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# Line 9 | Line 9
9   \usepackage{setspace}
10   \usepackage{tabularx}
11   \usepackage{graphicx}
12 + \usepackage{booktabs}
13   %\usepackage{berkeley}
14   \usepackage[ref]{overcite}
15   \pagestyle{plain}
# Line 22 | Line 23
23  
24   \begin{document}
25  
26 < \title{On the necessity of the Ewald Summation in molecular simulations.}
26 > \title{On the necessity of the Ewald Summation in molecular simulations: Alternatives to the accepted standard of cutoff policies}
27  
28   \author{Christopher J. Fennell and J. Daniel Gezelter \\
29   Department of Chemistry and Biochemistry\\
# Line 39 | Line 40 | Notre Dame, Indiana 46556}
40  
41   %\narrowtext
42  
43 < %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
43 > %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
44   %                              BODY OF TEXT
45 < %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
45 > %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
46  
47   \section{Introduction}
48  
# Line 49 | Line 50 | In this paper, a variety of simulation situations were
50   \begin{list}{-}{}
51   \item Liquid Water
52   \item Crystalline Water (Ice I$_\textrm{c}$)
53 + \item NaCl Crystal
54 + \item NaCl Melt
55   \item 1 M Solution of NaCl in Water
56   \item 10 M Solution of NaCl in Water
57   \item 6 \AA\  Radius Sphere of Argon in Water
55 – \item NaCl Crystal
56 – \item NaCl Melt
58   \end{list}
59   Additional discussion on the results from the individual systems was also performed to identify limitations of the considered methods in specific systems.
60  
# Line 74 | Line 75 | In order to evaluate the performance of the adapted Wo
75  
76   \section{Results and Discussion}
77  
78 < In order to evaluate the performance of the adapted Wolf Shifted Potential and Shifted Force electrostatic summation methods for Monte Carlo simulations, the energy differences between configurations need to be compared to the results using SPME.  Considering the SPME results to be the correct or desired behavior, ideal performance of a tested method is taken to be agreement between the energy differences calculated.  Linear least squares regression of the $\Delta$E values between configurations using SPME against $\Delta$E values using tested methods provides a quantitative comparison of this agreement.  Unitary results for both the correlation and correlation coefficient for these regressions indicate equivalent energetic results between the methods.  The correlation is the slope of the plotted data while the correlation coefficient ($R^2$) is a measure of the of the data scatter around the fitted line and gives an idea of the quality of the fit (Fig. \ref{linearFit}).
78 > In order to evaluate the performance of the adapted Wolf Shifted Potential and Shifted Force electrostatic summation methods for Monte Carlo simulations, the energy differences between configurations need to be compared to the results using SPME.  Considering the SPME results to be the correct or desired behavior, ideal performance of a tested method is taken to be agreement between the energy differences calculated.  Linear least squares regression of the $\Delta E$ values between configurations using SPME against $\Delta E$ values using tested methods provides a quantitative comparison of this agreement.  Unitary results for both the correlation and correlation coefficient for these regressions indicate equivalent energetic results between the methods.  The correlation is the slope of the plotted data while the correlation coefficient ($R^2$) is a measure of the of the data scatter around the fitted line and gives an idea of the quality of the fit (Fig. \ref{linearFit}).
79  
80   \begin{figure}
81   \centering
82   \includegraphics[width=3.25in]{./linearFit.pdf}
83 < \caption{Example least squares regression of the $\Delta$E between configurations for the SF method against SPME in the pure water system.  }
83 > \caption{Example least squares regression of the $\Delta E$ between configurations for the SF method against SPME in the pure water system.  }
84   \label{linearFit}
85   \end{figure}
86  
87 < With 500 independent configurations, 124,750 $\Delta$E data points are used in a regression of a single system.  Results and discussion for the individual analysis of each of the system types appear in the appendices of this paper.  To probe the applicability of each method in the general case, all the different system types were included in a single regression.  The results for this regression are shown in figure \ref{delE}.  
87 > With 500 independent configurations, 124,750 $\Delta E$ data points are used in a regression of a single system.  Results and discussion for the individual analysis of each of the system types appear in the appendices of this paper.  To probe the applicability of each method in the general case, all the different system types were included in a single regression.  The results for this regression are shown in figure \ref{delE}.  
88  
89   \begin{figure}
90   \centering
91   \includegraphics[width=3.25in]{./delEplot.pdf}
92 < \caption{The results from the statistical analysis of the $\Delta$E results for all the system types at 9 \AA\ (${\bullet}$), 12 \AA\ ($\blacksquare$), and 15 \AA\ ($\blacktriangledown$) cutoff radii.  Results close to a value of 1 (dashed line) indicate $\Delta$E values from that particular method (listed on the left) are nearly indistinguishable from those obtained from SPME.  Reaction Field results do not include NaCl crystal or melt configurations.}
92 > \caption{The results from the statistical analysis of the $\Delta$E results for all the system types at 9 \AA\ (${\bullet}$), 12 \AA\ ($\blacksquare$), and 15 \AA\ ($\blacktriangledown$) cutoff radii.  Results close to a value of 1 (dashed line) indicate $\Delta E$ values from that particular method (listed on the left) are nearly indistinguishable from those obtained from SPME.  Reaction Field results do not include NaCl crystal or melt configurations.}
93   \label{delE}
94   \end{figure}
95  
96   In figure \ref{delE}, it is readily apparent that it is unreasonable to expect realistic results using an unmodified cutoff.  This is not all that surprising since this results in large energy fluctuations as atoms move in and out of the cutoff radius.  These fluctuations can be alleviated to some degree by using group based cutoffs with a switching function.  The Group Switch Cutoff row doesn't show a significant improvement in this plot because the salt and salt solution systems contain non-neutral groups, see appendices \ref{app-water} and \ref{app-ice} for a comparison where all groups are neutral.  Correcting the resulting charged cutoff sphere is one of the purposes of the shifted potential proposed by Wolf \textit{et al.}, and this correction indeed improves the results as seen in the Shifted Potental rows.  While the undamped case of this method is a significant improvement over the pure cutoff, it still doesn't correlate that well with SPME.  Inclusion of potential damping improves the results, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows an excellent correlation and quality of fit with the SPME results, particularly with a cutoff radius greater than 12 \AA .  Use of a larger damping parameter is more helpful for the shortest cutoff shown, but it has a detrimental effect on simulations with larger cutoffs.  This trend is repeated in the Shifted Force rows, where increasing damping results in progressively poorer correlation; however, damping looks to be unnecessary with this method.  Overall, the undamped case is the best performing set, as the correlation and quality of fits are consistently superior regardless of the cutoff distance.  This result is beneficial in that the undamped case is less computationally prohibitive do to the lack of complimentary error function calculation when performing the electrostatic pair interaction.  The reaction field results illustrates some of that method's limitations, primarily that it was developed for use in homogenous systems; although it does provide results that are an improvement over those from an unmodified cutoff.
97  
98 < While studying the energy differences provides insight into how comparable these methods are energetically, if we want to use these methods in Molecular Dynamics simulations, we also need to consider their effect on forces and torques.  Both the magnitude and the direction of the force and torque vectors of each of the bodies in the system can be compared to those observed while using SPME.  Analysis of the magnitude of these vectors can be performed in the manner described previously for comparing $\Delta$E values, only instead of a single value between two system configurations, there is a value for each particle in each configuration.  For a system of 1000 water molecules and 40 ions, there are 1040 force vectors and 1000 torque vectors.  With 500 configurations, this results in excess of 500,000 data samples for each system type.  Figures \ref{frcMag} and \ref{trqMag} respectively show the force and torque vector magnitude results for the accumulated analysis over all the system types.
98 > While studying the energy differences provides insight into how comparable these methods are energetically, if we want to use these methods in Molecular Dynamics simulations, we also need to consider their effect on forces and torques.  Both the magnitude and the direction of the force and torque vectors of each of the bodies in the system can be compared to those observed while using SPME.  Analysis of the magnitude of these vectors can be performed in the manner described previously for comparing $\Delta E$ values, only instead of a single value between two system configurations, there is a value for each particle in each configuration.  For a system of 1000 water molecules and 40 ions, there are 1040 force vectors and 1000 torque vectors.  With 500 configurations, this results in excess of 500,000 data samples for each system type.  Figures \ref{frcMag} and \ref{trqMag} respectively show the force and torque vector magnitude results for the accumulated analysis over all the system types.
99  
100   \begin{figure}
101   \centering
# Line 103 | Line 104 | The results in figure \ref{frcMag} for the most part p
104   \label{frcMag}
105   \end{figure}
106  
107 < The results in figure \ref{frcMag} for the most part parallel those seen in the previous look at the $\Delta$E results.  The unmodified cutoff results are poor, but using group based cutoffs and a switching function provides a improvement much more significant than what was seen with $\Delta$E.  Looking at the Shifted Potential sets, the slope and R$^2$ improve with the use of damping to an optimal result of 0.2 \AA $^{-1}$ for the 12 and 15 \AA\ cutoffs.  Further increases in damping, while beneficial for simulations with a cutoff radius of 9 \AA\ , is detrimental to simulations with larger cutoff radii.  The undamped Shifted Force method gives forces in line with those obtained using SPME, and use of a damping function gives little to no gain.  The reaction field results are surprisingly good, considering the poor quality of the fits for the $\Delta$E results.  There is still a considerable degree of scatter in the data, but it correlates well in general.
107 > The results in figure \ref{frcMag} for the most part parallel those seen in the previous look at the $\Delta E$ results.  The unmodified cutoff results are poor, but using group based cutoffs and a switching function provides a improvement much more significant than what was seen with $\Delta E$.  Looking at the Shifted Potential sets, the slope and R$^2$ improve with the use of damping to an optimal result of 0.2 \AA $^{-1}$ for the 12 and 15 \AA\ cutoffs.  Further increases in damping, while beneficial for simulations with a cutoff radius of 9 \AA\ , is detrimental to simulations with larger cutoff radii.  The undamped Shifted Force method gives forces in line with those obtained using SPME, and use of a damping function gives little to no gain.  The reaction field results are surprisingly good, considering the poor quality of the fits for the $\Delta E$ results.  There is still a considerable degree of scatter in the data, but it correlates well in general.
108  
109   \begin{figure}
110   \centering
# Line 114 | Line 115 | Having force and torque vectors with magnitudes that a
115  
116   The torque vector magnitude results in figure \ref{trqMag} are similar to those seen for the forces, but more clearly show the improved behavior with increasing cutoff radius.  Moderate damping is beneficial to the Shifted Potential and unnecessary with the Shifted Force method, and they also show that over-damping adversely effects all cutoff radii rather than showing an improvement for systems with short cutoffs.  The reaction field method performs well when calculating the torques, better than the Shifted Force method over this limited data set.
117  
118 < Having force and torque vectors with magnitudes that are well correlated to SPME is good, but if they are not pointing in the proper direction the results will be incorrect.
118 > Having force and torque vectors with magnitudes that are well correlated to SPME is good, but if they are not pointing in the proper direction the results will be incorrect.  These vector directions were investigated through measurement of the angle formed between them and those from SPME.  The dot product of these unit vectors provides a theta value that is accumulated in a distribution function, weighted by the area on the unit sphere.  Narrow distributions of theta values indicates similar to identical results between the tested method and SPME.  To measure the narrowness of the resulting distributions, non-linear Gaussian fits were performed.
119  
120   \begin{figure}
121   \centering
# Line 123 | Line 124 | Having force and torque vectors with magnitudes that a
124   \label{gaussian}
125   \end{figure}
126  
127 + Figure \ref{gaussian} shows an example distribution and the non-linear fit applied.  The solid line is a Gaussian profile, while the dotted line is a Voigt profile, a convolution of a Gaussian and a Lorentzian profile.  Since this distribution is a measure of angular error between two different electrostatic summation methods, there is particular reason for it to adhere to a particular shape.  Because of this and the Gaussian profile's more statistically meaningful properties, Gaussian fitting was used to compare all the methods considered in this study.  The results (Fig. \ref{frcTrqAng}) are compared through the variance ($\sigma^2$) of these non-linear fits.  
128  
127 –
129   \begin{figure}
130   \centering
131   \includegraphics[width=3.25in]{./frcTrqAngplot.pdf}
# Line 132 | Line 133 | Having force and torque vectors with magnitudes that a
133   \label{frcTrqAng}
134   \end{figure}
135  
136 + Both the force and torque $\sigma^2$ results from the analysis of the total accumulated system data are tabulated in figure \ref{frcTrqAng}.  All of the sets, aside from the over-damped case show the improvement afforded by choosing a longer simulation cutoff.  Increasing the cutoff from 9 to 12 \AA\ typically results in a halving of $\sigma^2$, with a similar improvement going from 12 to 15 \AA .  The undamped Shifted Force, Group Based Cutoff, and Reaction Field methods all do equivalently well at capturing the direction of both the force and torque vectors.  Using damping improves the angular behavior significantly for the Shifted Potential and moderately for the Shifted Force methods.  Increasing the damping too far is destructive for both methods, particularly to the torque vectors.  Again it is important to recognize that the force vectors cover all particles in the systems, while torque vectors are only available for neutral molecular groups.  Damping appears to have a more beneficial non-neutral bodies, and this observation is investigated further in appendices \ref{app-melt}, \ref{app-salt}, \ref{app-sol1}, and \ref{app-sol10}.  
137  
138 + \begin{table}[htbp]
139 +   \centering
140 +   \caption{Variance ($\sigma^2$) of the force (top set) and torque (bottom set) vector angle difference distributions for the Shifted Potential and Shifted Force methods.  Calculations were performed both with (Y) and without (N) group based cutoffs and a switching function.  The $\alpha$ values have units of \AA$^{-1}$ and the variance values have units of degrees$^2$.}  
141 +   \begin{tabular}{@{} ccrrrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
142 +      \\
143 +      \toprule
144 +      & & \multicolumn{4}{c}{Shifted Potential} & \multicolumn{4}{c}{Shifted Force} \\
145 +      \cmidrule(lr){3-6}
146 +      \cmidrule(l){7-10}
147 +            Cutoff Radius    & Groups & $\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$ & $\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$\\
148 +      \midrule
149 +      
150 +        9 \AA   & N & 29.545 & 12.003 & 5.489 & 0.610 & 2.323 & 2.321 & 0.429 & 0.603 \\
151 +                & \textbf{Y} & \textbf{2.486} & \textbf{2.160} & \textbf{0.667} & \textbf{0.608} & \textbf{1.768} & \textbf{1.766} & \textbf{0.676} & \textbf{0.609} \\
152 +        12 \AA  & N & 19.381 & 3.097 & 0.190 & 0.608 & 0.920 & 0.736 & 0.133 & 0.612 \\
153 +                & \textbf{Y} & \textbf{0.515} & \textbf{0.288} & \textbf{0.127} & \textbf{0.586} & \textbf{0.308} & \textbf{0.249} & \textbf{0.127} & \textbf{0.586} \\
154 +        15 \AA  & N & 12.700 & 1.196 & 0.123 & 0.601 & 0.339 & 0.160 & 0.123 & 0.601 \\
155 +                & \textbf{Y} & \textbf{0.228} & \textbf{0.099} & \textbf{0.121} & \textbf{0.598} & \textbf{0.144} & \textbf{0.090} & \textbf{0.121} & \textbf{0.598} \\      
156  
157 +      \midrule
158 +      
159 +        9 \AA   & N & 262.716 & 116.585 & 5.234 & 5.103 & 2.392 & 2.350 & 1.770 & 5.122 \\
160 +                & \textbf{Y} & \textbf{2.115} & \textbf{1.914} & \textbf{1.878} & \textbf{5.142} & \textbf{2.076} & \textbf{2.039} & \textbf{1.972} & \textbf{5.146} \\
161 +        12 \AA  & N & 129.576 & 25.560 & 1.369 & 5.080 & 0.913 & 0.790 & 1.362 & 5.124 \\
162 +                & \textbf{Y} & \textbf{0.810} & \textbf{0.685} & \textbf{1.352} & \textbf{5.082} & \textbf{0.765} & \textbf{0.714} & \textbf{1.360} & \textbf{5.082} \\
163 +        15 \AA  & N & 87.275 & 4.473 & 1.271 & 5.000 & 0.372 & 0.312 & 1.271 & 5.000 \\
164 +                & \textbf{Y} & \textbf{0.282} & \textbf{0.294} & \textbf{1.272} & \textbf{4.999} & \textbf{0.324} & \textbf{0.318} & \textbf{1.272} & \textbf{4.999} \\
165 +
166 +      \bottomrule
167 +   \end{tabular}
168 +   \label{groupAngle}
169 + \end{table}
170 +
171 + Although not discussed previously, group based cutoffs can be applied to both the Shifted Potential and Force methods.  Use off a switching function corrects for the discontinuities that arise when atoms of a group exit the cutoff before the group's center of mass.  Though there are no significant benefit or drawbacks observed in $\Delta E$ and vector magnitude results when doing this, there is a measurable improvement in the vector angle results.  Table \ref{groupAngle} shows the angular variance values obtained using group based cutoffs and a switching function alongside the standard results seen in figure \ref{frcTrqAng} for comparison purposes.  The Shifted Potential shows much narrower angular distributions for both the force and torque vectors when using an $\alpha$ of 0.2 \AA$^{-1}$ or less, while Shifted Force shows improvements in the undamped and lightly damped cases.  Thus, by calculating the electrostatic interactions in terms of molecular pairs rather than atomic pairs, the direction of the force and torque vectors are determined more accurately.  
172 +
173 + One additional trend to recognize in table \ref{groupAngle} is that the $\sigma^2$ values for both Shifted Potential and Shifted Force converge as $\alpha$ increases, something that is easier to see when using group based cutoffs.  Looking back on figures \ref{delE}, \ref{frcMag}, and \ref{trqMag}, show this behavior clearly at large $\alpha$ and cutoff values.  The reason for this is that the complimentary error function inserted into the potential weakens the electrostatic interaction as $\alpha$ increases.  Thus, at larger values of $\alpha$, both the summation method types progress toward non-interacting functions, so care is required in choosing large damping functions lest one generate an undesirable loss in the pair interaction.  Kast \textit{et al.}  developed a method for choosing appropriate $\alpha$ values for these types of electrostatic summation methods by fitting to $g(r)$ data, and their methods indicate optimal values of 0.34, 0.25, and 0.16 \AA$^{-1}$ for cutoff values of 9, 12, and 15 \AA\ respectively.\cite{Kast03}  These appear to be reasonable choices to obtain proper MC behavior (Fig. \ref{delE}); however, based on these findings, choices this high would be introducing error in the molecular torques, particularly for the shorter cutoffs.  Based on the above findings, any empirical damping is arguably unnecessary with the choice of the Shifted Force method.
174 +
175   \section{Conclusions}
176  
177   \section{Acknowledgments}
# Line 142 | Line 180 | Having force and torque vectors with magnitudes that a
180  
181   \section{\label{app-water}Liquid Water}
182  
183 + \begin{table}[htbp]
184 +   \centering
185 +   \caption{Regression results for the liquid water system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).}  
186 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
187 +      \\
188 +      \toprule
189 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
190 +      \cmidrule(lr){3-4}
191 +      \cmidrule(lr){5-6}
192 +      \cmidrule(l){7-8}
193 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
194 +            \midrule
195 + PC  &     & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\
196 + SP  & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\
197 +    & 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\
198 +    & 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\
199 +    & 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\
200 + SF  & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\
201 +    & 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\
202 +    & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
203 +    & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
204 + GSC &     & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
205 + RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                              
206 +
207 +            \midrule
208 +
209 + PC  &     & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
210 + SP  & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
211 +    & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
212 +    & 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\
213 +    & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
214 + SF  & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\
215 +    & 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
216 +    & 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\
217 +    & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
218 + GSC &     & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
219 + RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\          
220 +
221 +            \midrule
222 +
223 + PC  &     & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
224 + SP  & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
225 +    & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
226 +    & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
227 +    & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
228 + SF  & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
229 +    & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
230 +    & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
231 +    & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
232 + GSC &     & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
233 + RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
234 +      \bottomrule
235 +   \end{tabular}
236 +   \label{spceTabTMag}
237 + \end{table}
238 +
239 + \begin{table}[htbp]
240 +   \centering
241 +   \caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the liquid water system.  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}  
242 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
243 +      \\
244 +      \toprule
245 +      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
246 +      \cmidrule(lr){3-5}
247 +      \cmidrule(l){6-8}
248 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
249 +            \midrule
250 + PC  &     & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
251 + SP  & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
252 +    & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
253 +    & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
254 +    & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
255 + SF  & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
256 +    & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
257 +    & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
258 +    & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
259 + GSC &     & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
260 + RF  &     & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\      
261 +                        \midrule
262 + GSSP  & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
263 +      & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
264 +      & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
265 +      & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
266 + GSSF  & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
267 +      & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
268 +      & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
269 +      & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
270 +      \bottomrule
271 +   \end{tabular}
272 +   \label{spceTabAng}
273 + \end{table}
274 +
275   \section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
276  
277 + \begin{table}[htbp]
278 +   \centering
279 +   \caption{Regression results for the ice I$_\textrm{c}$ system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).}    
280 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
281 +      \\
282 +      \toprule
283 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
284 +      \cmidrule(lr){3-4}
285 +      \cmidrule(lr){5-6}
286 +      \cmidrule(l){7-8}
287 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
288 +            \midrule
289 + PC  &     & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
290 + SP  & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
291 +    & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
292 +    & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
293 +    & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
294 + SF  & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
295 +    & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
296 +    & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
297 +    & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
298 + GSC &     & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
299 + RF  &     & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
300 +            \midrule
301 + PC  &     & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
302 + SP  & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
303 +    & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
304 +    & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
305 +    & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
306 + SF  & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
307 +    & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
308 +    & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
309 +    & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
310 + GSC &     & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
311 + RF  &     & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
312 +            \midrule
313 + PC  &     & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
314 + SP  & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
315 +    & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
316 +    & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
317 +    & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
318 + SF  & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
319 +    & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
320 +    & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
321 +    & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
322 + GSC &     & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
323 + RF  &     & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
324 +      \bottomrule
325 +   \end{tabular}
326 +   \label{iceTab}
327 + \end{table}
328 +
329 + \begin{table}[htbp]
330 +   \centering
331 +   \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.}    
332 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
333 +      \\
334 +      \toprule
335 +      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
336 +      \cmidrule(lr){3-5}
337 +      \cmidrule(l){6-8}
338 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
339 +            \midrule
340 + PC  &     & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
341 + SP  & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
342 +    & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
343 +    & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
344 +    & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
345 + SF  & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
346 +    & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
347 +    & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
348 +    & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
349 + GSC &     & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
350 + RF  &     & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
351 +                        \midrule
352 + GSSP  & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
353 +      & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
354 +      & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
355 +      & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
356 + GSSF  & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
357 +      & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
358 +      & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
359 +      & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
360 +      \bottomrule
361 +   \end{tabular}
362 +   \label{iceTabAng}
363 + \end{table}
364 +
365   \section{\label{app-melt}NaCl Melt}
366  
367 + \begin{table}[htbp]
368 +   \centering
369 +   \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.}  
370 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
371 +      \\
372 +      \toprule
373 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
374 +      \cmidrule(lr){3-4}
375 +      \cmidrule(lr){5-6}
376 +      \cmidrule(l){7-8}
377 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
378 +            \midrule
379 + PC  &     & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
380 + SP  & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\
381 +    & 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\
382 +    & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
383 +    & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
384 + SF  & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\
385 +    & 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\
386 +    & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
387 +    & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
388 +            \midrule
389 + PC  &     & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
390 + SP  & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\
391 +    & 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\
392 +    & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
393 +    & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
394 + SF  & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\
395 +    & 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\
396 +    & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
397 +    & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
398 +      \bottomrule
399 +   \end{tabular}
400 +   \label{meltTab}
401 + \end{table}
402 +
403 + \begin{table}[htbp]
404 +   \centering
405 +   \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.}    
406 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
407 +      \\
408 +      \toprule
409 +      & & \multicolumn{3}{c}{Force $\sigma^2$} \\
410 +      \cmidrule(lr){3-5}
411 +      \cmidrule(l){6-8}
412 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
413 +            \midrule
414 + PC  &     & 13.294 & 8.035 & 5.366 \\
415 + SP  & 0.0 & 13.316 & 8.037 & 5.385 \\
416 +    & 0.1 & 5.705 & 1.391 & 0.360 \\
417 +    & 0.2 & 2.415 & 7.534 & 13.927 \\
418 +    & 0.3 & 23.769 & 67.306 & 57.252 \\
419 + SF  & 0.0 & 1.693 & 0.603 & 0.256 \\
420 +    & 0.1 & 1.687 & 0.653 & 0.272 \\
421 +    & 0.2 & 2.598 & 7.523 & 13.930 \\
422 +    & 0.3 & 23.734 & 67.305 & 57.252 \\
423 +      \bottomrule
424 +   \end{tabular}
425 +   \label{meltTabAng}
426 + \end{table}
427 +
428   \section{\label{app-salt}NaCl Crystal}
429 +
430 + \begin{table}[htbp]
431 +   \centering
432 +   \caption{Regression results for the crystalline NaCl system. Tabulated results include $\Delta E$ values (top set) and force vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, and SF = Shifted Force.}    
433 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
434 +      \\
435 +      \toprule
436 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
437 +      \cmidrule(lr){3-4}
438 +      \cmidrule(lr){5-6}
439 +      \cmidrule(l){7-8}
440 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
441 +            \midrule
442 + PC  &     & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
443 + SP  & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
444 +    & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
445 +    & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
446 +    & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
447 + SF  & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
448 +    & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
449 +    & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
450 +    & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
451 +            \midrule
452 + PC  &     & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
453 + SP  & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
454 +    & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
455 +    & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
456 +    & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
457 + SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
458 +    & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
459 +    & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
460 +    & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
461 +      \bottomrule
462 +   \end{tabular}
463 +   \label{saltTab}
464 + \end{table}
465  
466 + \begin{table}[htbp]
467 +   \centering
468 +   \caption{Variance results from Gaussian fits to angular distributions of the force vectors in the crystalline NaCl system.  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).}        
469 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
470 +      \\
471 +      \toprule
472 +      & & \multicolumn{3}{c}{Force $\sigma^2$} \\
473 +      \cmidrule(lr){3-5}
474 +      \cmidrule(l){6-8}
475 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
476 +            \midrule
477 + PC  &     & 111.945 & 111.824 & 111.866 \\
478 + SP  & 0.0 & 112.414 & 152.215 & 38.087 \\
479 +    & 0.1 & 52.361 & 42.574 & 2.819 \\
480 +    & 0.2 & 10.847 & 9.709 & 9.686 \\
481 +    & 0.3 & 31.128 & 31.104 & 31.029 \\
482 + SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
483 +    & 0.1 & 9.462 & 3.303 & 1.721 \\
484 +    & 0.2 & 11.454 & 9.813 & 9.701 \\
485 +    & 0.3 & 31.120 & 31.105 & 31.029 \\
486 +      \bottomrule
487 +   \end{tabular}
488 +   \label{saltTabAng}
489 + \end{table}
490 +
491   \section{\label{app-sol1}1M NaCl Solution}
492  
493 + \begin{table}[htbp]
494 +   \centering
495 +   \caption{Regression results for the 1M NaCl solution system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}
496 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
497 +      \\
498 +      \toprule
499 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
500 +      \cmidrule(lr){3-4}
501 +      \cmidrule(lr){5-6}
502 +      \cmidrule(l){7-8}
503 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
504 +            \midrule
505 + PC  &     & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
506 + SP  & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
507 +    & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
508 +    & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
509 +    & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
510 + SF  & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
511 +    & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
512 +    & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
513 +    & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
514 + GSC &     & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
515 + RF  &     & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
516 +            \midrule
517 + PC  &     & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
518 + SP  & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
519 +    & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
520 +    & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
521 +    & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
522 + SF  & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
523 +    & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
524 +    & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
525 +    & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
526 + GSC &     & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
527 + RF  &     & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
528 +            \midrule
529 + PC  &     & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
530 + SP  & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
531 +    & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
532 +    & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
533 +    & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
534 + SF  & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
535 +    & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
536 +    & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
537 +    & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
538 + GSC &     & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
539 + RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
540 +      \bottomrule
541 +   \end{tabular}
542 +   \label{sol1Tab}
543 + \end{table}
544 +
545 + \begin{table}[htbp]
546 +   \centering
547 +   \caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the 1M 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.}      
548 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
549 +      \\
550 +      \toprule
551 +      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
552 +      \cmidrule(lr){3-5}
553 +      \cmidrule(l){6-8}
554 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
555 +            \midrule
556 + PC  &     & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
557 + SP  & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
558 +    & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
559 +    & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
560 +    & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
561 + SF  & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
562 +    & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
563 +    & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
564 +    & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
565 + GSC &     & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
566 + RF  &     & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
567 +                        \midrule
568 + GSSP  & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
569 +      & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
570 +      & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
571 +      & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
572 + GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
573 +      & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
574 +      & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
575 +      & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
576 +      \bottomrule
577 +   \end{tabular}
578 +   \label{sol1TabAng}
579 + \end{table}
580 +
581   \section{\label{app-sol10}10M NaCl Solution}
582  
583 + \begin{table}[htbp]
584 +   \centering
585 +   \caption{Regression results for the 10M NaCl solution system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).}      
586 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
587 +      \\
588 +      \toprule
589 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
590 +      \cmidrule(lr){3-4}
591 +      \cmidrule(lr){5-6}
592 +      \cmidrule(l){7-8}
593 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
594 +            \midrule
595 + PC  &     & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
596 + SP  & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
597 +    & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
598 +    & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
599 +    & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
600 + SF  & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
601 +    & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
602 +    & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
603 +    & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
604 + GSC &     & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
605 + RF  &     & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
606 +            \midrule
607 + PC  &     & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
608 + SP  & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
609 +    & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
610 +    & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
611 +    & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
612 + SF  & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
613 +    & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
614 +    & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
615 +    & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
616 + GSC &     & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
617 + RF  &     & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
618 +            \midrule
619 + PC  &     & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
620 + SP  & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
621 +    & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
622 +    & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
623 +    & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
624 + SF  & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
625 +    & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
626 +    & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
627 +    & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
628 + GSC &     & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
629 + RF  &     & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
630 +      \bottomrule
631 +   \end{tabular}
632 +   \label{sol10Tab}
633 + \end{table}
634 +
635 + \begin{table}[htbp]
636 +   \centering
637 +   \caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the 10M 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.}      
638 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
639 +      \\
640 +      \toprule
641 +      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
642 +      \cmidrule(lr){3-5}
643 +      \cmidrule(l){6-8}
644 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
645 +            \midrule
646 + PC  &     & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
647 + SP  & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
648 +    & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
649 +    & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
650 +    & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
651 + SF  & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
652 +    & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
653 +    & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
654 +    & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
655 + GSC &     & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
656 + RF  &     & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
657 +                        \midrule
658 + GSSP  & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
659 +      & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
660 +      & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
661 +      & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
662 + GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
663 +      & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
664 +      & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
665 +      & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
666 +      \bottomrule
667 +   \end{tabular}
668 +   \label{sol10TabAng}
669 + \end{table}
670 +
671   \section{\label{app-argon}Argon Sphere in Water}
672  
673 + \begin{table}[htbp]
674 +   \centering
675 +   \caption{Regression results for the 6 \AA\ argon sphere in liquid water system. Tabulated results include $\Delta E$ values (top set), force vector magnitudes (middle set) and torque vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx \infty$).}    
676 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
677 +      \\
678 +      \toprule
679 +      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
680 +      \cmidrule(lr){3-4}
681 +      \cmidrule(lr){5-6}
682 +      \cmidrule(l){7-8}
683 +            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
684 +            \midrule
685 + PC  &     & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
686 + SP  & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
687 +    & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
688 +    & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
689 +    & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
690 + SF  & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
691 +    & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
692 +    & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
693 +    & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
694 + GSC &     & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
695 + RF  &     & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
696 +            \midrule
697 + PC  &     & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
698 + SP  & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
699 +    & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
700 +    & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
701 +    & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
702 + SF  & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
703 +    & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
704 +    & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
705 +    & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
706 + GSC &     & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
707 + RF  &     & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
708 +            \midrule
709 + PC  &     & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
710 + SP  & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
711 +    & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
712 +    & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
713 +    & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
714 + SF  & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
715 +    & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
716 +    & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
717 +    & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
718 + GSC &     & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
719 + RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
720 +      \bottomrule
721 +   \end{tabular}
722 +   \label{argonTab}
723 + \end{table}
724 +
725 + \begin{table}[htbp]
726 +   \centering
727 +   \caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the 6 \AA\ sphere of argon in liquid water system.  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}
728 +   \begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space
729 +      \\
730 +      \toprule
731 +      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
732 +      \cmidrule(lr){3-5}
733 +      \cmidrule(l){6-8}
734 +            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
735 +            \midrule
736 + PC  &     & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
737 + SP  & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
738 +    & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
739 +    & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
740 +    & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
741 + SF  & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
742 +    & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
743 +    & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
744 +    & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
745 + GSC &     & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
746 + RF  &     & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
747 +                        \midrule
748 + GSSP  & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
749 +      & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
750 +      & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
751 +      & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
752 + GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
753 +      & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
754 +      & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
755 +      & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
756 +      \bottomrule
757 +   \end{tabular}
758 +   \label{argonTabAng}
759 + \end{table}
760 +
761   \newpage
762  
763   \bibliographystyle{achemso}

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