--- trunk/electrostaticMethodsPaper/SupportingInfo.tex	2006/03/22 21:00:07	2658
+++ trunk/electrostaticMethodsPaper/SupportingInfo.tex	2006/03/23 05:59:41	2660
@@ -29,7 +29,7 @@ vector magnitude, and force and torque vector directio
 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}
 
 500 liquid state configurations were generated as described in the
 Methods section using the SPC/E model of water.\cite{Berendsen87} The
@@ -66,10 +66,8 @@ GSC &     & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 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 \\
@@ -81,9 +79,7 @@ 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 \\
@@ -183,7 +179,7 @@ potentials.
 nothing for cases were error is introduced by overdamping the
 potentials.
 
-\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
+\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
@@ -306,7 +302,7 @@ sphere.  
 by actively enforcing a zeroed dipole moment within each cutoff
 sphere.  
 
-\section{\label{app-melt}NaCl Melt}
+\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
@@ -378,9 +374,18 @@ SF  & 0.0 & 1.693 & 0.603 & 0.256 \\
    \label{tab:meltAng}
 \end{table}
 
-The molten NaCl system shows the a 
+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}
+\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
@@ -459,8 +464,27 @@ SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
    \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 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
+$\alpha$.
+
+\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
@@ -571,17 +595,28 @@ GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 
    \label{tab:solnWeakAng}
 \end{table}
 
-\section{\label{app-sol10}Strong NaCl Solution}
+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.
 
+\section{\label{app:solnStr}Strong NaCl Solution}
+
 The bridging of the charged atomic and neutral molecular systems was
-furthered by considering a high ionic strength system consisting of 40
-ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for
-the energy gap comparisons and the force and torque vector magnitude
-comparisons are shown in table \ref{tab:solnWeak}.  The force and
-torque vector directionality results are displayed separately in table
-\ref{tab:solnWeakAng}, where the effect of group-based cutoffs and
-switching functions on the {\sc sp} and {\sc sf} potentials are
-investigated. 
+further developed by considering a high ionic strength system
+consisting of 40 ions in the 1000 SPC/E water solvent ($\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: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
@@ -676,8 +711,20 @@ GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 &
    \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 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.
 
+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 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
@@ -784,6 +831,8 @@ 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.
+
 \newpage 
 
 \bibliographystyle{jcp2}
