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\begin{document} |
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\title{Is the Ewald summation still necessary? \\ |
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Pairwise alternatives to the accepted standard for \\ |
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long-range electrostatics} |
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Pairwise alternatives to the accepted standard \\ |
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for long-range electrostatics} |
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\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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techniques. Comparisons were performed with this and other pairwise |
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methods against the smooth particle mesh Ewald ({\sc spme}) summation |
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to see how well they reproduce the energetics and dynamics of a |
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variety of simulation types. |
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variety of molecular simulations. |
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\end{abstract} |
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\newpage |
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conditions. However, in certain systems, such as vapor-liquid |
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interfaces and membranes, the intrinsic three-dimensional periodicity |
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can prove problematic. The Ewald sum has been reformulated to handle |
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2D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89}, but the |
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new methods are computationally expensive.\cite{Spohr97,Yeh99} More |
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2-D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} but these |
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methods are computationally expensive.\cite{Spohr97,Yeh99} More |
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recently, there have been several successful efforts toward reducing |
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the computational cost of 2D lattice summations, often enabling the |
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use of the mentioned |
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optimizations.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} |
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the computational cost of 2-D lattice |
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summations,\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} |
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bringing them more in line with the cost of the full 3-D summation. |
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Several studies have recognized that the inherent periodicity in the |
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Ewald sum can also have an effect on three-dimensional |
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systems.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00} |
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\begin{figure} |
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\centering |
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\includegraphics[width = 3.25in]{./dualLinear.pdf} |
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\includegraphics[width = \linewidth]{./dualLinear.pdf} |
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\caption{Example least squares regressions of the configuration energy |
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differences for SPC/E water systems. The upper plot shows a data set |
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with a poor correlation coefficient ($R^2$), while the lower plot |
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differences. |
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Results and discussion for the individual analysis of each of the |
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system types appear in the supporting information, while the |
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cumulative results over all the investigated systems appears below in |
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section \ref{sec:EnergyResults}. |
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system types appear in the supporting information,\cite{EPAPSdeposit} |
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while the cumulative results over all the investigated systems appears |
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below in section \ref{sec:EnergyResults}. |
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\subsection{Molecular Dynamics and the Force and Torque Vectors}\label{sec:MDMethods} |
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We evaluated the pairwise methods (outlined in section |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.25in]{./delEplot.pdf} |
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\includegraphics[width=5.5in]{./delEplot.pdf} |
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\caption{Statistical analysis of the quality of configurational energy |
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differences for a given electrostatic method compared with the |
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reference Ewald sum. Results with a value equal to 1 (dashed line) |
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significant improvement using the group-switched cutoff because the |
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salt and salt solution systems contain non-neutral groups. Interested |
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readers can consult the accompanying supporting information for a |
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comparison where all groups are neutral. |
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comparison where all groups are neutral.\cite{EPAPSdeposit} |
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For the {\sc sp} method, inclusion of electrostatic damping improves |
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the agreement with Ewald, and using an $\alpha$ of 0.2 \AA $^{-1}$ |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.25in]{./frcMagplot.pdf} |
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\includegraphics[width=5.5in]{./frcMagplot.pdf} |
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\caption{Statistical analysis of the quality of the force vector |
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magnitudes for a given electrostatic method compared with the |
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reference Ewald sum. Results with a value equal to 1 (dashed line) |
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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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\includegraphics[width=5.5in]{./trqMagplot.pdf} |
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\caption{Statistical analysis of the quality of the torque vector |
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magnitudes for a given electrostatic method compared with the |
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reference Ewald sum. Results with a value equal to 1 (dashed line) |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.25in]{./frcTrqAngplot.pdf} |
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\includegraphics[width=5.5in]{./frcTrqAngplot.pdf} |
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\caption{Statistical analysis of the width of the angular distribution |
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that the force and torque vectors from a given electrostatic method |
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make with their counterparts obtained using the reference Ewald sum. |
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particles in all seven systems, while torque vectors are only |
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available for neutral molecular groups. Damping is more beneficial to |
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charged bodies, and this observation is investigated further in the |
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accompanying supporting information. |
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accompanying supporting information.\cite{EPAPSdeposit} |
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Although not discussed previously, group based cutoffs can be applied |
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to both the {\sc sp} and {\sc sf} methods. The group-based cutoffs |
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\begin{figure} |
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\centering |
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\includegraphics[width = 3.25in]{./vCorrPlot.pdf} |
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\includegraphics[width = \linewidth]{./vCorrPlot.pdf} |
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\caption{Velocity autocorrelation functions of NaCl crystals at |
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1000 K using {\sc spme}, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc |
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sp} ($\alpha$ = 0.2). The inset is a magnification of the area around |
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\begin{figure} |
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\centering |
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\includegraphics[width = 3.25in]{./spectraSquare.pdf} |
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\includegraphics[width = \linewidth]{./spectraSquare.pdf} |
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\caption{Power spectra obtained from the velocity auto-correlation |
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functions of NaCl crystals at 1000 K while using {\sc spme}, {\sc sf} |
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($\alpha$ = 0, 0.1, \& 0.2), and {\sc sp} ($\alpha$ = 0.2). The inset |
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\begin{figure} |
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\centering |
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\includegraphics[width = 3.25in]{./increasedDamping.pdf} |
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\includegraphics[width = \linewidth]{./increasedDamping.pdf} |
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\caption{Effect of damping on the two lowest-frequency phonon modes in |
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the NaCl crystal at 1000~K. The undamped shifted force ({\sc sf}) |
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method is off by less than 10 cm$^{-1}$, and increasing the |
