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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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%\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 |
| 117 |
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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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but 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 |
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summations,\cite{Yeh99,Kawata01,Arnold02,deJoannis02,Brodka04} |
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bringing them more in line with the scaling for the full 3-D |
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treatments. The inherent periodicity in the Ewald’s method can also |
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be problematic for interfacial molecular |
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systems.\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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\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 |
