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%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
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\documentclass[12pt]{article} |
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\usepackage{endfloat} |
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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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\usepackage{epsf} |
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\begin{document} |
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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. |
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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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|
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\section{\label{app-water}Liquid Water} |
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\section{\label{app:water}Liquid Water} |
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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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\begin{table}[htbp] |
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\centering |
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\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 $\varepsilon \approx \infty$).} |
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\caption{Regression results for the liquid water system. Tabulated |
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results include $\Delta E$ values (top set), force vector magnitudes |
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(middle set) and torque vector magnitudes (bottom set). PC = Pure |
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Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group |
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Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
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\infty$).} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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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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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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RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\ |
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\bottomrule |
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\end{tabular} |
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\label{spceTabTMag} |
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\label{tab:spce} |
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\end{table} |
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|
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\begin{table}[htbp] |
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\centering |
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\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.} |
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\caption{Variance results from Gaussian fits to angular |
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distributions of the force and torque vectors in the liquid water |
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system. 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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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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& 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\ |
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\bottomrule |
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\end{tabular} |
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\label{spceTabAng} |
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\label{tab:spceAng} |
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\end{table} |
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|
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\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
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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 ({\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 ({\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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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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magnitude comparisons are shown in table \ref{tab:ice}. The force and |
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torque vector directionality results are displayed separately in table |
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\ref{tab:iceAng}, 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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|
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\begin{table}[htbp] |
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\centering |
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\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 $\varepsilon \approx \infty$).} |
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\caption{Regression results for the ice I$_\textrm{c}$ |
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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, and RF = Reaction Field (where |
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$\varepsilon \approx \infty$).} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\ |
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\bottomrule |
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\end{tabular} |
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\label{iceTab} |
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\label{tab:ice} |
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\end{table} |
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|
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\begin{table}[htbp] |
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& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
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\bottomrule |
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\end{tabular} |
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\label{iceTabAng} |
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\label{tab:iceAng} |
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\end{table} |
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|
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\section{\label{app-melt}NaCl Melt} |
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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 {\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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|
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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 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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\caption{Regression results for the molten 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.} |
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Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
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\midrule |
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PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
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SP & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\ |
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& 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\ |
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SP & 0.0 & 0.928 & 0.996 & 0.931 & 0.998 & 0.950 & 0.999 \\ |
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& 0.1 & 0.977 & 0.998 & 0.998 & 1.000 & 0.997 & 1.000 \\ |
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& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
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& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
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SF & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\ |
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& 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\ |
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SF & 0.0 & 0.996 & 1.000 & 0.995 & 1.000 & 0.997 & 1.000 \\ |
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& 0.1 & 1.021 & 1.000 & 1.024 & 1.000 & 1.007 & 1.000 \\ |
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& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
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& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
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\midrule |
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PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
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SP & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\ |
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& 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\ |
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SP & 0.0 & 0.973 & 0.981 & 0.975 & 0.988 & 0.979 & 0.992 \\ |
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& 0.1 & 0.987 & 0.992 & 0.993 & 0.998 & 0.997 & 0.999 \\ |
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& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
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& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
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SF & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\ |
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& 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\ |
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SF & 0.0 & 0.996 & 0.997 & 0.997 & 0.999 & 0.998 & 1.000 \\ |
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& 0.1 & 1.000 & 0.997 & 1.001 & 0.999 & 1.000 & 1.000 \\ |
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& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
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& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
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\bottomrule |
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|
\end{tabular} |
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\label{meltTab} |
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\label{tab:melt} |
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\end{table} |
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|
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\begin{table}[htbp] |
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& 0.3 & 23.734 & 67.305 & 57.252 \\ |
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\bottomrule |
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\end{tabular} |
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\label{meltTabAng} |
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\label{tab:meltAng} |
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|
\end{table} |
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|
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\section{\label{app-salt}NaCl Crystal} |
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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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|
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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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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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< |
\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.} |
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> |
\caption{Regression results for the crystalline NaCl |
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> |
system. Tabulated results include $\Delta E$ values (top set) and |
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> |
force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted |
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> |
Potential, and SF = Shifted Force.} |
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|
\begin{tabular}{@{} ccrrrrrr @{}} |
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|
\\ |
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|
\toprule |
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|
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 432 |
|
\bottomrule |
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|
\end{tabular} |
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< |
\label{saltTab} |
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> |
\label{tab:salt} |
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|
\end{table} |
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|
|
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|
\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 $\varepsilon \approx \infty$).} |
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> |
\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 |
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> |
Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
| 443 |
> |
\infty$).} |
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|
\begin{tabular}{@{} ccrrrrrr @{}} |
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|
\\ |
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|
\toprule |
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|
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
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|
\bottomrule |
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|
\end{tabular} |
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< |
\label{saltTabAng} |
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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 |
| 467 |
> |
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 |
| 470 |
> |
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} |
| 472 |
> |
method struggles in all cases, with the exception of good dynamics |
| 473 |
> |
reproduction when using weak electrostatic damping with a large cutoff |
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> |
radius. |
| 475 |
|
|
| 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 |
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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. |
| 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 |
| 547 |
|
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 548 |
|
\bottomrule |
| 549 |
|
\end{tabular} |
| 550 |
< |
\label{sol1Tab} |
| 550 |
> |
\label{tab:solnWeak} |
| 551 |
|
\end{table} |
| 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 |
| 588 |
|
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 589 |
|
\bottomrule |
| 590 |
|
\end{tabular} |
| 591 |
< |
\label{sol1TabAng} |
| 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 |
+ |
\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 |
| 667 |
|
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 668 |
|
\bottomrule |
| 669 |
|
\end{tabular} |
| 670 |
< |
\label{sol10Tab} |
| 670 |
> |
\label{tab:solnStr} |
| 671 |
|
\end{table} |
| 672 |
|
|
| 673 |
|
\begin{table}[htbp] |
| 703 |
|
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 704 |
|
\bottomrule |
| 705 |
|
\end{tabular} |
| 706 |
< |
\label{sol10TabAng} |
| 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 {\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 |
| 782 |
|
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 783 |
|
\bottomrule |
| 784 |
|
\end{tabular} |
| 785 |
< |
\label{argonTab} |
| 785 |
> |
\label{tab:argon} |
| 786 |
|
\end{table} |
| 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 |
| 823 |
|
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 824 |
|
\bottomrule |
| 825 |
|
\end{tabular} |
| 826 |
< |
\label{argonTabAng} |
| 826 |
> |
\label{tab:argonAng} |
| 827 |
|
\end{table} |
| 828 |
|
|
| 829 |
< |
\end{document} |
| 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} |
