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# User Rev Content
1 chrisfen 2599 %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
2     \documentclass[12pt]{article}
3 gezelter 2658 %\usepackage{endfloat}
4 chrisfen 2599 \usepackage{amsmath}
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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
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19     \textheight 9.0in \textwidth 6.5in
20     \brokenpenalty=10000
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22     \renewcommand\citemid{\ } % no comma in optional reference note
23    
24     \begin{document}
25    
26 gezelter 2658 This document includes individual system-based comparisons of the
27     studied methods with smooth particle-mesh Ewald. Each of the seven
28     systems comprises its own section and has its own discussion and
29     tabular listing of the results for the $\Delta E$, force and torque
30     vector magnitude, and force and torque vector direction comparisons.
31 chrisfen 2599
32     \section{\label{app-water}Liquid Water}
33    
34 gezelter 2658 500 liquid state configurations were generated as described in the
35     Methods section using the SPC/E model of water.\cite{Berendsen87} The
36     results for the energy gap comparisons and the force and torque vector
37     magnitude comparisons are shown in table \ref{tab:spce}. The force
38     and torque vector directionality results are displayed separately in
39     table \ref{tab:spceAng}, where the effect of group-based cutoffs and
40     switching functions on the {\sc sp} and {\sc sf} potentials are
41     investigated.
42 chrisfen 2599 \begin{table}[htbp]
43     \centering
44 gezelter 2658 \caption{Regression results for the liquid water system. Tabulated
45     results include $\Delta E$ values (top set), force vector magnitudes
46     (middle set) and torque vector magnitudes (bottom set). PC = Pure
47     Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
48     Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
49     \infty$).}
50 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
51     \\
52     \toprule
53     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
54     \cmidrule(lr){3-4}
55     \cmidrule(lr){5-6}
56     \cmidrule(l){7-8}
57     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
58     \midrule
59     PC & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\
60     SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\
61     & 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\
62     & 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\
63     & 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\
64     SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\
65     & 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\
66     & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
67     & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
68     GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
69     RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\
70    
71     \midrule
72    
73     PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
74     SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
75     & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
76     & 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\
77     & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
78     SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\
79     & 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
80     & 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\
81     & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
82     GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
83     RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
84    
85     \midrule
86    
87     PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
88     SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
89     & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
90     & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
91     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
92     SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
93     & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
94     & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
95     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
96     GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
97     RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
98     \bottomrule
99     \end{tabular}
100 chrisfen 2652 \label{tab:spce}
101 chrisfen 2599 \end{table}
102    
103     \begin{table}[htbp]
104     \centering
105 gezelter 2658 \caption{Variance results from Gaussian fits to angular
106     distributions of the force and torque vectors in the liquid water
107     system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
108     GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
109     \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
110     Group Switched Shifted Force.}
111 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
112     \\
113     \toprule
114     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
115     \cmidrule(lr){3-5}
116     \cmidrule(l){6-8}
117     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
118     \midrule
119     PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
120     SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
121     & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
122     & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
123     & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
124     SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
125     & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
126     & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
127     & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
128     GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
129     RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\
130     \midrule
131     GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
132     & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
133     & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
134     & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
135     GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
136     & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
137     & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
138     & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
139     \bottomrule
140     \end{tabular}
141 chrisfen 2642 \label{tab:spceAng}
142 chrisfen 2599 \end{table}
143    
144 gezelter 2658 For the most parts, the water results appear to parallel the combined
145     results seen in the discussion in the main paper. There is good
146     agreement with SPME in both energetic and dynamic behavior when using
147     the {\sc sf} method with and without damping. The {\sc sp} method does
148     well with an $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff
149     radii greater than 12 \AA. The results for both of these methods also
150     begin to decay as damping gets too large.
151 chrisfen 2642
152 gezelter 2658 The pure cutoff (PC) method performs poorly, as seen in the main
153     discussion section. In contrast to the combined values, however, the
154     use of a switching function and group based cutoffs really improves
155     the results for these neutral water molecules. The group switched
156     cutoff (GSC) shows mimics the energetics of SPME more poorly than the
157     {\sc sp} (with moderate damping) and {\sc sf} methods, but the
158     dynamics are quite good. The switching functions corrects
159     discontinuities in the potential and forces, leading to the improved
160     results. Such improvements with the use of a switching function has
161     been recognized in previous studies,\cite{Andrea83,Steinbach94} and it
162     is a useful tactic for stably incorporating local area electrostatic
163     effects.
164 chrisfen 2652
165 gezelter 2658 The reaction field (RF) method simply extends the results observed in
166     the GSC case. Both methods are similar in form (i.e. neutral groups,
167     switching function), but RF incorporates an added effect from the
168     external dielectric. This similarity translates into the same good
169     dynamic results and improved energetic results. These still fall
170     short of the moderately damped {\sc sp} and {\sc sf} methods, but they
171     display how incorporating some implicit properties of the surroundings
172     (i.e. $\epsilon_\textrm{S}$) can improve results.
173 chrisfen 2652
174 gezelter 2658 A final note for the liquid water system, use of group cutoffs and a
175     switching function also leads to noticeable improvements in the {\sc
176     sp} and {\sc sf} methods, primarily in directionality of the force and
177     torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant
178     narrowing of the angle distribution in the cases with little to no
179     damping and only modest improvement for the ideal conditions ($\alpha$
180     = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The {\sc sf}
181     method simply shows modest narrowing across all damping and cutoff
182     ranges of interest. Group cutoffs and the switching function do
183     nothing for cases were error is introduced by overdamping the
184     potentials.
185 chrisfen 2652
186 chrisfen 2599 \section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
187    
188 gezelter 2658 In addition to the disordered molecular system above, the ordered
189     molecular system of ice I$_\textrm{c}$ was also considered. The
190     results for the energy gap comparisons and the force and torque vector
191     magnitude comparisons are shown in table \ref{tab:ice}. The force and
192     torque vector directionality results are displayed separately in table
193     \ref{tab:iceAng}, where the effect of group-based cutoffs and
194     switching functions on the {\sc sp} and {\sc sf} potentials are
195     investigated.
196 chrisfen 2652
197 chrisfen 2599 \begin{table}[htbp]
198     \centering
199 gezelter 2658 \caption{Regression results for the ice I$_\textrm{c}$
200     system. Tabulated results include $\Delta E$ values (top set), force
201     vector magnitudes (middle set) and torque vector magnitudes (bottom
202     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
203     GSC = Group Switched Cutoff, and RF = Reaction Field (where
204     $\varepsilon \approx \infty$).}
205 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
206     \\
207     \toprule
208     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
209     \cmidrule(lr){3-4}
210     \cmidrule(lr){5-6}
211     \cmidrule(l){7-8}
212     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
213     \midrule
214     PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
215     SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
216     & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
217     & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
218     & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
219     SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
220     & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
221     & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
222     & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
223     GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
224     RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
225     \midrule
226     PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
227     SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
228     & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
229     & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
230     & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
231     SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
232     & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
233     & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
234     & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
235     GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
236     RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
237     \midrule
238     PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
239     SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
240     & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
241     & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
242     & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
243     SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
244     & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
245     & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
246     & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
247     GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
248     RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
249     \bottomrule
250     \end{tabular}
251 chrisfen 2652 \label{tab:ice}
252 chrisfen 2599 \end{table}
253    
254     \begin{table}[htbp]
255     \centering
256     \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.}
257     \begin{tabular}{@{} ccrrrrrr @{}}
258     \\
259     \toprule
260     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
261     \cmidrule(lr){3-5}
262     \cmidrule(l){6-8}
263     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
264     \midrule
265     PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
266     SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
267     & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
268     & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
269     & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
270     SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
271     & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
272     & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
273     & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
274     GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
275     RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
276     \midrule
277     GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
278     & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
279     & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
280     & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
281     GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
282     & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
283     & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
284     & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
285     \bottomrule
286     \end{tabular}
287 chrisfen 2652 \label{tab:iceAng}
288 chrisfen 2599 \end{table}
289    
290 gezelter 2658 Highly ordered systems are a difficult test for the pairwise systems
291     in that they lack the periodicity inherent to the Ewald summation. As
292     expected, the energy gap agreement with SPME reduces for the {\sc sp}
293     and {\sc sf} with parameters that were perfectly acceptable for the
294     disordered liquid system. Moving to higher $R_\textrm{c}$ remedies
295     this degraded performance, though at increase in computational cost.
296     However, the dynamics of this crystalline system (both in magnitude
297     and direction) are little affected. Both methods still reproduce the
298     Ewald behavior with the same parameter recommendations from the
299     previous section.
300 chrisfen 2652
301 gezelter 2658 It is also worth noting that RF exhibits a slightly improved energy
302     gap results over the liquid water system. One possible explanation is
303     that the ice I$_\textrm{c}$ crystal is ordered such that the net
304     dipole moment of the crystal is zero. With $\epsilon_\textrm{S} =
305     \infty$, the reaction field incorporates this structural organization
306     by actively enforcing a zeroed dipole moment within each cutoff
307     sphere.
308 chrisfen 2652
309 chrisfen 2599 \section{\label{app-melt}NaCl Melt}
310    
311 gezelter 2658 A high temperature NaCl melt was tested to gauge the accuracy of the
312     pairwise summation methods in a highly charge disordered system. The
313     results for the energy gap comparisons and the force and torque vector
314     magnitude comparisons are shown in table \ref{tab:melt}. The force
315     and torque vector directionality results are displayed separately in
316     table \ref{tab:meltAng}, where the effect of group-based cutoffs and
317     switching functions on the {\sc sp} and {\sc sf} potentials are
318     investigated.
319 chrisfen 2652
320 chrisfen 2599 \begin{table}[htbp]
321     \centering
322     \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.}
323     \begin{tabular}{@{} ccrrrrrr @{}}
324     \\
325     \toprule
326     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
327     \cmidrule(lr){3-4}
328     \cmidrule(lr){5-6}
329     \cmidrule(l){7-8}
330     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
331     \midrule
332     PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
333 chrisfen 2655 SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\
334     & 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\
335 chrisfen 2599 & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
336     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
337 chrisfen 2655 SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\
338     & 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\
339 chrisfen 2599 & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
340     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
341     \midrule
342     PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
343 chrisfen 2655 SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\
344     & 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\
345 chrisfen 2599 & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
346     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
347 chrisfen 2655 SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\
348     & 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\
349 chrisfen 2599 & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
350     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
351     \bottomrule
352     \end{tabular}
353 chrisfen 2652 \label{tab:melt}
354 chrisfen 2599 \end{table}
355    
356     \begin{table}[htbp]
357     \centering
358     \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.}
359     \begin{tabular}{@{} ccrrrrrr @{}}
360     \\
361     \toprule
362     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
363     \cmidrule(lr){3-5}
364     \cmidrule(l){6-8}
365     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
366     \midrule
367     PC & & 13.294 & 8.035 & 5.366 \\
368     SP & 0.0 & 13.316 & 8.037 & 5.385 \\
369     & 0.1 & 5.705 & 1.391 & 0.360 \\
370     & 0.2 & 2.415 & 7.534 & 13.927 \\
371     & 0.3 & 23.769 & 67.306 & 57.252 \\
372     SF & 0.0 & 1.693 & 0.603 & 0.256 \\
373     & 0.1 & 1.687 & 0.653 & 0.272 \\
374     & 0.2 & 2.598 & 7.523 & 13.930 \\
375     & 0.3 & 23.734 & 67.305 & 57.252 \\
376     \bottomrule
377     \end{tabular}
378 chrisfen 2652 \label{tab:meltAng}
379 chrisfen 2599 \end{table}
380    
381 chrisfen 2655 The molten NaCl system shows the a
382 chrisfen 2654
383 chrisfen 2599 \section{\label{app-salt}NaCl Crystal}
384    
385 gezelter 2658 A 1000K NaCl crystal was used to investigate the accuracy of the
386     pairwise summation methods in an ordered system of charged
387     particles. The results for the energy gap comparisons and the force
388     and torque vector magnitude comparisons are shown in table
389     \ref{tab:salt}. The force and torque vector directionality results
390     are displayed separately in table \ref{tab:saltAng}, where the effect
391     of group-based cutoffs and switching functions on the {\sc sp} and
392     {\sc sf} potentials are investigated.
393 chrisfen 2652
394 chrisfen 2599 \begin{table}[htbp]
395     \centering
396 gezelter 2658 \caption{Regression results for the crystalline NaCl
397     system. Tabulated results include $\Delta E$ values (top set) and
398     force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted
399     Potential, and SF = Shifted Force.}
400 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
401     \\
402     \toprule
403     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
404     \cmidrule(lr){3-4}
405     \cmidrule(lr){5-6}
406     \cmidrule(l){7-8}
407     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
408     \midrule
409     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
410     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
411     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
412     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
413     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
414     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
415     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
416     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
417     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
418     \midrule
419     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
420     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
421     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
422     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
423     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
424     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
425     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
426     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
427     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
428     \bottomrule
429     \end{tabular}
430 chrisfen 2652 \label{tab:salt}
431 chrisfen 2599 \end{table}
432    
433     \begin{table}[htbp]
434     \centering
435 gezelter 2658 \caption{Variance results from Gaussian fits to angular
436     distributions of the force vectors in the crystalline NaCl system. PC
437     = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
438     Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
439     \infty$).}
440 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
441     \\
442     \toprule
443     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
444     \cmidrule(lr){3-5}
445     \cmidrule(l){6-8}
446     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
447     \midrule
448     PC & & 111.945 & 111.824 & 111.866 \\
449     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
450     & 0.1 & 52.361 & 42.574 & 2.819 \\
451     & 0.2 & 10.847 & 9.709 & 9.686 \\
452     & 0.3 & 31.128 & 31.104 & 31.029 \\
453     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
454     & 0.1 & 9.462 & 3.303 & 1.721 \\
455     & 0.2 & 11.454 & 9.813 & 9.701 \\
456     & 0.3 & 31.120 & 31.105 & 31.029 \\
457     \bottomrule
458     \end{tabular}
459 chrisfen 2652 \label{tab:saltAng}
460 chrisfen 2599 \end{table}
461    
462     \section{\label{app-sol1}Weak NaCl Solution}
463    
464 gezelter 2658 In an effort to bridge the charged atomic and neutral molecular
465     systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
466     the liquid water system. This low ionic strength system consists of 4
467     ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
468     for the energy gap comparisons and the force and torque vector
469     magnitude comparisons are shown in table \ref{tab:solnWeak}. The
470     force and torque vector directionality results are displayed
471     separately in table \ref{tab:solnWeakAng}, where the effect of
472     group-based cutoffs and switching functions on the {\sc sp} and {\sc
473     sf} potentials are investigated.
474 chrisfen 2652
475 chrisfen 2599 \begin{table}[htbp]
476     \centering
477 gezelter 2658 \caption{Regression results for the weak NaCl solution
478     system. Tabulated results include $\Delta E$ values (top set), force
479     vector magnitudes (middle set) and torque vector magnitudes (bottom
480     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
481     GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
482     \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
483     Group Switched Shifted Force.}
484 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
485     \\
486     \toprule
487     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
488     \cmidrule(lr){3-4}
489     \cmidrule(lr){5-6}
490     \cmidrule(l){7-8}
491     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
492     \midrule
493     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
494     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
495     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
496     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
497     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
498     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
499     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
500     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
501     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
502     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
503     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
504     \midrule
505     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
506     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
507     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
508     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
509     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
510     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
511     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
512     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
513     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
514     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
515     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
516     \midrule
517     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
518     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
519     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
520     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
521     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
522     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
523     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
524     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
525     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
526     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
527     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
528     \bottomrule
529     \end{tabular}
530 chrisfen 2652 \label{tab:solnWeak}
531 chrisfen 2599 \end{table}
532    
533     \begin{table}[htbp]
534     \centering
535 gezelter 2658 \caption{Variance results from Gaussian fits to angular
536     distributions of the force and torque vectors in the weak NaCl
537     solution system. PC = Pure Cutoff, SP = Shifted Potential, SF =
538     Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where
539     $\varepsilon \approx \infty$), GSSP = Group Switched Shifted
540     Potential, and GSSF = Group Switched Shifted Force.}
541 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
542     \\
543     \toprule
544     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
545     \cmidrule(lr){3-5}
546     \cmidrule(l){6-8}
547     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
548     \midrule
549     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
550     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
551     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
552     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
553     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
554     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
555     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
556     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
557     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
558     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
559     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
560     \midrule
561     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
562     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
563     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
564     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
565     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
566     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
567     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
568     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
569     \bottomrule
570     \end{tabular}
571 chrisfen 2652 \label{tab:solnWeakAng}
572 chrisfen 2599 \end{table}
573    
574     \section{\label{app-sol10}Strong NaCl Solution}
575    
576 gezelter 2658 The bridging of the charged atomic and neutral molecular systems was
577     furthered by considering a high ionic strength system consisting of 40
578     ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for
579     the energy gap comparisons and the force and torque vector magnitude
580     comparisons are shown in table \ref{tab:solnWeak}. The force and
581     torque vector directionality results are displayed separately in table
582     \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and
583     switching functions on the {\sc sp} and {\sc sf} potentials are
584     investigated.
585 chrisfen 2652
586 chrisfen 2599 \begin{table}[htbp]
587     \centering
588 gezelter 2658 \caption{Regression results for the strong NaCl solution
589     system. Tabulated results include $\Delta E$ values (top set), force
590     vector magnitudes (middle set) and torque vector magnitudes (bottom
591     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
592     GSC = Group Switched Cutoff, and RF = Reaction Field (where
593     $\varepsilon \approx \infty$).}
594 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
595     \\
596     \toprule
597     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
598     \cmidrule(lr){3-4}
599     \cmidrule(lr){5-6}
600     \cmidrule(l){7-8}
601     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
602     \midrule
603     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
604     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
605     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
606     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
607     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
608     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
609     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
610     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
611     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
612     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
613     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
614     \midrule
615     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
616     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
617     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
618     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
619     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
620     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
621     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
622     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
623     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
624     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
625     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
626     \midrule
627     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
628     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
629     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
630     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
631     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
632     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
633     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
634     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
635     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
636     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
637     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
638     \bottomrule
639     \end{tabular}
640 chrisfen 2652 \label{tab:solnStr}
641 chrisfen 2599 \end{table}
642    
643     \begin{table}[htbp]
644     \centering
645     \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.}
646     \begin{tabular}{@{} ccrrrrrr @{}}
647     \\
648     \toprule
649     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
650     \cmidrule(lr){3-5}
651     \cmidrule(l){6-8}
652     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
653     \midrule
654     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
655     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
656     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
657     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
658     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
659     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
660     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
661     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
662     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
663     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
664     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
665     \midrule
666     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
667     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
668     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
669     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
670     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
671     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
672     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
673     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
674     \bottomrule
675     \end{tabular}
676 chrisfen 2652 \label{tab:solnStrAng}
677 chrisfen 2599 \end{table}
678    
679     \section{\label{app-argon}Argon Sphere in Water}
680    
681 gezelter 2658 The final model system studied was 6 \AA\ sphere of Argon solvated by
682     SPC/E water. The results for the energy gap comparisons and the force
683     and torque vector magnitude comparisons are shown in table
684     \ref{tab:solnWeak}. The force and torque vector directionality
685     results are displayed separately in table \ref{tab:solnWeakAng}, where
686     the effect of group-based cutoffs and switching functions on the {\sc
687     sp} and {\sc sf} potentials are investigated.
688 chrisfen 2652
689 chrisfen 2599 \begin{table}[htbp]
690     \centering
691 gezelter 2658 \caption{Regression results for the 6 \AA\ argon sphere in liquid
692     water system. Tabulated results include $\Delta E$ values (top set),
693     force vector magnitudes (middle set) and torque vector magnitudes
694     (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted
695     Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where
696     $\varepsilon \approx \infty$).}
697 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
698     \\
699     \toprule
700     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
701     \cmidrule(lr){3-4}
702     \cmidrule(lr){5-6}
703     \cmidrule(l){7-8}
704     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
705     \midrule
706     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
707     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
708     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
709     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
710     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
711     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
712     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
713     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
714     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
715     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
716     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
717     \midrule
718     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
719     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
720     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
721     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
722     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
723     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
724     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
725     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
726     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
727     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
728     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
729     \midrule
730     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
731     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
732     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
733     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
734     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
735     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
736     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
737     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
738     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
739     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
740     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
741     \bottomrule
742     \end{tabular}
743 chrisfen 2652 \label{tab:argon}
744 chrisfen 2599 \end{table}
745    
746     \begin{table}[htbp]
747     \centering
748 gezelter 2658 \caption{Variance results from Gaussian fits to angular
749     distributions of the force and torque vectors in the 6 \AA\ sphere of
750     argon in liquid water system. PC = Pure Cutoff, SP = Shifted
751     Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF =
752     Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group
753     Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}
754 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
755     \\
756     \toprule
757     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
758     \cmidrule(lr){3-5}
759     \cmidrule(l){6-8}
760     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
761     \midrule
762     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
763     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
764     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
765     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
766     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
767     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
768     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
769     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
770     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
771     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
772     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
773     \midrule
774     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
775     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
776     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
777     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
778     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
779     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
780     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
781     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
782     \bottomrule
783     \end{tabular}
784 chrisfen 2652 \label{tab:argonAng}
785 chrisfen 2599 \end{table}
786    
787 chrisfen 2641 \newpage
788    
789     \bibliographystyle{jcp2}
790     \bibliography{electrostaticMethods}
791    
792 gezelter 2658 \end{document}