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1   %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
2 < \documentclass[12pt]{article}
3 < \usepackage{endfloat}
2 > \documentclass[11pt]{article}
3 > %\usepackage{endfloat}
4   \usepackage{amsmath}
5   \usepackage{amssymb}
6   \usepackage{epsf}
# Line 23 | Line 23 | This document includes system based comparisons of the
23  
24   \begin{document}
25  
26 < 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.
26 > This document includes individual system-based comparisons of the
27 > studied methods with smooth particle mesh Ewald {\sc spme}.  Each of
28 > the seven systems comprises its own section and has its own discussion
29 > and tabular listing of the results for the $\Delta E$, force and
30 > torque vector magnitude, and force and torque vector direction
31 > comparisons.
32  
33 < \section{\label{app-water}Liquid Water}
33 > \section{\label{app:water}Liquid Water}
34  
35 < 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 switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
35 > The first system considered was liquid water at 300K using the SPC/E
36 > model of water.\cite{Berendsen87} The results for the energy gap
37 > comparisons and the force and torque vector magnitude comparisons are
38 > shown in table \ref{tab:spce}.  The force and torque vector
39 > directionality results are displayed separately in table
40 > \ref{tab:spceAng}, where the effect of group-based cutoffs and
41 > switching functions on the {\sc sp} and {\sc sf} potentials are
42 > investigated.
43   \begin{table}[htbp]
44     \centering
45 <   \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 $\varepsilon \approx \infty$).}  
45 >   \caption{Regression results for the liquid water system. Tabulated
46 > results include $\Delta E$ values (top set), force vector magnitudes
47 > (middle set) and torque vector magnitudes (bottom set).  PC = Pure
48 > Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
49 > Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
50 > \infty$).}      
51     \begin{tabular}{@{} ccrrrrrr @{}}
52        \\
53        \toprule
# Line 50 | Line 67 | RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.
67      & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
68      & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
69   GSC &     & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
70 < RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                              
54 <
70 > RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                
71              \midrule
56 –
72   PC  &     & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
73   SP  & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
74      & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
# Line 65 | Line 80 | RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.
80      & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
81   GSC &     & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
82   RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\          
68 –
83              \midrule
70 –
84   PC  &     & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
85   SP  & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
86      & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
# Line 86 | Line 99 | RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.
99  
100   \begin{table}[htbp]
101     \centering
102 <   \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.}  
102 >   \caption{Variance results from Gaussian fits to angular
103 > distributions of the force and torque vectors in the liquid water
104 > system.  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
105 > GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
106 > \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
107 > Group Switched Shifted Force.}  
108     \begin{tabular}{@{} ccrrrrrr @{}}
109        \\
110        \toprule
# Line 120 | Line 138 | For the most parts, the water results appear to parall
138     \label{tab:spceAng}
139   \end{table}
140  
141 < 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.
141 > The water results appear to parallel the combined results seen in the
142 > discussion section of the main paper.  There is good agreement with
143 > {\sc spme} in both energetic and dynamic behavior when using the {\sc sf}
144 > method with and without damping. The {\sc sp} method does well with an
145 > $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater
146 > than 12 \AA. Overdamping the electrostatics reduces the agreement between both these methods and {\sc spme}.
147  
148 < 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.
148 > The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
149 > observations in the main portion of this paper.  In contrast to the
150 > combined values, however, the use of a switching function and group
151 > based cutoffs greatly improves the results for these neutral water
152 > molecules.  The group switched cutoff ({\sc gsc}) does not mimic the
153 > energetics of {\sc spme} as well as the {\sc sp} (with moderate
154 > damping) and {\sc sf} methods, but the dynamics are quite good.  The
155 > switching functions correct discontinuities in the potential and
156 > forces, leading to these improved results.  Such improvements with the
157 > use of a switching function have been recognized in previous
158 > studies,\cite{Andrea83,Steinbach94} and this proves to be a useful
159 > tactic for stably incorporating local area electrostatic effects.
160  
161 < 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.
161 > The reaction field ({\sc rf}) method simply extends upon the results
162 > observed in the {\sc gsc} case.  Both methods are similar in form
163 > (i.e. neutral groups, switching function), but {\sc rf} incorporates
164 > an added effect from the external dielectric. This similarity
165 > translates into the same good dynamic results and improved energetic
166 > agreement with {\sc spme}.  Though this agreement is not to the level
167 > of the moderately damped {\sc sp} and {\sc sf} methods, these results
168 > show how incorporating some implicit properties of the surroundings
169 > (i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction.
170  
171 < 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.
172 <
173 < \section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
171 > As a final note for the liquid water system, use of group cutoffs and a
172 > switching function leads to noticeable improvements in the {\sc sp}
173 > and {\sc sf} methods, primarily in directionality of the force and
174 > torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows
175 > significant narrowing of the angle distribution when using little to
176 > no damping and only modest improvement for the recommended conditions
177 > ($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA).  The
178 > {\sc sf} method shows modest narrowing across all damping and cutoff
179 > ranges of interest.  When overdamping these methods, group cutoffs and
180 > the switching function do not improve the force and torque
181 > directionalities.
182  
183 < 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.
183 > \section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$}
184  
185 + In addition to the disordered molecular system above, the ordered
186 + molecular system of ice I$_\textrm{c}$ was also considered. The
187 + results for the energy gap comparisons and the force and torque vector
188 + magnitude comparisons are shown in table \ref{tab:ice}.  The force and
189 + torque vector directionality results are displayed separately in table
190 + \ref{tab:iceAng}, where the effect of group-based cutoffs and
191 + switching functions on the {\sc sp} and {\sc sf} potentials are
192 + investigated.
193 +
194   \begin{table}[htbp]
195     \centering
196 <   \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 $\varepsilon \approx \infty$).}    
196 >   \caption{Regression results for the ice I$_\textrm{c}$
197 > system. Tabulated results include $\Delta E$ values (top set), force
198 > vector magnitudes (middle set) and torque vector magnitudes (bottom
199 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
200 > GSC = Group Switched Cutoff, and RF = Reaction Field (where
201 > $\varepsilon \approx \infty$).}  
202     \begin{tabular}{@{} ccrrrrrr @{}}
203        \\
204        \toprule
# Line 220 | Line 284 | Highly ordered systems are a difficult test for the pa
284     \label{tab:iceAng}
285   \end{table}
286  
287 < 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.
287 > Highly ordered systems are a difficult test for the pairwise methods
288 > in that they lack the periodicity term of the Ewald summation.  As
289 > expected, the energy gap agreement with {\sc spme} is reduced for the
290 > {\sc sp} and {\sc sf} methods with parameters that were acceptable for
291 > the disordered liquid system.  Moving to higher $R_\textrm{c}$ helps
292 > improve the agreement, though at an increase in computational cost.
293 > The dynamics of this crystalline system (both in magnitude and
294 > direction) are little affected. Both methods still reproduce the Ewald
295 > behavior with the same parameter recommendations from the previous
296 > section.
297  
298 < It is also worth noting that RF exhibits a slightly 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.
298 > It is also worth noting that {\sc rf} exhibits improved energy gap
299 > results over the liquid water system.  One possible explanation is
300 > that the ice I$_\textrm{c}$ crystal is ordered such that the net
301 > dipole moment of the crystal is zero.  With $\epsilon_\textrm{S} =
302 > \infty$, the reaction field incorporates this structural organization
303 > by actively enforcing a zeroed dipole moment within each cutoff
304 > sphere.  
305  
306 < \section{\label{app-melt}NaCl Melt}
306 > \section{\label{app:melt}NaCl Melt}
307  
308 < 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.
308 > A high temperature NaCl melt was tested to gauge the accuracy of the
309 > pairwise summation methods in a charged disordered system. The results
310 > for the energy gap comparisons and the force vector magnitude
311 > comparisons are shown in table \ref{tab:melt}.  The force vector
312 > directionality results are displayed separately in table
313 > \ref{tab:meltAng}.
314  
315   \begin{table}[htbp]
316     \centering
# Line 289 | Line 373 | The molten NaCl system shows the a
373     \label{tab:meltAng}
374   \end{table}
375  
376 < The molten NaCl system shows the a
376 > The molten NaCl system shows more sensitivity to the electrostatic
377 > damping than the water systems. The most noticeable point is that the
378 > undamped {\sc sf} method does very well at replicating the {\sc spme}
379 > configurational energy differences and forces. Light damping appears
380 > to minimally improve the dynamics, but this comes with a deterioration
381 > of the energy gap results. In contrast, this light damping improves
382 > the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
383 > damping reduce the agreement with {\sc spme} for both methods. From
384 > these observations, the undamped {\sc sf} method is the best choice
385 > for disordered systems of charges.
386  
387 < \section{\label{app-salt}NaCl Crystal}
387 > \section{\label{app:salt}NaCl Crystal}
388  
389 < 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.
389 > A 1000K NaCl crystal was used to investigate the accuracy of the
390 > pairwise summation methods in an ordered system of charged
391 > particles. The results for the energy gap comparisons and the force
392 > vector magnitude comparisons are shown in table \ref{tab:salt}.  The
393 > force vector directionality results are displayed separately in table
394 > \ref{tab:saltAng}.
395  
396   \begin{table}[htbp]
397     \centering
398 <   \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.}    
398 >   \caption{Regression results for the crystalline NaCl
399 > system. Tabulated results include $\Delta E$ values (top set) and
400 > force vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted
401 > Potential, and SF = Shifted Force.}    
402     \begin{tabular}{@{} ccrrrrrr @{}}
403        \\
404        \toprule
# Line 333 | Line 434 | SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.
434  
435   \begin{table}[htbp]
436     \centering
437 <   \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 $\varepsilon \approx \infty$).}        
437 >   \caption{Variance results from Gaussian fits to angular
438 > distributions of the force vectors in the crystalline NaCl system.  PC
439 > = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
440 > Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
441 > \infty$).}      
442     \begin{tabular}{@{} ccrrrrrr @{}}
443        \\
444        \toprule
# Line 356 | Line 461 | SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
461     \label{tab:saltAng}
462   \end{table}
463  
464 < \section{\label{app-sol1}Weak NaCl Solution}
464 > The crystalline NaCl system is the most challenging test case for the
465 > pairwise summation methods, as evidenced by the results in tables
466 > \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
467 > {\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best
468 > choices. These methods match well with {\sc spme} across the energy
469 > gap, force magnitude, and force directionality tests.  The {\sc sp}
470 > method struggles in all cases, with the exception of good dynamics
471 > reproduction when using weak electrostatic damping with a large cutoff
472 > radius.
473  
474 < 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$0.11 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}.  The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
474 > The moderate electrostatic damping case is not as good as we would
475 > expect given the long-time dynamics results observed for this
476 > system. Since the data tabulated in tables \ref{tab:salt} and
477 > \ref{tab:saltAng} are a test of instantaneous dynamics, this indicates
478 > that good long-time dynamics comes in part at the expense of
479 > short-time dynamics.
480  
481 + \section{\label{app:solnWeak}Weak NaCl Solution}
482 +
483 + In an effort to bridge the charged atomic and neutral molecular
484 + systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
485 + the liquid water system. This low ionic strength system consists of 4
486 + ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
487 + for the energy gap comparisons and the force and torque vector
488 + magnitude comparisons are shown in table \ref{tab:solnWeak}.  The
489 + force and torque vector directionality results are displayed
490 + separately in table \ref{tab:solnWeakAng}, where the effect of
491 + group-based cutoffs and switching functions on the {\sc sp} and {\sc
492 + sf} potentials are investigated.
493 +
494   \begin{table}[htbp]
495     \centering
496 <   \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.}      
496 >   \caption{Regression results for the weak NaCl solution
497 > system. Tabulated results include $\Delta E$ values (top set), force
498 > vector magnitudes (middle set) and torque vector magnitudes (bottom
499 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
500 > GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon
501 > \approx \infty$).}      
502     \begin{tabular}{@{} ccrrrrrr @{}}
503        \\
504        \toprule
# Line 414 | Line 550 | RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.
550  
551   \begin{table}[htbp]
552     \centering
553 <   \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.}    
553 >   \caption{Variance results from Gaussian fits to angular
554 > distributions of the force and torque vectors in the weak NaCl
555 > solution system.  PC = Pure Cutoff, SP = Shifted Potential, SF =
556 > Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where
557 > $\varepsilon \approx \infty$), GSSP = Group Switched Shifted
558 > Potential, and GSSF = Group Switched Shifted Force.}    
559     \begin{tabular}{@{} ccrrrrrr @{}}
560        \\
561        \toprule
# Line 448 | Line 589 | GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 &
589     \label{tab:solnWeakAng}
590   \end{table}
591  
592 < \section{\label{app-sol10}Strong NaCl Solution}
592 > Because this system is a perturbation of the pure liquid water system,
593 > comparisons are best drawn between these two sets. The {\sc sp} and
594 > {\sc sf} methods are not significantly affected by the inclusion of a
595 > few ions. The aspect of cutoff sphere neutralization aids in the
596 > smooth incorporation of these ions; thus, all of the observations
597 > regarding these methods carry over from section \ref{app:water}. The
598 > differences between these systems are more visible for the {\sc rf}
599 > method. Though good force agreement is still maintained, the energy
600 > gaps show a significant increase in the data scatter. This foreshadows
601 > the breakdown of the method as we introduce charged inhomogeneities.
602  
603 < 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$1.1 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}.  The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
603 > \section{\label{app:solnStr}Strong NaCl Solution}
604  
605 + The bridging of the charged atomic and neutral molecular systems was
606 + further developed by considering a high ionic strength system
607 + consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
608 + M). The results for the energy gap comparisons and the force and
609 + torque vector magnitude comparisons are shown in table
610 + \ref{tab:solnStr}.  The force and torque vector directionality
611 + results are displayed separately in table \ref{tab:solnStrAng}, where
612 + the effect of group-based cutoffs and switching functions on the {\sc
613 + sp} and {\sc sf} potentials are investigated.
614 +
615   \begin{table}[htbp]
616     \centering
617 <   \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 $\varepsilon \approx \infty$).}  
617 >   \caption{Regression results for the strong NaCl solution
618 > system. Tabulated results include $\Delta E$ values (top set), force
619 > vector magnitudes (middle set) and torque vector magnitudes (bottom
620 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
621 > GSC = Group Switched Cutoff, and RF = Reaction Field (where
622 > $\varepsilon \approx \infty$).}        
623     \begin{tabular}{@{} ccrrrrrr @{}}
624        \\
625        \toprule
# Line 540 | Line 705 | GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 &
705     \label{tab:solnStrAng}
706   \end{table}
707  
708 < \section{\label{app-argon}Argon Sphere in Water}
708 > The {\sc rf} method struggles with the jump in ionic strength. The
709 > configuration energy differences degrade to unusable levels while the
710 > forces and torques show a more modest reduction in the agreement with
711 > {\sc spme}. The {\sc rf} method was designed for homogeneous systems,
712 > and this attribute is apparent in these results.
713  
714 < 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 effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
714 > The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
715 > their agreement with {\sc spme}. With these results, we still
716 > recommend no to moderate damping for the {\sc sf} method and moderate
717 > damping for the {\sc sp} method, both with cutoffs greater than 12
718 > \AA.
719  
720 + \section{\label{app:argon}Argon Sphere in Water}
721 +
722 + The final model system studied was a 6 \AA\ sphere of Argon solvated
723 + by SPC/E water. The results for the energy gap comparisons and the
724 + force and torque vector magnitude comparisons are shown in table
725 + \ref{tab:argon}.  The force and torque vector directionality
726 + results are displayed separately in table \ref{tab:argonAng}, where
727 + the effect of group-based cutoffs and switching functions on the {\sc
728 + sp} and {\sc sf} potentials are investigated.
729 +
730   \begin{table}[htbp]
731     \centering
732 <   \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 $\varepsilon \approx \infty$).}    
732 >   \caption{Regression results for the 6 \AA\ Argon sphere in liquid
733 > water system. Tabulated results include $\Delta E$ values (top set),
734 > force vector magnitudes (middle set) and torque vector magnitudes
735 > (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted
736 > Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where
737 > $\varepsilon \approx \infty$).}        
738     \begin{tabular}{@{} ccrrrrrr @{}}
739        \\
740        \toprule
# Line 598 | Line 786 | RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.
786  
787   \begin{table}[htbp]
788     \centering
789 <   \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.}
789 >   \caption{Variance results from Gaussian fits to angular
790 > distributions of the force and torque vectors in the 6 \AA\ sphere of
791 > Argon in liquid water system.  PC = Pure Cutoff, SP = Shifted
792 > Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF =
793 > Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group
794 > Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}  
795     \begin{tabular}{@{} ccrrrrrr @{}}
796        \\
797        \toprule
# Line 632 | Line 825 | GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 &
825     \label{tab:argonAng}
826   \end{table}
827  
828 + This system does not appear to show any significant deviations from
829 + the previously observed results. The {\sc sp} and {\sc sf} methods
830 + have aggrements similar to those observed in section
831 + \ref{app:water}. The only significant difference is the improvement
832 + in the configuration energy differences for the {\sc rf} method. This
833 + is surprising in that we are introducing an inhomogeneity to the
834 + system; however, this inhomogeneity is charge-neutral and does not
835 + result in charged cutoff spheres. The charge-neutrality of the cutoff
836 + spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce,
837 + seems to play a greater role in the stability of the {\sc rf} method
838 + than the required homogeneity of the environment.
839 +
840   \newpage
841  
842   \bibliographystyle{jcp2}
843   \bibliography{electrostaticMethods}
844  
845 < \end{document}
845 > \end{document}

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