp]
508     \centering
509     \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.}
510     \begin{tabular}{@{} ccrrrrrr @{}}
511     \\
512     \toprule
513     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
514     \cmidrule(lr){3-5}
515     \cmidrule(l){6-8}
516     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
517     \midrule
518     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
519     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
520     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
521     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
522     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
523     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
524     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
525     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
526     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
527     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
528     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
529     \midrule
530     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
531     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
532     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
533     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
534     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
535     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
536     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
537     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
538     \bottomrule
539     \end{tabular}
540 chrisfen 2652 \label{tab:solnStrAng}
541 chrisfen 2599 \end{table}
542    
543     \section{\label{app-argon}Argon Sphere in Water}
544    
545 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.
546    
547 chrisfen 2599 \begin{table}[htbp]
548     \centering
549     \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 $\varepsilon \approx \infty$).}
550     \begin{tabular}{@{} ccrrrrrr @{}}
551     \\
552     \toprule
553     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
554     \cmidrule(lr){3-4}
555     \cmidrule(lr){5-6}
556     \cmidrule(l){7-8}
557     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
558     \midrule
559     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
560     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
561     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
562     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
563     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
564     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
565     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
566     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
567     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
568     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
569     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
570     \midrule
571     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
572     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
573     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
574     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
575     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
576     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
577     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
578     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
579     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
580     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
581     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
582     \midrule
583     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
584     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
585     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
586     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
587     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
588     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
589     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
590     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
591     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
592     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
593     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
594     \bottomrule
595     \end{tabular}
596 chrisfen 2652 \label{tab:argon}
597 chrisfen 2599 \end{table}
598    
599     \begin{table}[htbp]
600     \centering
601     \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.}
602     \begin{tabular}{@{} ccrrrrrr @{}}
603     \\
604     \toprule
605     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
606     \cmidrule(lr){3-5}
607     \cmidrule(l){6-8}
608     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
609     \midrule
610     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
611     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
612     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
613     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
614     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
615     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
616     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
617     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
618     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
619     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
620     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
621     \midrule
622     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
623     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
624     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
625     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
626     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
627     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
628     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
629     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
630     \bottomrule
631     \end{tabular}
632 chrisfen 2652 \label{tab:argonAng}
633 chrisfen 2599 \end{table}
634    
635 chrisfen 2641 \newpage
636    
637     \bibliographystyle{jcp2}
638     \bibliography{electrostaticMethods}
639    
640 chrisfen 2599 \end{document}