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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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%\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 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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|
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\section{\label{app-water}Liquid Water} |
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\section{\label{app:water}Liquid Water} |
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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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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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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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|
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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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|
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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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|
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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 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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|
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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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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 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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|
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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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+ |
and torque vector magnitude comparisons are shown in table |
| 394 |
+ |
\ref{tab:salt}. The force and torque vector directionality results |
| 395 |
+ |
are displayed separately in table \ref{tab:saltAng}, where the effect |
| 396 |
+ |
of group-based cutoffs and switching functions on the {\sc sp} and |
| 397 |
+ |
{\sc sf} potentials are investigated. |
| 398 |
+ |
|
| 399 |
|
\begin{table}[htbp] |
| 400 |
|
\centering |
| 401 |
< |
\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 |
| 403 |
> |
force vector magnitudes (bottom set). PC = Pure Cutoff, SP = Shifted |
| 404 |
> |
Potential, and SF = Shifted Force.} |
| 405 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 406 |
|
\\ |
| 407 |
|
\toprule |
| 432 |
|
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 433 |
|
\bottomrule |
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|
\end{tabular} |
| 435 |
< |
\label{saltTab} |
| 435 |
> |
\label{tab:salt} |
| 436 |
|
\end{table} |
| 437 |
|
|
| 438 |
|
\begin{table}[htbp] |
| 439 |
|
\centering |
| 440 |
< |
\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 $\varepsilon \approx \infty$).} |
| 440 |
> |
\caption{Variance results from Gaussian fits to angular |
| 441 |
> |
distributions of the force vectors in the crystalline NaCl system. PC |
| 442 |
> |
= Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group |
| 443 |
> |
Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
| 444 |
> |
\infty$).} |
| 445 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 446 |
|
\\ |
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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 |
| 468 |
> |
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 |
| 470 |
> |
{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best |
| 471 |
> |
choices. These methods match well with {\sc spme} across the energy |
| 472 |
> |
gap, force magnitude, and force directionality tests. The {\sc sp} |
| 473 |
> |
method struggles in all cases with the exception of good dynamics |
| 474 |
> |
reproduction when using weak electrostatic damping with a large cutoff |
| 475 |
> |
radius. |
| 476 |
|
|
| 477 |
+ |
The moderate electrostatic damping case is not as good as we would |
| 478 |
+ |
expect given the good long-time dynamics results observed for this |
| 479 |
+ |
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 |
| 481 |
+ |
short-time dynamics. Further indication of this comes from the full |
| 482 |
+ |
power spectra shown in the main text. It appears as though a |
| 483 |
+ |
distortion is introduced between 200 to 300 cm$^{-1}$ with increased |
| 484 |
+ |
$\alpha$. |
| 485 |
+ |
|
| 486 |
+ |
\section{\label{app:solnWeak}Weak NaCl Solution} |
| 487 |
+ |
|
| 488 |
+ |
In an effort to bridge the charged atomic and neutral molecular |
| 489 |
+ |
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
| 490 |
+ |
the liquid water system. This low ionic strength system consists of 4 |
| 491 |
+ |
ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results |
| 492 |
+ |
for the energy gap comparisons and the force and torque vector |
| 493 |
+ |
magnitude comparisons are shown in table \ref{tab:solnWeak}. The |
| 494 |
+ |
force and torque vector directionality results are displayed |
| 495 |
+ |
separately in table \ref{tab:solnWeakAng}, where the effect of |
| 496 |
+ |
group-based cutoffs and switching functions on the {\sc sp} and {\sc |
| 497 |
+ |
sf} potentials are investigated. |
| 498 |
+ |
|
| 499 |
|
\begin{table}[htbp] |
| 500 |
|
\centering |
| 501 |
< |
\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.} |
| 501 |
> |
\caption{Regression results for the weak NaCl solution |
| 502 |
> |
system. Tabulated results include $\Delta E$ values (top set), force |
| 503 |
> |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 504 |
> |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 505 |
> |
GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon |
| 506 |
> |
\approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = |
| 507 |
> |
Group Switched Shifted Force.} |
| 508 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 509 |
|
\\ |
| 510 |
|
\toprule |
| 551 |
|
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 552 |
|
\bottomrule |
| 553 |
|
\end{tabular} |
| 554 |
< |
\label{sol1Tab} |
| 554 |
> |
\label{tab:solnWeak} |
| 555 |
|
\end{table} |
| 556 |
|
|
| 557 |
|
\begin{table}[htbp] |
| 558 |
|
\centering |
| 559 |
< |
\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.} |
| 559 |
> |
\caption{Variance results from Gaussian fits to angular |
| 560 |
> |
distributions of the force and torque vectors in the weak NaCl |
| 561 |
> |
solution system. PC = Pure Cutoff, SP = Shifted Potential, SF = |
| 562 |
> |
Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where |
| 563 |
> |
$\varepsilon \approx \infty$), GSSP = Group Switched Shifted |
| 564 |
> |
Potential, and GSSF = Group Switched Shifted Force.} |
| 565 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 566 |
|
\\ |
| 567 |
|
\toprule |
| 592 |
|
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 593 |
|
\bottomrule |
| 594 |
|
\end{tabular} |
| 595 |
< |
\label{sol1TabAng} |
| 595 |
> |
\label{tab:solnWeakAng} |
| 596 |
|
\end{table} |
| 597 |
|
|
| 598 |
< |
\section{\label{app-sol10}Strong NaCl Solution} |
| 598 |
> |
This weak ionic strength system can be considered as a perturbation of |
| 599 |
> |
the pure liquid water system. The {\sc sp} and {\sc sf} methods are |
| 600 |
> |
not significantly affected by the inclusion of a few ions. The aspect |
| 601 |
> |
of cutoff sphere neutralization aids in the smooth incorporation of |
| 602 |
> |
these ions; thus, all of the observations regarding these methods |
| 603 |
> |
carry over from section \ref{app:water}. The differences between these |
| 604 |
> |
systems are visible for the {\sc rf} method. Though good force |
| 605 |
> |
reproduction is still maintained, the energy gaps show a significant |
| 606 |
> |
increase in the data scatter. This foreshadows the breakdown of the |
| 607 |
> |
method as we introduce system inhomogeneities. |
| 608 |
|
|
| 609 |
+ |
\section{\label{app:solnStr}Strong NaCl Solution} |
| 610 |
+ |
|
| 611 |
+ |
The bridging of the charged atomic and neutral molecular systems was |
| 612 |
+ |
further developed by considering a high ionic strength system |
| 613 |
+ |
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
| 614 |
+ |
M). The results for the energy gap comparisons and the force and |
| 615 |
+ |
torque vector magnitude comparisons are shown in table |
| 616 |
+ |
\ref{tab:solnWeak}. The force and torque vector directionality |
| 617 |
+ |
results are displayed separately in table\ref{tab:solnWeakAng}, where |
| 618 |
+ |
the effect of group-based cutoffs and switching functions on the {\sc |
| 619 |
+ |
sp} and {\sc sf} potentials are investigated. |
| 620 |
+ |
|
| 621 |
|
\begin{table}[htbp] |
| 622 |
|
\centering |
| 623 |
< |
\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 $\varepsilon \approx \infty$).} |
| 623 |
> |
\caption{Regression results for the strong NaCl solution |
| 624 |
> |
system. Tabulated results include $\Delta E$ values (top set), force |
| 625 |
> |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 626 |
> |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 627 |
> |
GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 628 |
> |
$\varepsilon \approx \infty$).} |
| 629 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 630 |
|
\\ |
| 631 |
|
\toprule |
| 672 |
|
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 673 |
|
\bottomrule |
| 674 |
|
\end{tabular} |
| 675 |
< |
\label{sol10Tab} |
| 675 |
> |
\label{tab:solnStr} |
| 676 |
|
\end{table} |
| 677 |
|
|
| 678 |
|
\begin{table}[htbp] |
| 708 |
|
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 709 |
|
\bottomrule |
| 710 |
|
\end{tabular} |
| 711 |
< |
\label{sol10TabAng} |
| 711 |
> |
\label{tab:solnStrAng} |
| 712 |
|
\end{table} |
| 713 |
|
|
| 714 |
< |
\section{\label{app-argon}Argon Sphere in Water} |
| 714 |
> |
The {\sc rf} method struggles with the jump in ionic strength. The |
| 715 |
> |
configuration energy difference degrade to unuseable levels while the |
| 716 |
> |
forces and torques degrade in a more modest fashion. The {\sc rf} |
| 717 |
> |
method was designed for homogeneous systems, and this restriction is |
| 718 |
> |
apparent in these results. |
| 719 |
|
|
| 720 |
+ |
The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
| 721 |
+ |
their agreement with {\sc spme}. With these results, we still |
| 722 |
+ |
recommend no to moderate damping for the {\sc sf} method and moderate |
| 723 |
+ |
damping for the {\sc sp} method, both with cutoffs greater than 12 |
| 724 |
+ |
\AA. |
| 725 |
+ |
|
| 726 |
+ |
\section{\label{app:argon}Argon Sphere in Water} |
| 727 |
+ |
|
| 728 |
+ |
The final model system studied was 6 \AA\ sphere of Argon solvated by |
| 729 |
+ |
SPC/E water. The results for the energy gap comparisons and the force |
| 730 |
+ |
and torque vector magnitude comparisons are shown in table |
| 731 |
+ |
\ref{tab:solnWeak}. The force and torque vector directionality |
| 732 |
+ |
results are displayed separately in table \ref{tab:solnWeakAng}, where |
| 733 |
+ |
the effect of group-based cutoffs and switching functions on the {\sc |
| 734 |
+ |
sp} and {\sc sf} potentials are investigated. |
| 735 |
+ |
|
| 736 |
|
\begin{table}[htbp] |
| 737 |
|
\centering |
| 738 |
< |
\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 $\varepsilon \approx \infty$).} |
| 738 |
> |
\caption{Regression results for the 6 \AA\ argon sphere in liquid |
| 739 |
> |
water system. Tabulated results include $\Delta E$ values (top set), |
| 740 |
> |
force vector magnitudes (middle set) and torque vector magnitudes |
| 741 |
> |
(bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted |
| 742 |
> |
Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 743 |
> |
$\varepsilon \approx \infty$).} |
| 744 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 745 |
|
\\ |
| 746 |
|
\toprule |
| 787 |
|
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 788 |
|
\bottomrule |
| 789 |
|
\end{tabular} |
| 790 |
< |
\label{argonTab} |
| 790 |
> |
\label{tab:argon} |
| 791 |
|
\end{table} |
| 792 |
|
|
| 793 |
|
\begin{table}[htbp] |
| 794 |
|
\centering |
| 795 |
< |
\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.} |
| 795 |
> |
\caption{Variance results from Gaussian fits to angular |
| 796 |
> |
distributions of the force and torque vectors in the 6 \AA\ sphere of |
| 797 |
> |
argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
| 798 |
> |
Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = |
| 799 |
> |
Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group |
| 800 |
> |
Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
| 801 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 802 |
|
\\ |
| 803 |
|
\toprule |
| 828 |
|
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 829 |
|
\bottomrule |
| 830 |
|
\end{tabular} |
| 831 |
< |
\label{argonTabAng} |
| 831 |
> |
\label{tab:argonAng} |
| 832 |
|
\end{table} |
| 833 |
|
|
| 834 |
< |
\end{document} |
| 834 |
> |
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. |
| 835 |
> |
|
| 836 |
> |
\newpage |
| 837 |
> |
|
| 838 |
> |
\bibliographystyle{jcp2} |
| 839 |
> |
\bibliography{electrostaticMethods} |
| 840 |
> |
|
| 841 |
> |
\end{document} |
