--- trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/01/28 22:42:46	2575
+++ trunk/electrostaticMethodsPaper/electrostaticMethods.tex	2006/02/16 23:07:23	2594
@@ -2,12 +2,14 @@
 \documentclass[12pt]{article}
 \usepackage{endfloat}
 \usepackage{amsmath}
+\usepackage{amssymb}
 \usepackage{epsf}
 \usepackage{times}
 \usepackage{mathptm}
 \usepackage{setspace}
 \usepackage{tabularx}
 \usepackage{graphicx}
+%\usepackage{berkeley}
 \usepackage[ref]{overcite}
 \pagestyle{plain}
 \pagenumbering{arabic}
@@ -43,16 +45,117 @@ Notre Dame, Indiana 46556}
 
 \section{Introduction}
 
+In this paper, a variety of simulation situations were analyzed to determine the relative effectiveness of the adapted Wolf spherical truncation schemes at reproducing the results obtained using a smooth particle mesh Ewald (SPME) summation technique.  In addition to the Shifted-Potential and Shifted-Force adapted Wolf methods, both reaction field and uncorrected cutoff methods were included for comparison purposes.  The general usability of these methods in both Monte Carlo and Molecular Dynamics calculations was assessed through statistical analysis over the combined results from all of the following studied systems:
+\begin{list}{-}{}
+\item Liquid Water
+\item Crystalline Water (Ice I$_\textrm{c}$)
+\item 1 M Solution of NaCl in Water 
+\item 10 M Solution of NaCl in Water 
+\item 6 \AA\  Radius Sphere of Argon in Water 
+\item NaCl Crystal
+\item NaCl Melt
+\end{list}
+Additional discussion on the results from the individual systems was also performed to identify limitations of the considered methods in specific systems.
+
 \section{Methods}
 
+In each of the simulated systems, 500 distinct configurations were generated, and the electrostatic summation methods were compared via sequential application on each of these fixed configurations.  The methods compared include SPME, the aforementioned Shifted Potential and Shifted Force methods - both with damping parameters ($\alpha$) of 0, 0.1, 0.2, and 0.3 \AA$^{-1}$, reaction field with an infinite dielectric constant, and an unmodified cutoff.  Group-based cutoffs with a fifth-order polynomial switching function were necessary for the reaction field simulations and were utilized in the SP, SF, and pure cutoff methods for comparison to the standard lack of group-based cutoffs with a hard truncation.  
+
+Generation of the system configurations was dependent on the system type.  For the solid and liquid water configurations, configuration snapshots were taken at regular intervals from higher temperature 1000 SPC/E water molecule trajectories and individually equilibrated.  The solid and liquid NaCl systems consisted of 500 Na+ and 500 Cl- ions and were selected and equilibrated in the same fashion as the water systems.  For the 1 and 10 M NaCl solutions, 4 and 40 ions, respectively, were first solvated in a 1000 water molecule boxes.  Ion and water positions were then randomly swapped, and the resulting configurations were again individually equilibrated.  Finally, for the Argon/Water "charge void" systems, the identities of all the SPC/E waters within 6 \AA\ of the center of the equilibrated water configurations were converted to argon (Fig. \ref{argonSlice}).
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./slice.pdf}
+\caption{A slice from the center of a water box used in a charge void simulation.  The darkened region represents the boundary sphere within which the water molecules were converted to argon atoms.}
+\label{argonSlice}
+\end{figure}
+
+All of these comparisons were performed with three different cutoff radii (9, 12, and 15 \AA) to investigate the cutoff radius dependence of the various techniques.  It should be noted that the damping parameter chosen in SPME, or so called ``Ewald Coefficient", has a significant effect on the energies and forces calculated.  Typical molecular mechanics packages default this to a value dependent on the cutoff radius and a tolerance (typically less than $1 \times 10^{-5}$ kcal/mol).  We chose a tolerance of $1 \times 10^{-8}$, resulting in Ewald Coefficients of 0.4200, 0.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ respectively.
+
 \section{Results and Discussion}
 
+In order to evaluate the performance of the adapted Wolf Shifted Potential and Shifted Force electrostatic summation methods for Monte Carlo simulations, the energy differences between configurations need to be compared to the results using SPME.  Considering the SPME results to be the correct or desired behavior, ideal performance of a tested method is taken to be agreement between the energy differences calculated.  Linear least squares regression of the $\Delta$E values between configurations using SPME against $\Delta$E values using tested methods provides a quantitative comparison of this agreement.  Unitary results for both the correlation and correlation coefficient for these regressions indicate equivalent energetic results between the methods.  The correlation is the slope of the plotted data while the correlation coefficient ($R^2$) is a measure of the of the data scatter around the fitted line and gives an idea of the quality of the fit (Fig. \ref{linearFit}).
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./linearFit.pdf}
+\caption{Example least squares regression of the $\Delta$E between configurations for the SF method against SPME in the pure water system.  }
+\label{linearFit}
+\end{figure}
+
+With 500 independent configurations, 124,750 $\Delta$E data points are used in a regression of a single system.  Results and discussion for the individual analysis of each of the system types appear in the appendices of this paper.  To probe the applicability of each method in the general case, all the different system types were included in a single regression.  The results for this regression are shown in figure \ref{delE}.  
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./delEplot.pdf}
+\caption{The results from the statistical analysis of the $\Delta$E results for all the system types at 9 \AA\ (${\bullet}$), 12 \AA\ ($\blacksquare$), and 15 \AA\ ($\blacktriangledown$) cutoff radii.  Results close to a value of 1 (dashed line) indicate $\Delta$E values from that particular method (listed on the left) are nearly indistinguishable from those obtained from SPME.  Reaction Field results do not include NaCl crystal or melt configurations.}
+\label{delE}
+\end{figure}
+
+In figure \ref{delE}, it is readily apparent that it is unreasonable to expect realistic results using an unmodified cutoff.  This is not all that surprising since this results in large energy fluctuations as atoms move in and out of the cutoff radius.  These fluctuations can be alleviated to some degree by using group based cutoffs with a switching function.  The Group Switch Cutoff row doesn't show a significant improvement in this plot because the salt and salt solution systems contain non-neutral groups, see appendices \ref{app-water} and \ref{app-ice} for a comparison where all groups are neutral.  Correcting the resulting charged cutoff sphere is one of the purposes of the shifted potential proposed by Wolf \textit{et al.}, and this correction indeed improves the results as seen in the Shifted Potental rows.  While the undamped case of this method is a significant improvement over the pure cutoff, it still doesn't correlate that well with SPME.  Inclusion of potential damping improves the results, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows an excellent correlation and quality of fit with the SPME results, particularly with a cutoff radius greater than 12 \AA .  Use of a larger damping parameter is more helpful for the shortest cutoff shown, but it has a detrimental effect on simulations with larger cutoffs.  This trend is repeated in the Shifted Force rows, where increasing damping results in progressively poorer correlation; however, damping looks to be unnecessary with this method.  Overall, the undamped case is the best performing set, as the correlation and quality of fits are consistently superior regardless of the cutoff distance.  This result is beneficial in that the undamped case is less computationally prohibitive do to the lack of complimentary error function calculation when performing the electrostatic pair interaction.  The reaction field results illustrates some of that method's limitations, primarily that it was developed for use in homogenous systems; although it does provide results that are an improvement over those from an unmodified cutoff.
+
+While studying the energy differences provides insight into how comparable these methods are energetically, if we want to use these methods in Molecular Dynamics simulations, we also need to consider their effect on forces and torques.  Both the magnitude and the direction of the force and torque vectors of each of the bodies in the system can be compared to those observed while using SPME.  Analysis of the magnitude of these vectors can be performed in the manner described previously for comparing $\Delta$E values, only instead of a single value between two system configurations, there is a value for each particle in each configuration.  For a system of 1000 water molecules and 40 ions, there are 1040 force vectors and 1000 torque vectors.  With 500 configurations, this results in excess of 500,000 data samples for each system type.  Figures \ref{frcMag} and \ref{trqMag} respectively show the force and torque vector magnitude results for the accumulated analysis over all the system types.
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./frcMagplot.pdf}
+\caption{The results from the statistical analysis of the force vector magnitude results for all the system types at 9 \AA\ (${\bullet}$), 12 \AA\ ($\blacksquare$), and 15 \AA\ ($\blacktriangledown$) cutoff radii.  Results close to a value of 1 (dashed line) indicate force vector magnitude values from that particular method (listed on the left) are nearly indistinguishable from those obtained from SPME.}
+\label{frcMag}
+\end{figure}
+
+The results in figure \ref{frcMag} for the most part parallel those seen in the previous look at the $\Delta$E results.  The unmodified cutoff results are poor, but using group based cutoffs and a switching function provides a improvement much more significant than what was seen with $\Delta$E.  Looking at the Shifted Potential sets, the slope and R$^2$ improve with the use of damping to an optimal result of 0.2 \AA $^{-1}$ for the 12 and 15 \AA\ cutoffs.  Further increases in damping, while beneficial for simulations with a cutoff radius of 9 \AA\ , is detrimental to simulations with larger cutoff radii.  The undamped Shifted Force method gives forces in line with those obtained using SPME, and use of a damping function gives little to no gain.  The reaction field results are surprisingly good, considering the poor quality of the fits for the $\Delta$E results.  There is still a considerable degree of scatter in the data, but it correlates well in general.
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./trqMagplot.pdf}
+\caption{The results from the statistical analysis of the torque vector magnitude results for all the system types at 9 \AA\ (${\bullet}$), 12 \AA\ ($\blacksquare$), and 15 \AA\ ($\blacktriangledown$) cutoff radii.  Results close to a value of 1 (dashed line) indicate torque vector magnitude values from that particular method (listed on the left) are nearly indistinguishable from those obtained from SPME.  Torques are only accumulated on the rigid water molecules, so these results exclude NaCl the systems.}
+\label{trqMag}
+\end{figure}
+
+The torque vector magnitude results in figure \ref{trqMag} are similar to those seen for the forces, but more clearly show the improved behavior with increasing cutoff radius.  Moderate damping is beneficial to the Shifted Potential and unnecessary with the Shifted Force method, and they also show that over-damping adversely effects all cutoff radii rather than showing an improvement for systems with short cutoffs.  The reaction field method performs well when calculating the torques, better than the Shifted Force method over this limited data set.
+
+Having force and torque vectors with magnitudes that are well correlated to SPME is good, but if they are not pointing in the proper direction the results will be incorrect.
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./gaussFit.pdf}
+\caption{Example fitting of the angular distribution of the force vectors over all of the studied systems.  The solid and dotted lines show Gaussian and Voigt fits of the distribution data respectively.  Even though the Voigt profile make for a more accurate fit, the Gaussian was used due to more versatile statistical results.}
+\label{gaussian}
+\end{figure}
+
+
+
+\begin{figure}
+\centering
+\includegraphics[width=3.25in]{./frcTrqAngplot.pdf}
+\caption{The results from the statistical analysis of the force and torque vector angular distributions for all the system types at 9 \AA\ (${\bullet}$), 12 \AA\ ($\blacksquare$), and 15 \AA\ ($\blacktriangledown$) cutoff radii.  Plotted values are the variance ($\sigma^2$) of the Gaussian non-linear fits.  Results close to a value of 0 (dashed line) indicate force or torque vector directions from that particular method (listed on the left) are nearly indistinguishable from those obtained from SPME.  Torques are only accumulated on the rigid water molecules, so the torque vector angle results exclude NaCl the systems.}
+\label{frcTrqAng}
+\end{figure}
+
+
+
 \section{Conclusions}
 
 \section{Acknowledgments}
 
-\newpage
+\appendix
 
+\section{\label{app-water}Liquid Water}
+
+\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
+
+\section{\label{app-melt}NaCl Melt}
+
+\section{\label{app-salt}NaCl Crystal}
+
+\section{\label{app-sol1}1M NaCl Solution}
+
+\section{\label{app-sol10}10M NaCl Solution}
+
+\section{\label{app-argon}Argon Sphere in Water}
+
+\newpage 
+
 \bibliographystyle{achemso}
 \bibliography{electrostaticMethods}
 
