--- trunk/electrostaticMethodsPaper/SupportingInfo.tex	2006/02/28 14:09:55	2599
+++ trunk/electrostaticMethodsPaper/SupportingInfo.tex	2006/03/24 02:39:59	2667
@@ -1,6 +1,6 @@
 %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
-\documentclass[12pt]{article}
-\usepackage{endfloat}
+\documentclass[11pt]{article}
+%\usepackage{endfloat}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{epsf}
@@ -23,13 +23,31 @@ This document includes system based comparisons of the
 
 \begin{document}
 
-This document includes system based comparisons of the studied methods with smooth particle-mesh Ewald.  Each of the seven systems comprises it's own section and has it's own discussion and tabular listing of the results for the $\Delta E$, force and torque vector magnitude, and force and torque vector direction comparisons.
+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}
+\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 $\varepsilon \approx \infty$).}	
+   \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
@@ -49,10 +67,8 @@ RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.
     & 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 \\                              
-
+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 \\
@@ -64,9 +80,7 @@ RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.
     & 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 \\
@@ -80,12 +94,17 @@ RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.
 RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\
       \bottomrule
    \end{tabular}
-   \label{spceTabTMag}
+   \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.}	
+   \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
@@ -116,14 +135,70 @@ GSSF  & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 
       & 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\
       \bottomrule
    \end{tabular}
-   \label{spceTabAng}
+   \label{tab:spceAng}
 \end{table}
 
-\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
+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 $\varepsilon \approx \infty$).}	
+   \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
@@ -170,7 +245,7 @@ RF  &     & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.
 RF  &     & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\
       \bottomrule
    \end{tabular}
-   \label{iceTab}
+   \label{tab:ice}
 \end{table}
 
 \begin{table}[htbp]
@@ -206,11 +281,37 @@ GSSF  & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 
       & 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\
       \bottomrule
    \end{tabular}
-   \label{iceTabAng}
+   \label{tab:iceAng}
 \end{table}
 
-\section{\label{app-melt}NaCl Melt}
+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.}	
@@ -224,27 +325,27 @@ SP  & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.
             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 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\
-    & 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\
+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 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\
-    & 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\
+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 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\
-    & 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\
+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 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\
-    & 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\
+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{meltTab}
+   \label{tab:melt}
 \end{table}
 
 \begin{table}[htbp]
@@ -269,14 +370,35 @@ SF  & 0.0 & 1.693 & 0.603 & 0.256 \\
     & 0.3 & 23.734 & 67.305 & 57.252 \\
       \bottomrule
    \end{tabular}
-   \label{meltTabAng}
+   \label{tab:meltAng}
 \end{table}
 
-\section{\label{app-salt}NaCl Crystal}
+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.}	
+   \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
@@ -307,12 +429,16 @@ SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.
     & 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\
       \bottomrule
    \end{tabular}
-   \label{saltTab}
+   \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 $\varepsilon \approx \infty$).}	
+   \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
@@ -332,14 +458,47 @@ SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
     & 0.3 & 31.120 & 31.105 & 31.029 \\
       \bottomrule
    \end{tabular}
-   \label{saltTabAng}
+   \label{tab:saltAng}
 \end{table}
 
-\section{\label{app-sol1}Weak NaCl Solution}
+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, RF = Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}	
+   \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
@@ -386,12 +545,17 @@ RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.
 RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\
       \bottomrule
    \end{tabular}
-   \label{sol1Tab}
+   \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.}	
+   \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
@@ -422,14 +586,40 @@ GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 
       & 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\
       \bottomrule
    \end{tabular}
-   \label{sol1TabAng}
+   \label{tab:solnWeakAng}
 \end{table}
 
-\section{\label{app-sol10}Strong NaCl Solution}
+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 $\varepsilon \approx \infty$).}	
+   \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
@@ -476,7 +666,7 @@ RF  &     & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.
 RF  &     & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\
       \bottomrule
    \end{tabular}
-   \label{sol10Tab}
+   \label{tab:solnStr}
 \end{table}
 
 \begin{table}[htbp]
@@ -512,14 +702,39 @@ GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 &
       & 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\
       \bottomrule
    \end{tabular}
-   \label{sol10TabAng}
+   \label{tab:solnStrAng}
 \end{table}
 
-\section{\label{app-argon}Argon Sphere in Water}
+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 $\varepsilon \approx \infty$).}	
+   \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
@@ -566,12 +781,17 @@ RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.
 RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\
       \bottomrule
    \end{tabular}
-   \label{argonTab}
+   \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.}	
+   \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
@@ -602,7 +822,24 @@ GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 
       & 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\
       \bottomrule
    \end{tabular}
-   \label{argonTabAng}
+   \label{tab:argonAng}
 \end{table}
 
-\end{document}
\ No newline at end of file
+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}
