[ViewVC] Diff of: group/trunk/chrisDissertation/IndividualSystems.tex

Comparing trunk/chrisDissertation/IndividualSystems.tex (file contents):
Revision 2987 by chrisfen, Wed Aug 30 22:36:06 2006 UTC vs.
Revision 3023 by chrisfen, Tue Sep 26 03:07:59 2006 UTC

-<
+\chapter{\label{app:individualSystems} Individual System Analysis Results}
->
+\chapter{\label{app:IndividualResults} INDIVIDUAL SYSTEM ANALYSIS RESULTS}
-+
+The combined system results in chapter \ref{chap:electrostatics}
-+
+(sections \ref{sec:EnergyResults} through \ref{sec:FTDirResults}) show
-+
+how the pairwise methods compare to the Ewald summation in the general
-+
+sense over all of the system types.  It is also useful to consider
-+
+each of the studied systems in an individual fashion, so that we can
-+
+identify conditions that are particularly difficult for a selected
-+
+pairwise method to address. This allows us to further establish the
-+
+limitations of these pairwise techniques.  In this appendix, the
-+
+energy difference, force vector, and torque vector analyses are
-+
+presented on an individual system basis.
-+
-+
+\section{SPC/E Water Results}\label{sec:WaterResults}
-+
-+
+The first system considered was liquid water at 300~K 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 also
-+
+investigated.  In all of the individual results table, the method
-+
+abbreviations are as follows:
-+
-+
+\begin{itemize}[itemsep=0pt]
-+
+\item PC = Pure Cutoff,
-+
+\item SP = Shifted Potential,
-+
+\item SF = Shifted Force,
-+
+\item GSC = Group Switched Cutoff,
-+
+\item RF = Reaction Field (where $\varepsilon \approx\infty$),
-+
+\item GSSP = Group Switched Shifted Potential, and
-+
+\item GSSF = Group Switched Shifted Force.
-+
+\end{itemize}
-+
-+
+\begin{table}[htbp]
-+
+\centering
-+
+\caption{REGRESSION RESULTS OF THE LIQUID WATER SYSTEM FOR THE
-+
+$\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it middle})
-+
+AND TORQUE VECTOR MAGNITUDES ({\it lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 parallel the combined results seen in sections
-+
+\ref{sec:EnergyResults} through \ref{sec:FTDirResults}.  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. Over-damping the electrostatics
-+
+reduces the agreement between both these methods and {\sc spme}.
-+
-+
+The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
-+
+observations from the combined results.  In contrast to these results, 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 over-damping these methods, group cutoffs and
-+
+the switching function do not improve the force and torque
-+
+directionalities.
-+
-+
+\section{SPC/E Ice I$_\textrm{c}$ Results}\label{sec:IceResults}
-+
-+
+In addition to the disordered molecular system above, the ordered
-+
+molecular system of ice I$_\textrm{c}$ was also considered.  Ice
-+
+polymorph could have been used to fit this role; however, ice
-+
+I$_\textrm{c}$ was chosen because it can form an ideal periodic
-+
+lattice with the same number of water molecules used in the disordered
-+
+liquid state case.  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 also displayed.
-+
-+
+\begin{table}[htbp]
-+
+\centering
-+
+\caption{REGRESSION RESULTS OF THE ICE I$_\textrm{c}$ SYSTEM FOR
-+
+$\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES ({\it
-+
+middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 implicit periodicity 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 ideal 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{NaCl Melt Results}\label{sec:SaltMeltResults}
-+
-+
+A high temperature NaCl melt was tested to gauge the accuracy of the
-+
+pairwise summation methods in a disordered system of charges. 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 OF THE MOLTEN SODIUM CHLORIDE SYSTEM FOR
-+
+$\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES ({\it
-+
+lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 SODIUM CHLORIDE SYSTEM}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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{NaCl Crystal Results}\label{sec:SaltCrystalResults}
-+
-+
+Similar to the use of ice I$_\textrm{c}$ to investigate the role of
-+
+order in molecular systems on the effectiveness of the pairwise
-+
+methods, the 1000~K NaCl crystal system 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 OF THE CRYSTALLINE SODIUM CHLORIDE
-+
+SYSTEM FOR $\Delta E$ VALUES ({\it upper}) AND FORCE VECTOR MAGNITUDES
-+
+({\it lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 SODIUM CHLORIDE
-+
+SYSTEM}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 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
-+
+(see section \ref{sec:LongTimeDynamics}). Since the data 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{0.11M NaCl Solution Results}
-+
-+
+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 OF THE WEAK SODIUM CHLORIDE SOLUTION
-+
+SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES
-+
+({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 SODIUM
-+
+CHLORIDE SOLUTION SYSTEM}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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{sec:WaterResults}. 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 scatter
-+
+of the data.
-+
-+
+\section{1.1M NaCl Solution Results}
-+
-+
+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 OF THE STRONG SODIUM CHLORIDE SOLUTION
-+
+SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR MAGNITUDES
-+
+({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 SODIUM CHLORIDE SOLUTION
-+
+SYSTEM}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 undamped to moderate damping for the {\sc sf} method and
-+
+moderate damping for the {\sc sp} method, both with cutoffs greater
-+
+than 12~\AA.
-+
-+
+\section{6~\AA\ Argon Sphere in SPC/E Water Results}
-+
-+
+The final model system studied was a 6~\AA\ sphere of Argon solvated
-+
+by SPC/E water. This serves as a test case of a specifically sized
-+
+electrostatic defect in a disordered molecular system. 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 OF THE 6~\AA\ ARGON SPHERE IN LIQUID
-+
+WATER SYSTEM FOR $\Delta E$ VALUES ({\it upper}), FORCE VECTOR
-+
+MAGNITUDES ({\it middle}) AND TORQUE VECTOR MAGNITUDES ({\it lower})}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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}
-+
-+
+\footnotesize
-+
+\begin{tabular}{@{} ccrrrrrr @{}}
-+
+\toprule
-+
+\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 agreements similar to those observed in section
-+
+\ref{sec:WaterResults}. 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.

Diff Legend

-–
+Removed lines
-+
+Added lines
-<
+Changed lines
->
+Changed lines

Comparing trunk/chrisDissertation/IndividualSystems.tex (file contents): Revision 2987 by chrisfen, Wed Aug 30 22:36:06 2006 UTC vs. Revision 3023 by chrisfen, Tue Sep 26 03:07:59 2006 UTC

Diff Legend

Comparing trunk/chrisDissertation/IndividualSystems.tex (file contents):
Revision 2987 by chrisfen, Wed Aug 30 22:36:06 2006 UTC vs.
Revision 3023 by chrisfen, Tue Sep 26 03:07:59 2006 UTC