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%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
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%\documentclass[aps,prb,preprint]{revtex4} |
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\documentclass[11pt]{article} |
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\usepackage{endfloat} |
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\documentclass[10pt]{article} |
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%\usepackage{endfloat} |
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\usepackage{amsmath,bm} |
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\usepackage{amssymb} |
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\usepackage{epsf} |
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\date{\today} |
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\maketitle |
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\doublespacing |
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%\doublespacing |
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\begin{abstract} |
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We investigate pairwise electrostatic interaction methods and show |
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to the direct pairwise sum. They also lack the added periodicity of |
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the Ewald sum, so they can be used for systems which are non-periodic |
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or which have one- or two-dimensional periodicity. Below, these |
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methods are evaluated using a variety of model systems to establish |
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their usability in molecular simulations. |
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methods are evaluated using a variety of model systems to |
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establish their usability in molecular simulations. |
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\subsection{The Ewald Sum} |
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The complete accumulation of the electrostatic interactions in a system with |
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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} |
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Inclusion of a correction term in the Ewald summation is a possible |
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direction for handling 2D systems while still enabling the use of the |
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modern optimizations.\cite{Yeh99} |
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new 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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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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\begin{figure} |
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\centering |
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\includegraphics[width = \linewidth]{./dualLinear.pdf} |
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\includegraphics[width = 3.25in]{./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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\label{fig:linearFit} |
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\end{figure} |
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Each system type (detailed in section \ref{sec:RepSims}) was |
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represented using 500 independent configurations. Additionally, we |
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used seven different system types, so each of the alternative |
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(non-Ewald) electrostatic summation methods was evaluated using |
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873,250 configurational energy differences. |
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Each of the seven system types (detailed in section \ref{sec:RepSims}) |
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were represented using 500 independent configurations. Thus, each of |
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the alternative (non-Ewald) electrostatic summation methods was |
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evaluated using an accumulated 873,250 configurational energy |
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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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NaCl crystal is composed of two different atom types, the average of |
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the two resulting power spectra was used for comparisons. Simulations |
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were performed under the microcanonical ensemble, and velocity |
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information was saved every 5 fs over 100 ps trajectories. |
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information was saved every 5~fs over 100~ps trajectories. |
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\subsection{Representative Simulations}\label{sec:RepSims} |
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A variety of representative simulations were analyzed to determine the |
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relative effectiveness of the pairwise summation techniques in |
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reproducing the energetics and dynamics exhibited by {\sc spme}. We wanted |
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to span the space of modern simulations (i.e. from liquids of neutral |
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molecules to ionic crystals), so the systems studied were: |
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A variety of representative molecular simulations were analyzed to |
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determine the relative effectiveness of the pairwise summation |
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techniques in reproducing the energetics and dynamics exhibited by |
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{\sc spme}. We wanted to span the space of typical molecular |
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simulations (i.e. from liquids of neutral molecules to ionic |
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crystals), so the systems studied were: |
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\begin{enumerate} |
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\item liquid water (SPC/E),\cite{Berendsen87} |
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\item crystalline water (Ice I$_\textrm{c}$ crystals of SPC/E), |
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\begin{figure} |
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\centering |
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\includegraphics[width=5.5in]{./delEplot.pdf} |
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\includegraphics[width=3.25in]{./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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\begin{figure} |
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\centering |
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\includegraphics[width=5.5in]{./frcMagplot.pdf} |
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\includegraphics[width=3.25in]{./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=5.5in]{./trqMagplot.pdf} |
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\includegraphics[width=3.25in]{./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=5.5in]{./frcTrqAngplot.pdf} |
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\includegraphics[width=3.25in]{./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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\begin{figure} |
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\centering |
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\includegraphics[width = \linewidth]{./vCorrPlot.pdf} |
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\includegraphics[width = 3.25in]{./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 = \linewidth]{./spectraSquare.pdf} |
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\includegraphics[width = 3.25in]{./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 = \linewidth]{./increasedDamping.pdf} |
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\includegraphics[width = 3.25in]{./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 |
