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%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
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\documentclass[12pt]{article} |
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\documentclass[11pt]{article} |
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%\usepackage{endfloat} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\begin{document} |
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This document includes individual system-based comparisons of the |
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studied methods with smooth particle-mesh Ewald. Each of the seven |
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systems comprises its own section and has its own discussion and |
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tabular listing of the results for the $\Delta E$, force and torque |
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vector magnitude, and force and torque vector direction comparisons. |
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studied methods with smooth particle mesh Ewald {\sc spme}. Each of |
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the seven systems comprises its own section and has its own discussion |
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and tabular listing of the results for the $\Delta E$, force and |
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torque vector magnitude, and force and torque vector direction |
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comparisons. |
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|
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\section{\label{app-water}Liquid Water} |
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\section{\label{app:water}Liquid Water} |
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|
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500 liquid state configurations were generated as described in the |
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Methods section using the SPC/E model of water.\cite{Berendsen87} The |
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results for the energy gap comparisons and the force and torque vector |
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magnitude comparisons are shown in table \ref{tab:spce}. The force |
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and torque vector directionality results are displayed separately in |
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table \ref{tab:spceAng}, where the effect of group-based cutoffs and |
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The first system considered was liquid water at 300K using the SPC/E |
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model of water.\cite{Berendsen87} The results for the energy gap |
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comparisons and the force and torque vector magnitude comparisons are |
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shown in table \ref{tab:spce}. The force and torque vector |
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directionality results are displayed separately in table |
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\ref{tab:spceAng}, where the effect of group-based cutoffs and |
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switching functions on the {\sc sp} and {\sc sf} potentials are |
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investigated. |
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investigated. |
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\begin{table}[htbp] |
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\centering |
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\caption{Regression results for the liquid water system. Tabulated |
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& 0.2 & 0.992 & 0.989 & 1.001 & 0.995 & 0.994 & 0.996 \\ |
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& 0.3 & 0.984 & 0.980 & 0.996 & 0.985 & 0.982 & 0.987 \\ |
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GSC & & 0.918 & 0.862 & 0.852 & 0.756 & 0.801 & 0.700 \\ |
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RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
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|
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RF & & 0.971 & 0.958 & 0.975 & 0.987 & 0.959 & 0.983 \\ |
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\midrule |
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– |
|
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PC & & -1.647 & 0.000 & -0.127 & 0.000 & -0.979 & 0.000 \\ |
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SP & 0.0 & 0.735 & 0.368 & 0.813 & 0.537 & 0.865 & 0.659 \\ |
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& 0.1 & 0.850 & 0.612 & 0.956 & 0.887 & 0.992 & 0.979 \\ |
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& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
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GSC & & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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RF & & 0.999 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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– |
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\midrule |
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– |
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PC & & 2.387 & 0.000 & 0.183 & 0.000 & 1.282 & 0.000 \\ |
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SP & 0.0 & 0.847 & 0.543 & 0.904 & 0.694 & 0.935 & 0.786 \\ |
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& 0.1 & 0.922 & 0.749 & 0.980 & 0.934 & 0.996 & 0.988 \\ |
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\label{tab:spceAng} |
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\end{table} |
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|
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For the most parts, the water results appear to parallel the combined |
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results seen in the discussion in the main paper. There is good |
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agreement with SPME in both energetic and dynamic behavior when using |
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the {\sc sf} method with and without damping. The {\sc sp} method does |
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well with an $\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff |
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radii greater than 12 \AA. The results for both of these methods also |
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begin to decay as damping gets too large. |
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The water results appear to parallel the combined results seen in the |
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discussion section of the main paper. There is good agreement with |
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{\sc spme} in both energetic and dynamic behavior when using the {\sc sf} |
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method with and without damping. The {\sc sp} method does well with an |
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$\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater |
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than 12 \AA. Overdamping the electrostatics reduces the agreement between both these methods and {\sc spme}. |
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|
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The pure cutoff (PC) method performs poorly, as seen in the main |
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discussion section. In contrast to the combined values, however, the |
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use of a switching function and group based cutoffs really improves |
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the results for these neutral water molecules. The group switched |
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cutoff (GSC) shows mimics the energetics of SPME more poorly than the |
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{\sc sp} (with moderate damping) and {\sc sf} methods, but the |
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dynamics are quite good. The switching functions corrects |
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discontinuities in the potential and forces, leading to the improved |
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results. Such improvements with the use of a switching function has |
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been recognized in previous studies,\cite{Andrea83,Steinbach94} and it |
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is a useful tactic for stably incorporating local area electrostatic |
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effects. |
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|
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The reaction field (RF) method simply extends the results observed in |
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the GSC case. Both methods are similar in form (i.e. neutral groups, |
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switching function), but RF incorporates an added effect from the |
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external dielectric. This similarity translates into the same good |
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dynamic results and improved energetic results. These still fall |
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short of the moderately damped {\sc sp} and {\sc sf} methods, but they |
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display how incorporating some implicit properties of the surroundings |
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(i.e. $\epsilon_\textrm{S}$) can improve results. |
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The pure cutoff ({\sc pc}) method performs poorly, again mirroring the |
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observations in the main portion of this paper. In contrast to the |
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combined values, however, the use of a switching function and group |
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based cutoffs greatly improves the results for these neutral water |
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molecules. The group switched cutoff ({\sc gsc}) does not mimic the |
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energetics of {\sc spme} as well as the {\sc sp} (with moderate |
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damping) and {\sc sf} methods, but the dynamics are quite good. The |
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switching functions correct discontinuities in the potential and |
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forces, leading to these improved results. Such improvements with the |
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use of a switching function have been recognized in previous |
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studies,\cite{Andrea83,Steinbach94} and this proves to be a useful |
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tactic for stably incorporating local area electrostatic effects. |
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|
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A final note for the liquid water system, use of group cutoffs and a |
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switching function also leads to noticeable improvements in the {\sc |
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sp} and {\sc sf} methods, primarily in directionality of the force and |
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torque vectors (table \ref{tab:spceAng}). {\sc sp} shows significant |
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narrowing of the angle distribution in the cases with little to no |
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damping and only modest improvement for the ideal conditions ($\alpha$ |
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= 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The {\sc sf} |
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method simply shows modest narrowing across all damping and cutoff |
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ranges of interest. Group cutoffs and the switching function do |
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nothing for cases were error is introduced by overdamping the |
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potentials. |
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The reaction field ({\sc rf}) method simply extends upon the results |
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observed in the {\sc gsc} case. Both methods are similar in form |
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(i.e. neutral groups, switching function), but {\sc rf} incorporates |
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an added effect from the external dielectric. This similarity |
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translates into the same good dynamic results and improved energetic |
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agreement with {\sc spme}. Though this agreement is not to the level |
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of the moderately damped {\sc sp} and {\sc sf} methods, these results |
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show how incorporating some implicit properties of the surroundings |
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(i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction. |
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|
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\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
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As a final note for the liquid water system, use of group cutoffs and a |
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switching function leads to noticeable improvements in the {\sc sp} |
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and {\sc sf} methods, primarily in directionality of the force and |
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torque vectors (table \ref{tab:spceAng}). The {\sc sp} method shows |
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significant narrowing of the angle distribution when using little to |
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no damping and only modest improvement for the recommended conditions |
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($\alpha$ = 0.2 \AA${-1}$ and $R_\textrm{c} \geqslant 12$~\AA). The |
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{\sc sf} method shows modest narrowing across all damping and cutoff |
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ranges of interest. When overdamping these methods, group cutoffs and |
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the switching function do not improve the force and torque |
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directionalities. |
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|
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\section{\label{app:ice}Solid Water: Ice I$_\textrm{c}$} |
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|
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In addition to the disordered molecular system above, the ordered |
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molecular system of ice I$_\textrm{c}$ was also considered. The |
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results for the energy gap comparisons and the force and torque vector |
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\label{tab:iceAng} |
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\end{table} |
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|
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Highly ordered systems are a difficult test for the pairwise systems |
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in that they lack the periodicity inherent to the Ewald summation. As |
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expected, the energy gap agreement with SPME reduces for the {\sc sp} |
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and {\sc sf} with parameters that were perfectly acceptable for the |
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disordered liquid system. Moving to higher $R_\textrm{c}$ remedies |
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this degraded performance, though at increase in computational cost. |
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However, the dynamics of this crystalline system (both in magnitude |
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and direction) are little affected. Both methods still reproduce the |
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Ewald behavior with the same parameter recommendations from the |
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previous section. |
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Highly ordered systems are a difficult test for the pairwise methods |
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in that they lack the periodicity term of the Ewald summation. As |
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expected, the energy gap agreement with {\sc spme} is reduced for the |
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{\sc sp} and {\sc sf} methods with parameters that were acceptable for |
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the disordered liquid system. Moving to higher $R_\textrm{c}$ helps |
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improve the agreement, though at an increase in computational cost. |
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The dynamics of this crystalline system (both in magnitude and |
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direction) are little affected. Both methods still reproduce the Ewald |
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behavior with the same parameter recommendations from the previous |
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section. |
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|
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It is also worth noting that RF exhibits a slightly improved energy |
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gap results over the liquid water system. One possible explanation is |
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It is also worth noting that {\sc rf} exhibits improved energy gap |
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results over the liquid water system. One possible explanation is |
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that the ice I$_\textrm{c}$ crystal is ordered such that the net |
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dipole moment of the crystal is zero. With $\epsilon_\textrm{S} = |
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\infty$, the reaction field incorporates this structural organization |
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by actively enforcing a zeroed dipole moment within each cutoff |
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sphere. |
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|
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\section{\label{app-melt}NaCl Melt} |
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\section{\label{app:melt}NaCl Melt} |
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|
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A high temperature NaCl melt was tested to gauge the accuracy of the |
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pairwise summation methods in a highly charge disordered system. The |
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results for the energy gap comparisons and the force and torque vector |
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magnitude comparisons are shown in table \ref{tab:melt}. The force |
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and torque vector directionality results are displayed separately in |
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table \ref{tab:meltAng}, where the effect of group-based cutoffs and |
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switching functions on the {\sc sp} and {\sc sf} potentials are |
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investigated. |
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pairwise summation methods in a charged disordered system. The results |
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for the energy gap comparisons and the force vector magnitude |
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comparisons are shown in table \ref{tab:melt}. The force vector |
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directionality results are displayed separately in table |
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\ref{tab:meltAng}. |
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|
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\begin{table}[htbp] |
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\centering |
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\label{tab:meltAng} |
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\end{table} |
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|
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The molten NaCl system shows the a |
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The molten NaCl system shows more sensitivity to the electrostatic |
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damping than the water systems. The most noticeable point is that the |
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undamped {\sc sf} method does very well at replicating the {\sc spme} |
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configurational energy differences and forces. Light damping appears |
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to minimally improve the dynamics, but this comes with a deterioration |
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of the energy gap results. In contrast, this light damping improves |
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the {\sc sp} energy gaps and forces. Moderate and heavy electrostatic |
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damping reduce the agreement with {\sc spme} for both methods. From |
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these observations, the undamped {\sc sf} method is the best choice |
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for disordered systems of charges. |
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|
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\section{\label{app-salt}NaCl Crystal} |
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\section{\label{app:salt}NaCl Crystal} |
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|
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A 1000K NaCl crystal was used to investigate the accuracy of the |
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pairwise summation methods in an ordered system of charged |
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particles. The results for the energy gap comparisons and the force |
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and torque vector magnitude comparisons are shown in table |
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\ref{tab:salt}. The force and torque vector directionality results |
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are displayed separately in table \ref{tab:saltAng}, where the effect |
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of group-based cutoffs and switching functions on the {\sc sp} and |
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{\sc sf} potentials are investigated. |
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vector magnitude comparisons are shown in table \ref{tab:salt}. The |
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force vector directionality results are displayed separately in table |
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\ref{tab:saltAng}. |
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|
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\begin{table}[htbp] |
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\centering |
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\label{tab:saltAng} |
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\end{table} |
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|
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\section{\label{app-sol1}Weak NaCl Solution} |
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The crystalline NaCl system is the most challenging test case for the |
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pairwise summation methods, as evidenced by the results in tables |
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\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped |
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{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best |
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choices. These methods match well with {\sc spme} across the energy |
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gap, force magnitude, and force directionality tests. The {\sc sp} |
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method struggles in all cases, with the exception of good dynamics |
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reproduction when using weak electrostatic damping with a large cutoff |
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radius. |
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|
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The moderate electrostatic damping case is not as good as we would |
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expect given the long-time dynamics results observed for this |
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system. Since the data tabulated in tables \ref{tab:salt} and |
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\ref{tab:saltAng} are a test of instantaneous dynamics, this indicates |
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that good long-time dynamics comes in part at the expense of |
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short-time dynamics. |
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|
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\section{\label{app:solnWeak}Weak NaCl Solution} |
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|
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|
In an effort to bridge the charged atomic and neutral molecular |
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systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
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the liquid water system. This low ionic strength system consists of 4 |
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system. Tabulated results include $\Delta E$ values (top set), force |
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vector magnitudes (middle set) and torque vector magnitudes (bottom |
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set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
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< |
GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon |
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< |
\approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = |
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Group Switched Shifted Force.} |
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GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon |
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> |
\approx \infty$).} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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\label{tab:solnWeakAng} |
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|
\end{table} |
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|
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\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 |
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> |
{\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 |
+ |
\section{\label{app:solnStr}Strong NaCl Solution} |
| 604 |
+ |
|
| 605 |
|
The bridging of the charged atomic and neutral molecular systems was |
| 606 |
< |
furthered by considering a high ionic strength system consisting of 40 |
| 607 |
< |
ions in the 1000 SPC/E water solvent ($\approx$1.1 M). The results for |
| 608 |
< |
the energy gap comparisons and the force and torque vector magnitude |
| 609 |
< |
comparisons are shown in table \ref{tab:solnWeak}. The force and |
| 610 |
< |
torque vector directionality results are displayed separately in table |
| 611 |
< |
\ref{tab:solnWeakAng}, where the effect of group-based cutoffs and |
| 612 |
< |
switching functions on the {\sc sp} and {\sc sf} potentials are |
| 613 |
< |
investigated. |
| 606 |
> |
further developed by considering a high ionic strength system |
| 607 |
> |
consisting of 40 ions in the 1000 SPC/E water solvent ($\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 |
| 705 |
|
\label{tab:solnStrAng} |
| 706 |
|
\end{table} |
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|
|
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< |
\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 |
|
|
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< |
The final model system studied was 6 \AA\ sphere of Argon solvated by |
| 715 |
< |
SPC/E water. The results for the energy gap comparisons and the force |
| 716 |
< |
and torque vector magnitude comparisons are shown in table |
| 717 |
< |
\ref{tab:solnWeak}. The force and torque vector directionality |
| 718 |
< |
results are displayed separately in table \ref{tab:solnWeakAng}, where |
| 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 |
| 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 |
| 788 |
|
\centering |
| 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 |
| 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.} |
| 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} |
