--- trunk/electrostaticMethodsPaper/SupportingInfo.tex	2006/03/23 05:59:41	2660
+++ trunk/electrostaticMethodsPaper/SupportingInfo.tex	2006/03/24 17:28:09	2670
@@ -1,5 +1,5 @@
 %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
-\documentclass[12pt]{article}
+\documentclass[11pt]{article}
 %\usepackage{endfloat}
 \usepackage{amsmath}
 \usepackage{amssymb}
@@ -23,22 +23,24 @@
 
 \begin{document}
 
-This document includes individual system-based comparisons of the
-studied methods with smooth particle-mesh Ewald.  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.
+This document includes comparisons of the new pairwise electrostatic
+methods with {\sc spme} for each of the individual systems mentioned
+in paper. Each of the seven sections contains information about a
+single system type and has its own discussion and tabular listing of
+the results for the comparisons of $\Delta E$, the magnitudes of the
+forces and torques, and directionality of the force and torque
+vectors.
 
 \section{\label{app:water}Liquid Water}
 
-500 liquid state configurations were generated as described in the
-Methods section 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
+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. 
+investigated.
 \begin{table}[htbp]
    \centering
    \caption{Regression results for the liquid water system. Tabulated
@@ -137,47 +139,48 @@ GSSF  & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 
    \label{tab:spceAng}
 \end{table}
 
-For the most parts, the water results appear to parallel the combined
-results seen in the discussion in the main paper.  There is good
-agreement with 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. The results for both of these methods also
-begin to decay as damping gets too large.
+The water results 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 (PC) method performs poorly, as seen in the main
-discussion section.  In contrast to the combined values, however, the
-use of a switching function and group based cutoffs really improves
-the results for these neutral water molecules.  The group switched
-cutoff (GSC) shows mimics the energetics of SPME more poorly than the
-{\sc sp} (with moderate damping) and {\sc sf} methods, but the
-dynamics are quite good.  The switching functions corrects
-discontinuities in the potential and forces, leading to the improved
-results.  Such improvements with the use of a switching function has
-been recognized in previous studies,\cite{Andrea83,Steinbach94} and it
-is a useful tactic for stably incorporating local area electrostatic
-effects.
+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 (RF) method simply extends the results observed in
-the GSC case.  Both methods are similar in form (i.e. neutral groups,
-switching function), but RF incorporates an added effect from the
-external dielectric. This similarity translates into the same good
-dynamic results and improved energetic results.  These still fall
-short of the moderately damped {\sc sp} and {\sc sf} methods, but they
-display how incorporating some implicit properties of the surroundings
-(i.e. $\epsilon_\textrm{S}$) can improve results.
+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.
 
-A final note for the liquid water system, use of group cutoffs and a
-switching function also 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}).  {\sc sp} shows significant
-narrowing of the angle distribution in the cases with little to no
-damping and only modest improvement for the ideal conditions ($\alpha$
-= 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA).  The {\sc sf}
-method simply shows modest narrowing across all damping and cutoff
-ranges of interest.  Group cutoffs and the switching function do
-nothing for cases were error is introduced by overdamping the
-potentials.
+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$_\textrm{c}$}
 
@@ -283,19 +286,19 @@ GSSF  & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 
    \label{tab:iceAng}
 \end{table}
 
-Highly ordered systems are a difficult test for the pairwise systems
-in that they lack the periodicity inherent to the Ewald summation.  As
-expected, the energy gap agreement with SPME reduces for the {\sc sp}
-and {\sc sf} with parameters that were perfectly acceptable for the
-disordered liquid system.  Moving to higher $R_\textrm{c}$ remedies
-this degraded performance, though at increase in computational cost.
-However, 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. 
+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 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 RF exhibits a slightly improved energy
-gap results over the liquid water system.  One possible explanation is
+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
@@ -305,13 +308,11 @@ A high temperature NaCl melt was tested to gauge the a
 \section{\label{app:melt}NaCl Melt}
 
 A high temperature NaCl melt was tested to gauge the accuracy of the
-pairwise summation methods in a highly charge disordered system. The
-results for the energy gap comparisons and the force and torque vector
-magnitude comparisons are shown in table \ref{tab:melt}.  The force
-and torque vector directionality results are displayed separately in
-table \ref{tab:meltAng}, where the effect of group-based cutoffs and
-switching functions on the {\sc sp} and {\sc sf} potentials are
-investigated. 
+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
@@ -390,11 +391,9 @@ particles. The results for the energy gap comparisons 
 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
-and torque vector magnitude comparisons are shown in table
-\ref{tab:salt}.  The force and torque vector directionality results
-are displayed separately in table \ref{tab:saltAng}, where the effect
-of group-based cutoffs and switching functions on the {\sc sp} and
-{\sc sf} potentials are investigated. 
+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
@@ -470,18 +469,16 @@ gap, force magnitude, and force directionality tests. 
 {\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
+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 good long-time dynamics results observed for this
-system. Since these results are a test of instantaneous dynamics, this
-indicates that good long-time dynamics comes in part at the expense of
-short-time dynamics. Further indication of this comes from the full
-power spectra shown in the main text. It appears as though a
-distortion is introduced between 200 to 300 cm$^{-1}$ with increased
-$\alpha$.
+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}
 
@@ -502,9 +499,8 @@ set).  PC = Pure Cutoff, SP = Shifted Potential, SF = 
 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.}	 
+GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon
+\approx \infty$).}	 
    \begin{tabular}{@{} ccrrrrrr @{}}
       \\
       \toprule
@@ -595,16 +591,15 @@ GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 
    \label{tab:solnWeakAng}
 \end{table}
 
-This weak ionic strength system can be considered as a perturbation of
-the pure liquid water system. 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 visible for the {\sc rf} method. Though good force
-reproduction is still maintained, the energy gaps show a significant
-increase in the data scatter. This foreshadows the breakdown of the
-method as we introduce system inhomogeneities.
+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 scatter of the data.
 
 \section{\label{app:solnStr}Strong NaCl Solution}
 
@@ -613,8 +608,8 @@ torque vector magnitude comparisons are shown in table
 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:solnWeak}.  The force and torque vector directionality
-results are displayed separately in table\ref{tab:solnWeakAng}, where
+\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.
 
@@ -712,10 +707,10 @@ The {\sc rf} method struggles with the jump in ionic s
 \end{table}
 
 The {\sc rf} method struggles with the jump in ionic strength. The
-configuration energy difference degrade to unuseable levels while the
-forces and torques degrade in a more modest fashion. The {\sc rf}
-method was designed for homogeneous systems, and this restriction is
-apparent in these results.
+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
@@ -725,17 +720,17 @@ damping for the {\sc sp} method, both with cutoffs gre
 
 \section{\label{app:argon}Argon Sphere in Water}
 
-The final model system studied was 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:solnWeak}.  The force and torque vector directionality
-results are displayed separately in table \ref{tab:solnWeakAng}, where
+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
+   \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
@@ -794,7 +789,7 @@ distributions of the force and torque vectors in the 6
    \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
+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.} 	
@@ -831,7 +826,17 @@ GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 
    \label{tab:argonAng}
 \end{table}
 
-This system appears not to show in any significant deviation in the previously observed results. The {\sc sp} and {\sc sf} methods give result qualities similar to those observed in section \ref{app:water}. The only significant difference is the improvement for 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, which the {\sc sp} and {\sc sf} methods explicity enforce, seems to play a greater role in the stability of the {\sc rf} method than the necessity of a homogeneous environment.
+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 
 
