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\begin{document} |
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\title{Is the Ewald Summation necessary? \\ |
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Pairwise alternatives to the accepted standard for \\ |
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long-range electrostatics} |
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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 in molecular simulations} |
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|
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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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|
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\maketitle |
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\doublespacing |
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%\doublespacing |
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|
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\nobibliography{} |
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\begin{abstract} |
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We investigate pairwise electrostatic interaction methods and show |
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that there are viable and computationally efficient $(\mathscr{O}(N))$ |
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alternatives to the Ewald summation for typical modern molecular |
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simulations. These methods are extended from the damped and |
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cutoff-neutralized Coulombic sum originally proposed by Wolf |
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\textit{et al.} One of these, the damped shifted force method, shows |
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cutoff-neutralized Coulombic sum originally proposed by |
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[D. Wolf, P. Keblinski, S.~R. Phillpot, and J. Eggebrecht, {\it J. Chem. Phys.} {\bf 110}, 8255 (1999)] One of these, the damped shifted force method, shows |
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a remarkable ability to reproduce the energetic and dynamic |
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characteristics exhibited by simulations employing lattice summation |
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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 to see |
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how well they reproduce the energetics and dynamics of a variety of |
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simulation types. |
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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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\end{abstract} |
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|
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\newpage |
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regarding possible artifacts caused by the inherent periodicity of the |
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explicit Ewald summation.\cite{Tobias01} |
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|
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In this paper, we focus on a new set of shifted methods devised by |
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In this paper, we focus on a new set of pairwise methods devised by |
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Wolf {\it et al.},\cite{Wolf99} which we further extend. These |
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methods along with a few other mixed methods (i.e. reaction field) are |
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compared with the smooth particle mesh Ewald |
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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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|
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\subsection{The Ewald Sum} |
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The complete accumulation electrostatic interactions in a system with |
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The complete accumulation of the electrostatic interactions in a system with |
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periodic boundary conditions (PBC) requires the consideration of the |
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effect of all charges within a (cubic) simulation box as well as those |
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in the periodic replicas, |
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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. 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. Now, much larger systems of charges are investigated |
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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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|
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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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|
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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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charge contained within the cutoff radius is crucial for potential |
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stability. They devised a pairwise summation method that ensures |
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charge neutrality and gives results similar to those obtained with the |
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Ewald summation. The resulting shifted Coulomb potential |
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(Eq. \ref{eq:WolfPot}) includes image-charges subtracted out through |
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placement on the cutoff sphere and a distance-dependent damping |
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function (identical to that seen in the real-space portion of the |
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Ewald sum) to aid convergence |
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Ewald summation. The resulting shifted Coulomb potential includes |
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image-charges subtracted out through placement on the cutoff sphere |
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and a distance-dependent damping function (identical to that seen in |
| 241 |
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the real-space portion of the Ewald sum) to aid convergence |
| 242 |
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\begin{equation} |
| 243 |
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V_{\textrm{Wolf}}(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}\right\}. |
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\label{eq:WolfPot} |
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\label{fig:linearFit} |
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\end{figure} |
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|
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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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|
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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 |
| 558 |
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below in section \ref{sec:EnergyResults}. |
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|
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\subsection{Molecular Dynamics and the Force and Torque Vectors}\label{sec:MDMethods} |
| 561 |
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We evaluated the pairwise methods (outlined in section |
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between two different electrostatic summation methods, there is no |
| 588 |
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{\it a priori} reason for the profile to adhere to any specific |
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shape. Thus, gaussian fits were used to measure the width of the |
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resulting distributions. |
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% |
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%\begin{figure} |
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%\centering |
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%\includegraphics[width = \linewidth]{./gaussFit.pdf} |
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%\caption{Sample fit of the angular distribution of the force vectors |
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%accumulated using all of the studied systems. Gaussian fits were used |
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%to obtain values for the variance in force and torque vectors.} |
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%\label{fig:gaussian} |
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%\end{figure} |
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% |
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%Figure \ref{fig:gaussian} shows an example distribution with applied |
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%non-linear fits. The solid line is a Gaussian profile, while the |
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%dotted line is a Voigt profile, a convolution of a Gaussian and a |
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%Lorentzian. |
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%Since this distribution is a measure of angular error between two |
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%different electrostatic summation methods, there is no {\it a priori} |
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%reason for the profile to adhere to any specific shape. |
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%Gaussian fits was used to compare all the tested methods. |
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The variance ($\sigma^2$) was extracted from each of these fits and |
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was used to compare distribution widths. Values of $\sigma^2$ near |
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zero indicate vector directions indistinguishable from those |
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calculated when using the reference method ({\sc spme}). |
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resulting distributions. The variance ($\sigma^2$) was extracted from |
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each of these fits and was used to compare distribution widths. |
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Values of $\sigma^2$ near zero indicate vector directions |
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indistinguishable from those calculated when using the reference |
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method ({\sc spme}). |
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|
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\subsection{Short-time Dynamics} |
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|
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|
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The effects of the same subset of alternative electrostatic methods on |
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the {\it long-time} dynamics of charged systems were evaluated using |
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the same model system (NaCl crystals at 1000K). The power spectrum |
| 622 |
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the same model system (NaCl crystals at 1000~K). The power spectrum |
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($I(\omega)$) was obtained via Fourier transform of the velocity |
| 624 |
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autocorrelation function, \begin{equation} I(\omega) = |
| 625 |
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\frac{1}{2\pi}\int^{\infty}_{-\infty}C_v(t)e^{-i\omega t}dt, |
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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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|
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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 |
| 640 |
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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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the crystal). The solid and liquid NaCl systems consisted of 500 |
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$\textrm{Na}^{+}$ and 500 $\textrm{Cl}^{-}$ ions. Configurations for |
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these systems were selected and equilibrated in the same manner as the |
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water systems. The equilibrated temperatures were 1000~K for the NaCl |
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crystal and 7000~K for the liquid. The ionic solutions were made by |
| 665 |
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solvating 4 (or 40) ions in a periodic box containing 1000 SPC/E water |
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molecules. Ion and water positions were then randomly swapped, and |
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the resulting configurations were again equilibrated individually. |
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Finally, for the Argon / Water ``charge void'' systems, the identities |
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of all the SPC/E waters within 6 \AA\ of the center of the |
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equilibrated water configurations were converted to argon. |
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%(Fig. \ref{fig:argonSlice}). |
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water systems. In order to introduce measurable fluctuations in the |
| 664 |
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configuration energy differences, the crystalline simulations were |
| 665 |
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equilibrated at 1000~K, near the $T_\textrm{m}$ for NaCl. The liquid |
| 666 |
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NaCl configurations needed to represent a fully disordered array of |
| 667 |
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point charges, so the high temperature of 7000~K was selected for |
| 668 |
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equilibration. The ionic solutions were made by solvating 4 (or 40) |
| 669 |
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ions in a periodic box containing 1000 SPC/E water molecules. Ion and |
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water positions were then randomly swapped, and the resulting |
| 671 |
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configurations were again equilibrated individually. Finally, for the |
| 672 |
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Argon / Water ``charge void'' systems, the identities of all the SPC/E |
| 673 |
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waters within 6 \AA\ of the center of the equilibrated water |
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configurations were converted to argon. |
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|
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These procedures guaranteed us a set of representative configurations |
| 677 |
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from chemically-relevant systems sampled from appropriate |
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ensembles. Force field parameters for the ions and Argon were taken |
| 679 |
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from the force field utilized by {\sc oopse}.\cite{Meineke05} |
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|
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– |
%\begin{figure} |
| 690 |
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%\centering |
| 691 |
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%\includegraphics[width = \linewidth]{./slice.pdf} |
| 692 |
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%\caption{A slice from the center of a water box used in a charge void |
| 693 |
– |
%simulation. The darkened region represents the boundary sphere within |
| 694 |
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%which the water molecules were converted to argon atoms.} |
| 695 |
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%\label{fig:argonSlice} |
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%\end{figure} |
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|
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\subsection{Comparison of Summation Methods}\label{sec:ESMethods} |
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We compared the following alternative summation methods with results |
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from the reference method ({\sc spme}): |
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(i.e. Lennard-Jones interactions) were handled in exactly the same |
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manner across all systems and configurations. |
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|
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The althernative methods were also evaluated with three different |
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The alternative methods were also evaluated with three different |
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cutoff radii (9, 12, and 15 \AA). As noted previously, the |
| 704 |
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convergence parameter ($\alpha$) plays a role in the balance of the |
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real-space and reciprocal-space portions of the Ewald calculation. |
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significant improvement using the group-switched cutoff because the |
| 752 |
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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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|
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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 $^{-1}$ |
| 899 |
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particles in all seven systems, while torque vectors are only |
| 900 |
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available for neutral molecular groups. Damping is more beneficial to |
| 901 |
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charged bodies, and this observation is investigated further in the |
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< |
accompanying supporting information. |
| 902 |
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accompanying supporting information.\cite{EPAPSdeposit} |
| 903 |
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|
| 904 |
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Although not discussed previously, group based cutoffs can be applied |
| 905 |
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to both the {\sc sp} and {\sc sf} methods. The group-based cutoffs |
| 1079 |
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\centering |
| 1080 |
|
\includegraphics[width = \linewidth]{./increasedDamping.pdf} |
| 1081 |
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\caption{Effect of damping on the two lowest-frequency phonon modes in |
| 1082 |
< |
the NaCl crystal at 1000K. The undamped shifted force ({\sc sf}) |
| 1082 |
> |
the NaCl crystal at 1000~K. The undamped shifted force ({\sc sf}) |
| 1083 |
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method is off by less than 10 cm$^{-1}$, and increasing the |
| 1084 |
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electrostatic damping to 0.25 \AA$^{-1}$ gives quantitative agreement |
| 1085 |
|
with the power spectrum obtained using the Ewald sum. Overdamping can |
