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\begin{document} |
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\preprint{AIP/123-QED} |
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%\preprint{AIP/123-QED} |
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\title[Taylor-shifted and Gradient-shifted electrostatics for multipoles] |
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{Real space alternatives to the Ewald |
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\title{Real space alternatives to the Ewald |
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Sum. I. Taylor-shifted and Gradient-shifted electrostatics for multipoles} |
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\author{Madan Lamichhane} |
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results for ordered arrays of multipoles. |
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\end{abstract} |
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\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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% Classification Scheme. |
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\keywords{Suggested keywords}%Use showkeys class option if keyword |
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%\keywords{Suggested keywords}%Use showkeys class option if keyword |
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%display desired |
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\maketitle |
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the number of dipoles per unit volume, and $\mu$ is the strength of |
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the dipole. |
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To test the new electrostatic methods, we have constructed very large |
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($N = 8,000, 16,000, 32,000$) arrays of dipoles in the orientations |
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described in \ref{tab:LT}. For the purposes of this paper, the primary |
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quantity of interest is the analytic energy constant for the perfect |
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arrays. Convergence to these constants are shown as a function of both |
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the cutoff radius, $r_c$, and the damping parameter, $\alpha$ in Fig. |
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XXX. We have simultaneously tested a hard cutoff (where the kernel is |
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simply truncated at the cutoff radius), as well as a shifted potential |
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(SP) form which includes a potential-shifting and self-interaction |
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term, but does not shift the forces and torques smoothly at the cutoff |
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To test the new electrostatic methods, we have constructed very large, |
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$N$ = 8,000~(sc), 16,000~(bcc), or 32,000~(fcc) arrays of dipoles in |
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the orientations described in table \ref{tab:LT}. For the purposes of |
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testing the energy expressions and the self-neutralization schemes, |
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the primary quantity of interest is the analytic energy constant for |
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the perfect arrays. Convergence to these constants are shown as a |
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function of both the cutoff radius, $r_c$, and the damping parameter, |
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$\alpha$ in Figs. \ref{fig:energyConstVsCutoff} and XXX. We have |
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simultaneously tested a hard cutoff (where the kernel is simply |
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truncated at the cutoff radius), as well as a shifted potential (SP) |
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form which includes a potential-shifting and self-interaction term, |
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but does not shift the forces and torques smoothly at the cutoff |
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radius. |
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\begin{figure} |
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\includegraphics[width=4.5in]{energyConstVsCutoff} |
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\caption{Convergence to the analytic energy constants as a function of |
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cutoff radius (normalized by the lattice constant) for the different |
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real-space methods. The two crystals shown here are the ``B'' array |
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for bcc crystals with the dipoles along the 001 direction (upper), |
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as well as the minimum energy bcc lattice (lower). The analytic |
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energy constants are shown as a grey dashed line. The left panel |
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shows results for the undamped kernel ($1/r$), while the damped |
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error function kernel, $B_0(r)$ was used in the right panel. } |
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\label{fig:energyConstVsCutoff} |
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\end{figure} |
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The Hard cutoff exhibits oscillations around the analytic energy |
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constants, and converges to incorrect energies when the complementary |
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error function damping kernel is used. The shifted potential (SP) and |
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gradient-shifted force (GSF) approximations converge to the correct |
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energy smoothly by $r_c / 6 a$ even for the undamped case. This |
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indicates that the correction provided by the self term is required |
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for obtaining accurate energies. The Taylor-shifted force (TSF) |
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approximation appears to perturb the potential too much inside the |
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cutoff region to provide accurate measures of the energy constants. |
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{\it Quadrupolar} analogues to the Madelung constants were first |
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worked out by Nagai and Nakamura who computed the energies of selected |
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quadrupole arrays based on extensions to the Luttinger and Tisza |
