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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 {\sc spme}. Each of |
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the seven systems comprises its own section and has its own discussion |
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and tabular listing of the results for the $\Delta E$, force and |
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torque vector magnitude, and force and torque vector direction |
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comparisons. |
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|
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\section{\label{app-water}Liquid Water} |
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\section{\label{app:water}Liquid Water} |
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500 liquid state configurations were generated as described in the Methods section using the SPC/E model of water.\cite{Berendsen87} The results for the energy gap comparisons and the force and torque vector magnitude comparisons are shown in table \ref{tab:spceMag}. The force and torque vector directionality results are displayed separately in table \ref{tab:spceAng}, where the effect of group-based cutoffs and switching functions on the {\sc sp} and {\sc sf} potentials are investigated. |
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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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\midrule |
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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{tab:spceMag} |
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\label{tab:spce} |
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\end{table} |
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|
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Unless there is a significant change in result in any of the further systems, we are going to neglect to comment on the pure cutoff (PC) system. It is unreasonable to expect it to perform well in either energetic or dynamic studies using molecular groups, as evidenced in previous studies and in the results displayed here and in the rest of this paper.\cite{Adams79,Steinbach94} In contrast to PC, the {\sc sp} method shows variety in the results. In the weakly and undamped cases, the results are poor for both the energy gap and dynamics, and this is not surprising considering the energy oscillations observed by Wolf {\it et al.} and the discontinuity in the forces discussed in the main portion of this paper.\cite{Wolf99} Long cutoff radii, moderate damping, or a combination of the two are required for {\sc sp} to perform respectably. With a cutoff greater than 12 \AA\ and $\alpha$ of 0.2 \AA$^{-1}$, {\sc sp} provides result right in line with SPME. |
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The {\sc sf} method displays energetic and dynamic results very similar to SPME under undamped to moderately damped conditions. The quality seems to degrade in the overdamped case ($\alpha = 0.3 \AA^{-1}$) to values identical to {\sc sp}, so it is important not to get carried away with the use of damping. A cutoff radius choice of 12 \AA\ or higher is recommended, primarily due to the energy gap results of interest in Monte Carlo (MC) calculations. |
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The group switched cutoff (GSC) and reaction field (RF) methods seem to have very similar behavior, with the preference given to RF for the improved energy gap results. Neither mimics the energetics of SPME as well as the {\sc sp} (with moderate damping) and {\sc sf} methods, and the results seem relatively independent of cutoff radius. The dynamics for both methods, however, are quite good. Both methods utilize switching functions, which correct and discontinuities in the potential and forces, a possible reason for the improved results. It is interesting to compare the PC with the GSC cases, and recognize the significant improvement that group based cutoffs and switching functions provide. This as been recognized in previous studies,\cite{Andrea83,Steinbach94} and is a useful tactic for stably incorporating local area electrostatic effects. |
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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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\label{tab:spceAng} |
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\end{table} |
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|
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The directionality of the force and torque vectors show a lot of parallels with the magnitude results in table \ref{tab:spceMag}. |
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The water results appear to parallel the combined results seen in the |
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discussion section of the main paper. There is good agreement with |
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{\sc spme} in both energetic and dynamic behavior when using the {\sc sf} |
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method with and without damping. The {\sc sp} method does well with an |
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$\alpha$ around 0.2 \AA$^{-1}$, particularly with cutoff radii greater |
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than 12 \AA. Overdamping the electrostatics reduces the agreement between both these methods and {\sc spme}. |
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|
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\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
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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{tab: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{tab: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 periodicity term of the Ewald summation. As |
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expected, the energy gap agreement with {\sc spme} is reduced for the |
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{\sc sp} and {\sc sf} methods with parameters that were acceptable for |
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the disordered liquid system. Moving to higher $R_\textrm{c}$ helps |
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improve the agreement, though at an increase in computational cost. |
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The dynamics of this crystalline system (both in magnitude and |
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direction) are little affected. Both methods still reproduce the Ewald |
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behavior with the same parameter recommendations from the previous |
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section. |
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|
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It is also worth noting that {\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 charged disordered system. The results |
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for the energy gap comparisons and the force vector magnitude |
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comparisons are shown in table \ref{tab:melt}. The force vector |
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directionality results are displayed separately in table |
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\ref{tab:meltAng}. |
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|
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\begin{table}[htbp] |
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\centering |
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\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{tab: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{tab: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 \\ |
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|
\bottomrule |
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|
\end{tabular} |
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\label{tab: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] |
| 436 |
|
\centering |
| 437 |
< |
\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 |
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> |
distributions of the force vectors in the crystalline NaCl system. PC |
| 439 |
> |
= Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, GSC = Group |
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> |
Switched Cutoff, and RF = Reaction Field (where $\varepsilon \approx |
| 441 |
> |
\infty$).} |
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|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 443 |
|
\\ |
| 444 |
|
\toprule |
| 458 |
|
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
| 459 |
|
\bottomrule |
| 460 |
|
\end{tabular} |
| 461 |
< |
\label{tab:saltTabAng} |
| 461 |
> |
\label{tab:saltAng} |
| 462 |
|
\end{table} |
| 463 |
|
|
| 464 |
< |
\section{\label{app-sol1}Weak NaCl Solution} |
| 464 |
> |
The crystalline NaCl system is the most challenging test case for the |
| 465 |
> |
pairwise summation methods, as evidenced by the results in tables |
| 466 |
> |
\ref{tab:salt} and \ref{tab:saltAng}. The undamped and weakly damped |
| 467 |
> |
{\sc sf} methods with a 12 \AA\ cutoff radius seem to be the best |
| 468 |
> |
choices. These methods match well with {\sc spme} across the energy |
| 469 |
> |
gap, force magnitude, and force directionality tests. The {\sc sp} |
| 470 |
> |
method struggles in all cases, with the exception of good dynamics |
| 471 |
> |
reproduction when using weak electrostatic damping with a large cutoff |
| 472 |
> |
radius. |
| 473 |
|
|
| 474 |
+ |
The moderate electrostatic damping case is not as good as we would |
| 475 |
+ |
expect given the long-time dynamics results observed for this |
| 476 |
+ |
system. Since the data tabulated in tables \ref{tab:salt} and |
| 477 |
+ |
\ref{tab:saltAng} are a test of instantaneous dynamics, this indicates |
| 478 |
+ |
that good long-time dynamics comes in part at the expense of |
| 479 |
+ |
short-time dynamics. |
| 480 |
+ |
|
| 481 |
+ |
\section{\label{app:solnWeak}Weak NaCl Solution} |
| 482 |
+ |
|
| 483 |
+ |
In an effort to bridge the charged atomic and neutral molecular |
| 484 |
+ |
systems, Na$^+$ and Cl$^-$ ion charge defects were incorporated into |
| 485 |
+ |
the liquid water system. This low ionic strength system consists of 4 |
| 486 |
+ |
ions in the 1000 SPC/E water solvent ($\approx$0.11 M). The results |
| 487 |
+ |
for the energy gap comparisons and the force and torque vector |
| 488 |
+ |
magnitude comparisons are shown in table \ref{tab:solnWeak}. The |
| 489 |
+ |
force and torque vector directionality results are displayed |
| 490 |
+ |
separately in table \ref{tab:solnWeakAng}, where the effect of |
| 491 |
+ |
group-based cutoffs and switching functions on the {\sc sp} and {\sc |
| 492 |
+ |
sf} potentials are investigated. |
| 493 |
+ |
|
| 494 |
|
\begin{table}[htbp] |
| 495 |
|
\centering |
| 496 |
< |
\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.} |
| 496 |
> |
\caption{Regression results for the weak NaCl solution |
| 497 |
> |
system. Tabulated results include $\Delta E$ values (top set), force |
| 498 |
> |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 499 |
> |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 500 |
> |
GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon |
| 501 |
> |
\approx \infty$).} |
| 502 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 503 |
|
\\ |
| 504 |
|
\toprule |
| 545 |
|
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 546 |
|
\bottomrule |
| 547 |
|
\end{tabular} |
| 548 |
< |
\label{tab:sol1Tab} |
| 548 |
> |
\label{tab:solnWeak} |
| 549 |
|
\end{table} |
| 550 |
|
|
| 551 |
|
\begin{table}[htbp] |
| 552 |
|
\centering |
| 553 |
< |
\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.} |
| 553 |
> |
\caption{Variance results from Gaussian fits to angular |
| 554 |
> |
distributions of the force and torque vectors in the weak NaCl |
| 555 |
> |
solution system. PC = Pure Cutoff, SP = Shifted Potential, SF = |
| 556 |
> |
Shifted Force, GSC = Group Switched Cutoff, RF = Reaction Field (where |
| 557 |
> |
$\varepsilon \approx \infty$), GSSP = Group Switched Shifted |
| 558 |
> |
Potential, and GSSF = Group Switched Shifted Force.} |
| 559 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 560 |
|
\\ |
| 561 |
|
\toprule |
| 586 |
|
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 587 |
|
\bottomrule |
| 588 |
|
\end{tabular} |
| 589 |
< |
\label{tab:sol1TabAng} |
| 589 |
> |
\label{tab:solnWeakAng} |
| 590 |
|
\end{table} |
| 591 |
|
|
| 592 |
< |
\section{\label{app-sol10}Strong NaCl Solution} |
| 592 |
> |
Because this system is a perturbation of the pure liquid water system, |
| 593 |
> |
comparisons are best drawn between these two sets. The {\sc sp} and |
| 594 |
> |
{\sc sf} methods are not significantly affected by the inclusion of a |
| 595 |
> |
few ions. The aspect of cutoff sphere neutralization aids in the |
| 596 |
> |
smooth incorporation of these ions; thus, all of the observations |
| 597 |
> |
regarding these methods carry over from section \ref{app:water}. The |
| 598 |
> |
differences between these systems are more visible for the {\sc rf} |
| 599 |
> |
method. Though good force agreement is still maintained, the energy |
| 600 |
> |
gaps show a significant increase in the data scatter. This foreshadows |
| 601 |
> |
the breakdown of the method as we introduce charged inhomogeneities. |
| 602 |
|
|
| 603 |
+ |
\section{\label{app:solnStr}Strong NaCl Solution} |
| 604 |
+ |
|
| 605 |
+ |
The bridging of the charged atomic and neutral molecular systems was |
| 606 |
+ |
further developed by considering a high ionic strength system |
| 607 |
+ |
consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
| 608 |
+ |
M). The results for the energy gap comparisons and the force and |
| 609 |
+ |
torque vector magnitude comparisons are shown in table |
| 610 |
+ |
\ref{tab:solnStr}. The force and torque vector directionality |
| 611 |
+ |
results are displayed separately in table \ref{tab:solnStrAng}, where |
| 612 |
+ |
the effect of group-based cutoffs and switching functions on the {\sc |
| 613 |
+ |
sp} and {\sc sf} potentials are investigated. |
| 614 |
+ |
|
| 615 |
|
\begin{table}[htbp] |
| 616 |
|
\centering |
| 617 |
< |
\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 $\varepsilon \approx \infty$).} |
| 617 |
> |
\caption{Regression results for the strong NaCl solution |
| 618 |
> |
system. Tabulated results include $\Delta E$ values (top set), force |
| 619 |
> |
vector magnitudes (middle set) and torque vector magnitudes (bottom |
| 620 |
> |
set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
| 621 |
> |
GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 622 |
> |
$\varepsilon \approx \infty$).} |
| 623 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 624 |
|
\\ |
| 625 |
|
\toprule |
| 666 |
|
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 667 |
|
\bottomrule |
| 668 |
|
\end{tabular} |
| 669 |
< |
\label{tab:sol10Tab} |
| 669 |
> |
\label{tab:solnStr} |
| 670 |
|
\end{table} |
| 671 |
|
|
| 672 |
|
\begin{table}[htbp] |
| 702 |
|
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 703 |
|
\bottomrule |
| 704 |
|
\end{tabular} |
| 705 |
< |
\label{tab:sol10TabAng} |
| 705 |
> |
\label{tab:solnStrAng} |
| 706 |
|
\end{table} |
| 707 |
|
|
| 708 |
< |
\section{\label{app-argon}Argon Sphere in Water} |
| 708 |
> |
The {\sc rf} method struggles with the jump in ionic strength. The |
| 709 |
> |
configuration energy differences degrade to unusable levels while the |
| 710 |
> |
forces and torques show a more modest reduction in the agreement with |
| 711 |
> |
{\sc spme}. The {\sc rf} method was designed for homogeneous systems, |
| 712 |
> |
and this attribute is apparent in these results. |
| 713 |
|
|
| 714 |
+ |
The {\sc sp} and {\sc sf} methods require larger cutoffs to maintain |
| 715 |
+ |
their agreement with {\sc spme}. With these results, we still |
| 716 |
+ |
recommend no to moderate damping for the {\sc sf} method and moderate |
| 717 |
+ |
damping for the {\sc sp} method, both with cutoffs greater than 12 |
| 718 |
+ |
\AA. |
| 719 |
+ |
|
| 720 |
+ |
\section{\label{app:argon}Argon Sphere in Water} |
| 721 |
+ |
|
| 722 |
+ |
The final model system studied was a 6 \AA\ sphere of Argon solvated |
| 723 |
+ |
by SPC/E water. The results for the energy gap comparisons and the |
| 724 |
+ |
force and torque vector magnitude comparisons are shown in table |
| 725 |
+ |
\ref{tab:argon}. The force and torque vector directionality |
| 726 |
+ |
results are displayed separately in table \ref{tab:argonAng}, where |
| 727 |
+ |
the effect of group-based cutoffs and switching functions on the {\sc |
| 728 |
+ |
sp} and {\sc sf} potentials are investigated. |
| 729 |
+ |
|
| 730 |
|
\begin{table}[htbp] |
| 731 |
|
\centering |
| 732 |
< |
\caption{Regression results for the 6 \AA\ argon sphere in liquid 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 $\varepsilon \approx \infty$).} |
| 732 |
> |
\caption{Regression results for the 6 \AA\ Argon sphere in liquid |
| 733 |
> |
water system. Tabulated results include $\Delta E$ values (top set), |
| 734 |
> |
force vector magnitudes (middle set) and torque vector magnitudes |
| 735 |
> |
(bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted |
| 736 |
> |
Force, GSC = Group Switched Cutoff, and RF = Reaction Field (where |
| 737 |
> |
$\varepsilon \approx \infty$).} |
| 738 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 739 |
|
\\ |
| 740 |
|
\toprule |
| 781 |
|
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 782 |
|
\bottomrule |
| 783 |
|
\end{tabular} |
| 784 |
< |
\label{tab:argonTab} |
| 784 |
> |
\label{tab:argon} |
| 785 |
|
\end{table} |
| 786 |
|
|
| 787 |
|
\begin{table}[htbp] |
| 788 |
|
\centering |
| 789 |
< |
\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.} |
| 789 |
> |
\caption{Variance results from Gaussian fits to angular |
| 790 |
> |
distributions of the force and torque vectors in the 6 \AA\ sphere of |
| 791 |
> |
Argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
| 792 |
> |
Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = |
| 793 |
> |
Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group |
| 794 |
> |
Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
| 795 |
|
\begin{tabular}{@{} ccrrrrrr @{}} |
| 796 |
|
\\ |
| 797 |
|
\toprule |
| 822 |
|
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 823 |
|
\bottomrule |
| 824 |
|
\end{tabular} |
| 825 |
< |
\label{tab:argonTabAng} |
| 825 |
> |
\label{tab:argonAng} |
| 826 |
|
\end{table} |
| 827 |
|
|
| 828 |
+ |
This system does not appear to show any significant deviations from |
| 829 |
+ |
the previously observed results. The {\sc sp} and {\sc sf} methods |
| 830 |
+ |
have aggrements similar to those observed in section |
| 831 |
+ |
\ref{app:water}. The only significant difference is the improvement |
| 832 |
+ |
in the configuration energy differences for the {\sc rf} method. This |
| 833 |
+ |
is surprising in that we are introducing an inhomogeneity to the |
| 834 |
+ |
system; however, this inhomogeneity is charge-neutral and does not |
| 835 |
+ |
result in charged cutoff spheres. The charge-neutrality of the cutoff |
| 836 |
+ |
spheres, which the {\sc sp} and {\sc sf} methods explicitly enforce, |
| 837 |
+ |
seems to play a greater role in the stability of the {\sc rf} method |
| 838 |
+ |
than the required homogeneity of the environment. |
| 839 |
+ |
|
| 840 |
|
\newpage |
| 841 |
|
|
| 842 |
|
\bibliographystyle{jcp2} |
| 843 |
|
\bibliography{electrostaticMethods} |
| 844 |
|
|
| 845 |
< |
\end{document} |
| 845 |
> |
\end{document} |
