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\usepackage{endfloat} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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%\usepackage{ifsym} |
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\usepackage{epsf} |
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\usepackage{times} |
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\usepackage{mathptm} |
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\begin{document} |
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\title{On the necessity of the Ewald Summation in molecular simulations: Alternatives to the accepted standard of cutoff policies} |
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\title{Is the Ewald Summation necessary in typical molecular simulations: Alternatives to the accepted standard of cutoff policies} |
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\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry\\ |
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\section{Introduction} |
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In this paper, a variety of simulation situations were analyzed to determine the relative effectiveness of the adapted Wolf spherical truncation schemes at reproducing the results obtained using a smooth particle mesh Ewald (SPME) summation technique. In addition to the Shifted-Potential and Shifted-Force adapted Wolf methods, both reaction field and uncorrected cutoff methods were included for comparison purposes. The general usability of these methods in both Monte Carlo and Molecular Dynamics calculations was assessed through statistical analysis over the combined results from all of the following studied systems: |
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\begin{list}{-}{} |
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\begin{enumerate} |
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\item Liquid Water |
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\item Crystalline Water (Ice I$_\textrm{c}$) |
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\item NaCl Crystal |
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\item NaCl Melt |
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\item 1 M Solution of NaCl in Water |
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\item 10 M Solution of NaCl in Water |
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\item Low Ionic Strength Solution of NaCl in Water |
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\item High Ionic Strength Solution of NaCl in Water |
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\item 6 \AA\ Radius Sphere of Argon in Water |
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\end{list} |
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\end{enumerate} |
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Additional discussion on the results from the individual systems was also performed to identify limitations of the considered methods in specific systems. |
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\section{Methods} |
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In each of the simulated systems, 500 distinct configurations were generated, and the electrostatic summation methods were compared via sequential application on each of these fixed configurations. The methods compared include SPME, the aforementioned Shifted Potential and Shifted Force methods - both with damping parameters ($\alpha$) of 0, 0.1, 0.2, and 0.3 \AA$^{-1}$, reaction field with an infinite dielectric constant, and an unmodified cutoff. Group-based cutoffs with a fifth-order polynomial switching function were necessary for the reaction field simulations and were utilized in the SP, SF, and pure cutoff methods for comparison to the standard lack of group-based cutoffs with a hard truncation. |
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Generation of the system configurations was dependent on the system type. For the solid and liquid water configurations, configuration snapshots were taken at regular intervals from higher temperature 1000 SPC/E water molecule trajectories and individually equilibrated. The solid and liquid NaCl systems consisted of 500 Na+ and 500 Cl- ions and were selected and equilibrated in the same fashion as the water systems. For the 1 and 10 M NaCl solutions, 4 and 40 ions, respectively, were first solvated in a 1000 water molecule boxes. Ion and water positions were then randomly swapped, and the resulting configurations were again individually equilibrated. Finally, for the Argon/Water "charge void" systems, the identities of all the SPC/E waters within 6 \AA\ of the center of the equilibrated water configurations were converted to argon (Fig. \ref{argonSlice}). |
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Generation of the system configurations was dependent on the system type. For the solid and liquid water configurations, configuration snapshots were taken at regular intervals from higher temperature 1000 SPC/E water molecule trajectories and each equilibrated individually. The solid and liquid NaCl systems consisted of 500 Na+ and 500 Cl- ions and were selected and equilibrated in the same fashion as the water systems. For the low and high ionic strength NaCl solutions, 4 and 40 ions were first solvated in a 1000 water molecule boxes respectively. Ion and water positions were then randomly swapped, and the resulting configurations were again equilibrated individually. Finally, for the Argon/Water "charge void" systems, the identities of all the SPC/E waters within 6 \AA\ of the center of the equilibrated water configurations were converted to argon (Fig. \ref{argonSlice}). |
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\begin{figure} |
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\centering |
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\section{Results and Discussion} |
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\subsection{$\Delta E$ Comparison} |
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In order to evaluate the performance of the adapted Wolf Shifted Potential and Shifted Force electrostatic summation methods for Monte Carlo simulations, the energy differences between configurations need to be compared to the results using SPME. Considering the SPME results to be the correct or desired behavior, ideal performance of a tested method is taken to be agreement between the energy differences calculated. Linear least squares regression of the $\Delta E$ values between configurations using SPME against $\Delta E$ values using tested methods provides a quantitative comparison of this agreement. Unitary results for both the correlation and correlation coefficient for these regressions indicate equivalent energetic results between the methods. The correlation is the slope of the plotted data while the correlation coefficient ($R^2$) is a measure of the of the data scatter around the fitted line and gives an idea of the quality of the fit (Fig. \ref{linearFit}). |
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\begin{figure} |
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\label{linearFit} |
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\end{figure} |
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With 500 independent configurations, 124,750 $\Delta E$ data points are used in a regression of a single system. Results and discussion for the individual analysis of each of the system types appear in the appendices of this paper. To probe the applicability of each method in the general case, all the different system types were included in a single regression. The results for this regression are shown in figure \ref{delE}. |
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With 500 independent configurations, 124,750 $\Delta E$ data points are used in a regression of a single system. Results and discussion for the individual analysis of each of the system types appear in the supporting information. To probe the applicability of each method in the general case, all the different system types were included in a single regression. The results for this regression are shown in figure \ref{delE}. |
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\begin{figure} |
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\centering |
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\label{delE} |
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\end{figure} |
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In figure \ref{delE}, it is readily apparent that it is unreasonable to expect realistic results using an unmodified cutoff. This is not all that surprising since this results in large energy fluctuations as atoms move in and out of the cutoff radius. These fluctuations can be alleviated to some degree by using group based cutoffs with a switching function. The Group Switch Cutoff row doesn't show a significant improvement in this plot because the salt and salt solution systems contain non-neutral groups, see appendices \ref{app-water} and \ref{app-ice} for a comparison where all groups are neutral. Correcting the resulting charged cutoff sphere is one of the purposes of the shifted potential proposed by Wolf \textit{et al.}, and this correction indeed improves the results as seen in the Shifted Potental rows. While the undamped case of this method is a significant improvement over the pure cutoff, it still doesn't correlate that well with SPME. Inclusion of potential damping improves the results, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows an excellent correlation and quality of fit with the SPME results, particularly with a cutoff radius greater than 12 \AA . Use of a larger damping parameter is more helpful for the shortest cutoff shown, but it has a detrimental effect on simulations with larger cutoffs. This trend is repeated in the Shifted Force rows, where increasing damping results in progressively poorer correlation; however, damping looks to be unnecessary with this method. Overall, the undamped case is the best performing set, as the correlation and quality of fits are consistently superior regardless of the cutoff distance. This result is beneficial in that the undamped case is less computationally prohibitive do to the lack of complimentary error function calculation when performing the electrostatic pair interaction. The reaction field results illustrates some of that method's limitations, primarily that it was developed for use in homogenous systems; although it does provide results that are an improvement over those from an unmodified cutoff. |
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In figure \ref{delE}, it is apparent that it is unreasonable to expect realistic results using an unmodified cutoff. This is not all that surprising since this results in large energy fluctuations as atoms move in and out of the cutoff radius. These fluctuations can be alleviated to some degree by using group based cutoffs with a switching function. The Group Switch Cutoff row doesn't show a significant improvement in this plot because the salt and salt solution systems contain non-neutral groups, see the accompanying supporting information for a comparison where all groups are neutral. Correcting the resulting charged cutoff sphere is one of the purposes of the shifted potential proposed by Wolf \textit{et al.}, and this correction indeed improves the results as seen in the Shifted Potental rows. While the undamped case of this method is a significant improvement over the pure cutoff, it still doesn't correlate that well with SPME. Inclusion of potential damping improves the results, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows an excellent correlation and quality of fit with the SPME results, particularly with a cutoff radius greater than 12 \AA . Use of a larger damping parameter is more helpful for the shortest cutoff shown, but it has a detrimental effect on simulations with larger cutoffs. This trend is repeated in the Shifted Force rows, where increasing damping results in progressively poorer correlation; however, damping looks to be unnecessary with this method. Overall, the undamped case is the best performing set, as the correlation and quality of fits are consistently superior regardless of the cutoff distance. This result is beneficial in that the undamped case is less computationally prohibitive do to the lack of complimentary error function calculation when performing the electrostatic pair interaction. The reaction field results illustrates some of that method's limitations, primarily that it was developed for use in homogenous systems; although it does provide results that are an improvement over those from an unmodified cutoff. |
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\subsection{Force Magnitude Comparison} |
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While studying the energy differences provides insight into how comparable these methods are energetically, if we want to use these methods in Molecular Dynamics simulations, we also need to consider their effect on forces and torques. Both the magnitude and the direction of the force and torque vectors of each of the bodies in the system can be compared to those observed while using SPME. Analysis of the magnitude of these vectors can be performed in the manner described previously for comparing $\Delta E$ values, only instead of a single value between two system configurations, there is a value for each particle in each configuration. For a system of 1000 water molecules and 40 ions, there are 1040 force vectors and 1000 torque vectors. With 500 configurations, this results in excess of 500,000 data samples for each system type. Figures \ref{frcMag} and \ref{trqMag} respectively show the force and torque vector magnitude results for the accumulated analysis over all the system types. |
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\begin{figure} |
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The results in figure \ref{frcMag} for the most part parallel those seen in the previous look at the $\Delta E$ results. The unmodified cutoff results are poor, but using group based cutoffs and a switching function provides a improvement much more significant than what was seen with $\Delta E$. Looking at the Shifted Potential sets, the slope and R$^2$ improve with the use of damping to an optimal result of 0.2 \AA $^{-1}$ for the 12 and 15 \AA\ cutoffs. Further increases in damping, while beneficial for simulations with a cutoff radius of 9 \AA\ , is detrimental to simulations with larger cutoff radii. The undamped Shifted Force method gives forces in line with those obtained using SPME, and use of a damping function gives little to no gain. The reaction field results are surprisingly good, considering the poor quality of the fits for the $\Delta E$ results. There is still a considerable degree of scatter in the data, but it correlates well in general. |
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\subsection{Torque Magnitude Comparison} |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.25in]{./trqMagplot.pdf} |
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The torque vector magnitude results in figure \ref{trqMag} are similar to those seen for the forces, but more clearly show the improved behavior with increasing cutoff radius. Moderate damping is beneficial to the Shifted Potential and unnecessary with the Shifted Force method, and they also show that over-damping adversely effects all cutoff radii rather than showing an improvement for systems with short cutoffs. The reaction field method performs well when calculating the torques, better than the Shifted Force method over this limited data set. |
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\subsection{Force and Torque Direction Comparison} |
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Having force and torque vectors with magnitudes that are well correlated to SPME is good, but if they are not pointing in the proper direction the results will be incorrect. These vector directions were investigated through measurement of the angle formed between them and those from SPME. The dot product of these unit vectors provides a theta value that is accumulated in a distribution function, weighted by the area on the unit sphere. Narrow distributions of theta values indicates similar to identical results between the tested method and SPME. To measure the narrowness of the resulting distributions, non-linear Gaussian fits were performed. |
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\begin{figure} |
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\label{frcTrqAng} |
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\end{figure} |
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Both the force and torque $\sigma^2$ results from the analysis of the total accumulated system data are tabulated in figure \ref{frcTrqAng}. All of the sets, aside from the over-damped case show the improvement afforded by choosing a longer simulation cutoff. Increasing the cutoff from 9 to 12 \AA\ typically results in a halving of $\sigma^2$, with a similar improvement going from 12 to 15 \AA . The undamped Shifted Force, Group Based Cutoff, and Reaction Field methods all do equivalently well at capturing the direction of both the force and torque vectors. Using damping improves the angular behavior significantly for the Shifted Potential and moderately for the Shifted Force methods. Increasing the damping too far is destructive for both methods, particularly to the torque vectors. Again it is important to recognize that the force vectors cover all particles in the systems, while torque vectors are only available for neutral molecular groups. Damping appears to have a more beneficial non-neutral bodies, and this observation is investigated further in appendices \ref{app-melt}, \ref{app-salt}, \ref{app-sol1}, and \ref{app-sol10}. |
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Both the force and torque $\sigma^2$ results from the analysis of the total accumulated system data are tabulated in figure \ref{frcTrqAng}. All of the sets, aside from the over-damped case show the improvement afforded by choosing a longer simulation cutoff. Increasing the cutoff from 9 to 12 \AA\ typically results in a halving of $\sigma^2$, with a similar improvement going from 12 to 15 \AA . The undamped Shifted Force, Group Based Cutoff, and Reaction Field methods all do equivalently well at capturing the direction of both the force and torque vectors. Using damping improves the angular behavior significantly for the Shifted Potential and moderately for the Shifted Force methods. Increasing the damping too far is destructive for both methods, particularly to the torque vectors. Again it is important to recognize that the force vectors cover all particles in the systems, while torque vectors are only available for neutral molecular groups. Damping appears to have a more beneficial non-neutral bodies, and this observation is investigated further in the accompanying supporting information. |
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\begin{table}[htbp] |
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\centering |
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\caption{Variance ($\sigma^2$) of the force (top set) and torque (bottom set) vector angle difference distributions for the Shifted Potential and Shifted Force methods. Calculations were performed both with (Y) and without (N) group based cutoffs and a switching function. The $\alpha$ values have units of \AA$^{-1}$ and the variance values have units of degrees$^2$.} |
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\begin{tabular}{@{} ccrrrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
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\begin{tabular}{@{} ccrrrrrrrr @{}} |
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\\ |
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\toprule |
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& & \multicolumn{4}{c}{Shifted Potential} & \multicolumn{4}{c}{Shifted Force} \\ |
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\cmidrule(lr){3-6} |
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\cmidrule(l){7-10} |
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Cutoff Radius & Groups & $\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$ & $\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$\\ |
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$R_\textrm{c}$ & Groups & $\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$ & $\alpha = 0$ & $\alpha = 0.1$ & $\alpha = 0.2$ & $\alpha = 0.3$\\ |
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\midrule |
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9 \AA & N & 29.545 & 12.003 & 5.489 & 0.610 & 2.323 & 2.321 & 0.429 & 0.603 \\ |
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& \textbf{Y} & \textbf{2.486} & \textbf{2.160} & \textbf{0.667} & \textbf{0.608} & \textbf{1.768} & \textbf{1.766} & \textbf{0.676} & \textbf{0.609} \\ |
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12 \AA & N & 19.381 & 3.097 & 0.190 & 0.608 & 0.920 & 0.736 & 0.133 & 0.612 \\ |
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& \textbf{Y} & \textbf{0.515} & \textbf{0.288} & \textbf{0.127} & \textbf{0.586} & \textbf{0.308} & \textbf{0.249} & \textbf{0.127} & \textbf{0.586} \\ |
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15 \AA & N & 12.700 & 1.196 & 0.123 & 0.601 & 0.339 & 0.160 & 0.123 & 0.601 \\ |
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& \textbf{Y} & \textbf{0.228} & \textbf{0.099} & \textbf{0.121} & \textbf{0.598} & \textbf{0.144} & \textbf{0.090} & \textbf{0.121} & \textbf{0.598} \\ |
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9 \AA & N & 29.545 & 12.003 & 5.489 & 0.610 & 2.323 & 2.321 & 0.429 & 0.603 \\ |
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& \textbf{Y} & \textbf{2.486} & \textbf{2.160} & \textbf{0.667} & \textbf{0.608} & \textbf{1.768} & \textbf{1.766} & \textbf{0.676} & \textbf{0.609} \\ |
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12 \AA & N & 19.381 & 3.097 & 0.190 & 0.608 & 0.920 & 0.736 & 0.133 & 0.612 \\ |
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& \textbf{Y} & \textbf{0.515} & \textbf{0.288} & \textbf{0.127} & \textbf{0.586} & \textbf{0.308} & \textbf{0.249} & \textbf{0.127} & \textbf{0.586} \\ |
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15 \AA & N & 12.700 & 1.196 & 0.123 & 0.601 & 0.339 & 0.160 & 0.123 & 0.601 \\ |
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& \textbf{Y} & \textbf{0.228} & \textbf{0.099} & \textbf{0.121} & \textbf{0.598} & \textbf{0.144} & \textbf{0.090} & \textbf{0.121} & \textbf{0.598} \\ |
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\midrule |
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9 \AA & N & 262.716 & 116.585 & 5.234 & 5.103 & 2.392 & 2.350 & 1.770 & 5.122 \\ |
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& \textbf{Y} & \textbf{2.115} & \textbf{1.914} & \textbf{1.878} & \textbf{5.142} & \textbf{2.076} & \textbf{2.039} & \textbf{1.972} & \textbf{5.146} \\ |
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12 \AA & N & 129.576 & 25.560 & 1.369 & 5.080 & 0.913 & 0.790 & 1.362 & 5.124 \\ |
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& \textbf{Y} & \textbf{0.810} & \textbf{0.685} & \textbf{1.352} & \textbf{5.082} & \textbf{0.765} & \textbf{0.714} & \textbf{1.360} & \textbf{5.082} \\ |
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15 \AA & N & 87.275 & 4.473 & 1.271 & 5.000 & 0.372 & 0.312 & 1.271 & 5.000 \\ |
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& \textbf{Y} & \textbf{0.282} & \textbf{0.294} & \textbf{1.272} & \textbf{4.999} & \textbf{0.324} & \textbf{0.318} & \textbf{1.272} & \textbf{4.999} \\ |
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9 \AA & N & 262.716 & 116.585 & 5.234 & 5.103 & 2.392 & 2.350 & 1.770 & 5.122 \\ |
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& \textbf{Y} & \textbf{2.115} & \textbf{1.914} & \textbf{1.878} & \textbf{5.142} & \textbf{2.076} & \textbf{2.039} & \textbf{1.972} & \textbf{5.146} \\ |
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12 \AA & N & 129.576 & 25.560 & 1.369 & 5.080 & 0.913 & 0.790 & 1.362 & 5.124 \\ |
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& \textbf{Y} & \textbf{0.810} & \textbf{0.685} & \textbf{1.352} & \textbf{5.082} & \textbf{0.765} & \textbf{0.714} & \textbf{1.360} & \textbf{5.082} \\ |
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15 \AA & N & 87.275 & 4.473 & 1.271 & 5.000 & 0.372 & 0.312 & 1.271 & 5.000 \\ |
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& \textbf{Y} & \textbf{0.282} & \textbf{0.294} & \textbf{1.272} & \textbf{4.999} & \textbf{0.324} & \textbf{0.318} & \textbf{1.272} & \textbf{4.999} \\ |
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\bottomrule |
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\end{tabular} |
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\section{Conclusions} |
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\section{Acknowledgments} |
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\appendix |
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\section{\label{app-water}Liquid Water} |
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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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\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
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\\ |
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\toprule |
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& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
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\cmidrule(lr){3-4} |
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\cmidrule(lr){5-6} |
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\cmidrule(l){7-8} |
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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 & & 3.046 & 0.002 & -3.018 & 0.002 & 4.719 & 0.005 \\ |
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SP & 0.0 & 1.035 & 0.218 & 0.908 & 0.313 & 1.037 & 0.470 \\ |
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& 0.1 & 1.021 & 0.387 & 0.965 & 0.752 & 1.006 & 0.947 \\ |
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& 0.2 & 0.997 & 0.962 & 1.001 & 0.994 & 0.994 & 0.996 \\ |
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& 0.3 & 0.984 & 0.980 & 0.997 & 0.985 & 0.982 & 0.987 \\ |
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SF & 0.0 & 0.977 & 0.974 & 0.996 & 0.992 & 0.991 & 0.997 \\ |
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& 0.1 & 0.983 & 0.974 & 1.001 & 0.994 & 0.996 & 0.998 \\ |
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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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\midrule |
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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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& 0.2 & 0.996 & 0.989 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
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& 0.3 & 0.996 & 0.998 & 0.997 & 0.998 & 0.996 & 0.998 \\ |
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SF & 0.0 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 0.999 \\ |
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& 0.1 & 0.998 & 0.995 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
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& 0.2 & 0.999 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
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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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& 0.2 & 0.987 & 0.985 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
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& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 228 |
– |
SF & 0.0 & 0.978 & 0.990 & 0.988 & 0.997 & 0.993 & 0.999 \\ |
| 229 |
– |
& 0.1 & 0.983 & 0.991 & 0.993 & 0.997 & 0.997 & 0.999 \\ |
| 230 |
– |
& 0.2 & 0.986 & 0.989 & 0.989 & 0.992 & 0.990 & 0.993 \\ |
| 231 |
– |
& 0.3 & 0.965 & 0.973 & 0.967 & 0.975 & 0.967 & 0.976 \\ |
| 232 |
– |
GSC & & 0.995 & 0.981 & 0.999 & 0.991 & 1.001 & 0.994 \\ |
| 233 |
– |
RF & & 0.993 & 0.989 & 0.998 & 0.996 & 1.000 & 0.999 \\ |
| 234 |
– |
\bottomrule |
| 235 |
– |
\end{tabular} |
| 236 |
– |
\label{spceTabTMag} |
| 237 |
– |
\end{table} |
| 238 |
– |
|
| 239 |
– |
\begin{table}[htbp] |
| 240 |
– |
\centering |
| 241 |
– |
\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.} |
| 242 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 243 |
– |
\\ |
| 244 |
– |
\toprule |
| 245 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 246 |
– |
\cmidrule(lr){3-5} |
| 247 |
– |
\cmidrule(l){6-8} |
| 248 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 249 |
– |
\midrule |
| 250 |
– |
PC & & 783.759 & 481.353 & 332.677 & 248.674 & 144.382 & 98.535 \\ |
| 251 |
– |
SP & 0.0 & 659.440 & 380.699 & 250.002 & 235.151 & 134.661 & 88.135 \\ |
| 252 |
– |
& 0.1 & 293.849 & 67.772 & 11.609 & 105.090 & 23.813 & 4.369 \\ |
| 253 |
– |
& 0.2 & 5.975 & 0.136 & 0.094 & 5.553 & 1.784 & 1.536 \\ |
| 254 |
– |
& 0.3 & 0.725 & 0.707 & 0.693 & 7.293 & 6.933 & 6.748 \\ |
| 255 |
– |
SF & 0.0 & 2.238 & 0.713 & 0.292 & 3.290 & 1.090 & 0.416 \\ |
| 256 |
– |
& 0.1 & 2.238 & 0.524 & 0.115 & 3.184 & 0.945 & 0.326 \\ |
| 257 |
– |
& 0.2 & 0.374 & 0.102 & 0.094 & 2.598 & 1.755 & 1.537 \\ |
| 258 |
– |
& 0.3 & 0.721 & 0.707 & 0.693 & 7.322 & 6.933 & 6.748 \\ |
| 259 |
– |
GSC & & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
| 260 |
– |
RF & & 2.091 & 0.403 & 0.113 & 3.583 & 1.071 & 0.399 \\ |
| 261 |
– |
\midrule |
| 262 |
– |
GSSP & 0.0 & 2.431 & 0.614 & 0.274 & 5.135 & 2.133 & 1.339 \\ |
| 263 |
– |
& 0.1 & 1.879 & 0.291 & 0.057 & 3.983 & 1.117 & 0.370 \\ |
| 264 |
– |
& 0.2 & 0.443 & 0.103 & 0.093 & 2.821 & 1.794 & 1.532 \\ |
| 265 |
– |
& 0.3 & 0.728 & 0.694 & 0.692 & 7.387 & 6.942 & 6.748 \\ |
| 266 |
– |
GSSF & 0.0 & 1.298 & 0.270 & 0.083 & 3.098 & 0.992 & 0.375 \\ |
| 267 |
– |
& 0.1 & 1.296 & 0.210 & 0.044 & 3.055 & 0.922 & 0.330 \\ |
| 268 |
– |
& 0.2 & 0.433 & 0.104 & 0.093 & 2.895 & 1.797 & 1.532 \\ |
| 269 |
– |
& 0.3 & 0.728 & 0.694 & 0.692 & 7.410 & 6.942 & 6.748 \\ |
| 270 |
– |
\bottomrule |
| 271 |
– |
\end{tabular} |
| 272 |
– |
\label{spceTabAng} |
| 273 |
– |
\end{table} |
| 274 |
– |
|
| 275 |
– |
\section{\label{app-ice}Solid Water: Ice I$_\textrm{c}$} |
| 276 |
– |
|
| 277 |
– |
\begin{table}[htbp] |
| 278 |
– |
\centering |
| 279 |
– |
\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 $\varepsilon \approx \infty$).} |
| 280 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 281 |
– |
\\ |
| 282 |
– |
\toprule |
| 283 |
– |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 284 |
– |
\cmidrule(lr){3-4} |
| 285 |
– |
\cmidrule(lr){5-6} |
| 286 |
– |
\cmidrule(l){7-8} |
| 287 |
– |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 288 |
– |
\midrule |
| 289 |
– |
PC & & 19.897 & 0.047 & -29.214 & 0.048 & -3.771 & 0.001 \\ |
| 290 |
– |
SP & 0.0 & -0.014 & 0.000 & 2.135 & 0.347 & 0.457 & 0.045 \\ |
| 291 |
– |
& 0.1 & 0.321 & 0.017 & 1.490 & 0.584 & 0.886 & 0.796 \\ |
| 292 |
– |
& 0.2 & 0.896 & 0.872 & 1.011 & 0.998 & 0.997 & 0.999 \\ |
| 293 |
– |
& 0.3 & 0.983 & 0.997 & 0.992 & 0.997 & 0.991 & 0.997 \\ |
| 294 |
– |
SF & 0.0 & 0.943 & 0.979 & 1.048 & 0.978 & 0.995 & 0.999 \\ |
| 295 |
– |
& 0.1 & 0.948 & 0.979 & 1.044 & 0.983 & 1.000 & 0.999 \\ |
| 296 |
– |
& 0.2 & 0.982 & 0.997 & 0.969 & 0.960 & 0.997 & 0.999 \\ |
| 297 |
– |
& 0.3 & 0.985 & 0.997 & 0.961 & 0.961 & 0.991 & 0.997 \\ |
| 298 |
– |
GSC & & 0.983 & 0.985 & 0.966 & 0.994 & 1.003 & 0.999 \\ |
| 299 |
– |
RF & & 0.924 & 0.944 & 0.990 & 0.996 & 0.991 & 0.998 \\ |
| 300 |
– |
\midrule |
| 301 |
– |
PC & & -4.375 & 0.000 & 6.781 & 0.000 & -3.369 & 0.000 \\ |
| 302 |
– |
SP & 0.0 & 0.515 & 0.164 & 0.856 & 0.426 & 0.743 & 0.478 \\ |
| 303 |
– |
& 0.1 & 0.696 & 0.405 & 0.977 & 0.817 & 0.974 & 0.964 \\ |
| 304 |
– |
& 0.2 & 0.981 & 0.980 & 1.001 & 1.000 & 1.000 & 1.000 \\ |
| 305 |
– |
& 0.3 & 0.996 & 0.998 & 0.997 & 0.999 & 0.997 & 0.999 \\ |
| 306 |
– |
SF & 0.0 & 0.991 & 0.995 & 1.003 & 0.998 & 0.999 & 1.000 \\ |
| 307 |
– |
& 0.1 & 0.992 & 0.995 & 1.003 & 0.998 & 1.000 & 1.000 \\ |
| 308 |
– |
& 0.2 & 0.998 & 0.998 & 0.981 & 0.962 & 1.000 & 1.000 \\ |
| 309 |
– |
& 0.3 & 0.996 & 0.998 & 0.976 & 0.957 & 0.997 & 0.999 \\ |
| 310 |
– |
GSC & & 0.997 & 0.996 & 0.998 & 0.999 & 1.000 & 1.000 \\ |
| 311 |
– |
RF & & 0.988 & 0.989 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 312 |
– |
\midrule |
| 313 |
– |
PC & & -6.367 & 0.000 & -3.552 & 0.000 & -3.447 & 0.000 \\ |
| 314 |
– |
SP & 0.0 & 0.643 & 0.409 & 0.833 & 0.607 & 0.961 & 0.805 \\ |
| 315 |
– |
& 0.1 & 0.791 & 0.683 & 0.957 & 0.914 & 1.000 & 0.989 \\ |
| 316 |
– |
& 0.2 & 0.974 & 0.991 & 0.993 & 0.998 & 0.993 & 0.998 \\ |
| 317 |
– |
& 0.3 & 0.976 & 0.992 & 0.977 & 0.992 & 0.977 & 0.992 \\ |
| 318 |
– |
SF & 0.0 & 0.979 & 0.997 & 0.992 & 0.999 & 0.994 & 1.000 \\ |
| 319 |
– |
& 0.1 & 0.984 & 0.997 & 0.996 & 0.999 & 0.998 & 1.000 \\ |
| 320 |
– |
& 0.2 & 0.991 & 0.997 & 0.974 & 0.958 & 0.993 & 0.998 \\ |
| 321 |
– |
& 0.3 & 0.977 & 0.992 & 0.956 & 0.948 & 0.977 & 0.992 \\ |
| 322 |
– |
GSC & & 0.999 & 0.997 & 0.996 & 0.999 & 1.002 & 1.000 \\ |
| 323 |
– |
RF & & 0.994 & 0.997 & 0.997 & 0.999 & 1.000 & 1.000 \\ |
| 324 |
– |
\bottomrule |
| 325 |
– |
\end{tabular} |
| 326 |
– |
\label{iceTab} |
| 327 |
– |
\end{table} |
| 328 |
– |
|
| 329 |
– |
\begin{table}[htbp] |
| 330 |
– |
\centering |
| 331 |
– |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the ice I$_\textrm{c}$ 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.} |
| 332 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 333 |
– |
\\ |
| 334 |
– |
\toprule |
| 335 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 336 |
– |
\cmidrule(lr){3-5} |
| 337 |
– |
\cmidrule(l){6-8} |
| 338 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 339 |
– |
\midrule |
| 340 |
– |
PC & & 2128.921 & 603.197 & 715.579 & 329.056 & 221.397 & 81.042 \\ |
| 341 |
– |
SP & 0.0 & 1429.341 & 470.320 & 447.557 & 301.678 & 197.437 & 73.840 \\ |
| 342 |
– |
& 0.1 & 590.008 & 107.510 & 18.883 & 118.201 & 32.472 & 3.599 \\ |
| 343 |
– |
& 0.2 & 10.057 & 0.105 & 0.038 & 2.875 & 0.572 & 0.518 \\ |
| 344 |
– |
& 0.3 & 0.245 & 0.260 & 0.262 & 2.365 & 2.396 & 2.327 \\ |
| 345 |
– |
SF & 0.0 & 1.745 & 1.161 & 0.212 & 1.135 & 0.426 & 0.155 \\ |
| 346 |
– |
& 0.1 & 1.721 & 0.868 & 0.082 & 1.118 & 0.358 & 0.118 \\ |
| 347 |
– |
& 0.2 & 0.201 & 0.040 & 0.038 & 0.786 & 0.555 & 0.518 \\ |
| 348 |
– |
& 0.3 & 0.241 & 0.260 & 0.262 & 2.368 & 2.400 & 2.327 \\ |
| 349 |
– |
GSC & & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 350 |
– |
RF & & 2.887 & 0.217 & 0.107 & 1.006 & 0.281 & 0.085 \\ |
| 351 |
– |
\midrule |
| 352 |
– |
GSSP & 0.0 & 1.483 & 0.261 & 0.099 & 0.926 & 0.295 & 0.095 \\ |
| 353 |
– |
& 0.1 & 1.341 & 0.123 & 0.037 & 0.835 & 0.234 & 0.085 \\ |
| 354 |
– |
& 0.2 & 0.558 & 0.040 & 0.037 & 0.823 & 0.557 & 0.519 \\ |
| 355 |
– |
& 0.3 & 0.250 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 356 |
– |
GSSF & 0.0 & 2.124 & 0.132 & 0.069 & 0.919 & 0.263 & 0.099 \\ |
| 357 |
– |
& 0.1 & 2.165 & 0.101 & 0.035 & 0.895 & 0.244 & 0.096 \\ |
| 358 |
– |
& 0.2 & 0.706 & 0.040 & 0.037 & 0.870 & 0.559 & 0.519 \\ |
| 359 |
– |
& 0.3 & 0.251 & 0.251 & 0.259 & 2.387 & 2.395 & 2.328 \\ |
| 360 |
– |
\bottomrule |
| 361 |
– |
\end{tabular} |
| 362 |
– |
\label{iceTabAng} |
| 363 |
– |
\end{table} |
| 364 |
– |
|
| 365 |
– |
\section{\label{app-melt}NaCl Melt} |
| 366 |
– |
|
| 367 |
– |
\begin{table}[htbp] |
| 368 |
– |
\centering |
| 369 |
– |
\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.} |
| 370 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 371 |
– |
\\ |
| 372 |
– |
\toprule |
| 373 |
– |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 374 |
– |
\cmidrule(lr){3-4} |
| 375 |
– |
\cmidrule(lr){5-6} |
| 376 |
– |
\cmidrule(l){7-8} |
| 377 |
– |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 378 |
– |
\midrule |
| 379 |
– |
PC & & -0.008 & 0.000 & -0.049 & 0.005 & -0.136 & 0.020 \\ |
| 380 |
– |
SP & 0.0 & 0.937 & 0.996 & 0.880 & 0.995 & 0.971 & 0.999 \\ |
| 381 |
– |
& 0.1 & 1.004 & 0.999 & 0.958 & 1.000 & 0.928 & 0.994 \\ |
| 382 |
– |
& 0.2 & 0.960 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 383 |
– |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 384 |
– |
SF & 0.0 & 1.001 & 1.000 & 0.949 & 1.000 & 1.008 & 1.000 \\ |
| 385 |
– |
& 0.1 & 1.025 & 1.000 & 0.960 & 1.000 & 0.929 & 0.994 \\ |
| 386 |
– |
& 0.2 & 0.966 & 1.000 & 0.813 & 0.996 & 0.811 & 0.954 \\ |
| 387 |
– |
& 0.3 & 0.671 & 0.994 & 0.439 & 0.929 & 0.535 & 0.831 \\ |
| 388 |
– |
\midrule |
| 389 |
– |
PC & & 1.103 & 0.000 & 0.989 & 0.000 & 0.802 & 0.000 \\ |
| 390 |
– |
SP & 0.0 & 0.976 & 0.983 & 1.001 & 0.991 & 0.985 & 0.995 \\ |
| 391 |
– |
& 0.1 & 0.996 & 0.997 & 0.997 & 0.998 & 0.996 & 0.996 \\ |
| 392 |
– |
& 0.2 & 0.993 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 393 |
– |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 394 |
– |
SF & 0.0 & 0.997 & 0.998 & 0.995 & 0.999 & 0.999 & 1.000 \\ |
| 395 |
– |
& 0.1 & 1.001 & 0.997 & 0.997 & 0.999 & 0.996 & 0.996 \\ |
| 396 |
– |
& 0.2 & 0.994 & 0.996 & 0.985 & 0.988 & 0.986 & 0.981 \\ |
| 397 |
– |
& 0.3 & 0.956 & 0.956 & 0.940 & 0.912 & 0.948 & 0.929 \\ |
| 398 |
– |
\bottomrule |
| 399 |
– |
\end{tabular} |
| 400 |
– |
\label{meltTab} |
| 401 |
– |
\end{table} |
| 402 |
– |
|
| 403 |
– |
\begin{table}[htbp] |
| 404 |
– |
\centering |
| 405 |
– |
\caption{Variance results from Gaussian fits to angular distributions of the force vectors in the molten NaCl system. PC = Pure Cutoff, SP = Shifted Potential, and SF = Shifted Force.} |
| 406 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 407 |
– |
\\ |
| 408 |
– |
\toprule |
| 409 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 410 |
– |
\cmidrule(lr){3-5} |
| 411 |
– |
\cmidrule(l){6-8} |
| 412 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 413 |
– |
\midrule |
| 414 |
– |
PC & & 13.294 & 8.035 & 5.366 \\ |
| 415 |
– |
SP & 0.0 & 13.316 & 8.037 & 5.385 \\ |
| 416 |
– |
& 0.1 & 5.705 & 1.391 & 0.360 \\ |
| 417 |
– |
& 0.2 & 2.415 & 7.534 & 13.927 \\ |
| 418 |
– |
& 0.3 & 23.769 & 67.306 & 57.252 \\ |
| 419 |
– |
SF & 0.0 & 1.693 & 0.603 & 0.256 \\ |
| 420 |
– |
& 0.1 & 1.687 & 0.653 & 0.272 \\ |
| 421 |
– |
& 0.2 & 2.598 & 7.523 & 13.930 \\ |
| 422 |
– |
& 0.3 & 23.734 & 67.305 & 57.252 \\ |
| 423 |
– |
\bottomrule |
| 424 |
– |
\end{tabular} |
| 425 |
– |
\label{meltTabAng} |
| 426 |
– |
\end{table} |
| 427 |
– |
|
| 428 |
– |
\section{\label{app-salt}NaCl Crystal} |
| 429 |
– |
|
| 430 |
– |
\begin{table}[htbp] |
| 431 |
– |
\centering |
| 432 |
– |
\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.} |
| 433 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 434 |
– |
\\ |
| 435 |
– |
\toprule |
| 436 |
– |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 437 |
– |
\cmidrule(lr){3-4} |
| 438 |
– |
\cmidrule(lr){5-6} |
| 439 |
– |
\cmidrule(l){7-8} |
| 440 |
– |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 441 |
– |
\midrule |
| 442 |
– |
PC & & -20.241 & 0.228 & -20.248 & 0.229 & -20.239 & 0.228 \\ |
| 443 |
– |
SP & 0.0 & 1.039 & 0.733 & 2.037 & 0.565 & 1.225 & 0.743 \\ |
| 444 |
– |
& 0.1 & 1.049 & 0.865 & 1.424 & 0.784 & 1.029 & 0.980 \\ |
| 445 |
– |
& 0.2 & 0.982 & 0.976 & 0.969 & 0.980 & 0.960 & 0.980 \\ |
| 446 |
– |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.945 \\ |
| 447 |
– |
SF & 0.0 & 1.041 & 0.967 & 0.994 & 0.989 & 0.957 & 0.993 \\ |
| 448 |
– |
& 0.1 & 1.050 & 0.968 & 0.996 & 0.991 & 0.972 & 0.995 \\ |
| 449 |
– |
& 0.2 & 0.982 & 0.975 & 0.959 & 0.980 & 0.960 & 0.980 \\ |
| 450 |
– |
& 0.3 & 0.873 & 0.944 & 0.872 & 0.945 & 0.872 & 0.944 \\ |
| 451 |
– |
\midrule |
| 452 |
– |
PC & & 0.795 & 0.000 & 0.792 & 0.000 & 0.793 & 0.000 \\ |
| 453 |
– |
SP & 0.0 & 0.916 & 0.829 & 1.086 & 0.791 & 1.010 & 0.936 \\ |
| 454 |
– |
& 0.1 & 0.958 & 0.917 & 1.049 & 0.943 & 1.001 & 0.995 \\ |
| 455 |
– |
& 0.2 & 0.981 & 0.981 & 0.982 & 0.984 & 0.981 & 0.984 \\ |
| 456 |
– |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 457 |
– |
SF & 0.0 & 1.002 & 0.983 & 0.997 & 0.994 & 0.991 & 0.997 \\ |
| 458 |
– |
& 0.1 & 1.003 & 0.984 & 0.996 & 0.995 & 0.993 & 0.997 \\ |
| 459 |
– |
& 0.2 & 0.983 & 0.980 & 0.981 & 0.984 & 0.981 & 0.984 \\ |
| 460 |
– |
& 0.3 & 0.950 & 0.952 & 0.950 & 0.953 & 0.950 & 0.953 \\ |
| 461 |
– |
\bottomrule |
| 462 |
– |
\end{tabular} |
| 463 |
– |
\label{saltTab} |
| 464 |
– |
\end{table} |
| 186 |
|
|
| 466 |
– |
\begin{table}[htbp] |
| 467 |
– |
\centering |
| 468 |
– |
\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 $\varepsilon \approx \infty$).} |
| 469 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 470 |
– |
\\ |
| 471 |
– |
\toprule |
| 472 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} \\ |
| 473 |
– |
\cmidrule(lr){3-5} |
| 474 |
– |
\cmidrule(l){6-8} |
| 475 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA \\ |
| 476 |
– |
\midrule |
| 477 |
– |
PC & & 111.945 & 111.824 & 111.866 \\ |
| 478 |
– |
SP & 0.0 & 112.414 & 152.215 & 38.087 \\ |
| 479 |
– |
& 0.1 & 52.361 & 42.574 & 2.819 \\ |
| 480 |
– |
& 0.2 & 10.847 & 9.709 & 9.686 \\ |
| 481 |
– |
& 0.3 & 31.128 & 31.104 & 31.029 \\ |
| 482 |
– |
SF & 0.0 & 10.025 & 3.555 & 1.648 \\ |
| 483 |
– |
& 0.1 & 9.462 & 3.303 & 1.721 \\ |
| 484 |
– |
& 0.2 & 11.454 & 9.813 & 9.701 \\ |
| 485 |
– |
& 0.3 & 31.120 & 31.105 & 31.029 \\ |
| 486 |
– |
\bottomrule |
| 487 |
– |
\end{tabular} |
| 488 |
– |
\label{saltTabAng} |
| 489 |
– |
\end{table} |
| 490 |
– |
|
| 491 |
– |
\section{\label{app-sol1}1M NaCl Solution} |
| 492 |
– |
|
| 493 |
– |
\begin{table}[htbp] |
| 494 |
– |
\centering |
| 495 |
– |
\caption{Regression results for the 1M 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 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 497 |
– |
\\ |
| 498 |
– |
\toprule |
| 499 |
– |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 500 |
– |
\cmidrule(lr){3-4} |
| 501 |
– |
\cmidrule(lr){5-6} |
| 502 |
– |
\cmidrule(l){7-8} |
| 503 |
– |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 504 |
– |
\midrule |
| 505 |
– |
PC & & 0.247 & 0.000 & -1.103 & 0.001 & 5.480 & 0.015 \\ |
| 506 |
– |
SP & 0.0 & 0.935 & 0.388 & 0.984 & 0.541 & 1.010 & 0.685 \\ |
| 507 |
– |
& 0.1 & 0.951 & 0.603 & 0.993 & 0.875 & 1.001 & 0.979 \\ |
| 508 |
– |
& 0.2 & 0.969 & 0.968 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 509 |
– |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 510 |
– |
SF & 0.0 & 0.963 & 0.971 & 0.989 & 0.996 & 0.991 & 0.998 \\ |
| 511 |
– |
& 0.1 & 0.970 & 0.971 & 0.995 & 0.997 & 0.997 & 0.999 \\ |
| 512 |
– |
& 0.2 & 0.972 & 0.975 & 0.996 & 0.997 & 0.994 & 0.997 \\ |
| 513 |
– |
& 0.3 & 0.955 & 0.966 & 0.984 & 0.992 & 0.978 & 0.991 \\ |
| 514 |
– |
GSC & & 0.964 & 0.731 & 0.984 & 0.704 & 1.005 & 0.770 \\ |
| 515 |
– |
RF & & 0.968 & 0.605 & 0.974 & 0.541 & 1.014 & 0.614 \\ |
| 516 |
– |
\midrule |
| 517 |
– |
PC & & 1.354 & 0.000 & -1.190 & 0.000 & -0.314 & 0.000 \\ |
| 518 |
– |
SP & 0.0 & 0.720 & 0.338 & 0.808 & 0.523 & 0.860 & 0.643 \\ |
| 519 |
– |
& 0.1 & 0.839 & 0.583 & 0.955 & 0.882 & 0.992 & 0.978 \\ |
| 520 |
– |
& 0.2 & 0.995 & 0.987 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 521 |
– |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 522 |
– |
SF & 0.0 & 0.998 & 0.994 & 1.000 & 0.998 & 1.000 & 0.999 \\ |
| 523 |
– |
& 0.1 & 0.997 & 0.994 & 1.000 & 0.999 & 1.000 & 1.000 \\ |
| 524 |
– |
& 0.2 & 0.999 & 0.998 & 0.999 & 1.000 & 0.999 & 1.000 \\ |
| 525 |
– |
& 0.3 & 0.995 & 0.996 & 0.996 & 0.998 & 0.996 & 0.998 \\ |
| 526 |
– |
GSC & & 0.995 & 0.990 & 0.998 & 0.997 & 0.998 & 0.996 \\ |
| 527 |
– |
RF & & 0.998 & 0.993 & 0.999 & 0.998 & 0.999 & 0.996 \\ |
| 528 |
– |
\midrule |
| 529 |
– |
PC & & 2.437 & 0.000 & -1.872 & 0.000 & 2.138 & 0.000 \\ |
| 530 |
– |
SP & 0.0 & 0.838 & 0.525 & 0.901 & 0.686 & 0.932 & 0.779 \\ |
| 531 |
– |
& 0.1 & 0.914 & 0.733 & 0.979 & 0.932 & 0.995 & 0.987 \\ |
| 532 |
– |
& 0.2 & 0.977 & 0.969 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 533 |
– |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 534 |
– |
SF & 0.0 & 0.969 & 0.977 & 0.987 & 0.996 & 0.993 & 0.998 \\ |
| 535 |
– |
& 0.1 & 0.975 & 0.978 & 0.993 & 0.996 & 0.997 & 0.998 \\ |
| 536 |
– |
& 0.2 & 0.976 & 0.973 & 0.988 & 0.990 & 0.989 & 0.990 \\ |
| 537 |
– |
& 0.3 & 0.952 & 0.950 & 0.964 & 0.971 & 0.965 & 0.970 \\ |
| 538 |
– |
GSC & & 0.980 & 0.959 & 0.990 & 0.983 & 0.992 & 0.989 \\ |
| 539 |
– |
RF & & 0.984 & 0.975 & 0.996 & 0.995 & 0.998 & 0.998 \\ |
| 540 |
– |
\bottomrule |
| 541 |
– |
\end{tabular} |
| 542 |
– |
\label{sol1Tab} |
| 543 |
– |
\end{table} |
| 544 |
– |
|
| 545 |
– |
\begin{table}[htbp] |
| 546 |
– |
\centering |
| 547 |
– |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the 1M 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.} |
| 548 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 549 |
– |
\\ |
| 550 |
– |
\toprule |
| 551 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 552 |
– |
\cmidrule(lr){3-5} |
| 553 |
– |
\cmidrule(l){6-8} |
| 554 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 555 |
– |
\midrule |
| 556 |
– |
PC & & 882.863 & 510.435 & 344.201 & 277.691 & 154.231 & 100.131 \\ |
| 557 |
– |
SP & 0.0 & 732.569 & 405.704 & 257.756 & 261.445 & 142.245 & 91.497 \\ |
| 558 |
– |
& 0.1 & 329.031 & 70.746 & 12.014 & 118.496 & 25.218 & 4.711 \\ |
| 559 |
– |
& 0.2 & 6.772 & 0.153 & 0.118 & 9.780 & 2.101 & 2.102 \\ |
| 560 |
– |
& 0.3 & 0.951 & 0.774 & 0.784 & 12.108 & 7.673 & 7.851 \\ |
| 561 |
– |
SF & 0.0 & 2.555 & 0.762 & 0.313 & 6.590 & 1.328 & 0.558 \\ |
| 562 |
– |
& 0.1 & 2.561 & 0.560 & 0.123 & 6.464 & 1.162 & 0.457 \\ |
| 563 |
– |
& 0.2 & 0.501 & 0.118 & 0.118 & 5.698 & 2.074 & 2.099 \\ |
| 564 |
– |
& 0.3 & 0.943 & 0.774 & 0.784 & 12.118 & 7.674 & 7.851 \\ |
| 565 |
– |
GSC & & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 566 |
– |
RF & & 2.415 & 0.452 & 0.130 & 6.915 & 1.423 & 0.507 \\ |
| 567 |
– |
\midrule |
| 568 |
– |
GSSP & 0.0 & 2.915 & 0.643 & 0.261 & 9.576 & 3.133 & 1.812 \\ |
| 569 |
– |
& 0.1 & 2.251 & 0.324 & 0.064 & 7.628 & 1.639 & 0.497 \\ |
| 570 |
– |
& 0.2 & 0.590 & 0.118 & 0.116 & 6.080 & 2.096 & 2.103 \\ |
| 571 |
– |
& 0.3 & 0.953 & 0.759 & 0.780 & 12.347 & 7.683 & 7.849 \\ |
| 572 |
– |
GSSF & 0.0 & 1.541 & 0.301 & 0.096 & 6.407 & 1.316 & 0.496 \\ |
| 573 |
– |
& 0.1 & 1.541 & 0.237 & 0.050 & 6.356 & 1.202 & 0.457 \\ |
| 574 |
– |
& 0.2 & 0.568 & 0.118 & 0.116 & 6.166 & 2.105 & 2.105 \\ |
| 575 |
– |
& 0.3 & 0.954 & 0.759 & 0.780 & 12.337 & 7.684 & 7.849 \\ |
| 576 |
– |
\bottomrule |
| 577 |
– |
\end{tabular} |
| 578 |
– |
\label{sol1TabAng} |
| 579 |
– |
\end{table} |
| 580 |
– |
|
| 581 |
– |
\section{\label{app-sol10}10M NaCl Solution} |
| 582 |
– |
|
| 583 |
– |
\begin{table}[htbp] |
| 584 |
– |
\centering |
| 585 |
– |
\caption{Regression results for the 10M 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 $\varepsilon \approx \infty$).} |
| 586 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 587 |
– |
\\ |
| 588 |
– |
\toprule |
| 589 |
– |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 590 |
– |
\cmidrule(lr){3-4} |
| 591 |
– |
\cmidrule(lr){5-6} |
| 592 |
– |
\cmidrule(l){7-8} |
| 593 |
– |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 594 |
– |
\midrule |
| 595 |
– |
PC & & -0.081 & 0.000 & 0.945 & 0.001 & 0.073 & 0.000 \\ |
| 596 |
– |
SP & 0.0 & 0.978 & 0.469 & 0.996 & 0.672 & 0.975 & 0.668 \\ |
| 597 |
– |
& 0.1 & 0.944 & 0.645 & 0.997 & 0.886 & 0.991 & 0.978 \\ |
| 598 |
– |
& 0.2 & 0.873 & 0.896 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 599 |
– |
& 0.3 & 0.831 & 0.860 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 600 |
– |
SF & 0.0 & 0.858 & 0.905 & 0.985 & 0.970 & 0.990 & 0.998 \\ |
| 601 |
– |
& 0.1 & 0.865 & 0.907 & 0.992 & 0.974 & 0.994 & 0.999 \\ |
| 602 |
– |
& 0.2 & 0.862 & 0.894 & 0.985 & 0.993 & 0.980 & 0.993 \\ |
| 603 |
– |
& 0.3 & 0.831 & 0.859 & 0.960 & 0.979 & 0.955 & 0.977 \\ |
| 604 |
– |
GSC & & 1.985 & 0.152 & 0.760 & 0.031 & 1.106 & 0.062 \\ |
| 605 |
– |
RF & & 2.414 & 0.116 & 0.813 & 0.017 & 1.434 & 0.047 \\ |
| 606 |
– |
\midrule |
| 607 |
– |
PC & & -7.028 & 0.000 & -9.364 & 0.000 & 0.925 & 0.865 \\ |
| 608 |
– |
SP & 0.0 & 0.701 & 0.319 & 0.909 & 0.773 & 0.861 & 0.665 \\ |
| 609 |
– |
& 0.1 & 0.824 & 0.565 & 0.970 & 0.930 & 0.990 & 0.979 \\ |
| 610 |
– |
& 0.2 & 0.988 & 0.981 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 611 |
– |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 612 |
– |
SF & 0.0 & 0.993 & 0.988 & 0.992 & 0.984 & 0.998 & 0.999 \\ |
| 613 |
– |
& 0.1 & 0.993 & 0.989 & 0.993 & 0.986 & 0.998 & 1.000 \\ |
| 614 |
– |
& 0.2 & 0.993 & 0.992 & 0.995 & 0.998 & 0.991 & 0.998 \\ |
| 615 |
– |
& 0.3 & 0.983 & 0.985 & 0.985 & 0.991 & 0.978 & 0.990 \\ |
| 616 |
– |
GSC & & 0.964 & 0.897 & 0.970 & 0.917 & 0.925 & 0.865 \\ |
| 617 |
– |
RF & & 0.994 & 0.864 & 0.988 & 0.865 & 0.980 & 0.784 \\ |
| 618 |
– |
\midrule |
| 619 |
– |
PC & & -2.212 & 0.000 & -0.588 & 0.000 & 0.953 & 0.925 \\ |
| 620 |
– |
SP & 0.0 & 0.800 & 0.479 & 0.930 & 0.804 & 0.924 & 0.759 \\ |
| 621 |
– |
& 0.1 & 0.883 & 0.694 & 0.976 & 0.942 & 0.993 & 0.986 \\ |
| 622 |
– |
& 0.2 & 0.952 & 0.943 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 623 |
– |
& 0.3 & 0.914 & 0.909 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 624 |
– |
SF & 0.0 & 0.945 & 0.953 & 0.980 & 0.984 & 0.991 & 0.998 \\ |
| 625 |
– |
& 0.1 & 0.951 & 0.954 & 0.987 & 0.986 & 0.995 & 0.998 \\ |
| 626 |
– |
& 0.2 & 0.951 & 0.946 & 0.980 & 0.984 & 0.980 & 0.983 \\ |
| 627 |
– |
& 0.3 & 0.914 & 0.908 & 0.943 & 0.948 & 0.944 & 0.946 \\ |
| 628 |
– |
GSC & & 0.882 & 0.818 & 0.939 & 0.902 & 0.953 & 0.925 \\ |
| 629 |
– |
RF & & 0.949 & 0.939 & 0.988 & 0.988 & 0.992 & 0.993 \\ |
| 630 |
– |
\bottomrule |
| 631 |
– |
\end{tabular} |
| 632 |
– |
\label{sol10Tab} |
| 633 |
– |
\end{table} |
| 634 |
– |
|
| 635 |
– |
\begin{table}[htbp] |
| 636 |
– |
\centering |
| 637 |
– |
\caption{Variance results from Gaussian fits to angular distributions of the force and torque vectors in the 10M 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.} |
| 638 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 639 |
– |
\\ |
| 640 |
– |
\toprule |
| 641 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 642 |
– |
\cmidrule(lr){3-5} |
| 643 |
– |
\cmidrule(l){6-8} |
| 644 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 645 |
– |
\midrule |
| 646 |
– |
PC & & 957.784 & 513.373 & 2.260 & 340.043 & 179.443 & 13.079 \\ |
| 647 |
– |
SP & 0.0 & 786.244 & 139.985 & 259.289 & 311.519 & 90.280 & 105.187 \\ |
| 648 |
– |
& 0.1 & 354.697 & 38.614 & 12.274 & 144.531 & 23.787 & 5.401 \\ |
| 649 |
– |
& 0.2 & 7.674 & 0.363 & 0.215 & 16.655 & 3.601 & 3.634 \\ |
| 650 |
– |
& 0.3 & 1.745 & 1.456 & 1.449 & 23.669 & 14.376 & 14.240 \\ |
| 651 |
– |
SF & 0.0 & 3.282 & 8.567 & 0.369 & 11.904 & 6.589 & 0.717 \\ |
| 652 |
– |
& 0.1 & 3.263 & 7.479 & 0.142 & 11.634 & 5.750 & 0.591 \\ |
| 653 |
– |
& 0.2 & 0.686 & 0.324 & 0.215 & 10.809 & 3.580 & 3.635 \\ |
| 654 |
– |
& 0.3 & 1.749 & 1.456 & 1.449 & 23.635 & 14.375 & 14.240 \\ |
| 655 |
– |
GSC & & 6.181 & 2.904 & 2.263 & 44.349 & 19.442 & 12.873 \\ |
| 656 |
– |
RF & & 3.891 & 0.847 & 0.323 & 18.628 & 3.995 & 2.072 \\ |
| 657 |
– |
\midrule |
| 658 |
– |
GSSP & 0.0 & 6.197 & 2.929 & 2.290 & 44.441 & 19.442 & 12.873 \\ |
| 659 |
– |
& 0.1 & 4.688 & 1.064 & 0.260 & 31.208 & 6.967 & 2.303 \\ |
| 660 |
– |
& 0.2 & 1.021 & 0.218 & 0.213 & 14.425 & 3.629 & 3.649 \\ |
| 661 |
– |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.540 & 14.390 & 14.245 \\ |
| 662 |
– |
GSSF & 0.0 & 2.494 & 0.546 & 0.217 & 16.391 & 3.230 & 1.613 \\ |
| 663 |
– |
& 0.1 & 2.448 & 0.429 & 0.106 & 16.390 & 2.827 & 1.159 \\ |
| 664 |
– |
& 0.2 & 0.899 & 0.214 & 0.213 & 13.542 & 3.583 & 3.645 \\ |
| 665 |
– |
& 0.3 & 1.752 & 1.454 & 1.451 & 23.587 & 14.390 & 14.245 \\ |
| 666 |
– |
\bottomrule |
| 667 |
– |
\end{tabular} |
| 668 |
– |
\label{sol10TabAng} |
| 669 |
– |
\end{table} |
| 670 |
– |
|
| 671 |
– |
\section{\label{app-argon}Argon Sphere in Water} |
| 672 |
– |
|
| 673 |
– |
\begin{table}[htbp] |
| 674 |
– |
\centering |
| 675 |
– |
\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 $\varepsilon \approx \infty$).} |
| 676 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 677 |
– |
\\ |
| 678 |
– |
\toprule |
| 679 |
– |
& & \multicolumn{2}{c}{9 \AA} & \multicolumn{2}{c}{12 \AA} & \multicolumn{2}{c}{15 \AA}\\ |
| 680 |
– |
\cmidrule(lr){3-4} |
| 681 |
– |
\cmidrule(lr){5-6} |
| 682 |
– |
\cmidrule(l){7-8} |
| 683 |
– |
Method & $\alpha$ & slope & $R^2$ & slope & $R^2$ & slope & $R^2$ \\ |
| 684 |
– |
\midrule |
| 685 |
– |
PC & & 2.320 & 0.008 & -0.650 & 0.001 & 3.848 & 0.029 \\ |
| 686 |
– |
SP & 0.0 & 1.053 & 0.711 & 0.977 & 0.820 & 0.974 & 0.882 \\ |
| 687 |
– |
& 0.1 & 1.032 & 0.846 & 0.989 & 0.965 & 0.992 & 0.994 \\ |
| 688 |
– |
& 0.2 & 0.993 & 0.995 & 0.982 & 0.998 & 0.986 & 0.998 \\ |
| 689 |
– |
& 0.3 & 0.968 & 0.995 & 0.954 & 0.992 & 0.961 & 0.994 \\ |
| 690 |
– |
SF & 0.0 & 0.982 & 0.996 & 0.992 & 0.999 & 0.993 & 1.000 \\ |
| 691 |
– |
& 0.1 & 0.987 & 0.996 & 0.996 & 0.999 & 0.997 & 1.000 \\ |
| 692 |
– |
& 0.2 & 0.989 & 0.998 & 0.984 & 0.998 & 0.989 & 0.998 \\ |
| 693 |
– |
& 0.3 & 0.971 & 0.995 & 0.957 & 0.992 & 0.965 & 0.994 \\ |
| 694 |
– |
GSC & & 1.002 & 0.983 & 0.992 & 0.973 & 0.996 & 0.971 \\ |
| 695 |
– |
RF & & 0.998 & 0.995 & 0.999 & 0.998 & 0.998 & 0.998 \\ |
| 696 |
– |
\midrule |
| 697 |
– |
PC & & -36.559 & 0.002 & -44.917 & 0.004 & -52.945 & 0.006 \\ |
| 698 |
– |
SP & 0.0 & 0.890 & 0.786 & 0.927 & 0.867 & 0.949 & 0.909 \\ |
| 699 |
– |
& 0.1 & 0.942 & 0.895 & 0.984 & 0.974 & 0.997 & 0.995 \\ |
| 700 |
– |
& 0.2 & 0.999 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 701 |
– |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 702 |
– |
SF & 0.0 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 703 |
– |
& 0.1 & 1.000 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 704 |
– |
& 0.2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 705 |
– |
& 0.3 & 1.001 & 0.999 & 1.001 & 1.000 & 1.001 & 1.000 \\ |
| 706 |
– |
GSC & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 707 |
– |
RF & & 0.999 & 0.999 & 1.000 & 1.000 & 1.000 & 1.000 \\ |
| 708 |
– |
\midrule |
| 709 |
– |
PC & & 1.984 & 0.000 & 0.012 & 0.000 & 1.357 & 0.000 \\ |
| 710 |
– |
SP & 0.0 & 0.850 & 0.552 & 0.907 & 0.703 & 0.938 & 0.793 \\ |
| 711 |
– |
& 0.1 & 0.924 & 0.755 & 0.980 & 0.936 & 0.995 & 0.988 \\ |
| 712 |
– |
& 0.2 & 0.985 & 0.983 & 0.986 & 0.988 & 0.987 & 0.988 \\ |
| 713 |
– |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 714 |
– |
SF & 0.0 & 0.977 & 0.989 & 0.987 & 0.995 & 0.992 & 0.998 \\ |
| 715 |
– |
& 0.1 & 0.982 & 0.989 & 0.992 & 0.996 & 0.997 & 0.998 \\ |
| 716 |
– |
& 0.2 & 0.984 & 0.987 & 0.986 & 0.987 & 0.987 & 0.988 \\ |
| 717 |
– |
& 0.3 & 0.961 & 0.966 & 0.959 & 0.964 & 0.960 & 0.966 \\ |
| 718 |
– |
GSC & & 0.995 & 0.981 & 0.999 & 0.990 & 1.000 & 0.993 \\ |
| 719 |
– |
RF & & 0.993 & 0.988 & 0.997 & 0.995 & 0.999 & 0.998 \\ |
| 720 |
– |
\bottomrule |
| 721 |
– |
\end{tabular} |
| 722 |
– |
\label{argonTab} |
| 723 |
– |
\end{table} |
| 724 |
– |
|
| 725 |
– |
\begin{table}[htbp] |
| 726 |
– |
\centering |
| 727 |
– |
\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.} |
| 728 |
– |
\begin{tabular}{@{} ccrrrrrr @{}} % Column formatting, @{} suppresses leading/trailing space |
| 729 |
– |
\\ |
| 730 |
– |
\toprule |
| 731 |
– |
& & \multicolumn{3}{c}{Force $\sigma^2$} & \multicolumn{3}{c}{Torque $\sigma^2$} \\ |
| 732 |
– |
\cmidrule(lr){3-5} |
| 733 |
– |
\cmidrule(l){6-8} |
| 734 |
– |
Method & $\alpha$ & 9 \AA & 12 \AA & 15 \AA & 9 \AA & 12 \AA & 15 \AA \\ |
| 735 |
– |
\midrule |
| 736 |
– |
PC & & 568.025 & 265.993 & 195.099 & 246.626 & 138.600 & 91.654 \\ |
| 737 |
– |
SP & 0.0 & 504.578 & 251.694 & 179.932 & 231.568 & 131.444 & 85.119 \\ |
| 738 |
– |
& 0.1 & 224.886 & 49.746 & 9.346 & 104.482 & 23.683 & 4.480 \\ |
| 739 |
– |
& 0.2 & 4.889 & 0.197 & 0.155 & 6.029 & 2.507 & 2.269 \\ |
| 740 |
– |
& 0.3 & 0.817 & 0.833 & 0.812 & 8.286 & 8.436 & 8.135 \\ |
| 741 |
– |
SF & 0.0 & 1.924 & 0.675 & 0.304 & 3.658 & 1.448 & 0.600 \\ |
| 742 |
– |
& 0.1 & 1.937 & 0.515 & 0.143 & 3.565 & 1.308 & 0.546 \\ |
| 743 |
– |
& 0.2 & 0.407 & 0.166 & 0.156 & 3.086 & 2.501 & 2.274 \\ |
| 744 |
– |
& 0.3 & 0.815 & 0.833 & 0.812 & 8.330 & 8.437 & 8.135 \\ |
| 745 |
– |
GSC & & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 746 |
– |
RF & & 1.822 & 0.408 & 0.142 & 3.799 & 1.362 & 0.550 \\ |
| 747 |
– |
\midrule |
| 748 |
– |
GSSP & 0.0 & 2.098 & 0.584 & 0.284 & 5.391 & 2.414 & 1.501 \\ |
| 749 |
– |
& 0.1 & 1.652 & 0.309 & 0.087 & 4.197 & 1.401 & 0.590 \\ |
| 750 |
– |
& 0.2 & 0.465 & 0.165 & 0.153 & 3.323 & 2.529 & 2.273 \\ |
| 751 |
– |
& 0.3 & 0.813 & 0.825 & 0.816 & 8.316 & 8.447 & 8.132 \\ |
| 752 |
– |
GSSF & 0.0 & 1.173 & 0.292 & 0.113 & 3.452 & 1.347 & 0.583 \\ |
| 753 |
– |
& 0.1 & 1.166 & 0.240 & 0.076 & 3.381 & 1.281 & 0.575 \\ |
| 754 |
– |
& 0.2 & 0.459 & 0.165 & 0.153 & 3.430 & 2.542 & 2.273 \\ |
| 755 |
– |
& 0.3 & 0.814 & 0.825 & 0.816 & 8.325 & 8.447 & 8.132 \\ |
| 756 |
– |
\bottomrule |
| 757 |
– |
\end{tabular} |
| 758 |
– |
\label{argonTabAng} |
| 759 |
– |
\end{table} |
| 760 |
– |
|
| 187 |
|
\newpage |
| 188 |
|
|
| 189 |
|
\bibliographystyle{achemso} |
