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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 really 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 corrects discontinuities in the potential and
156 > forces, leading to these improved results.  Such improvements with the
157 > use of a switching function has 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 > 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} reduces 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 and torque vector
311 > magnitude comparisons are shown in table \ref{tab:melt}.  The force
312 > and torque vector directionality results are displayed separately in
313 > table \ref{tab:meltAng}, where the effect of group-based cutoffs and
314 > switching functions on the {\sc sp} and {\sc sf} potentials are
315 > investigated.
316  
317   \begin{table}[htbp]
318     \centering
# Line 289 | Line 375 | The molten NaCl system shows the a
375     \label{tab:meltAng}
376   \end{table}
377  
378 < The molten NaCl system shows the a
378 > The molten NaCl system shows more sensitivity to the electrostatic
379 > damping than the water systems. The most noticeable point is that the
380 > undamped {\sc sf} method does very well at replicating the {\sc spme}
381 > configurational energy differences and forces. Light damping appears
382 > to minimally improve the dynamics, but this comes with a deterioration
383 > of the energy gap results. In contrast, this light damping improves
384 > the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic
385 > damping reduce the agreement with {\sc spme} for both methods. From
386 > these observations, the undamped {\sc sf} method is the best choice
387 > for disordered systems of charges.
388  
389 < \section{\label{app-salt}NaCl Crystal}
389 > \section{\label{app:salt}NaCl Crystal}
390  
391 < 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.
391 > A 1000K NaCl crystal was used to investigate the accuracy of the
392 > pairwise summation methods in an ordered system of charged
393 > particles. The results for the energy gap comparisons and the force
394 > and torque vector magnitude comparisons are shown in table
395 > \ref{tab:salt}.  The force and torque vector directionality results
396 > are displayed separately in table \ref{tab:saltAng}, where the effect
397 > of group-based cutoffs and switching functions on the {\sc sp} and
398 > {\sc sf} potentials are investigated.
399  
400   \begin{table}[htbp]
401     \centering
402 <   \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.}    
402 >   \caption{Regression results for the crystalline NaCl
403 > system. Tabulated results include $\Delta E$ values (top set) and
404 > force vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted
405 > Potential, and SF = Shifted Force.}    
406     \begin{tabular}{@{} ccrrrrrr @{}}
407        \\
408        \toprule
# Line 333 | Line 438 | SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.
438  
439   \begin{table}[htbp]
440     \centering
441 <   \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 $\varepsilon \approx \infty$).}        
441 >   \caption{Variance results from Gaussian fits to angular
442 > distributions of the force vectors in the crystalline NaCl system.  PC
443 > = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
444 > Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
445 > \infty$).}      
446     \begin{tabular}{@{} ccrrrrrr @{}}
447        \\
448        \toprule
# Line 356 | Line 465 | SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
465     \label{tab:saltAng}
466   \end{table}
467  
468 < \section{\label{app-sol1}Weak NaCl Solution}
468 > The crystalline NaCl system is the most challenging test case for the
469 > pairwise summation methods, as evidenced by the results in tables
470 > \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
471 > {\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best
472 > choices. These methods match well with {\sc spme} across the energy
473 > gap, force magnitude, and force directionality tests.  The {\sc sp}
474 > method struggles in all cases, with the exception of good dynamics
475 > reproduction when using weak electrostatic damping with a large cutoff
476 > radius.
477  
478 < 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$0.11 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}.  The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
478 > The moderate electrostatic damping case is not as good as we would
479 > expect given the good long-time dynamics results observed for this
480 > system. Since the data tabulated in table \ref{tab:salt} and
481 > \ref{tab:saltAng} are a test of instantaneous dynamics, this indicates
482 > that good long-time dynamics comes in part at the expense of
483 > short-time dynamics. Further indication of this comes from the full
484 > power spectra shown in the main text. It appears as though a
485 > distortion is introduced between 200 to 350 cm$^{-1}$ with increased
486 > $\alpha$.
487  
488 + \section{\label{app:solnWeak}Weak NaCl Solution}
489 +
490 + In an effort to bridge the charged atomic and neutral molecular
491 + systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
492 + the liquid water system. This low ionic strength system consists of 4
493 + ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
494 + for the energy gap comparisons and the force and torque vector
495 + magnitude comparisons are shown in table \ref{tab:solnWeak}.  The
496 + force and torque vector directionality results are displayed
497 + separately in table \ref{tab:solnWeakAng}, where the effect of
498 + group-based cutoffs and switching functions on the {\sc sp} and {\sc
499 + sf} potentials are investigated.
500 +
501   \begin{table}[htbp]
502     \centering
503 <   \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.}      
503 >   \caption{Regression results for the weak NaCl solution
504 > system. Tabulated results include $\Delta E$ values (top set), force
505 > vector magnitudes (middle set) and torque vector magnitudes (bottom
506 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
507 > GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
508 > \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
509 > Group Switched Shifted Force.}  
510     \begin{tabular}{@{} ccrrrrrr @{}}
511        \\
512        \toprule
# Line 414 | Line 558 | RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.
558  
559   \begin{table}[htbp]
560     \centering
561 <   \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.}    
561 >   \caption{Variance results from Gaussian fits to angular
562 > distributions of the force and torque vectors in the weak NaCl
563 > solution system.  PC = Pure Cutoff, SP = Shifted Potential, SF =
564 > Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where
565 > $\varepsilon \approx \infty$), GSSP = Group Switched Shifted
566 > Potential, and GSSF = Group Switched Shifted Force.}    
567     \begin{tabular}{@{} ccrrrrrr @{}}
568        \\
569        \toprule
# Line 448 | Line 597 | GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 &
597     \label{tab:solnWeakAng}
598   \end{table}
599  
600 < \section{\label{app-sol10}Strong NaCl Solution}
600 > Because this system is a perturbation of the pure liquid water system,
601 > comparisons are best drawn between these two sets. The {\sc sp} and
602 > {\sc sf} methods are not significantly affected by the inclusion of a
603 > few ions. The aspect of cutoff sphere neutralization aids in the
604 > smooth incorporation of these ions; thus, all of the observations
605 > regarding these methods carry over from section \ref{app:water}. The
606 > differences between these systems are more visible for the {\sc rf}
607 > method. Though good force agreement is still maintained, the energy
608 > gaps show a significant increase in the data scatter. This foreshadows
609 > the breakdown of the method as we introduce charged inhomogeneities.
610  
611 < 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$1.1 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}.  The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
611 > \section{\label{app:solnStr}Strong NaCl Solution}
612  
613 + The bridging of the charged atomic and neutral molecular systems was
614 + further developed by considering a high ionic strength system
615 + consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
616 + M). The results for the energy gap comparisons and the force and
617 + torque vector magnitude comparisons are shown in table
618 + \ref{tab:solnWeak}.  The force and torque vector directionality
619 + results are displayed separately in table \ref{tab:solnWeakAng}, where
620 + the effect of group-based cutoffs and switching functions on the {\sc
621 + sp} and {\sc sf} potentials are investigated.
622 +
623   \begin{table}[htbp]
624     \centering
625 <   \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 $\varepsilon \approx \infty$).}  
625 >   \caption{Regression results for the strong NaCl solution
626 > system. Tabulated results include $\Delta E$ values (top set), force
627 > vector magnitudes (middle set) and torque vector magnitudes (bottom
628 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
629 > GSC = Group Switched Cutoff, and RF = Reaction Field (where
630 > $\varepsilon \approx \infty$).}        
631     \begin{tabular}{@{} ccrrrrrr @{}}
632        \\
633        \toprule
# Line 540 | Line 713 | GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 &
713     \label{tab:solnStrAng}
714   \end{table}
715  
716 < \section{\label{app-argon}Argon Sphere in Water}
716 > The {\sc rf} method struggles with the jump in ionic strength. The
717 > configuration energy difference degrade to unusable levels while the
718 > forces and torques show a more modest reduction in the agreement with
719 > {\sc spme}. The {\sc rf} method was designed for homogeneous systems,
720 > and this attribute is apparent in these results.
721  
722 < 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.
722 > The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
723 > their agreement with {\sc spme}. With these results, we still
724 > recommend no to moderate damping for the {\sc sf} method and moderate
725 > damping for the {\sc sp} method, both with cutoffs greater than 12
726 > \AA.
727  
728 + \section{\label{app:argon}Argon Sphere in Water}
729 +
730 + The final model system studied was 6 \AA\ sphere of Argon solvated by
731 + SPC/E water. The results for the energy gap comparisons and the force
732 + and torque vector magnitude comparisons are shown in table
733 + \ref{tab:solnWeak}.  The force and torque vector directionality
734 + results are displayed separately in table \ref{tab:solnWeakAng}, where
735 + the effect of group-based cutoffs and switching functions on the {\sc
736 + sp} and {\sc sf} potentials are investigated.
737 +
738   \begin{table}[htbp]
739     \centering
740 <   \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 $\varepsilon \approx \infty$).}    
740 >   \caption{Regression results for the 6 \AA\ argon sphere in liquid
741 > water system. Tabulated results include $\Delta E$ values (top set),
742 > force vector magnitudes (middle set) and torque vector magnitudes
743 > (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted
744 > Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where
745 > $\varepsilon \approx \infty$).}        
746     \begin{tabular}{@{} ccrrrrrr @{}}
747        \\
748        \toprule
# Line 598 | Line 794 | RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.
794  
795   \begin{table}[htbp]
796     \centering
797 <   \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.}
797 >   \caption{Variance results from Gaussian fits to angular
798 > distributions of the force and torque vectors in the 6 \AA\ sphere of
799 > argon in liquid water system.  PC = Pure Cutoff, SP = Shifted
800 > Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF =
801 > Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group
802 > Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}  
803     \begin{tabular}{@{} ccrrrrrr @{}}
804        \\
805        \toprule
# Line 632 | Line 833 | GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 &
833     \label{tab:argonAng}
834   \end{table}
835  
836 + This system appears not to show in any significant deviation in the
837 + previously observed results. The {\sc sp} and {\sc sf} methods give
838 + result qualities similar to those observed in section
839 + \ref{app:water}. The only significant difference is the improvement
840 + for the configuration energy differences for the {\sc rf} method. This
841 + is surprising in that we are introducing an inhomogeneity to the
842 + system; however, this inhomogeneity is charge-neutral and does not
843 + result in charged cutoff spheres. The charge-neutrality of the cutoff
844 + spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce,
845 + seems to play a greater role in the stability of the {\sc rf} method
846 + than the required homogeneity of the environment.
847 +
848   \newpage
849  
850   \bibliographystyle{jcp2}
851   \bibliography{electrostaticMethods}
852  
853 < \end{document}
853 > \end{document}

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