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1 chrisfen 2599 %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
2 chrisfen 2666 \documentclass[11pt]{article}
3 gezelter 2658 %\usepackage{endfloat}
4 chrisfen 2599 \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
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22     \renewcommand\citemid{\ } % no comma in optional reference note
23    
24     \begin{document}
25    
26 gezelter 2670 This document includes comparisons of the new pairwise electrostatic
27     methods with {\sc spme} for each of the individual systems mentioned
28     in paper. Each of the seven sections contains information about a
29     single system type and has its own discussion and tabular listing of
30     the results for the comparisons of $\Delta E$, the magnitudes of the
31     forces and torques, and directionality of the force and torque
32     vectors.
33 chrisfen 2599
34 chrisfen 2660 \section{\label{app:water}Liquid Water}
35 chrisfen 2599
36 chrisfen 2666 The first system considered was liquid water at 300K using the SPC/E
37     model of water.\cite{Berendsen87} The results for the energy gap
38     comparisons and the force and torque vector magnitude comparisons are
39     shown in table \ref{tab:spce}. The force and torque vector
40     directionality results are displayed separately in table
41     \ref{tab:spceAng}, where the effect of group-based cutoffs and
42 gezelter 2658 switching functions on the {\sc sp} and {\sc sf} potentials are
43 chrisfen 2666 investigated.
44 chrisfen 2599 \begin{table}[htbp]
45     \centering
46 gezelter 2658 \caption{Regression results for the liquid water system. Tabulated
47     results include $\Delta E$ values (top set), force vector magnitudes
48     (middle set) and torque vector magnitudes (bottom set). PC = Pure
49     Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
50     Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
51     \infty$).}
52 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
53     \\
54     \toprule
55     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
56     \cmidrule(lr){3-4}
57     \cmidrule(lr){5-6}
58     \cmidrule(l){7-8}
59     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
60     \midrule
61     PC & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\
62     SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\
63     & 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\
64     & 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\
65     & 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\
66     SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\
67     & 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\
68     & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
69     & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
70     GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
71 chrisfen 2660 RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\
72 chrisfen 2599 \midrule
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     \midrule
85     PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
86     SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
87     & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
88     & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
89     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
90     SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
91     & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
92     & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
93     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
94     GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
95     RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
96     \bottomrule
97     \end{tabular}
98 chrisfen 2652 \label{tab:spce}
99 chrisfen 2599 \end{table}
100    
101     \begin{table}[htbp]
102     \centering
103 gezelter 2658 \caption{Variance results from Gaussian fits to angular
104     distributions of the force and torque vectors in the liquid water
105     system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
106     GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
107     \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
108     Group Switched Shifted Force.}
109 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
110     \\
111     \toprule
112     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
113     \cmidrule(lr){3-5}
114     \cmidrule(l){6-8}
115     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
116     \midrule
117     PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
118     SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
119     & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
120     & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
121     & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
122     SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
123     & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
124     & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
125     & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
126     GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
127     RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\
128     \midrule
129     GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
130     & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
131     & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
132     & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
133     GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
134     & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
135     & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
136     & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
137     \bottomrule
138     \end{tabular}
139 chrisfen 2642 \label{tab:spceAng}
140 chrisfen 2599 \end{table}
141    
142 gezelter 2670 The water results parallel the combined results seen in the discussion
143     section of the main paper. There is good agreement with {\sc spme} in
144     both energetic and dynamic behavior when using the {\sc sf} method
145     with and without damping. The {\sc sp} method does well with an
146 chrisfen 2666 $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater
147 gezelter 2670 than 12 \AA. Overdamping the electrostatics reduces the agreement
148     between both these methods and {\sc spme}.
149 chrisfen 2642
150 chrisfen 2666 The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
151     observations in the main portion of this paper. In contrast to the
152     combined values, however, the use of a switching function and group
153 chrisfen 2667 based cutoffs greatly improves the results for these neutral water
154 chrisfen 2666 molecules. The group switched cutoff ({\sc gsc}) does not mimic the
155     energetics of {\sc spme} as well as the {\sc sp} (with moderate
156     damping) and {\sc sf} methods, but the dynamics are quite good. The
157 chrisfen 2667 switching functions correct discontinuities in the potential and
158 chrisfen 2666 forces, leading to these improved results. Such improvements with the
159 chrisfen 2667 use of a switching function have been recognized in previous
160 chrisfen 2666 studies,\cite{Andrea83,Steinbach94} and this proves to be a useful
161     tactic for stably incorporating local area electrostatic effects.
162 chrisfen 2652
163 chrisfen 2666 The reaction field ({\sc rf}) method simply extends upon the results
164     observed in the {\sc gsc} case. Both methods are similar in form
165     (i.e. neutral groups, switching function), but {\sc rf} incorporates
166     an added effect from the external dielectric. This similarity
167     translates into the same good dynamic results and improved energetic
168     agreement with {\sc spme}. Though this agreement is not to the level
169     of the moderately damped {\sc sp} and {\sc sf} methods, these results
170     show how incorporating some implicit properties of the surroundings
171     (i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction.
172 chrisfen 2652
173 chrisfen 2667 As a final note for the liquid water system, use of group cutoffs and a
174 chrisfen 2666 switching function leads to noticeable improvements in the {\sc sp}
175     and {\sc sf} methods, primarily in directionality of the force and
176     torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows
177     significant narrowing of the angle distribution when using little to
178     no damping and only modest improvement for the recommended conditions
179     ($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The
180     {\sc sf} method shows modest narrowing across all damping and cutoff
181     ranges of interest. When overdamping these methods, group cutoffs and
182     the switching function do not improve the force and torque
183     directionalities.
184 chrisfen 2652
185 chrisfen 2660 \section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$}
186 chrisfen 2599
187 gezelter 2658 In addition to the disordered molecular system above, the ordered
188     molecular system of ice I$_\textrm{c}$ was also considered. The
189     results for the energy gap comparisons and the force and torque vector
190     magnitude comparisons are shown in table \ref{tab:ice}. The force and
191     torque vector directionality results are displayed separately in table
192     \ref{tab:iceAng}, where the effect of group-based cutoffs and
193     switching functions on the {\sc sp} and {\sc sf} potentials are
194     investigated.
195 chrisfen 2652
196 chrisfen 2599 \begin{table}[htbp]
197     \centering
198 gezelter 2658 \caption{Regression results for the ice I$_\textrm{c}$
199     system. Tabulated results include $\Delta E$ values (top set), force
200     vector magnitudes (middle set) and torque vector magnitudes (bottom
201     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
202     GSC = Group Switched Cutoff, and RF = Reaction Field (where
203     $\varepsilon \approx \infty$).}
204 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
205     \\
206     \toprule
207     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
208     \cmidrule(lr){3-4}
209     \cmidrule(lr){5-6}
210     \cmidrule(l){7-8}
211     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
212     \midrule
213     PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
214     SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
215     & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
216     & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
217     & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
218     SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
219     & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
220     & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
221     & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
222     GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
223     RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
224     \midrule
225     PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
226     SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
227     & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
228     & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
229     & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
230     SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
231     & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
232     & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
233     & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
234     GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
235     RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
236     \midrule
237     PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
238     SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
239     & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
240     & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
241     & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
242     SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
243     & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
244     & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
245     & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
246     GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
247     RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
248     \bottomrule
249     \end{tabular}
250 chrisfen 2652 \label{tab:ice}
251 chrisfen 2599 \end{table}
252    
253     \begin{table}[htbp]
254     \centering
255     \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.}
256     \begin{tabular}{@{} ccrrrrrr @{}}
257     \\
258     \toprule
259     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
260     \cmidrule(lr){3-5}
261     \cmidrule(l){6-8}
262     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
263     \midrule
264     PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
265     SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
266     & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
267     & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
268     & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
269     SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
270     & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
271     & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
272     & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
273     GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
274     RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
275     \midrule
276     GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
277     & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
278     & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
279     & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
280     GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
281     & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
282     & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
283     & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
284     \bottomrule
285     \end{tabular}
286 chrisfen 2652 \label{tab:iceAng}
287 chrisfen 2599 \end{table}
288    
289 chrisfen 2666 Highly ordered systems are a difficult test for the pairwise methods
290 gezelter 2670 in that they lack the implicit periodicity of the Ewald summation. As
291 chrisfen 2667 expected, the energy gap agreement with {\sc spme} is reduced for the
292 chrisfen 2666 {\sc sp} and {\sc sf} methods with parameters that were acceptable for
293     the disordered liquid system. Moving to higher $R_\textrm{c}$ helps
294     improve the agreement, though at an increase in computational cost.
295     The dynamics of this crystalline system (both in magnitude and
296     direction) are little affected. Both methods still reproduce the Ewald
297     behavior with the same parameter recommendations from the previous
298     section.
299 chrisfen 2652
300 chrisfen 2666 It is also worth noting that {\sc rf} exhibits improved energy gap
301     results over the liquid water system. One possible explanation is
302 gezelter 2658 that the ice I$_\textrm{c}$ crystal is ordered such that the net
303     dipole moment of the crystal is zero. With $\epsilon_\textrm{S} =
304     \infty$, the reaction field incorporates this structural organization
305     by actively enforcing a zeroed dipole moment within each cutoff
306     sphere.
307 chrisfen 2652
308 chrisfen 2660 \section{\label{app:melt}NaCl Melt}
309 chrisfen 2599
310 gezelter 2658 A high temperature NaCl melt was tested to gauge the accuracy of the
311 gezelter 2670 pairwise summation methods in a disordered system of charges. The
312     results for the energy gap comparisons and the force vector magnitude
313 chrisfen 2667 comparisons are shown in table \ref{tab:melt}. The force vector
314     directionality results are displayed separately in table
315     \ref{tab:meltAng}.
316 chrisfen 2652
317 chrisfen 2599 \begin{table}[htbp]
318     \centering
319     \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.}
320     \begin{tabular}{@{} ccrrrrrr @{}}
321     \\
322     \toprule
323     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
324     \cmidrule(lr){3-4}
325     \cmidrule(lr){5-6}
326     \cmidrule(l){7-8}
327     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
328     \midrule
329     PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
330 chrisfen 2655 SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\
331     & 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\
332 chrisfen 2599 & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
333     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
334 chrisfen 2655 SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\
335     & 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\
336 chrisfen 2599 & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
337     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
338     \midrule
339     PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
340 chrisfen 2655 SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\
341     & 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\
342 chrisfen 2599 & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
343     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
344 chrisfen 2655 SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\
345     & 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\
346 chrisfen 2599 & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
347     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
348     \bottomrule
349     \end{tabular}
350 chrisfen 2652 \label{tab:melt}
351 chrisfen 2599 \end{table}
352    
353     \begin{table}[htbp]
354     \centering
355     \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.}
356     \begin{tabular}{@{} ccrrrrrr @{}}
357     \\
358     \toprule
359     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
360     \cmidrule(lr){3-5}
361     \cmidrule(l){6-8}
362     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
363     \midrule
364     PC & & 13.294 & 8.035 & 5.366 \\
365     SP & 0.0 & 13.316 & 8.037 & 5.385 \\
366     & 0.1 & 5.705 & 1.391 & 0.360 \\
367     & 0.2 & 2.415 & 7.534 & 13.927 \\
368     & 0.3 & 23.769 & 67.306 & 57.252 \\
369     SF & 0.0 & 1.693 & 0.603 & 0.256 \\
370     & 0.1 & 1.687 & 0.653 & 0.272 \\
371     & 0.2 & 2.598 & 7.523 & 13.930 \\
372     & 0.3 & 23.734 & 67.305 & 57.252 \\
373     \bottomrule
374     \end{tabular}
375 chrisfen 2652 \label{tab:meltAng}
376 chrisfen 2599 \end{table}
377    
378 chrisfen 2660 The molten NaCl system shows more sensitivity to the electrostatic
379     damping than the water systems. The most noticeable point is that the
380     undamped {\sc sf} method does very well at replicating the {\sc spme}
381     configurational energy differences and forces. Light damping appears
382     to minimally improve the dynamics, but this comes with a deterioration
383     of the energy gap results. In contrast, this light damping improves
384     the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
385     damping reduce the agreement with {\sc spme} for both methods. From
386     these observations, the undamped {\sc sf} method is the best choice
387     for disordered systems of charges.
388 chrisfen 2654
389 chrisfen 2660 \section{\label{app:salt}NaCl Crystal}
390 chrisfen 2599
391 gezelter 2658 A 1000K NaCl crystal was used to investigate the accuracy of the
392     pairwise summation methods in an ordered system of charged
393     particles. The results for the energy gap comparisons and the force
394 chrisfen 2667 vector magnitude comparisons are shown in table \ref{tab:salt}. The
395     force vector directionality results are displayed separately in table
396     \ref{tab:saltAng}.
397 chrisfen 2652
398 chrisfen 2599 \begin{table}[htbp]
399     \centering
400 gezelter 2658 \caption{Regression results for the crystalline NaCl
401     system. Tabulated results include $\Delta E$ values (top set) and
402     force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted
403     Potential, and SF = Shifted Force.}
404 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
405     \\
406     \toprule
407     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
408     \cmidrule(lr){3-4}
409     \cmidrule(lr){5-6}
410     \cmidrule(l){7-8}
411     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
412     \midrule
413     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
414     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
415     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
416     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
417     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
418     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
419     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
420     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
421     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
422     \midrule
423     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
424     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
425     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
426     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
427     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
428     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
429     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
430     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
431     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
432     \bottomrule
433     \end{tabular}
434 chrisfen 2652 \label{tab:salt}
435 chrisfen 2599 \end{table}
436    
437     \begin{table}[htbp]
438     \centering
439 gezelter 2658 \caption{Variance results from Gaussian fits to angular
440     distributions of the force vectors in the crystalline NaCl system. PC
441     = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
442     Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
443     \infty$).}
444 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
445     \\
446     \toprule
447     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
448     \cmidrule(lr){3-5}
449     \cmidrule(l){6-8}
450     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
451     \midrule
452     PC & & 111.945 & 111.824 & 111.866 \\
453     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
454     & 0.1 & 52.361 & 42.574 & 2.819 \\
455     & 0.2 & 10.847 & 9.709 & 9.686 \\
456     & 0.3 & 31.128 & 31.104 & 31.029 \\
457     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
458     & 0.1 & 9.462 & 3.303 & 1.721 \\
459     & 0.2 & 11.454 & 9.813 & 9.701 \\
460     & 0.3 & 31.120 & 31.105 & 31.029 \\
461     \bottomrule
462     \end{tabular}
463 chrisfen 2652 \label{tab:saltAng}
464 chrisfen 2599 \end{table}
465    
466 chrisfen 2660 The crystalline NaCl system is the most challenging test case for the
467     pairwise summation methods, as evidenced by the results in tables
468     \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
469     {\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best
470     choices. These methods match well with {\sc spme} across the energy
471     gap, force magnitude, and force directionality tests. The {\sc sp}
472 chrisfen 2666 method struggles in all cases, with the exception of good dynamics
473 chrisfen 2660 reproduction when using weak electrostatic damping with a large cutoff
474     radius.
475 chrisfen 2599
476 chrisfen 2660 The moderate electrostatic damping case is not as good as we would
477 chrisfen 2667 expect given the long-time dynamics results observed for this
478     system. Since the data tabulated in tables \ref{tab:salt} and
479 chrisfen 2666 \ref{tab:saltAng} are a test of instantaneous dynamics, this indicates
480     that good long-time dynamics comes in part at the expense of
481 chrisfen 2667 short-time dynamics.
482 chrisfen 2660
483     \section{\label{app:solnWeak}Weak NaCl Solution}
484    
485 gezelter 2658 In an effort to bridge the charged atomic and neutral molecular
486     systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
487     the liquid water system. This low ionic strength system consists of 4
488     ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
489     for the energy gap comparisons and the force and torque vector
490     magnitude comparisons are shown in table \ref{tab:solnWeak}. The
491     force and torque vector directionality results are displayed
492     separately in table \ref{tab:solnWeakAng}, where the effect of
493     group-based cutoffs and switching functions on the {\sc sp} and {\sc
494     sf} potentials are investigated.
495 chrisfen 2652
496 chrisfen 2599 \begin{table}[htbp]
497     \centering
498 gezelter 2658 \caption{Regression results for the weak NaCl solution
499     system. Tabulated results include $\Delta E$ values (top set), force
500     vector magnitudes (middle set) and torque vector magnitudes (bottom
501     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
502 chrisfen 2667 GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon
503     \approx \infty$).}
504 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
505     \\
506     \toprule
507     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
508     \cmidrule(lr){3-4}
509     \cmidrule(lr){5-6}
510     \cmidrule(l){7-8}
511     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
512     \midrule
513     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
514     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
515     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
516     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
517     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
518     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
519     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
520     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
521     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
522     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
523     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
524     \midrule
525     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
526     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
527     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
528     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
529     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
530     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
531     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
532     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
533     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
534     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
535     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
536     \midrule
537     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
538     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
539     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
540     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
541     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
542     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
543     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
544     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
545     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
546     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
547     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
548     \bottomrule
549     \end{tabular}
550 chrisfen 2652 \label{tab:solnWeak}
551 chrisfen 2599 \end{table}
552    
553     \begin{table}[htbp]
554     \centering
555 gezelter 2658 \caption{Variance results from Gaussian fits to angular
556     distributions of the force and torque vectors in the weak NaCl
557     solution system. PC = Pure Cutoff, SP = Shifted Potential, SF =
558     Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where
559     $\varepsilon \approx \infty$), GSSP = Group Switched Shifted
560     Potential, and GSSF = Group Switched Shifted Force.}
561 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
562     \\
563     \toprule
564     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
565     \cmidrule(lr){3-5}
566     \cmidrule(l){6-8}
567     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
568     \midrule
569     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
570     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
571     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
572     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
573     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
574     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
575     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
576     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
577     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
578     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
579     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
580     \midrule
581     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
582     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
583     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
584     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
585     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
586     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
587     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
588     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
589     \bottomrule
590     \end{tabular}
591 chrisfen 2652 \label{tab:solnWeakAng}
592 chrisfen 2599 \end{table}
593    
594 chrisfen 2666 Because this system is a perturbation of the pure liquid water system,
595     comparisons are best drawn between these two sets. The {\sc sp} and
596     {\sc sf} methods are not significantly affected by the inclusion of a
597     few ions. The aspect of cutoff sphere neutralization aids in the
598     smooth incorporation of these ions; thus, all of the observations
599     regarding these methods carry over from section \ref{app:water}. The
600     differences between these systems are more visible for the {\sc rf}
601     method. Though good force agreement is still maintained, the energy
602 gezelter 2670 gaps show a significant increase in the scatter of the data.
603 chrisfen 2599
604 chrisfen 2660 \section{\label{app:solnStr}Strong NaCl Solution}
605    
606 gezelter 2658 The bridging of the charged atomic and neutral molecular systems was
607 chrisfen 2660 further developed by considering a high ionic strength system
608     consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
609     M). The results for the energy gap comparisons and the force and
610     torque vector magnitude comparisons are shown in table
611 chrisfen 2667 \ref{tab:solnStr}. The force and torque vector directionality
612     results are displayed separately in table \ref{tab:solnStrAng}, where
613 chrisfen 2660 the effect of group-based cutoffs and switching functions on the {\sc
614     sp} and {\sc sf} potentials are investigated.
615 chrisfen 2652
616 chrisfen 2599 \begin{table}[htbp]
617     \centering
618 gezelter 2658 \caption{Regression results for the strong NaCl solution
619     system. Tabulated results include $\Delta E$ values (top set), force
620     vector magnitudes (middle set) and torque vector magnitudes (bottom
621     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
622     GSC = Group Switched Cutoff, and RF = Reaction Field (where
623     $\varepsilon \approx \infty$).}
624 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
625     \\
626     \toprule
627     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
628     \cmidrule(lr){3-4}
629     \cmidrule(lr){5-6}
630     \cmidrule(l){7-8}
631     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
632     \midrule
633     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
634     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
635     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
636     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
637     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
638     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
639     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
640     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
641     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
642     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
643     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
644     \midrule
645     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
646     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
647     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
648     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
649     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
650     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
651     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
652     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
653     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
654     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
655     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
656     \midrule
657     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
658     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
659     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
660     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
661     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
662     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
663     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
664     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
665     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
666     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
667     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
668     \bottomrule
669     \end{tabular}
670 chrisfen 2652 \label{tab:solnStr}
671 chrisfen 2599 \end{table}
672    
673     \begin{table}[htbp]
674     \centering
675     \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.}
676     \begin{tabular}{@{} ccrrrrrr @{}}
677     \\
678     \toprule
679     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
680     \cmidrule(lr){3-5}
681     \cmidrule(l){6-8}
682     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
683     \midrule
684     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
685     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
686     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
687     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
688     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
689     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
690     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
691     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
692     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
693     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
694     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
695     \midrule
696     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
697     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
698     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
699     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
700     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
701     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
702     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
703     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
704     \bottomrule
705     \end{tabular}
706 chrisfen 2652 \label{tab:solnStrAng}
707 chrisfen 2599 \end{table}
708    
709 chrisfen 2660 The {\sc rf} method struggles with the jump in ionic strength. The
710 chrisfen 2667 configuration energy differences degrade to unusable levels while the
711 chrisfen 2666 forces and torques show a more modest reduction in the agreement with
712     {\sc spme}. The {\sc rf} method was designed for homogeneous systems,
713     and this attribute is apparent in these results.
714 chrisfen 2599
715 chrisfen 2660 The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
716     their agreement with {\sc spme}. With these results, we still
717     recommend no to moderate damping for the {\sc sf} method and moderate
718     damping for the {\sc sp} method, both with cutoffs greater than 12
719     \AA.
720    
721     \section{\label{app:argon}Argon Sphere in Water}
722    
723 chrisfen 2667 The final model system studied was a 6 \AA\ sphere of Argon solvated
724     by SPC/E water. The results for the energy gap comparisons and the
725     force and torque vector magnitude comparisons are shown in table
726     \ref{tab:argon}. The force and torque vector directionality
727     results are displayed separately in table \ref{tab:argonAng}, where
728 gezelter 2658 the effect of group-based cutoffs and switching functions on the {\sc
729     sp} and {\sc sf} potentials are investigated.
730 chrisfen 2652
731 chrisfen 2599 \begin{table}[htbp]
732     \centering
733 chrisfen 2667 \caption{Regression results for the 6 \AA\ Argon sphere in liquid
734 gezelter 2658 water system. Tabulated results include $\Delta E$ values (top set),
735     force vector magnitudes (middle set) and torque vector magnitudes
736     (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted
737     Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where
738     $\varepsilon \approx \infty$).}
739 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
740     \\
741     \toprule
742     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
743     \cmidrule(lr){3-4}
744     \cmidrule(lr){5-6}
745     \cmidrule(l){7-8}
746     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
747     \midrule
748     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
749     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
750     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
751     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
752     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
753     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
754     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
755     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
756     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
757     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
758     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
759     \midrule
760     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
761     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
762     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
763     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
764     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
765     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
766     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
767     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
768     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
769     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
770     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
771     \midrule
772     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
773     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
774     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
775     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
776     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
777     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
778     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
779     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
780     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
781     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
782     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
783     \bottomrule
784     \end{tabular}
785 chrisfen 2652 \label{tab:argon}
786 chrisfen 2599 \end{table}
787    
788     \begin{table}[htbp]
789     \centering
790 gezelter 2658 \caption{Variance results from Gaussian fits to angular
791     distributions of the force and torque vectors in the 6 \AA\ sphere of
792 chrisfen 2667 Argon in liquid water system. PC = Pure Cutoff, SP = Shifted
793 gezelter 2658 Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF =
794     Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group
795     Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}
796 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
797     \\
798     \toprule
799     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
800     \cmidrule(lr){3-5}
801     \cmidrule(l){6-8}
802     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
803     \midrule
804     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
805     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
806     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
807     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
808     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
809     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
810     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
811     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
812     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
813     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
814     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
815     \midrule
816     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
817     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
818     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
819     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
820     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
821     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
822     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
823     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
824     \bottomrule
825     \end{tabular}
826 chrisfen 2652 \label{tab:argonAng}
827 chrisfen 2599 \end{table}
828    
829 chrisfen 2667 This system does not appear to show any significant deviations from
830     the previously observed results. The {\sc sp} and {\sc sf} methods
831     have aggrements similar to those observed in section
832 chrisfen 2666 \ref{app:water}. The only significant difference is the improvement
833 chrisfen 2667 in the configuration energy differences for the {\sc rf} method. This
834 chrisfen 2666 is surprising in that we are introducing an inhomogeneity to the
835     system; however, this inhomogeneity is charge-neutral and does not
836     result in charged cutoff spheres. The charge-neutrality of the cutoff
837     spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce,
838     seems to play a greater role in the stability of the {\sc rf} method
839     than the required homogeneity of the environment.
840 chrisfen 2660
841 chrisfen 2641 \newpage
842    
843     \bibliographystyle{jcp2}
844     \bibliography{electrostaticMethods}
845    
846 gezelter 2658 \end{document}