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Revision: 2660
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updated plot and supporting information

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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}
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 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 chrisfen 2660 \section{\label{app:water}Liquid Water}
33 chrisfen 2599
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 chrisfen 2660 RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\
70 chrisfen 2599 \midrule
71     PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
72     SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
73     & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
74     & 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\
75     & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
76     SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\
77     & 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
78     & 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\
79     & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
80     GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
81     RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
82     \midrule
83     PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
84     SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
85     & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
86     & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
87     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
88     SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
89     & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
90     & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
91     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
92     GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
93     RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
94     \bottomrule
95     \end{tabular}
96 chrisfen 2652 \label{tab:spce}
97 chrisfen 2599 \end{table}
98    
99     \begin{table}[htbp]
100     \centering
101 gezelter 2658 \caption{Variance results from Gaussian fits to angular
102     distributions of the force and torque vectors in the liquid water
103     system. PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
104     GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
105     \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
106     Group Switched Shifted Force.}
107 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
108     \\
109     \toprule
110     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
111     \cmidrule(lr){3-5}
112     \cmidrule(l){6-8}
113     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
114     \midrule
115     PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
116     SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
117     & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
118     & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
119     & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
120     SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
121     & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
122     & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
123     & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
124     GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
125     RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\
126     \midrule
127     GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
128     & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
129     & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
130     & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
131     GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
132     & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
133     & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
134     & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
135     \bottomrule
136     \end{tabular}
137 chrisfen 2642 \label{tab:spceAng}
138 chrisfen 2599 \end{table}
139    
140 gezelter 2658 For the most parts, the water results appear to parallel the combined
141     results seen in the discussion in the main paper. There is good
142     agreement with SPME in both energetic and dynamic behavior when using
143     the {\sc sf} method with and without damping. The {\sc sp} method does
144     well with an $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff
145     radii greater than 12 \AA. The results for both of these methods also
146     begin to decay as damping gets too large.
147 chrisfen 2642
148 gezelter 2658 The pure cutoff (PC) method performs poorly, as seen in the main
149     discussion section. In contrast to the combined values, however, the
150     use of a switching function and group based cutoffs really improves
151     the results for these neutral water molecules. The group switched
152     cutoff (GSC) shows mimics the energetics of SPME more poorly than the
153     {\sc sp} (with moderate damping) and {\sc sf} methods, but the
154     dynamics are quite good. The switching functions corrects
155     discontinuities in the potential and forces, leading to the improved
156     results. Such improvements with the use of a switching function has
157     been recognized in previous studies,\cite{Andrea83,Steinbach94} and it
158     is a useful tactic for stably incorporating local area electrostatic
159     effects.
160 chrisfen 2652
161 gezelter 2658 The reaction field (RF) method simply extends the results observed in
162     the GSC case. Both methods are similar in form (i.e. neutral groups,
163     switching function), but RF incorporates an added effect from the
164     external dielectric. This similarity translates into the same good
165     dynamic results and improved energetic results. These still fall
166     short of the moderately damped {\sc sp} and {\sc sf} methods, but they
167     display how incorporating some implicit properties of the surroundings
168     (i.e. $\epsilon_\textrm{S}$) can improve results.
169 chrisfen 2652
170 gezelter 2658 A final note for the liquid water system, use of group cutoffs and a
171     switching function also leads to noticeable improvements in the {\sc
172     sp} and {\sc sf} methods, primarily in directionality of the force and
173     torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant
174     narrowing of the angle distribution in the cases with little to no
175     damping and only modest improvement for the ideal conditions ($\alpha$
176     = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The {\sc sf}
177     method simply shows modest narrowing across all damping and cutoff
178     ranges of interest. Group cutoffs and the switching function do
179     nothing for cases were error is introduced by overdamping the
180     potentials.
181 chrisfen 2652
182 chrisfen 2660 \section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$}
183 chrisfen 2599
184 gezelter 2658 In addition to the disordered molecular system above, the ordered
185     molecular system of ice I$_\textrm{c}$ was also considered. The
186     results for the energy gap comparisons and the force and torque vector
187     magnitude comparisons are shown in table \ref{tab:ice}. The force and
188     torque vector directionality results are displayed separately in table
189     \ref{tab:iceAng}, where the effect of group-based cutoffs and
190     switching functions on the {\sc sp} and {\sc sf} potentials are
191     investigated.
192 chrisfen 2652
193 chrisfen 2599 \begin{table}[htbp]
194     \centering
195 gezelter 2658 \caption{Regression results for the ice I$_\textrm{c}$
196     system. Tabulated results include $\Delta E$ values (top set), force
197     vector magnitudes (middle set) and torque vector magnitudes (bottom
198     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
199     GSC = Group Switched Cutoff, and RF = Reaction Field (where
200     $\varepsilon \approx \infty$).}
201 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
202     \\
203     \toprule
204     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
205     \cmidrule(lr){3-4}
206     \cmidrule(lr){5-6}
207     \cmidrule(l){7-8}
208     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
209     \midrule
210     PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
211     SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
212     & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
213     & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
214     & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
215     SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
216     & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
217     & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
218     & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
219     GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
220     RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
221     \midrule
222     PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
223     SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
224     & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
225     & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
226     & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
227     SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
228     & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
229     & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
230     & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
231     GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
232     RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
233     \midrule
234     PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
235     SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
236     & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
237     & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
238     & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
239     SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
240     & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
241     & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
242     & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
243     GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
244     RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
245     \bottomrule
246     \end{tabular}
247 chrisfen 2652 \label{tab:ice}
248 chrisfen 2599 \end{table}
249    
250     \begin{table}[htbp]
251     \centering
252     \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.}
253     \begin{tabular}{@{} ccrrrrrr @{}}
254     \\
255     \toprule
256     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
257     \cmidrule(lr){3-5}
258     \cmidrule(l){6-8}
259     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
260     \midrule
261     PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
262     SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
263     & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
264     & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
265     & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
266     SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
267     & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
268     & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
269     & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
270     GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
271     RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
272     \midrule
273     GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
274     & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
275     & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
276     & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
277     GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
278     & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
279     & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
280     & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
281     \bottomrule
282     \end{tabular}
283 chrisfen 2652 \label{tab:iceAng}
284 chrisfen 2599 \end{table}
285    
286 gezelter 2658 Highly ordered systems are a difficult test for the pairwise systems
287     in that they lack the periodicity inherent to the Ewald summation. As
288     expected, the energy gap agreement with SPME reduces for the {\sc sp}
289     and {\sc sf} with parameters that were perfectly acceptable for the
290     disordered liquid system. Moving to higher $R_\textrm{c}$ remedies
291     this degraded performance, though at increase in computational cost.
292     However, the dynamics of this crystalline system (both in magnitude
293     and direction) are little affected. Both methods still reproduce the
294     Ewald behavior with the same parameter recommendations from the
295     previous section.
296 chrisfen 2652
297 gezelter 2658 It is also worth noting that RF exhibits a slightly improved energy
298     gap results over the liquid water system. One possible explanation is
299     that the ice I$_\textrm{c}$ crystal is ordered such that the net
300     dipole moment of the crystal is zero. With $\epsilon_\textrm{S} =
301     \infty$, the reaction field incorporates this structural organization
302     by actively enforcing a zeroed dipole moment within each cutoff
303     sphere.
304 chrisfen 2652
305 chrisfen 2660 \section{\label{app:melt}NaCl Melt}
306 chrisfen 2599
307 gezelter 2658 A high temperature NaCl melt was tested to gauge the accuracy of the
308     pairwise summation methods in a highly charge disordered system. The
309     results for the energy gap comparisons and the force and torque vector
310     magnitude comparisons are shown in table \ref{tab:melt}. The force
311     and torque vector directionality results are displayed separately in
312     table \ref{tab:meltAng}, where the effect of group-based cutoffs and
313     switching functions on the {\sc sp} and {\sc sf} potentials are
314     investigated.
315 chrisfen 2652
316 chrisfen 2599 \begin{table}[htbp]
317     \centering
318     \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.}
319     \begin{tabular}{@{} ccrrrrrr @{}}
320     \\
321     \toprule
322     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
323     \cmidrule(lr){3-4}
324     \cmidrule(lr){5-6}
325     \cmidrule(l){7-8}
326     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
327     \midrule
328     PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
329 chrisfen 2655 SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\
330     & 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\
331 chrisfen 2599 & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
332     & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
333 chrisfen 2655 SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\
334     & 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\
335 chrisfen 2599 & 0.2 & 0.966 & 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     \midrule
338     PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
339 chrisfen 2655 SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\
340     & 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\
341 chrisfen 2599 & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
342     & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
343 chrisfen 2655 SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\
344     & 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\
345 chrisfen 2599 & 0.2 & 0.994 & 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     \bottomrule
348     \end{tabular}
349 chrisfen 2652 \label{tab:melt}
350 chrisfen 2599 \end{table}
351    
352     \begin{table}[htbp]
353     \centering
354     \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.}
355     \begin{tabular}{@{} ccrrrrrr @{}}
356     \\
357     \toprule
358     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
359     \cmidrule(lr){3-5}
360     \cmidrule(l){6-8}
361     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
362     \midrule
363     PC & & 13.294 & 8.035 & 5.366 \\
364     SP & 0.0 & 13.316 & 8.037 & 5.385 \\
365     & 0.1 & 5.705 & 1.391 & 0.360 \\
366     & 0.2 & 2.415 & 7.534 & 13.927 \\
367     & 0.3 & 23.769 & 67.306 & 57.252 \\
368     SF & 0.0 & 1.693 & 0.603 & 0.256 \\
369     & 0.1 & 1.687 & 0.653 & 0.272 \\
370     & 0.2 & 2.598 & 7.523 & 13.930 \\
371     & 0.3 & 23.734 & 67.305 & 57.252 \\
372     \bottomrule
373     \end{tabular}
374 chrisfen 2652 \label{tab:meltAng}
375 chrisfen 2599 \end{table}
376    
377 chrisfen 2660 The molten NaCl system shows more sensitivity to the electrostatic
378     damping than the water systems. The most noticeable point is that the
379     undamped {\sc sf} method does very well at replicating the {\sc spme}
380     configurational energy differences and forces. Light damping appears
381     to minimally improve the dynamics, but this comes with a deterioration
382     of the energy gap results. In contrast, this light damping improves
383     the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
384     damping reduce the agreement with {\sc spme} for both methods. From
385     these observations, the undamped {\sc sf} method is the best choice
386     for disordered systems of charges.
387 chrisfen 2654
388 chrisfen 2660 \section{\label{app:salt}NaCl Crystal}
389 chrisfen 2599
390 gezelter 2658 A 1000K NaCl crystal was used to investigate the accuracy of the
391     pairwise summation methods in an ordered system of charged
392     particles. The results for the energy gap comparisons and the force
393     and torque vector magnitude comparisons are shown in table
394     \ref{tab:salt}. The force and torque vector directionality results
395     are displayed separately in table \ref{tab:saltAng}, where the effect
396     of group-based cutoffs and switching functions on the {\sc sp} and
397     {\sc sf} potentials are investigated.
398 chrisfen 2652
399 chrisfen 2599 \begin{table}[htbp]
400     \centering
401 gezelter 2658 \caption{Regression results for the crystalline NaCl
402     system. Tabulated results include $\Delta E$ values (top set) and
403     force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted
404     Potential, and SF = Shifted Force.}
405 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
406     \\
407     \toprule
408     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
409     \cmidrule(lr){3-4}
410     \cmidrule(lr){5-6}
411     \cmidrule(l){7-8}
412     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
413     \midrule
414     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
415     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
416     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
417     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
418     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
419     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
420     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
421     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
422     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
423     \midrule
424     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
425     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
426     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
427     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
428     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
429     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
430     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
431     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
432     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
433     \bottomrule
434     \end{tabular}
435 chrisfen 2652 \label{tab:salt}
436 chrisfen 2599 \end{table}
437    
438     \begin{table}[htbp]
439     \centering
440 gezelter 2658 \caption{Variance results from Gaussian fits to angular
441     distributions of the force vectors in the crystalline NaCl system. PC
442     = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
443     Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
444     \infty$).}
445 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
446     \\
447     \toprule
448     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
449     \cmidrule(lr){3-5}
450     \cmidrule(l){6-8}
451     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
452     \midrule
453     PC & & 111.945 & 111.824 & 111.866 \\
454     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
455     & 0.1 & 52.361 & 42.574 & 2.819 \\
456     & 0.2 & 10.847 & 9.709 & 9.686 \\
457     & 0.3 & 31.128 & 31.104 & 31.029 \\
458     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
459     & 0.1 & 9.462 & 3.303 & 1.721 \\
460     & 0.2 & 11.454 & 9.813 & 9.701 \\
461     & 0.3 & 31.120 & 31.105 & 31.029 \\
462     \bottomrule
463     \end{tabular}
464 chrisfen 2652 \label{tab:saltAng}
465 chrisfen 2599 \end{table}
466    
467 chrisfen 2660 The crystalline NaCl system is the most challenging test case for the
468     pairwise summation methods, as evidenced by the results in tables
469     \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
470     {\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best
471     choices. These methods match well with {\sc spme} across the energy
472     gap, force magnitude, and force directionality tests. The {\sc sp}
473     method struggles in all cases with the exception of good dynamics
474     reproduction when using weak electrostatic damping with a large cutoff
475     radius.
476 chrisfen 2599
477 chrisfen 2660 The moderate electrostatic damping case is not as good as we would
478     expect given the good long-time dynamics results observed for this
479     system. Since these results are a test of instantaneous dynamics, this
480     indicates that good long-time dynamics comes in part at the expense of
481     short-time dynamics. Further indication of this comes from the full
482     power spectra shown in the main text. It appears as though a
483     distortion is introduced between 200 to 300 cm$^{-1}$ with increased
484     $\alpha$.
485    
486     \section{\label{app:solnWeak}Weak NaCl Solution}
487    
488 gezelter 2658 In an effort to bridge the charged atomic and neutral molecular
489     systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
490     the liquid water system. This low ionic strength system consists of 4
491     ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
492     for the energy gap comparisons and the force and torque vector
493     magnitude comparisons are shown in table \ref{tab:solnWeak}. The
494     force and torque vector directionality results are displayed
495     separately in table \ref{tab:solnWeakAng}, where the effect of
496     group-based cutoffs and switching functions on the {\sc sp} and {\sc
497     sf} potentials are investigated.
498 chrisfen 2652
499 chrisfen 2599 \begin{table}[htbp]
500     \centering
501 gezelter 2658 \caption{Regression results for the weak NaCl solution
502     system. Tabulated results include $\Delta E$ values (top set), force
503     vector magnitudes (middle set) and torque vector magnitudes (bottom
504     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
505     GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
506     \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
507     Group Switched Shifted Force.}
508 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
509     \\
510     \toprule
511     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
512     \cmidrule(lr){3-4}
513     \cmidrule(lr){5-6}
514     \cmidrule(l){7-8}
515     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
516     \midrule
517     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
518     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
519     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
520     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
521     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
522     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
523     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
524     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
525     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
526     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
527     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
528     \midrule
529     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
530     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
531     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
532     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
533     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
534     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
535     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
536     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
537     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
538     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
539     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
540     \midrule
541     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
542     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
543     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
544     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
545     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
546     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
547     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
548     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
549     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
550     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
551     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
552     \bottomrule
553     \end{tabular}
554 chrisfen 2652 \label{tab:solnWeak}
555 chrisfen 2599 \end{table}
556    
557     \begin{table}[htbp]
558     \centering
559 gezelter 2658 \caption{Variance results from Gaussian fits to angular
560     distributions of the force and torque vectors in the weak NaCl
561     solution system. PC = Pure Cutoff, SP = Shifted Potential, SF =
562     Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where
563     $\varepsilon \approx \infty$), GSSP = Group Switched Shifted
564     Potential, and GSSF = Group Switched Shifted Force.}
565 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
566     \\
567     \toprule
568     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
569     \cmidrule(lr){3-5}
570     \cmidrule(l){6-8}
571     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
572     \midrule
573     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
574     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
575     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
576     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
577     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
578     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
579     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
580     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
581     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
582     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
583     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
584     \midrule
585     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
586     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
587     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
588     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
589     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
590     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
591     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
592     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
593     \bottomrule
594     \end{tabular}
595 chrisfen 2652 \label{tab:solnWeakAng}
596 chrisfen 2599 \end{table}
597    
598 chrisfen 2660 This weak ionic strength system can be considered as a perturbation of
599     the pure liquid water system. The {\sc sp} and {\sc sf} methods are
600     not significantly affected by the inclusion of a few ions. The aspect
601     of cutoff sphere neutralization aids in the smooth incorporation of
602     these ions; thus, all of the observations regarding these methods
603     carry over from section \ref{app:water}. The differences between these
604     systems are visible for the {\sc rf} method. Though good force
605     reproduction is still maintained, the energy gaps show a significant
606     increase in the data scatter. This foreshadows the breakdown of the
607     method as we introduce system inhomogeneities.
608 chrisfen 2599
609 chrisfen 2660 \section{\label{app:solnStr}Strong NaCl Solution}
610    
611 gezelter 2658 The bridging of the charged atomic and neutral molecular systems was
612 chrisfen 2660 further developed by considering a high ionic strength system
613     consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
614     M). The results for the energy gap comparisons and the force and
615     torque vector magnitude comparisons are shown in table
616     \ref{tab:solnWeak}. The force and torque vector directionality
617     results are displayed separately in table\ref{tab:solnWeakAng}, where
618     the effect of group-based cutoffs and switching functions on the {\sc
619     sp} and {\sc sf} potentials are investigated.
620 chrisfen 2652
621 chrisfen 2599 \begin{table}[htbp]
622     \centering
623 gezelter 2658 \caption{Regression results for the strong NaCl solution
624     system. Tabulated results include $\Delta E$ values (top set), force
625     vector magnitudes (middle set) and torque vector magnitudes (bottom
626     set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
627     GSC = Group Switched Cutoff, and RF = Reaction Field (where
628     $\varepsilon \approx \infty$).}
629 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
630     \\
631     \toprule
632     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
633     \cmidrule(lr){3-4}
634     \cmidrule(lr){5-6}
635     \cmidrule(l){7-8}
636     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
637     \midrule
638     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
639     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
640     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
641     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
642     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
643     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
644     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
645     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
646     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
647     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
648     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
649     \midrule
650     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
651     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
652     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
653     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
654     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
655     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
656     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
657     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
658     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
659     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
660     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
661     \midrule
662     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
663     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
664     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
665     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
666     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
667     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
668     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
669     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
670     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
671     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
672     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
673     \bottomrule
674     \end{tabular}
675 chrisfen 2652 \label{tab:solnStr}
676 chrisfen 2599 \end{table}
677    
678     \begin{table}[htbp]
679     \centering
680     \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.}
681     \begin{tabular}{@{} ccrrrrrr @{}}
682     \\
683     \toprule
684     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
685     \cmidrule(lr){3-5}
686     \cmidrule(l){6-8}
687     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
688     \midrule
689     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
690     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
691     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
692     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
693     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
694     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
695     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
696     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
697     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
698     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
699     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
700     \midrule
701     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
702     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
703     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
704     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
705     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
706     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
707     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
708     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
709     \bottomrule
710     \end{tabular}
711 chrisfen 2652 \label{tab:solnStrAng}
712 chrisfen 2599 \end{table}
713    
714 chrisfen 2660 The {\sc rf} method struggles with the jump in ionic strength. The
715     configuration energy difference degrade to unuseable levels while the
716     forces and torques degrade in a more modest fashion. The {\sc rf}
717     method was designed for homogeneous systems, and this restriction is
718     apparent in these results.
719 chrisfen 2599
720 chrisfen 2660 The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
721     their agreement with {\sc spme}. With these results, we still
722     recommend no to moderate damping for the {\sc sf} method and moderate
723     damping for the {\sc sp} method, both with cutoffs greater than 12
724     \AA.
725    
726     \section{\label{app:argon}Argon Sphere in Water}
727    
728 gezelter 2658 The final model system studied was 6 \AA\ sphere of Argon solvated by
729     SPC/E water. The results for the energy gap comparisons and the force
730     and torque vector magnitude comparisons are shown in table
731     \ref{tab:solnWeak}. The force and torque vector directionality
732     results are displayed separately in table \ref{tab:solnWeakAng}, where
733     the effect of group-based cutoffs and switching functions on the {\sc
734     sp} and {\sc sf} potentials are investigated.
735 chrisfen 2652
736 chrisfen 2599 \begin{table}[htbp]
737     \centering
738 gezelter 2658 \caption{Regression results for the 6 \AA\ argon sphere in liquid
739     water system. Tabulated results include $\Delta E$ values (top set),
740     force vector magnitudes (middle set) and torque vector magnitudes
741     (bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted
742     Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where
743     $\varepsilon \approx \infty$).}
744 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
745     \\
746     \toprule
747     & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
748     \cmidrule(lr){3-4}
749     \cmidrule(lr){5-6}
750     \cmidrule(l){7-8}
751     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
752     \midrule
753     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
754     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
755     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
756     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
757     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
758     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
759     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
760     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
761     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
762     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
763     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
764     \midrule
765     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
766     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
767     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
768     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
769     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
770     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
771     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
772     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
773     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
774     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
775     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
776     \midrule
777     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
778     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
779     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
780     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
781     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
782     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
783     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
784     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
785     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
786     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
787     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
788     \bottomrule
789     \end{tabular}
790 chrisfen 2652 \label{tab:argon}
791 chrisfen 2599 \end{table}
792    
793     \begin{table}[htbp]
794     \centering
795 gezelter 2658 \caption{Variance results from Gaussian fits to angular
796     distributions of the force and torque vectors in the 6 \AA\ sphere of
797     argon in liquid water system. PC = Pure Cutoff, SP = Shifted
798     Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF =
799     Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group
800     Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}
801 chrisfen 2599 \begin{tabular}{@{} ccrrrrrr @{}}
802     \\
803     \toprule
804     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
805     \cmidrule(lr){3-5}
806     \cmidrule(l){6-8}
807     Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
808     \midrule
809     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
810     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
811     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
812     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
813     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
814     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
815     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
816     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
817     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
818     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
819     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
820     \midrule
821     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
822     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
823     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
824     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
825     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
826     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
827     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
828     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
829     \bottomrule
830     \end{tabular}
831 chrisfen 2652 \label{tab:argonAng}
832 chrisfen 2599 \end{table}
833    
834 chrisfen 2660 This system appears not to show in any significant deviation in the previously observed results. The {\sc sp} and {\sc sf} methods give result qualities similar to those observed in section \ref{app:water}. The only significant difference is the improvement for the configuration energy differences for the {\sc rf} method. This is surprising in that we are introducing an inhomogeneity to the system; however, this inhomogeneity is charge-neutral and does not result in charged cutoff spheres. The charge-neutrality, which the {\sc sp} and {\sc sf} methods explicity enforce, seems to play a greater role in the stability of the {\sc rf} method than the necessity of a homogeneous environment.
835    
836 chrisfen 2641 \newpage
837    
838     \bibliographystyle{jcp2}
839     \bibliography{electrostaticMethods}
840    
841 gezelter 2658 \end{document}