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%\preprint{AIP/123-QED} |
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|
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\title{Real space alternatives to the Ewald Sum. II. Comparison of Methods} |
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\title{Real space electrostatics for multipoles. II. Comparisons with |
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the Ewald Sum} |
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|
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University of Notre Dame, Notre Dame, IN 46556} |
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\date{\today} |
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\begin{abstract} |
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We report on tests of the real-space shifted potential (SP), |
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gradient-shifted force (GSF), and Taylor-shifted force (TSF) methods |
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for multipole interactions developed in the first paper in this |
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series, using the multipolar Ewald sum as a reference method. The |
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tests were carried out in a variety of condensed-phase environments |
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designed to test up to quadrupole-quadrupole interactions. |
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Comparisons of the energy differences between configurations, |
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molecular forces, and torques were used to analyze how well the |
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real-space models perform relative to the more computationally |
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expensive Ewald treatment. We have also investigated the energy |
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conservation properties of the new methods in molecular dynamics |
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simulations. The SP method shows excellent agreement with |
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configurational energy differences, forces, and torques, and would |
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be suitable for use in Monte Carlo calculations. Of the two new |
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shifted-force methods, the GSF approach shows the best agreement |
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with Ewald-derived energies, forces, and torques and also exhibits |
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energy conservation properties that make it an excellent choice for |
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efficient computation of electrostatic interactions in molecular |
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dynamics simulations. |
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We report on tests of the shifted potential (SP), gradient shifted |
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force (GSF), and Taylor shifted force (TSF) real-space methods for |
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multipole interactions developed in the first paper in this series, |
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using the multipolar Ewald sum as a reference method. The tests were |
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carried out in a variety of condensed-phase environments designed to |
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test up to quadrupole-quadrupole interactions. Comparisons of the |
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energy differences between configurations, molecular forces, and |
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torques were used to analyze how well the real-space models perform |
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relative to the more computationally expensive Ewald treatment. We |
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have also investigated the energy conservation properties of the new |
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methods in molecular dynamics simulations. The SP method shows |
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excellent agreement with configurational energy differences, forces, |
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and torques, and would be suitable for use in Monte Carlo |
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calculations. Of the two new shifted-force methods, the GSF |
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approach shows the best agreement with Ewald-derived energies, |
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forces, and torques and also exhibits energy conservation properties |
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that make it an excellent choice for efficient computation of |
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electrostatic interactions in molecular dynamics simulations. |
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\end{abstract} |
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|
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%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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the conditionally convergent electrostatic energy is converted into |
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two absolutely convergent contributions, one which is carried out in |
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real space with a cutoff radius, and one in reciprocal |
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space. BETTER CITATIONS\cite{Clarke:1986eu,Woodcock75} |
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space.\cite{Ewald21,deLeeuw80,Smith81,Allen87} |
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|
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When carried out as originally formulated, the reciprocal-space |
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portion of the Ewald sum exhibits relatively poor computational |
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scaling, making it prohibitive for large systems. By utilizing |
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particle meshes and three dimensional fast Fourier transforms (FFT), |
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the particle-mesh Ewald (PME), particle-particle particle-mesh Ewald |
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scaling, making it prohibitive for large systems. By utilizing a |
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particle mesh and three dimensional fast Fourier transforms (FFT), the |
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particle-mesh Ewald (PME), particle-particle particle-mesh Ewald |
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(P\textsuperscript{3}ME), and smooth particle mesh Ewald (SPME) |
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methods can decrease the computational cost from $O(N^2)$ down to $O(N |
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\log |
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N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb}. |
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N)$.\cite{Takada93,Gunsteren94,Gunsteren95,Darden:1993pd,Essmann:1995pb} |
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|
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Because of the artificial periodicity required for the Ewald sum, |
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interfacial molecular systems such as membranes and liquid-vapor |
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interfaces require modifications to the |
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method.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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Parry's extension of the three dimensional Ewald sum is appropriate |
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for slab geometries.\cite{Parry:1975if} Modified Ewald methods that |
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were developed to handle two-dimensional (2D) electrostatic |
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interactions in interfacial systems have not seen similar |
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particle-mesh treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77, |
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Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly |
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with system size. The inherent periodicity in the Ewald’s method can |
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also be problematic for interfacial molecular |
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systems.\cite{Fennell:2006lq} |
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interfaces require modifications to the method. Parry's extension of |
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the three dimensional Ewald sum is appropriate for slab |
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geometries.\cite{Parry:1975if} Modified Ewald methods that were |
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developed to handle two-dimensional (2-D) electrostatic |
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interactions.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl} |
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These methods were originally quite computationally |
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expensive.\cite{Spohr97,Yeh99} There have been several successful |
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efforts that reduced the computational cost of 2-D lattice summations, |
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bringing them more in line with the scaling for the full 3-D |
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treatments.\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} The |
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inherent periodicity required by the Ewald method can also be |
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problematic in a number of protein/solvent and ionic solution |
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environments.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00,Fennell:2006lq} |
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|
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\subsection{Real-space methods} |
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Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$ |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{schematic.pdf} |
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\includegraphics[width=\linewidth]{schematic.eps} |
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\caption{Top: Ionic systems exhibit local clustering of dissimilar |
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charges (in the smaller grey circle), so interactions are |
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effectively charge-multipole at longer distances. With hard |
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and have been compared with the values obtained from the multipolar |
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Ewald sum. In Monte Carlo (MC) simulations, the energy differences |
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between two configurations is the primary quantity that governs how |
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the simulation proceeds. These differences are the most imporant |
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the simulation proceeds. These differences are the most important |
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indicators of the reliability of a method even if the absolute |
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energies are not exact. For each of the multipolar systems listed |
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above, we have compared the change in electrostatic potential energy |
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\subsection{Implementation} |
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The real-space methods developed in the first paper in this series |
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have been implemented in our group's open source molecular simulation |
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program, OpenMD,\cite{openmd} which was used for all calculations in |
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program, OpenMD,\cite{Meineke05,openmd} which was used for all calculations in |
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this work. The complementary error function can be a relatively slow |
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function on some processors, so all of the radial functions are |
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precomputed on a fine grid and are spline-interpolated to provide |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6\linewidth]{energyPlot_slopeCorrelation_combined-crop.pdf} |
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\includegraphics[width=0.85\linewidth]{energyPlot_slopeCorrelation_combined.eps} |
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\caption{Statistical analysis of the quality of configurational |
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energy differences for the real-space electrostatic methods |
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compared with the reference Ewald sum. Results with a value equal |
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|
| 867 |
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined-crop.pdf} |
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\includegraphics[width=0.6\linewidth]{forcePlot_slopeCorrelation_combined.eps} |
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\caption{Statistical analysis of the quality of the force vector |
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magnitudes for the real-space electrostatic methods compared with |
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the reference Ewald sum. Results with a value equal to 1 (dashed |
| 880 |
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|
| 881 |
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined-crop.pdf} |
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\includegraphics[width=0.6\linewidth]{torquePlot_slopeCorrelation_combined.eps} |
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\caption{Statistical analysis of the quality of the torque vector |
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magnitudes for the real-space electrostatic methods compared with |
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the reference Ewald sum. Results with a value equal to 1 (dashed |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=0.6 \linewidth]{Variance_forceNtorque_modified-crop.pdf} |
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\includegraphics[width=0.65\linewidth]{Variance_forceNtorque_modified.eps} |
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\caption{The circular variance of the direction of the force and |
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torque vectors obtained from the real-space methods around the |
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reference Ewald vectors. A variance equal to 0 (dashed line) |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\textwidth]{newDrift_12.pdf} |
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\includegraphics[width=\textwidth]{newDrift_12.eps} |
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\label{fig:energyDrift} |
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\caption{Analysis of the energy conservation of the real-space |
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electrostatic methods. $\delta \mathrm{E}_1$ is the linear drift in |
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utilized the same real-space cutoff, $r_c = 12$\AA.} |
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\end{figure} |
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|
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\subsection{Reproduction of Structural Features\label{sec:structure}} |
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One of the best tests of modified interaction potentials is the |
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fidelity with which they can reproduce structural features in a |
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liquid. One commonly-utilized measure of structural ordering is the |
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pair distribution function, $g(r)$, which measures local density |
| 1005 |
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deviations in relation to the bulk density. In the electrostatic |
| 1006 |
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approaches studied here, the short-range repulsion from the |
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Lennard-Jones potential is identical for the various electrostatic |
| 1008 |
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methods, and since short range repulsion determines much of the local |
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liquid ordering, one would not expect to see any differences in |
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$g(r)$. Indeed, the pair distributions are essentially identical for |
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all of the electrostatic methods studied (for each of the different |
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systems under investigation). Interested readers may consult the |
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supplementary information for plots of these pair distribution |
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functions. |
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|
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A direct measure of the structural features that is a more |
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enlightening test of the modified electrostatic methods is the average |
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value of the electrostatic energy $\langle U_\mathrm{elect} \rangle$ |
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which is obtained by sampling the liquid-state configurations |
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experienced by a liquid evolving entirely under the influence of the |
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methods being investigated. In figure \ref{fig:Uelect} we show how |
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$\langle U_\mathrm{elect} \rangle$ for varies with the damping parameter, |
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$\alpha$, for each of the methods. |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\textwidth]{averagePotentialEnergy_r9_12.eps} |
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\label{fig:Uelect} |
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\caption{The average electrostatic potential energy, |
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$\langle U_\mathrm{elect} \rangle$ for the SSDQ water with ions as a function |
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of the damping parameter, $\alpha$, for each of the real-space |
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electrostatic methods. Top panel: simulations run with a real-space |
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cutoff, $r_c = 9$\AA. Bottom panel: the same quantity, but with a |
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larger cutoff, $r_c = 12$\AA.} |
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\end{figure} |
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|
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It is clear that moderate damping is important for converging the mean |
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potential energy values, particularly for the two shifted force |
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methods (GSF and TSF). A value of $\alpha \approx 0.18$ \AA$^{-1}$ is |
| 1040 |
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sufficient to converge the SP and GSF energies with a cutoff of 12 |
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\AA, while shorter cutoffs require more dramatic damping ($\alpha |
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\approx 0.36$ \AA$^{-1}$ for $r_c = 9$ \AA). It is also clear from |
| 1043 |
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fig. \ref{fig:Uelect} that it is possible to overdamp the real-space |
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electrostatic methods, causing the estimate of the energy to drop |
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below the Ewald results. |
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|
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These ``optimal'' values of the damping coefficient are slightly |
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larger than what were observed for DSF electrostatics for purely |
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point-charge systems, although a value of $\alpha=0.18$ \AA$^{-1}$ for |
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$r_c = 12$\AA appears to be an excellent compromise for mixed charge |
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multipole systems. |
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|
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\subsection{Reproduction of Dynamic Properties\label{sec:structure}} |
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To test the fidelity of the electrostatic methods at reproducing |
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dynamics in a multipolar liquid, it is also useful to look at |
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transport properties, particularly the diffusion constant, |
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\begin{equation} |
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D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle \left| |
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\mathbf{r}(t) -\mathbf{r}(0) \right|^2 \rangle |
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\label{eq:diff} |
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\end{equation} |
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which measures long-time behavior and is sensitive to the forces on |
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the multipoles. For the soft dipolar fluid, and the SSDQ liquid |
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systems, the self-diffusion constants (D) were calculated from linear |
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fits to the long-time portion of the mean square displacement |
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($\langle r^{2}(t) \rangle$).\cite{Allen87} |
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|
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In addition to translational diffusion, orientational relaxation times |
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were calculated for comparisons with the Ewald simulations and with |
| 1070 |
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experiments. These values were determined from the same 1~ns $NVE$ |
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trajectories used for translational diffusion by calculating the |
| 1072 |
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orientational time correlation function, |
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\begin{equation} |
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C_l^\alpha(t) = \left\langle P_l\left[\hat{\mathbf{u}}_i^\gamma(t) |
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\cdot\hat{\mathbf{u}}_i^\gamma(0)\right]\right\rangle, |
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\label{eq:OrientCorr} |
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\end{equation} |
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where $P_l$ is the Legendre polynomial of order $l$ and |
| 1079 |
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$\hat{\mathbf{u}}_i^\alpha$ is the unit vector of molecule $i$ along |
| 1080 |
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axis $\gamma$. The body-fixed reference frame used for our |
| 1081 |
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orientational correlation functions has the $z$-axis running along the |
| 1082 |
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dipoles, and for the SSDQ water model, the $y$-axis connects the two |
| 1083 |
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implied hydrogen atoms. |
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|
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From the orientation autocorrelation functions, we can obtain time |
| 1086 |
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constants for rotational relaxation either by fitting an exponential |
| 1087 |
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function or by integrating the entire correlation function. These |
| 1088 |
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decay times are directly comparable to water orientational relaxation |
| 1089 |
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times from nuclear magnetic resonance (NMR). The relaxation constant |
| 1090 |
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obtained from $C_2^y(t)$ is normally of experimental interest because |
| 1091 |
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it describes the relaxation of the principle axis connecting the |
| 1092 |
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hydrogen atoms. Thus, $C_2^y(t)$ can be compared to the intermolecular |
| 1093 |
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portion of the dipole-dipole relaxation from a proton NMR signal and |
| 1094 |
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should provide an estimate of the NMR relaxation time |
| 1095 |
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constant.\cite{Impey82} |
| 1096 |
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|
| 1097 |
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Results for the diffusion constants and orientational relaxation times |
| 1098 |
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are shown in figure \ref{fig:dynamics}. From this data, it is apparent |
| 1099 |
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that the values for both $D$ and $\tau_2$ using the Ewald sum are |
| 1100 |
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reproduced with high fidelity by the GSF method. |
| 1101 |
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|
| 1102 |
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The $\tau_2$ results in \ref{fig:dynamics} show a much greater |
| 1103 |
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difference between the real-space and the Ewald results. |
| 1104 |
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|
| 1105 |
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|
| 1106 |
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\section{CONCLUSION} |
| 1107 |
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In the first paper in this series, we generalized the |
| 1108 |
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charge-neutralized electrostatic energy originally developed by Wolf |
