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\begin{document} |
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This document includes individual system-based comparisons of the |
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studied methods with smooth particle mesh Ewald {\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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This document includes comparisons of the new pairwise electrostatic |
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methods with {\sc spme} for each of the individual systems mentioned |
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in paper. Each of the seven sections contains information about a |
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single system type and has its own discussion and tabular listing of |
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the results for the comparisons of $\Delta E$, the magnitudes of the |
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forces and torques, and directionality of the force and torque |
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vectors. |
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\section{\label{app:water}Liquid Water} |
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|
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\label{tab:spceAng} |
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\end{table} |
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|
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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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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 between both these methods and {\sc spme}. |
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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 really improves the results for these neutral water |
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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 corrects discontinuities in the potential and |
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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 has been recognized in previous |
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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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show how incorporating some implicit properties of the surroundings |
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(i.e. $\epsilon_\textrm{S}$) can improve the solvent depiction. |
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|
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A final note for the liquid water system, use of group cutoffs and a |
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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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\end{table} |
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|
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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} reduces for the |
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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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\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 and torque vector |
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magnitude comparisons are shown in table \ref{tab:melt}. The force |
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and torque vector directionality results are displayed separately in |
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table \ref{tab:meltAng}, where the effect of group-based cutoffs and |
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switching functions on the {\sc sp} and {\sc sf} potentials are |
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investigated. |
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pairwise summation methods in a disordered system of charges. The |
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results for the energy gap comparisons and the force vector magnitude |
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comparisons are shown in table \ref{tab:melt}. The force vector |
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directionality results are displayed separately in table |
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\ref{tab:meltAng}. |
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|
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\begin{table}[htbp] |
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\centering |
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A 1000K NaCl crystal was used to investigate the accuracy of the |
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pairwise summation methods in an ordered system of charged |
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particles. The results for the energy gap comparisons and the force |
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and torque vector magnitude comparisons are shown in table |
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\ref{tab:salt}. The force and torque vector directionality results |
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are displayed separately in table \ref{tab:saltAng}, where the effect |
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of group-based cutoffs and switching functions on the {\sc sp} and |
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{\sc sf} potentials are investigated. |
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vector magnitude comparisons are shown in table \ref{tab:salt}. The |
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force vector directionality results are displayed separately in table |
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\ref{tab:saltAng}. |
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|
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\begin{table}[htbp] |
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\centering |
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radius. |
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|
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The moderate electrostatic damping case is not as good as we would |
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expect given the good long-time dynamics results observed for this |
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system. Since the data tabulated in table \ref{tab:salt} and |
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expect given the long-time dynamics results observed for this |
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system. Since the data tabulated in tables \ref{tab:salt} and |
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\ref{tab:saltAng} are a test of instantaneous dynamics, this indicates |
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that good long-time dynamics comes in part at the expense of |
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short-time dynamics. Further indication of this comes from the full |
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power spectra shown in the main text. It appears as though a |
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distortion is introduced between 200 to 350 cm$^{-1}$ with increased |
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$\alpha$. |
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short-time dynamics. |
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|
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\section{\label{app:solnWeak}Weak NaCl Solution} |
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|
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system. Tabulated results include $\Delta E$ values (top set), force |
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vector magnitudes (middle set) and torque vector magnitudes (bottom |
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set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted Force, |
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GSC = Group Switched Cutoff, RF = Reaction Field (where $\varepsilon |
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\approx \infty$), GSSP = Group Switched Shifted Potential, and GSSF = |
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Group Switched Shifted Force.} |
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GSC = Group Switched Cutoff, and RF = Reaction Field (where $\varepsilon |
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\approx \infty$).} |
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\begin{tabular}{@{} ccrrrrrr @{}} |
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\\ |
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\toprule |
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regarding these methods carry over from section \ref{app:water}. The |
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differences between these systems are more visible for the {\sc rf} |
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method. Though good force agreement is still maintained, the energy |
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gaps show a significant increase in the data scatter. This foreshadows |
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the breakdown of the method as we introduce charged inhomogeneities. |
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gaps show a significant increase in the scatter of the data. |
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|
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\section{\label{app:solnStr}Strong NaCl Solution} |
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|
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consisting of 40 ions in the 1000 SPC/E water solvent ($\approx$1.1 |
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M). The results for the energy gap comparisons and the force and |
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torque vector magnitude comparisons are shown in table |
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\ref{tab:solnWeak}. The force and torque vector directionality |
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results are displayed separately in table \ref{tab:solnWeakAng}, where |
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\ref{tab:solnStr}. The force and torque vector directionality |
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results are displayed separately in table \ref{tab:solnStrAng}, where |
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the effect of group-based cutoffs and switching functions on the {\sc |
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sp} and {\sc sf} potentials are investigated. |
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|
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\end{table} |
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The {\sc rf} method struggles with the jump in ionic strength. The |
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configuration energy difference degrade to unusable levels while the |
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configuration energy differences degrade to unusable levels while the |
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forces and torques show a more modest reduction in the agreement with |
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{\sc spme}. The {\sc rf} method was designed for homogeneous systems, |
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and this attribute is apparent in these results. |
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|
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\section{\label{app:argon}Argon Sphere in Water} |
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The final model system studied was 6 \AA\ sphere of Argon solvated by |
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SPC/E water. The results for the energy gap comparisons and the force |
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and torque vector magnitude comparisons are shown in table |
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\ref{tab:solnWeak}. The force and torque vector directionality |
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results are displayed separately in table \ref{tab:solnWeakAng}, where |
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The final model system studied was a 6 \AA\ sphere of Argon solvated |
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by SPC/E water. The results for the energy gap comparisons and the |
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force and torque vector magnitude comparisons are shown in table |
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\ref{tab:argon}. The force and torque vector directionality |
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results are displayed separately in table \ref{tab:argonAng}, where |
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the effect of group-based cutoffs and switching functions on the {\sc |
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sp} and {\sc sf} potentials are investigated. |
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|
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\begin{table}[htbp] |
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\centering |
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\caption{Regression results for the 6 \AA\ argon sphere in liquid |
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\caption{Regression results for the 6 \AA\ Argon sphere in liquid |
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water system. Tabulated results include $\Delta E$ values (top set), |
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force vector magnitudes (middle set) and torque vector magnitudes |
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(bottom set). PC = Pure Cutoff, SP = Shifted Potential, SF = Shifted |
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\centering |
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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 6 \AA\ sphere of |
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argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
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Argon in liquid water system. PC = Pure Cutoff, SP = Shifted |
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Potential, SF = Shifted Force, GSC = Group Switched Cutoff, RF = |
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Reaction Field (where $\varepsilon \approx \infty$), GSSP = Group |
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Switched Shifted Potential, and GSSF = Group Switched Shifted Force.} |
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\label{tab:argonAng} |
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\end{table} |
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|
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This system appears not to show in any significant deviation in the |
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previously observed results. The {\sc sp} and {\sc sf} methods give |
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result qualities similar to those observed in section |
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This system does not appear to show any significant deviations from |
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the previously observed results. The {\sc sp} and {\sc sf} methods |
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have aggrements similar to those observed in section |
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\ref{app:water}. The only significant difference is the improvement |
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for the configuration energy differences for the {\sc rf} method. This |
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in the configuration energy differences for the {\sc rf} method. This |
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is surprising in that we are introducing an inhomogeneity to the |
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system; however, this inhomogeneity is charge-neutral and does not |
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result in charged cutoff spheres. The charge-neutrality of the cutoff |
