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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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This document includes comparisons of the new pairwise electrostatic |
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methods with {\sc spme} for each of the individual systems mentioned |
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in paper. Each of the seven sections contains information about a |
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single system type and has its own discussion and tabular listing of |
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the results for the comparisons of $\Delta E$, the magnitudes of the |
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forces and torques, and directionality of the force and torque |
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vectors. |
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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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\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 parallel the combined results seen in the discussion |
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section of the main paper. There is good agreement with {\sc spme} in |
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both energetic and dynamic behavior when using the {\sc sf} method |
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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 |
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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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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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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 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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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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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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\label{tab:iceAng} |
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\end{table} |
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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 implicit periodicity 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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\section{\label{app:melt}NaCl Melt} |
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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 disordered system of charges. The |
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results 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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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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{\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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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 good long-time dynamics results observed for this |
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system. Since these results are a test of instantaneous dynamics, this |
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indicates that good long-time dynamics comes in part at the expense of |
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short-time dynamics. Further indication of this comes from the full |
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power spectra shown in the main text. It appears as though a |
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distortion is introduced between 200 to 300 cm$^{-1}$ with increased |
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$\alpha$. |
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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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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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This weak ionic strength system can be considered as a perturbation of |
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the pure liquid water system. The {\sc sp} and {\sc sf} methods are |
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not significantly affected by the inclusion of a few ions. The aspect |
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of cutoff sphere neutralization aids in the smooth incorporation of |
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these ions; thus, all of the observations regarding these methods |
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carry over from section \ref{app:water}. The differences between these |
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systems are visible for the {\sc rf} method. Though good force |
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reproduction is still maintained, the energy gaps show a significant |
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increase in the data scatter. This foreshadows the breakdown of the |
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method as we introduce system inhomogeneities. |
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Because this system is a perturbation of the pure liquid water system, |
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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 |
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few ions. The aspect of cutoff sphere neutralization aids in the |
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smooth incorporation of these ions; thus, all of the observations |
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regarding these methods carry over from section \ref{app:water}. The |
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differences between these systems are more visible for the {\sc rf} |
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method. Though good force agreement is still maintained, the energy |
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gaps show a significant increase in the scatter of the data. |
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|
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\section{\label{app:solnStr}Strong NaCl Solution} |
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|
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consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
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M). The results for the energy gap comparisons and the force and |
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torque vector magnitude comparisons are shown in table |
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\ref{tab:solnWeak}. The force and torque vector directionality |
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results are displayed separately in table\ref{tab:solnWeakAng}, where |
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\ref{tab:solnStr}. The force and torque vector directionality |
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results are displayed separately in table \ref{tab:solnStrAng}, where |
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the effect of group-based cutoffs and switching functions on the {\sc |
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sp} and {\sc sf} potentials are investigated. |
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|
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\end{table} |
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|
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The {\sc rf} method struggles with the jump in ionic strength. The |
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configuration energy difference degrade to unuseable levels while the |
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forces and torques degrade in a more modest fashion. The {\sc rf} |
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method was designed for homogeneous systems, and this restriction is |
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apparent in these results. |
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configuration energy differences degrade to unusable levels while the |
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forces and torques show a more modest reduction in the agreement with |
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{\sc spme}. The {\sc rf} method was designed for homogeneous systems, |
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and this attribute is apparent in these results. |
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|
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The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
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their agreement with {\sc spme}. With these results, we still |
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|
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\section{\label{app:argon}Argon Sphere in Water} |
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|
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The final model system studied was 6 \AA\ sphere of Argon solvated by |
| 724 |
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SPC/E water. The results for the energy gap comparisons and the force |
| 725 |
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and torque vector magnitude comparisons are shown in table |
| 726 |
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\ref{tab:solnWeak}. The force and torque vector directionality |
| 727 |
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results are displayed separately in table \ref{tab:solnWeakAng}, where |
| 723 |
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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 |
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force and torque vector magnitude comparisons are shown in table |
| 726 |
> |
\ref{tab:argon}. The force and torque vector directionality |
| 727 |
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results are displayed separately in table \ref{tab:argonAng}, where |
| 728 |
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the effect of group-based cutoffs and switching functions on the {\sc |
| 729 |
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sp} and {\sc sf} potentials are investigated. |
| 730 |
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|
| 731 |
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\begin{table}[htbp] |
| 732 |
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\centering |
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\caption{Regression results for the 6 \AA\ argon sphere in liquid |
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\caption{Regression results for the 6 \AA\ Argon sphere in liquid |
| 734 |
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water system. Tabulated results include $\Delta E$ values (top set), |
| 735 |
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force vector magnitudes (middle set) and torque vector magnitudes |
| 736 |
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(bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted |
| 789 |
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\centering |
| 790 |
|
\caption{Variance results from Gaussian fits to angular |
| 791 |
|
distributions of the force and torque vectors in the 6 \AA\ sphere of |
| 792 |
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argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
| 792 |
> |
Argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
| 793 |
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Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = |
| 794 |
|
Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group |
| 795 |
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Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
| 826 |
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\label{tab:argonAng} |
| 827 |
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\end{table} |
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|
| 829 |
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This system appears not to show in any significant deviation in the previously observed results. The {\sc sp} and {\sc sf} methods give result qualities similar to those observed in section \ref{app:water}. The only significant difference is the improvement for the configuration energy differences for the {\sc rf} method. This is surprising in that we are introducing an inhomogeneity to the system; however, this inhomogeneity is charge-neutral and does not result in charged cutoff spheres. The charge-neutrality, which the {\sc sp} and {\sc sf} methods explicity enforce, seems to play a greater role in the stability of the {\sc rf} method than the necessity of a homogeneous environment. |
| 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 |
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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 |
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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. |
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|
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\newpage |
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|
