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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 comparisons of the new pairwise electrostatic
27 > methods with {\sc spme} for each of the individual systems mentioned
28 > in paper. Each of the seven sections contains information about a
29 > single system type and has its own discussion and tabular listing of
30 > the results for the comparisons of $\Delta E$, the magnitudes of the
31 > forces and torques, and directionality of the force and torque
32 > vectors.
33  
34 < \section{\label{app-water}Liquid Water}
34 > \section{\label{app:water}Liquid Water}
35  
36 < 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.
36 > The first system considered was liquid water at 300K using the SPC/E
37 > model of water.\cite{Berendsen87} The results for the energy gap
38 > comparisons and the force and torque vector magnitude comparisons are
39 > shown in table \ref{tab:spce}.  The force and torque vector
40 > directionality results are displayed separately in table
41 > \ref{tab:spceAng}, where the effect of group-based cutoffs and
42 > switching functions on the {\sc sp} and {\sc sf} potentials are
43 > investigated.
44   \begin{table}[htbp]
45     \centering
46 <   \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 $\varepsilon \approx \infty$).}  
46 >   \caption{Regression results for the liquid water system. Tabulated
47 > results include $\Delta E$ values (top set), force vector magnitudes
48 > (middle set) and torque vector magnitudes (bottom set).  PC = Pure
49 > Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
50 > Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
51 > \infty$).}      
52     \begin{tabular}{@{} ccrrrrrr @{}}
53        \\
54        \toprule
# Line 50 | Line 68 | RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.
68      & 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\
69      & 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\
70   GSC &     & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\
71 < RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                              
54 <
71 > RF  &     & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\                
72              \midrule
56 –
73   PC  &     & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\
74   SP  & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\
75      & 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\
# Line 65 | Line 81 | RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.
81      & 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\
82   GSC &     & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\
83   RF  &     & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\          
68 –
84              \midrule
70 –
85   PC  &     & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\
86   SP  & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\
87      & 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\
# Line 86 | Line 100 | RF  &     & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.
100  
101   \begin{table}[htbp]
102     \centering
103 <   \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.}  
103 >   \caption{Variance results from Gaussian fits to angular
104 > distributions of the force and torque vectors in the liquid water
105 > system.  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
106 > GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon
107 > \approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF =
108 > Group Switched Shifted Force.}  
109     \begin{tabular}{@{} ccrrrrrr @{}}
110        \\
111        \toprule
# Line 120 | Line 139 | For the most parts, the water results appear to parall
139     \label{tab:spceAng}
140   \end{table}
141  
142 < 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.
142 > The water results parallel the combined results seen in the discussion
143 > section of the main paper.  There is good agreement with {\sc spme} in
144 > both energetic and dynamic behavior when using the {\sc sf} method
145 > with and without damping. The {\sc sp} method does well with an
146 > $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater
147 > than 12 \AA. Overdamping the electrostatics reduces the agreement
148 > between both these methods and {\sc spme}.
149  
150 < 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.
150 > The pure cutoff ({\sc pc}) method performs poorly, again mirroring the
151 > observations in the main portion of this paper.  In contrast to the
152 > combined values, however, the use of a switching function and group
153 > based cutoffs greatly improves the results for these neutral water
154 > molecules.  The group switched cutoff ({\sc gsc}) does not mimic the
155 > energetics of {\sc spme} as well as the {\sc sp} (with moderate
156 > damping) and {\sc sf} methods, but the dynamics are quite good.  The
157 > switching functions correct discontinuities in the potential and
158 > forces, leading to these improved results.  Such improvements with the
159 > use of a switching function have been recognized in previous
160 > studies,\cite{Andrea83,Steinbach94} and this proves to be a useful
161 > tactic for stably incorporating local area electrostatic effects.
162  
163 < 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.
163 > The reaction field ({\sc rf}) method simply extends upon the results
164 > observed in the {\sc gsc} case.  Both methods are similar in form
165 > (i.e. neutral groups, switching function), but {\sc rf} incorporates
166 > an added effect from the external dielectric. This similarity
167 > translates into the same good dynamic results and improved energetic
168 > agreement with {\sc spme}.  Though this agreement is not to the level
169 > of the moderately damped {\sc sp} and {\sc sf} methods, these results
170 > show how incorporating some implicit properties of the surroundings
171 > (i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction.
172  
173 < 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.
174 <
175 < \section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$}
173 > As a final note for the liquid water system, use of group cutoffs and a
174 > switching function leads to noticeable improvements in the {\sc sp}
175 > and {\sc sf} methods, primarily in directionality of the force and
176 > torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows
177 > significant narrowing of the angle distribution when using little to
178 > no damping and only modest improvement for the recommended conditions
179 > ($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA).  The
180 > {\sc sf} method shows modest narrowing across all damping and cutoff
181 > ranges of interest.  When overdamping these methods, group cutoffs and
182 > the switching function do not improve the force and torque
183 > directionalities.
184  
185 < 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.
185 > \section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$}
186  
187 + In addition to the disordered molecular system above, the ordered
188 + molecular system of ice I$_\textrm{c}$ was also considered. The
189 + results for the energy gap comparisons and the force and torque vector
190 + magnitude comparisons are shown in table \ref{tab:ice}.  The force and
191 + torque vector directionality results are displayed separately in table
192 + \ref{tab:iceAng}, where the effect of group-based cutoffs and
193 + switching functions on the {\sc sp} and {\sc sf} potentials are
194 + investigated.
195 +
196   \begin{table}[htbp]
197     \centering
198 <   \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 $\varepsilon \approx \infty$).}    
198 >   \caption{Regression results for the ice I$_\textrm{c}$
199 > system. Tabulated results include $\Delta E$ values (top set), force
200 > vector magnitudes (middle set) and torque vector magnitudes (bottom
201 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
202 > GSC = Group Switched Cutoff, and RF = Reaction Field (where
203 > $\varepsilon \approx \infty$).}  
204     \begin{tabular}{@{} ccrrrrrr @{}}
205        \\
206        \toprule
# Line 220 | Line 286 | Highly ordered systems are a difficult test for the pa
286     \label{tab:iceAng}
287   \end{table}
288  
289 < 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.
289 > Highly ordered systems are a difficult test for the pairwise methods
290 > in that they lack the implicit periodicity of the Ewald summation.  As
291 > expected, the energy gap agreement with {\sc spme} is reduced for the
292 > {\sc sp} and {\sc sf} methods with parameters that were acceptable for
293 > the disordered liquid system.  Moving to higher $R_\textrm{c}$ helps
294 > improve the agreement, though at an increase in computational cost.
295 > The dynamics of this crystalline system (both in magnitude and
296 > direction) are little affected. Both methods still reproduce the Ewald
297 > behavior with the same parameter recommendations from the previous
298 > section.
299  
300 < It is also worth noting that RF exhibits a slightly improved energy gap results over the liquid water system.  This can be rationalized by noting that the ice I$_\textrm{c}$ is
300 > It is also worth noting that {\sc rf} exhibits improved energy gap
301 > results over the liquid water system.  One possible explanation is
302 > that the ice I$_\textrm{c}$ crystal is ordered such that the net
303 > dipole moment of the crystal is zero.  With $\epsilon_\textrm{S} =
304 > \infty$, the reaction field incorporates this structural organization
305 > by actively enforcing a zeroed dipole moment within each cutoff
306 > sphere.  
307  
308 < \section{\label{app-melt}NaCl Melt}
308 > \section{\label{app:melt}NaCl Melt}
309  
310 < 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.
310 > A high temperature NaCl melt was tested to gauge the accuracy of the
311 > pairwise summation methods in a disordered system of charges. The
312 > results for the energy gap comparisons and the force vector magnitude
313 > comparisons are shown in table \ref{tab:melt}.  The force vector
314 > directionality results are displayed separately in table
315 > \ref{tab:meltAng}.
316  
317   \begin{table}[htbp]
318     \centering
# Line 241 | Line 327 | SP  & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.
327              Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\
328              \midrule
329   PC  &     & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\
330 < SP  & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\
331 <    & 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\
330 > SP  & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\
331 >    & 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\
332      & 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
333      & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
334 < SF  & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\
335 <    & 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\
334 > SF  & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\
335 >    & 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\
336      & 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\
337      & 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\
338              \midrule
339   PC  &     & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\
340 < SP  & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\
341 <    & 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\
340 > SP  & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\
341 >    & 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\
342      & 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
343      & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
344 < SF  & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\
345 <    & 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\
344 > SF  & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\
345 >    & 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\
346      & 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\
347      & 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\
348        \bottomrule
# Line 289 | Line 375 | SF  & 0.0 & 1.693 & 0.603 & 0.256 \\
375     \label{tab:meltAng}
376   \end{table}
377  
378 < \section{\label{app-salt}NaCl Crystal}
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 < 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 > \section{\label{app:salt}NaCl Crystal}
390  
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 + vector magnitude comparisons are shown in table \ref{tab:salt}.  The
395 + force vector directionality results are displayed separately in table
396 + \ref{tab:saltAng}.
397 +
398   \begin{table}[htbp]
399     \centering
400 <   \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.}    
400 >   \caption{Regression results for the crystalline NaCl
401 > system. Tabulated results include $\Delta E$ values (top set) and
402 > force vector magnitudes (bottom set).  PC = Pure Cutoff, SP = Shifted
403 > Potential, and SF = Shifted Force.}    
404     \begin{tabular}{@{} ccrrrrrr @{}}
405        \\
406        \toprule
# Line 331 | Line 436 | SF  & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.
436  
437   \begin{table}[htbp]
438     \centering
439 <   \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 $\varepsilon \approx \infty$).}        
439 >   \caption{Variance results from Gaussian fits to angular
440 > distributions of the force vectors in the crystalline NaCl system.  PC
441 > = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group
442 > Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx
443 > \infty$).}      
444     \begin{tabular}{@{} ccrrrrrr @{}}
445        \\
446        \toprule
# Line 354 | Line 463 | SF  & 0.0 & 10.025 & 3.555 & 1.648 \\
463     \label{tab:saltAng}
464   \end{table}
465  
466 < \section{\label{app-sol1}Weak NaCl Solution}
466 > The crystalline NaCl system is the most challenging test case for the
467 > pairwise summation methods, as evidenced by the results in tables
468 > \ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped
469 > {\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best
470 > choices. These methods match well with {\sc spme} across the energy
471 > gap, force magnitude, and force directionality tests.  The {\sc sp}
472 > method struggles in all cases, with the exception of good dynamics
473 > reproduction when using weak electrostatic damping with a large cutoff
474 > radius.
475  
476 < 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$0.11 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}.  The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
476 > The moderate electrostatic damping case is not as good as we would
477 > expect given the long-time dynamics results observed for this
478 > system. Since the data tabulated in tables \ref{tab:salt} and
479 > \ref{tab:saltAng} are a test of instantaneous dynamics, this indicates
480 > that good long-time dynamics comes in part at the expense of
481 > short-time dynamics.
482  
483 + \section{\label{app:solnWeak}Weak NaCl Solution}
484 +
485 + In an effort to bridge the charged atomic and neutral molecular
486 + systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into
487 + the liquid water system. This low ionic strength system consists of 4
488 + ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results
489 + for the energy gap comparisons and the force and torque vector
490 + magnitude comparisons are shown in table \ref{tab:solnWeak}.  The
491 + force and torque vector directionality results are displayed
492 + separately in table \ref{tab:solnWeakAng}, where the effect of
493 + group-based cutoffs and switching functions on the {\sc sp} and {\sc
494 + sf} potentials are investigated.
495 +
496   \begin{table}[htbp]
497     \centering
498 <   \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.}      
498 >   \caption{Regression results for the weak NaCl solution
499 > system. Tabulated results include $\Delta E$ values (top set), force
500 > vector magnitudes (middle set) and torque vector magnitudes (bottom
501 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
502 > GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon
503 > \approx \infty$).}      
504     \begin{tabular}{@{} ccrrrrrr @{}}
505        \\
506        \toprule
# Line 412 | Line 552 | RF  &     & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.
552  
553   \begin{table}[htbp]
554     \centering
555 <   \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.}    
555 >   \caption{Variance results from Gaussian fits to angular
556 > distributions of the force and torque vectors in the weak NaCl
557 > solution system.  PC = Pure Cutoff, SP = Shifted Potential, SF =
558 > Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where
559 > $\varepsilon \approx \infty$), GSSP = Group Switched Shifted
560 > Potential, and GSSF = Group Switched Shifted Force.}    
561     \begin{tabular}{@{} ccrrrrrr @{}}
562        \\
563        \toprule
# Line 446 | Line 591 | GSSF  & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 &
591     \label{tab:solnWeakAng}
592   \end{table}
593  
594 < \section{\label{app-sol10}Strong NaCl Solution}
594 > Because this system is a perturbation of the pure liquid water system,
595 > comparisons are best drawn between these two sets. The {\sc sp} and
596 > {\sc sf} methods are not significantly affected by the inclusion of a
597 > few ions. The aspect of cutoff sphere neutralization aids in the
598 > smooth incorporation of these ions; thus, all of the observations
599 > regarding these methods carry over from section \ref{app:water}. The
600 > differences between these systems are more visible for the {\sc rf}
601 > method. Though good force agreement is still maintained, the energy
602 > gaps show a significant increase in the scatter of the data.
603  
604 < 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$1.1 M). The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:solnWeak}.  The force and torque vector directionality results are displayed separately in table \ref{tab:solnWeakAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated.
604 > \section{\label{app:solnStr}Strong NaCl Solution}
605  
606 + The bridging of the charged atomic and neutral molecular systems was
607 + further developed by considering a high ionic strength system
608 + consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1
609 + M). The results for the energy gap comparisons and the force and
610 + torque vector magnitude comparisons are shown in table
611 + \ref{tab:solnStr}.  The force and torque vector directionality
612 + results are displayed separately in table \ref{tab:solnStrAng}, where
613 + the effect of group-based cutoffs and switching functions on the {\sc
614 + sp} and {\sc sf} potentials are investigated.
615 +
616   \begin{table}[htbp]
617     \centering
618 <   \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 $\varepsilon \approx \infty$).}  
618 >   \caption{Regression results for the strong NaCl solution
619 > system. Tabulated results include $\Delta E$ values (top set), force
620 > vector magnitudes (middle set) and torque vector magnitudes (bottom
621 > set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force,
622 > GSC = Group Switched Cutoff, and RF = Reaction Field (where
623 > $\varepsilon \approx \infty$).}        
624     \begin{tabular}{@{} ccrrrrrr @{}}
625        \\
626        \toprule
# Line 538 | Line 706 | GSSF  & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 &
706     \label{tab:solnStrAng}
707   \end{table}
708  
709 < \section{\label{app-argon}Argon Sphere in Water}
709 > The {\sc rf} method struggles with the jump in ionic strength. The
710 > configuration energy differences degrade to unusable levels while the
711 > forces and torques show a more modest reduction in the agreement with
712 > {\sc spme}. The {\sc rf} method was designed for homogeneous systems,
713 > and this attribute is apparent in these results.
714  
715 < 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.
715 > The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain
716 > their agreement with {\sc spme}. With these results, we still
717 > recommend no to moderate damping for the {\sc sf} method and moderate
718 > damping for the {\sc sp} method, both with cutoffs greater than 12
719 > \AA.
720  
721 + \section{\label{app:argon}Argon Sphere in Water}
722 +
723 + The final model system studied was a 6 \AA\ sphere of Argon solvated
724 + by SPC/E water. The results for the energy gap comparisons and the
725 + force and torque vector magnitude comparisons are shown in table
726 + \ref{tab:argon}.  The force and torque vector directionality
727 + results are displayed separately in table \ref{tab:argonAng}, where
728 + the effect of group-based cutoffs and switching functions on the {\sc
729 + sp} and {\sc sf} potentials are investigated.
730 +
731   \begin{table}[htbp]
732     \centering
733 <   \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 $\varepsilon \approx \infty$).}    
733 >   \caption{Regression results for the 6 \AA\ Argon sphere in liquid
734 > water system. Tabulated results include $\Delta E$ values (top set),
735 > force vector magnitudes (middle set) and torque vector magnitudes
736 > (bottom set).  PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted
737 > Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where
738 > $\varepsilon \approx \infty$).}        
739     \begin{tabular}{@{} ccrrrrrr @{}}
740        \\
741        \toprule
# Line 596 | Line 787 | RF  &     & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.
787  
788   \begin{table}[htbp]
789     \centering
790 <   \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.}
790 >   \caption{Variance results from Gaussian fits to angular
791 > distributions of the force and torque vectors in the 6 \AA\ sphere of
792 > Argon in liquid water system.  PC = Pure Cutoff, SP = Shifted
793 > Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF =
794 > Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group
795 > Switched Shifted Potential, and GSSF = Group Switched Shifted Force.}  
796     \begin{tabular}{@{} ccrrrrrr @{}}
797        \\
798        \toprule
# Line 630 | Line 826 | GSSF  & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 &
826     \label{tab:argonAng}
827   \end{table}
828  
829 + This system does not appear to show any significant deviations from
830 + the previously observed results. The {\sc sp} and {\sc sf} methods
831 + have aggrements similar to those observed in section
832 + \ref{app:water}. The only significant difference is the improvement
833 + in the configuration energy differences for the {\sc rf} method. This
834 + is surprising in that we are introducing an inhomogeneity to the
835 + system; however, this inhomogeneity is charge-neutral and does not
836 + result in charged cutoff spheres. The charge-neutrality of the cutoff
837 + spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce,
838 + seems to play a greater role in the stability of the {\sc rf} method
839 + than the required homogeneity of the environment.
840 +
841   \newpage
842  
843   \bibliographystyle{jcp2}
844   \bibliography{electrostaticMethods}
845  
846 < \end{document}
846 > \end{document}

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