trunk/electrostaticMethodsPaper/SupportingInfo.tex

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\begin{document}

This document includes individual system-based comparisons of the
studied methods with smooth particle mesh Ewald {\sc spme}.  Each of
the seven systems comprises its own section and has its own discussion
and tabular listing of the results for the $\Delta E$, force and
torque vector magnitude, and force and torque vector direction
comparisons.

\section{\label{app:water}Liquid Water}

The first system considered was liquid water at 300K using the SPC/E
model of water.\cite{Berendsen87} The results for the energy gap
comparisons and the force and torque vector magnitude comparisons are
shown in table \ref{tab:spce}.  The force and torque vector
directionality results are displayed separately in table
\ref{tab:spceAng}, where the effect of group-based cutoffs and
switching functions on the {\sc sp} and {\sc sf} potentials are
investigated.
\begin{table}[htbp]
   \centering
   \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 $\varepsilon \approx
\infty$).}      
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\
SP  & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\
    & 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\
    & 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\
    & 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\
SF  & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\
    & 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\
    & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
    & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
GSC &     & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                 
            \midrule
PC  &     & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
SP  & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
    & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
    & 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\
    & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
SF  & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\
    & 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
    & 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\
    & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
GSC &     & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\          
            \midrule
PC  &     & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
SP  & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
    & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
    & 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\
    & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
SF  & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\
    & 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\
    & 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\
    & 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\
GSC &     & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\
RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
      \bottomrule
   \end{tabular}
   \label{tab:spce}
\end{table}

\begin{table}[htbp]
   \centering
   \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.}  
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\
SP  & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\
    & 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\
    & 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\
    & 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\
SF  & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\
    & 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\
    & 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\
    & 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\
GSC &     & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
RF  &     & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\       
                        \midrule
GSSP  & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\
      & 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\
      & 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\
      & 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\
GSSF  & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\
      & 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\
      & 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\
      & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
      \bottomrule
   \end{tabular}
   \label{tab:spceAng}
\end{table}

The water results appear to parallel the combined results seen in the
discussion section of the main paper.  There is good agreement with
{\sc spme} in both energetic and dynamic behavior when using the {\sc sf}
method with and without damping. The {\sc sp} method does well with an
$\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater
than 12 \AA. Overdamping the electrostatics reduces the agreement between both these methods and {\sc spme}.

The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
observations in the main portion of this paper.  In contrast to the
combined values, however, the use of a switching function and group
based cutoffs greatly improves the results for these neutral water
molecules.  The group switched cutoff ({\sc gsc}) does not mimic the
energetics of {\sc spme} as well as the {\sc sp} (with moderate
damping) and {\sc sf} methods, but the dynamics are quite good.  The
switching functions correct discontinuities in the potential and
forces, leading to these improved results.  Such improvements with the
use of a switching function have been recognized in previous
studies,\cite{Andrea83,Steinbach94} and this proves to be a useful
tactic for stably incorporating local area electrostatic effects.

The reaction field ({\sc rf}) method simply extends upon the results
observed in the {\sc gsc} case.  Both methods are similar in form
(i.e. neutral groups, switching function), but {\sc rf} incorporates
an added effect from the external dielectric. This similarity
translates into the same good dynamic results and improved energetic
agreement with {\sc spme}.  Though this agreement is not to the level
of the moderately damped {\sc sp} and {\sc sf} methods, these results
show how incorporating some implicit properties of the surroundings
(i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction.

As a final note for the liquid water system, use of group cutoffs and a
switching function leads to noticeable improvements in the {\sc sp}
and {\sc sf} methods, primarily in directionality of the force and
torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows
significant narrowing of the angle distribution when using little to
no damping and only modest improvement for the recommended conditions
($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA).  The
{\sc sf} method shows modest narrowing across all damping and cutoff
ranges of interest.  When overdamping these methods, group cutoffs and
the switching function do not improve the force and torque
directionalities.

\section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$}

In addition to the disordered molecular system above, the ordered
molecular system of ice I$_\textrm{c}$ was also considered. The
results for the energy gap comparisons and the force and torque vector
magnitude comparisons are shown in table \ref{tab:ice}.  The force and
torque vector directionality results are displayed separately in table
\ref{tab:iceAng}, where the effect of group-based cutoffs and
switching functions on the {\sc sp} and {\sc sf} potentials are
investigated. 

\begin{table}[htbp]
   \centering
   \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
$\varepsilon \approx \infty$).}  
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\
SP  & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\
    & 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\
    & 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\
    & 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\
SF  & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\
    & 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\
    & 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\
    & 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\
GSC &     & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\
RF  &     & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\
            \midrule
PC  &     & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\
SP  & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\
    & 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\
    & 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\
    & 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\
SF  & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\
    & 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\
    & 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\
    & 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\
GSC &     & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\
RF  &     & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\
            \midrule
PC  &     & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\
SP  & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\
    & 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\
    & 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\
    & 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\
SF  & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\
    & 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\
    & 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\
    & 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\
GSC &     & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\
RF  &     & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
      \bottomrule
   \end{tabular}
   \label{tab:ice}
\end{table}

\begin{table}[htbp]
   \centering
   \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.}     
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\
SP  & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\
    & 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\
    & 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\
    & 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\
SF  & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\
    & 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\
    & 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\
    & 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\
GSC &     & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
RF  &     & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\
                        \midrule
GSSP  & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\
      & 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\
      & 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\
      & 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
GSSF  & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\
      & 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\
      & 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\
      & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
      \bottomrule
   \end{tabular}
   \label{tab:iceAng}
\end{table}

Highly ordered systems are a difficult test for the pairwise methods
in that they lack the periodicity term of the Ewald summation.  As
expected, the energy gap agreement with {\sc spme} is reduced for the
{\sc sp} and {\sc sf} methods with parameters that were acceptable for
the disordered liquid system.  Moving to higher $R_\textrm{c}$ helps
improve the agreement, though at an increase in computational cost.
The dynamics of this crystalline system (both in magnitude and
direction) are little affected. Both methods still reproduce the Ewald
behavior with the same parameter recommendations from the previous
section.

It is also worth noting that {\sc rf} exhibits improved energy gap
results over the liquid water system.  One possible explanation is
that the ice I$_\textrm{c}$ crystal is ordered such that the net
dipole moment of the crystal is zero.  With $\epsilon_\textrm{S} =
\infty$, the reaction field incorporates this structural organization
by actively enforcing a zeroed dipole moment within each cutoff
sphere.  

\section{\label{app:melt}NaCl Melt}

A high temperature NaCl melt was tested to gauge the accuracy of the
pairwise summation methods in a charged disordered system. The results
for the energy gap comparisons and the force vector magnitude
comparisons are shown in table \ref{tab:melt}.  The force vector
directionality results are displayed separately in table
\ref{tab:meltAng}.

\begin{table}[htbp]
   \centering
   \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.}  
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
SP  & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\
    & 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\
    & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
    & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
SF  & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\
    & 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\
    & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
    & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
            \midrule
PC  &     & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
SP  & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\
    & 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\
    & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
    & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
SF  & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\
    & 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\
    & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
    & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
      \bottomrule
   \end{tabular}
   \label{tab:melt}
\end{table}

\begin{table}[htbp]
   \centering
   \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.}    
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 13.294 & 8.035 & 5.366 \\
SP  & 0.0 & 13.316 & 8.037 & 5.385 \\
    & 0.1 & 5.705 & 1.391 & 0.360 \\
    & 0.2 & 2.415 & 7.534 & 13.927 \\
    & 0.3 & 23.769 & 67.306 & 57.252 \\
SF  & 0.0 & 1.693 & 0.603 & 0.256 \\
    & 0.1 & 1.687 & 0.653 & 0.272 \\
    & 0.2 & 2.598 & 7.523 & 13.930 \\
    & 0.3 & 23.734 & 67.305 & 57.252 \\
      \bottomrule
   \end{tabular}
   \label{tab:meltAng}
\end{table}

The molten NaCl system shows more sensitivity to the electrostatic
damping than the water systems. The most noticeable point is that the
undamped {\sc sf} method does very well at replicating the {\sc spme}
configurational energy differences and forces. Light damping appears
to minimally improve the dynamics, but this comes with a deterioration
of the energy gap results. In contrast, this light damping improves
the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
damping reduce the agreement with {\sc spme} for both methods. From
these observations, the undamped {\sc sf} method is the best choice
for disordered systems of charges.

\section{\label{app:salt}NaCl Crystal}

A 1000K NaCl crystal was used to investigate the accuracy of the
pairwise summation methods in an ordered system of charged
particles. The results for the energy gap comparisons and the force
vector magnitude comparisons are shown in table \ref{tab:salt}.  The
force vector directionality results are displayed separately in table
\ref{tab:saltAng}.

\begin{table}[htbp]
   \centering
   \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.}     
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\
SP  & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\
    & 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\
    & 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\
    & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\
SF  & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\
    & 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\
    & 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\
    & 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\
            \midrule
PC  &     & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\
SP  & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\
    & 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\
    & 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\
    & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\
    & 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\
    & 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\
    & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
      \bottomrule
   \end{tabular}
   \label{tab:salt}
\end{table}

\begin{table}[htbp]
   \centering
   \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 $\varepsilon \approx
\infty$).}       
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 111.945 & 111.824 & 111.866 \\
SP  & 0.0 & 112.414 & 152.215 & 38.087 \\
    & 0.1 & 52.361 & 42.574 & 2.819 \\
    & 0.2 & 10.847 & 9.709 & 9.686 \\
    & 0.3 & 31.128 & 31.104 & 31.029 \\
SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
    & 0.1 & 9.462 & 3.303 & 1.721 \\
    & 0.2 & 11.454 & 9.813 & 9.701 \\
    & 0.3 & 31.120 & 31.105 & 31.029 \\
      \bottomrule
   \end{tabular}
   \label{tab:saltAng}
\end{table}

The crystalline NaCl system is the most challenging test case for the
pairwise summation methods, as evidenced by the results in tables
\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best
choices. These methods match well with {\sc spme} across the energy
gap, force magnitude, and force directionality tests.  The {\sc sp}
method struggles in all cases, with the exception of good dynamics
reproduction when using weak electrostatic damping with a large cutoff
radius.

The moderate electrostatic damping case is not as good as we would
expect given the long-time dynamics results observed for this
system. Since the data tabulated in tables \ref{tab:salt} and
\ref{tab:saltAng} are a test of instantaneous dynamics, this indicates
that good long-time dynamics comes in part at the expense of
short-time dynamics.

\section{\label{app:solnWeak}Weak NaCl Solution}

In an effort to bridge the charged atomic and neutral molecular
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
the liquid water system. This low ionic strength system consists of 4
ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
for the energy gap comparisons and the force and torque vector
magnitude comparisons are shown in table \ref{tab:solnWeak}.  The
force and torque vector directionality results are displayed
separately in table \ref{tab:solnWeakAng}, where the effect of
group-based cutoffs and switching functions on the {\sc sp} and {\sc
sf} potentials are investigated. 

\begin{table}[htbp]
   \centering
   \caption{Regression results for the weak 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 $\varepsilon
\approx \infty$).}       
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\
SP  & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\
    & 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\
    & 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\
    & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
SF  & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\
    & 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\
    & 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\
    & 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\
GSC &     & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\
RF  &     & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\
            \midrule
PC  &     & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\
SP  & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\
    & 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\
    & 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\
    & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
SF  & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\
    & 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\
    & 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\
    & 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\
GSC &     & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\
RF  &     & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\
            \midrule
PC  &     & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\
SP  & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\
    & 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\
    & 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\
    & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
SF  & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\
    & 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\
    & 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\
    & 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\
GSC &     & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\
RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
      \bottomrule
   \end{tabular}
   \label{tab:solnWeak}
\end{table}

\begin{table}[htbp]
   \centering
   \caption{Variance results from Gaussian fits to angular
distributions of the force and torque vectors in the weak 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.}     
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\
SP  & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\
    & 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\
    & 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\
    & 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\
SF  & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\
    & 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\
    & 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\
    & 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\
GSC &     & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
RF  &     & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\
                        \midrule
GSSP  & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\
      & 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\
      & 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\
      & 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\
GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\
      & 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\
      & 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\
      & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
      \bottomrule
   \end{tabular}
   \label{tab:solnWeakAng}
\end{table}

Because this system is a perturbation of the pure liquid water system,
comparisons are best drawn between these two sets. The {\sc sp} and
{\sc sf} methods are not significantly affected by the inclusion of a
few ions. The aspect of cutoff sphere neutralization aids in the
smooth incorporation of these ions; thus, all of the observations
regarding these methods carry over from section \ref{app:water}. The
differences between these systems are more visible for the {\sc rf}
method. Though good force agreement is still maintained, the energy
gaps show a significant increase in the data scatter. This foreshadows
the breakdown of the method as we introduce charged inhomogeneities.

\section{\label{app:solnStr}Strong NaCl Solution}

The bridging of the charged atomic and neutral molecular systems was
further developed by considering a high ionic strength system
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
M). The results for the energy gap comparisons and the force and
torque vector magnitude comparisons are shown in table
\ref{tab:solnStr}.  The force and torque vector directionality
results are displayed separately in table \ref{tab:solnStrAng}, where
the effect of group-based cutoffs and switching functions on the {\sc
sp} and {\sc sf} potentials are investigated.

\begin{table}[htbp]
   \centering
   \caption{Regression results for the strong 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
$\varepsilon \approx \infty$).}         
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\
SP  & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\
    & 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\
    & 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\
    & 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\
SF  & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\
    & 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\
    & 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\
    & 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\
GSC &     & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\
RF  &     & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\
            \midrule
PC  &     & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\
SP  & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\
    & 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\
    & 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\
    & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
SF  & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\
    & 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\
    & 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\
    & 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\
GSC &     & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\
RF  &     & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\
            \midrule
PC  &     & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\
SP  & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\
    & 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\
    & 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\
    & 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\
SF  & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\
    & 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\
    & 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\
    & 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\
GSC &     & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\
RF  &     & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
      \bottomrule
   \end{tabular}
   \label{tab:solnStr}
\end{table}

\begin{table}[htbp]
   \centering
   \caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the strong NaCl solution system.  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}   
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\
SP  & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\
    & 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\
    & 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\
    & 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\
SF  & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\
    & 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\
    & 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\
    & 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\
GSC &     & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\
RF  &     & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\
                        \midrule
GSSP  & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\
      & 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\
      & 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\
      & 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\
GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\
      & 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\
      & 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\
      & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
      \bottomrule
   \end{tabular}
   \label{tab:solnStrAng}
\end{table}

The {\sc rf} method struggles with the jump in ionic strength. The
configuration energy differences degrade to unusable levels while the
forces and torques show a more modest reduction in the agreement with
{\sc spme}. The {\sc rf} method was designed for homogeneous systems,
and this attribute is apparent in these results.

The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
their agreement with {\sc spme}. With these results, we still
recommend no to moderate damping for the {\sc sf} method and moderate
damping for the {\sc sp} method, both with cutoffs greater than 12
\AA.

\section{\label{app:argon}Argon Sphere in Water}

The final model system studied was a 6 \AA\ sphere of Argon solvated
by SPC/E water. The results for the energy gap comparisons and the
force and torque vector magnitude comparisons are shown in table
\ref{tab:argon}.  The force and torque vector directionality
results are displayed separately in table \ref{tab:argonAng}, where
the effect of group-based cutoffs and switching functions on the {\sc
sp} and {\sc sf} potentials are investigated. 

\begin{table}[htbp]
   \centering
   \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
$\varepsilon \approx \infty$).}         
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\
      \cmidrule(lr){3-4} 
      \cmidrule(lr){5-6} 
      \cmidrule(l){7-8}
            Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
            \midrule
PC  &     & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\
SP  & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\
    & 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\
    & 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\
    & 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\
SF  & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\
    & 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\
    & 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\
    & 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\
GSC &     & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\
RF  &     & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\
            \midrule
PC  &     & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\
SP  & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\
    & 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\
    & 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\
    & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
SF  & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
    & 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
    & 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
    & 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\
GSC &     & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
RF  &     & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\
            \midrule
PC  &     & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\
SP  & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\
    & 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\
    & 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\
    & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
SF  & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\
    & 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\
    & 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\
    & 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\
GSC &     & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\
RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
      \bottomrule
   \end{tabular}
   \label{tab:argon}
\end{table}

\begin{table}[htbp]
   \centering
   \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.}   
   \begin{tabular}{@{} ccrrrrrr @{}}
      \\
      \toprule
      & & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\
      \cmidrule(lr){3-5} 
      \cmidrule(l){6-8}
            Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\
            \midrule
PC  &     & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\
SP  & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\
    & 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\
    & 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\
    & 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\
SF  & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\
    & 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\
    & 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\
    & 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\
GSC &     & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
RF  &     & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\
                        \midrule
GSSP  & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\
      & 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\
      & 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\
      & 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\
GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\
      & 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\
      & 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\
      & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
      \bottomrule
   \end{tabular}
   \label{tab:argonAng}
\end{table}

This system does not appear to show any significant deviations from
the previously observed results. The {\sc sp} and {\sc sf} methods
have aggrements similar to those observed in section
\ref{app:water}. The only significant difference is the improvement
in the configuration energy differences for the {\sc rf} method. This
is surprising in that we are introducing an inhomogeneity to the
system; however, this inhomogeneity is charge-neutral and does not
result in charged cutoff spheres. The charge-neutrality of the cutoff
spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce,
seems to play a greater role in the stability of the {\sc rf} method
than the required homogeneity of the environment.

\newpage 

\bibliographystyle{jcp2}
\bibliography{electrostaticMethods}

\end{document}
Revision:	2667
Committed:	Fri Mar 24 02:39:59 2006 UTC (18 years, 3 months ago) by chrisfen
Content type:	application/x-tex
File size:	41698 byte(s)
Log Message:	Figure adjusted and Steve's recommendations incorporated