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1 chrisfen 3001 \chapter{\label{app:IndividualResults} INDIVIDUAL SYSTEM ANALYSIS RESULTS}
2 chrisfen 2987
3 chrisfen 3001 The combined system results in chapter \ref{chap:electrostatics}
4     (sections \ref{sec:EnergyResults} through \ref{sec:FTDirResults}) show
5     how the pairwise methods compare to the Ewald summation in the general
6 chrisfen 3029 sense over all of the system types investigated. It is also useful to
7     consider each of the studied systems in an individual fashion, so that
8     we can identify conditions that are particularly difficult for a
9     selected pairwise method to address. This allows us to further
10     establish the limitations of these pairwise techniques. In this
11     appendix, the energy difference, force vector, and torque vector
12     analyses are presented on an individual system basis.
13 chrisfen 3001
14     \section{SPC/E Water Results}\label{sec:WaterResults}
15    
16     The first system considered was liquid water at 300~K using the SPC/E
17     model of water.\cite{Berendsen87} The results for the energy gap
18     comparisons and the force and torque vector magnitude comparisons are
19     shown in table \ref{tab:spce}. The force and torque vector
20     directionality results are displayed separately in table
21     \ref{tab:spceAng}, where the effect of group-based cutoffs and
22     switching functions on the {\sc sp} and {\sc sf} potentials are also
23     investigated. In all of the individual results table, the method
24     abbreviations are as follows:
25    
26     \begin{itemize}[itemsep=0pt]
27     \item PC = Pure Cutoff,
28     \item SP = Shifted Potential,
29     \item SF = Shifted Force,
30     \item GSC = Group Switched Cutoff,
31     \item RF = Reaction Field (where $\varepsilon \approx\infty$),
32     \item GSSP = Group Switched Shifted Potential, and
33     \item GSSF = Group Switched Shifted Force.
34     \end{itemize}
35    
36     \begin{table}[htbp]
37     \centering
38     \caption{REGRESSION RESULTS OF THE LIQUID WATER SYSTEM FOR THE
39     $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it middle})
40     AND TORQUE VECTOR MAGNITUDES ({\it lower})}
41    
42     \footnotesize
43     \begin{tabular}{@{} ccrrrrrr @{}}
44     \toprule
45     \toprule
46     & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
47     \cmidrule(lr){3-4}
48     \cmidrule(lr){5-6}
49     \cmidrule(l){7-8}
50     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
51     \midrule
52     PC & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\
53     SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\
54     & 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\
55     & 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\
56     & 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\
57     SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\
58     & 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\
59     & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
60     & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
61     GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
62     RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\
63     \midrule
64     PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
65     SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
66     & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
67     & 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\
68     & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
69     SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\
70     & 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
71     & 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\
72     & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
73     GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
74     RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
75     \midrule
76     PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
77     SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
78     & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
79     & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
80     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
81     SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
82     & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
83     & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
84     & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
85     GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
86     RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
87     \bottomrule
88     \end{tabular}
89     \label{tab:spce}
90     \end{table}
91    
92     \begin{table}[htbp]
93     \centering
94     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
95     DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE LIQUID WATER
96     SYSTEM}
97    
98     \footnotesize
99     \begin{tabular}{@{} ccrrrrrr @{}}
100     \toprule
101     \toprule
102     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
103     \cmidrule(lr){3-5}
104     \cmidrule(l){6-8}
105     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
106     \midrule
107     PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
108     SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
109     & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
110     & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
111     & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
112     SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
113     & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
114     & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
115     & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
116     GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
117     RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\
118     \midrule
119     GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
120     & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
121     & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
122     & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
123     GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
124     & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
125     & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
126     & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
127     \bottomrule
128     \end{tabular}
129     \label{tab:spceAng}
130     \end{table}
131    
132     The water results parallel the combined results seen in sections
133     \ref{sec:EnergyResults} through \ref{sec:FTDirResults}. There is good
134     agreement with {\sc spme} in both energetic and dynamic behavior when
135     using the {\sc sf} method with and without damping. The {\sc sp}
136     method does well with an $\alpha$ around 0.2~\AA$^{-1}$, particularly
137     with cutoff radii greater than 12~\AA. Over-damping the electrostatics
138     reduces the agreement between both these methods and {\sc spme}.
139    
140     The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
141     observations from the combined results. In contrast to these results, however, the use of a switching function and group
142     based cutoffs greatly improves the results for these neutral water
143     molecules. The group switched cutoff ({\sc gsc}) does not mimic the
144     energetics of {\sc spme} as well as the {\sc sp} (with moderate
145     damping) and {\sc sf} methods, but the dynamics are quite good. The
146     switching functions correct discontinuities in the potential and
147     forces, leading to these improved results. Such improvements with the
148     use of a switching function have been recognized in previous
149     studies,\cite{Andrea83,Steinbach94} and this proves to be a useful
150     tactic for stably incorporating local area electrostatic effects.
151    
152     The reaction field ({\sc rf}) method simply extends upon the results
153     observed in the {\sc gsc} case. Both methods are similar in form
154     (i.e. neutral groups, switching function), but {\sc rf} incorporates
155     an added effect from the external dielectric. This similarity
156     translates into the same good dynamic results and improved energetic
157     agreement with {\sc spme}. Though this agreement is not to the level
158     of the moderately damped {\sc sp} and {\sc sf} methods, these results
159     show how incorporating some implicit properties of the surroundings
160     (i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction.
161    
162     As a final note for the liquid water system, use of group cutoffs and a
163     switching function leads to noticeable improvements in the {\sc sp}
164     and {\sc sf} methods, primarily in directionality of the force and
165     torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows
166     significant narrowing of the angle distribution when using little to
167     no damping and only modest improvement for the recommended conditions
168     ($\alpha = 0.2$~\AA$^{-1}$ and $R_\textrm{c}~\geqslant~12$~\AA). The
169     {\sc sf} method shows modest narrowing across all damping and cutoff
170     ranges of interest. When over-damping these methods, group cutoffs and
171     the switching function do not improve the force and torque
172     directionalities.
173    
174     \section{SPC/E Ice I$_\textrm{c}$ Results}\label{sec:IceResults}
175    
176     In addition to the disordered molecular system above, the ordered
177 chrisfen 3029 molecular system of ice I$_\textrm{c}$ was also considered. Any ice
178 chrisfen 3001 polymorph could have been used to fit this role; however, ice
179     I$_\textrm{c}$ was chosen because it can form an ideal periodic
180     lattice with the same number of water molecules used in the disordered
181 chrisfen 3029 liquid state system. The results for the energy gap comparisons and
182     the force and torque vector magnitude comparisons are shown in table
183 chrisfen 3001 \ref{tab:ice}. The force and torque vector directionality results are
184     displayed separately in table \ref{tab:iceAng}, where the effect of
185     group-based cutoffs and switching functions on the {\sc sp} and {\sc
186     sf} potentials are also displayed.
187    
188     \begin{table}[htbp]
189     \centering
190     \caption{REGRESSION RESULTS OF THE ICE I$_\textrm{c}$ SYSTEM FOR
191     $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it
192     middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
193    
194     \footnotesize
195     \begin{tabular}{@{} ccrrrrrr @{}}
196     \toprule
197     \toprule
198     & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
199     \cmidrule(lr){3-4}
200     \cmidrule(lr){5-6}
201     \cmidrule(l){7-8}
202     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
203     \midrule
204     PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
205     SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
206     & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
207     & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
208     & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
209     SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
210     & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
211     & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
212     & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
213     GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
214     RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
215     \midrule
216     PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
217     SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
218     & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
219     & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
220     & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
221     SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
222     & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
223     & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
224     & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
225     GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
226     RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
227     \midrule
228     PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
229     SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
230     & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
231     & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
232     & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
233     SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
234     & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
235     & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
236     & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
237     GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
238     RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
239     \bottomrule
240     \end{tabular}
241     \label{tab:ice}
242     \end{table}
243    
244     \begin{table}[htbp]
245     \centering
246     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS
247     OF THE FORCE AND TORQUE VECTORS IN THE ICE I$_\textrm{c}$ SYSTEM}
248    
249     \footnotesize
250     \begin{tabular}{@{} ccrrrrrr @{}}
251     \toprule
252     \toprule
253     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque
254     $\sigma^2$} \\
255     \cmidrule(lr){3-5}
256     \cmidrule(l){6-8}
257     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
258     \midrule
259     PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
260     SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
261     & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
262     & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
263     & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
264     SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
265     & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
266     & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
267     & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
268     GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
269     RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
270     \midrule
271     GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
272     & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
273     & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
274     & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
275     GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
276     & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
277     & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
278     & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
279     \bottomrule
280     \end{tabular}
281     \label{tab:iceAng}
282     \end{table}
283    
284     Highly ordered systems are a difficult test for the pairwise methods
285     in that they lack the implicit periodicity of the Ewald summation. As
286     expected, the energy gap agreement with {\sc spme} is reduced for the
287     {\sc sp} and {\sc sf} methods with parameters that were ideal for the
288     disordered liquid system. Moving to higher $R_\textrm{c}$ helps
289     improve the agreement, though at an increase in computational cost.
290     The dynamics of this crystalline system (both in magnitude and
291     direction) are little affected. Both methods still reproduce the Ewald
292     behavior with the same parameter recommendations from the previous
293     section.
294    
295     It is also worth noting that {\sc rf} exhibits improved energy gap
296     results over the liquid water system. One possible explanation is
297     that the ice I$_\textrm{c}$ crystal is ordered such that the net
298     dipole moment of the crystal is zero. With $\epsilon_\textrm{S} =
299     \infty$, the reaction field incorporates this structural organization
300     by actively enforcing a zeroed dipole moment within each cutoff
301     sphere.
302    
303     \section{NaCl Melt Results}\label{sec:SaltMeltResults}
304    
305     A high temperature NaCl melt was tested to gauge the accuracy of the
306     pairwise summation methods in a disordered system of charges. The
307     results for the energy gap comparisons and the force vector magnitude
308     comparisons are shown in table \ref{tab:melt}. The force vector
309     directionality results are displayed separately in table
310     \ref{tab:meltAng}.
311    
312     \begin{table}[htbp]
313     \centering
314     \caption{REGRESSION RESULTS OF THE MOLTEN SODIUM CHLORIDE SYSTEM FOR
315     $\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES ({\it
316     lower})}
317    
318     \footnotesize
319     \begin{tabular}{@{} ccrrrrrr @{}}
320     \toprule
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     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     & 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     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     & 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     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     & 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     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     & 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     \label{tab:melt}
350     \end{table}
351    
352     \begin{table}[htbp]
353     \centering
354     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS
355     OF THE FORCE VECTORS IN THE MOLTEN SODIUM CHLORIDE SYSTEM}
356    
357     \footnotesize
358     \begin{tabular}{@{} ccrrrrrr @{}}
359     \toprule
360     \toprule
361     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
362     \cmidrule(lr){3-5}
363     \cmidrule(l){6-8}
364     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA \\
365     \midrule
366     PC & & 13.294 & 8.035 & 5.366 \\
367     SP & 0.0 & 13.316 & 8.037 & 5.385 \\
368     & 0.1 & 5.705 & 1.391 & 0.360 \\
369     & 0.2 & 2.415 & 7.534 & 13.927 \\
370     & 0.3 & 23.769 & 67.306 & 57.252 \\
371     SF & 0.0 & 1.693 & 0.603 & 0.256 \\
372     & 0.1 & 1.687 & 0.653 & 0.272 \\
373     & 0.2 & 2.598 & 7.523 & 13.930 \\
374     & 0.3 & 23.734 & 67.305 & 57.252 \\
375     \bottomrule
376     \end{tabular}
377     \label{tab:meltAng}
378     \end{table}
379    
380     The molten NaCl system shows more sensitivity to the electrostatic
381     damping than the water systems. The most noticeable point is that the
382     undamped {\sc sf} method does very well at replicating the {\sc spme}
383     configurational energy differences and forces. Light damping appears
384     to minimally improve the dynamics, but this comes with a deterioration
385     of the energy gap results. In contrast, this light damping improves
386     the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
387     damping reduce the agreement with {\sc spme} for both methods. From
388     these observations, the undamped {\sc sf} method is the best choice
389     for disordered systems of charges.
390    
391     \section{NaCl Crystal Results}\label{sec:SaltCrystalResults}
392    
393     Similar to the use of ice I$_\textrm{c}$ to investigate the role of
394     order in molecular systems on the effectiveness of the pairwise
395 chrisfen 3029 methods, the 1000~K NaCl crystal system was sampled to investigate the
396 chrisfen 3001 accuracy of the pairwise summation methods in an ordered system of
397     charged particles. The results for the energy gap comparisons and the
398     force vector magnitude comparisons are shown in table \ref{tab:salt}.
399     The force vector directionality results are displayed separately in
400     table \ref{tab:saltAng}.
401    
402     \begin{table}[htbp]
403     \centering
404     \caption{REGRESSION RESULTS OF THE CRYSTALLINE SODIUM CHLORIDE
405     SYSTEM FOR $\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES
406     ({\it lower})}
407    
408     \footnotesize
409     \begin{tabular}{@{} ccrrrrrr @{}}
410     \toprule
411     \toprule
412     & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
413     \cmidrule(lr){3-4}
414     \cmidrule(lr){5-6}
415     \cmidrule(l){7-8}
416     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
417     \midrule
418     PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
419     SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
420     & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
421     & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
422     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
423     SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
424     & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
425     & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
426     & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
427     \midrule
428     PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
429     SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
430     & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
431     & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
432     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
433     SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
434     & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
435     & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
436     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
437     \bottomrule
438     \end{tabular}
439     \label{tab:salt}
440     \end{table}
441    
442     \begin{table}[htbp]
443     \centering
444     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
445     DISTRIBUTIONS OF THE FORCE VECTORS IN THE CRYSTALLINE SODIUM CHLORIDE
446     SYSTEM}
447    
448     \footnotesize
449     \begin{tabular}{@{} ccrrrrrr @{}}
450     \toprule
451     \toprule
452     & & \multicolumn{3}{c}{Force $\sigma^2$} \\
453     \cmidrule(lr){3-5}
454     \cmidrule(l){6-8}
455     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA \\
456     \midrule
457     PC & & 111.945 & 111.824 & 111.866 \\
458     SP & 0.0 & 112.414 & 152.215 & 38.087 \\
459     & 0.1 & 52.361 & 42.574 & 2.819 \\
460     & 0.2 & 10.847 & 9.709 & 9.686 \\
461     & 0.3 & 31.128 & 31.104 & 31.029 \\
462     SF & 0.0 & 10.025 & 3.555 & 1.648 \\
463     & 0.1 & 9.462 & 3.303 & 1.721 \\
464     & 0.2 & 11.454 & 9.813 & 9.701 \\
465     & 0.3 & 31.120 & 31.105 & 31.029 \\
466     \bottomrule
467     \end{tabular}
468     \label{tab:saltAng}
469     \end{table}
470    
471     The crystalline NaCl system is the most challenging test case for the
472     pairwise summation methods, as evidenced by the results in tables
473     \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
474     {\sc sf} methods seem to be the best choices. These methods match well
475     with {\sc spme} across the energy gap, force magnitude, and force
476     directionality tests. The {\sc sp} method struggles in all cases,
477     with the exception of good dynamics reproduction when using weak
478     electrostatic damping with a large cutoff radius.
479    
480     The moderate electrostatic damping case is not as good as we would
481     expect given the long-time dynamics results observed for this system
482 chrisfen 3023 (see section \ref{sec:LongTimeDynamics}). Since the data in tables
483     \ref{tab:salt} and \ref{tab:saltAng} are a test of instantaneous
484     dynamics, this indicates that good long-time dynamics comes in part at
485     the expense of short-time dynamics.
486 chrisfen 3001
487 chrisfen 3029 \section{0.11 M NaCl Solution Results}
488 chrisfen 3001
489     In an effort to bridge the charged atomic and neutral molecular
490 chrisfen 3029 systems, Na$^+$ and Cl$^-$ ions were incorporated into the liquid
491     water system. This low ionic strength system consists of 4 ions in the
492     1000 SPC/E water solvent ($\approx$0.11 M). The results for the energy
493     gap comparisons and the force and torque vector magnitude comparisons
494     are shown in table \ref{tab:solnWeak}. The force and torque vector
495     directionality results are displayed separately in table
496     \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and
497     switching functions on the {\sc sp} and {\sc sf} potentials are
498     investigated.
499 chrisfen 3001
500     \begin{table}[htbp]
501     \centering
502     \caption{REGRESSION RESULTS OF THE WEAK SODIUM CHLORIDE SOLUTION
503     SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES
504     ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
505    
506     \footnotesize
507     \begin{tabular}{@{} ccrrrrrr @{}}
508     \toprule
509     \toprule
510     & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
511     \cmidrule(lr){3-4}
512     \cmidrule(lr){5-6}
513     \cmidrule(l){7-8}
514     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
515     \midrule
516     PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
517     SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
518     & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
519     & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
520     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
521     SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
522     & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
523     & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
524     & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
525     GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
526     RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
527     \midrule
528     PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
529     SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
530     & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
531     & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
532     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
533     SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
534     & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
535     & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
536     & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
537     GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
538     RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
539     \midrule
540     PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
541     SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
542     & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
543     & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
544     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
545     SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
546     & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
547     & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
548     & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
549     GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
550     RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
551     \bottomrule
552     \end{tabular}
553     \label{tab:solnWeak}
554     \end{table}
555    
556     \begin{table}[htbp]
557     \centering
558     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
559     DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE WEAK SODIUM
560     CHLORIDE SOLUTION SYSTEM}
561    
562     \footnotesize
563     \begin{tabular}{@{} ccrrrrrr @{}}
564     \toprule
565     \toprule
566     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
567     \cmidrule(lr){3-5}
568     \cmidrule(l){6-8}
569     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
570     \midrule
571     PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
572     SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
573     & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
574     & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
575     & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
576     SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
577     & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
578     & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
579     & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
580     GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
581     RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
582     \midrule
583     GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
584     & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
585     & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
586     & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
587     GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
588     & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
589     & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
590     & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
591     \bottomrule
592     \end{tabular}
593     \label{tab:solnWeakAng}
594     \end{table}
595    
596     Because this system is a perturbation of the pure liquid water system,
597     comparisons are best drawn between these two sets. The {\sc sp} and
598     {\sc sf} methods are not significantly affected by the inclusion of a
599     few ions. The aspect of cutoff sphere neutralization aids in the
600     smooth incorporation of these ions; thus, all of the observations
601     regarding these methods carry over from section
602     \ref{sec:WaterResults}. The differences between these systems are more
603 chrisfen 3029 visible for the {\sc rf} method. Good force agreement is still
604     maintained; however, the energy gaps show a significant increase in
605     the scatter of the data.
606 chrisfen 3001
607 chrisfen 3029 \section{1.1 M NaCl Solution Results}
608 chrisfen 3001
609     The bridging of the charged atomic and neutral molecular systems was
610     further developed by considering a high ionic strength system
611 chrisfen 3029 consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
612 chrisfen 3001 M). The results for the energy gap comparisons and the force and
613     torque vector magnitude comparisons are shown in table
614     \ref{tab:solnStr}. The force and torque vector directionality
615     results are displayed separately in table \ref{tab:solnStrAng}, where
616     the effect of group-based cutoffs and switching functions on the {\sc
617     sp} and {\sc sf} potentials are investigated.
618    
619     \begin{table}[htbp]
620     \centering
621     \caption{REGRESSION RESULTS OF THE STRONG SODIUM CHLORIDE SOLUTION
622     SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES
623     ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
624    
625     \footnotesize
626     \begin{tabular}{@{} ccrrrrrr @{}}
627     \toprule
628     \toprule
629     & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
630     \cmidrule(lr){3-4}
631     \cmidrule(lr){5-6}
632     \cmidrule(l){7-8}
633     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
634     \midrule
635     PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
636     SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
637     & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
638     & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
639     & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
640     SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
641     & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
642     & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
643     & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
644     GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
645     RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
646     \midrule
647     PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
648     SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
649     & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
650     & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
651     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
652     SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
653     & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
654     & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
655     & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
656     GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
657     RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
658     \midrule
659     PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
660     SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
661     & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
662     & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
663     & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
664     SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
665     & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
666     & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
667     & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
668     GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
669     RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
670     \bottomrule
671     \end{tabular}
672     \label{tab:solnStr}
673     \end{table}
674    
675     \begin{table}[htbp]
676     \centering
677     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR DISTRIBUTIONS
678     OF THE FORCE AND TORQUE VECTORS IN THE STRONG SODIUM CHLORIDE SOLUTION
679     SYSTEM}
680    
681     \footnotesize
682     \begin{tabular}{@{} ccrrrrrr @{}}
683     \toprule
684     \toprule
685     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
686     \cmidrule(lr){3-5}
687     \cmidrule(l){6-8}
688     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
689     \midrule
690     PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
691     SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
692     & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
693     & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
694     & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
695     SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
696     & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
697     & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
698     & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
699     GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
700     RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
701     \midrule
702     GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
703     & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
704     & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
705     & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
706     GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
707     & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
708     & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
709     & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
710     \bottomrule
711     \end{tabular}
712     \label{tab:solnStrAng}
713     \end{table}
714    
715     The {\sc rf} method struggles with the jump in ionic strength. The
716     configuration energy differences degrade to unusable levels while the
717     forces and torques show a more modest reduction in the agreement with
718     {\sc spme}. The {\sc rf} method was designed for homogeneous systems,
719     and this attribute is apparent in these results.
720    
721     The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
722     their agreement with {\sc spme}. With these results, we still
723     recommend undamped to moderate damping for the {\sc sf} method and
724     moderate damping for the {\sc sp} method, both with cutoffs greater
725     than 12~\AA.
726    
727     \section{6~\AA\ Argon Sphere in SPC/E Water Results}
728    
729 chrisfen 3029 The final model system studied was a 6~\AA\ sphere of argon solvated
730 chrisfen 3001 by SPC/E water. This serves as a test case of a specifically sized
731     electrostatic defect in a disordered molecular system. The results for
732     the energy gap comparisons and the force and torque vector magnitude
733     comparisons are shown in table \ref{tab:argon}. The force and torque
734     vector directionality results are displayed separately in table
735     \ref{tab:argonAng}, where the effect of group-based cutoffs and
736     switching functions on the {\sc sp} and {\sc sf} potentials are
737     investigated.
738    
739     \begin{table}[htbp]
740     \centering
741     \caption{REGRESSION RESULTS OF THE 6~\AA\ ARGON SPHERE IN LIQUID
742     WATER SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR
743     MAGNITUDES ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
744    
745     \footnotesize
746     \begin{tabular}{@{} ccrrrrrr @{}}
747     \toprule
748     \toprule
749     & & \multicolumn{2}{c}{9~\AA} & \multicolumn{2}{c}{12~\AA} & \multicolumn{2}{c}{15~\AA}\\
750     \cmidrule(lr){3-4}
751     \cmidrule(lr){5-6}
752     \cmidrule(l){7-8}
753     Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
754     \midrule
755     PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
756     SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
757     & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
758     & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
759     & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
760     SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
761     & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
762     & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
763     & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
764     GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
765     RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
766     \midrule
767     PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
768     SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
769     & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
770     & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
771     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
772     SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
773     & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
774     & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
775     & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
776     GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
777     RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
778     \midrule
779     PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
780     SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
781     & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
782     & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
783     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
784     SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
785     & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
786     & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
787     & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
788     GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
789     RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
790     \bottomrule
791     \end{tabular}
792     \label{tab:argon}
793     \end{table}
794    
795     \begin{table}[htbp]
796     \centering
797     \caption{VARIANCE RESULTS FROM GAUSSIAN FITS TO ANGULAR
798     DISTRIBUTIONS OF THE FORCE AND TORQUE VECTORS IN THE 6~\AA\ SPHERE OF
799     ARGON IN LIQUID WATER SYSTEM}
800    
801     \footnotesize
802     \begin{tabular}{@{} ccrrrrrr @{}}
803     \toprule
804     \toprule
805     & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
806     \cmidrule(lr){3-5}
807     \cmidrule(l){6-8}
808     Method & $\alpha$ & 9~\AA & 12~\AA & 15~\AA & 9~\AA & 12~\AA & 15~\AA \\
809     \midrule
810     PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
811     SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
812     & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
813     & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
814     & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
815     SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
816     & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
817     & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
818     & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
819     GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
820     RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
821     \midrule
822     GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
823     & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
824     & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
825     & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
826     GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
827     & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
828     & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
829     & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
830     \bottomrule
831     \end{tabular}
832     \label{tab:argonAng}
833     \end{table}
834    
835     This system does not appear to show any significant deviations from
836     the previously observed results. The {\sc sp} and {\sc sf} methods
837     have agreements similar to those observed in section
838     \ref{sec:WaterResults}. The only significant difference is the
839     improvement in the configuration energy differences for the {\sc rf}
840     method. This is surprising in that we are introducing an inhomogeneity
841     to the system; however, this inhomogeneity is charge-neutral and does
842     not result in charged cutoff spheres. The charge-neutrality of the
843     cutoff spheres, which the {\sc sp} and {\sc sf} methods explicitly
844     enforce, seems to play a greater role in the stability of the {\sc rf}
845     method than the required homogeneity of the environment.