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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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\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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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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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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\begin{figure} |
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\centering |
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\includegraphics[width = \linewidth]{./ewaldProgression.pdf} |
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\caption{The change in the application of the Ewald sum with |
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increasing computational power. A:~Initially, only small systems could |
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be studied, and the Ewald sum replicated the simulation box to |
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convergence. B:~Now, much larger systems of charges can be |
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investigated with fixed-distance cutoffs.} |
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\caption{The change in the need for the Ewald sum with |
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increasing computational power. A:~Initially, only small systems |
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could be studied, and the Ewald sum replicated the simulation box to |
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convergence. B:~Now, radial cutoff methods should be able to reach |
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convergence for the larger systems of charges that are common today.} |
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\label{fig:ewaldTime} |
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\end{figure} |
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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} |
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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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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 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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\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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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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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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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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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 |
