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%\preprint{AIP/123-QED} |
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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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Sum. I. Shifted electrostatics for multipoles} |
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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self-neutralization schemes, the primary quantity of interest is the |
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analytic energy constant for the perfect arrays. Convergence to these |
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constants are shown as a function of both the cutoff radius, $r_c$, |
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and the damping parameter, $\alpha$ in Figs. \ref{fig:Dipoles_rCut} |
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and the damping parameter, $\alpha$ in Figs.\ref{fig:Dipoles_rCut} |
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and \ref{fig:Dipoles_alpha}. We have simultaneously tested a hard |
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cutoff (where the kernel is simply truncated at the cutoff radius), as |
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well as a shifted potential (SP) form which includes a |
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forces and torques smoothly at the cutoff radius. The SP method is |
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essentially an extension of the original Wolf method for multipoles. |
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The Hard cutoff exhibits oscillations around the analytic energy |
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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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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 |
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approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
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energy constants for the lowest energy configurations for linear |
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quadrupoles. |
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approach.\cite{Nagai01081960,Nagai01091963} |
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In analogy to the dipolar arrays, the total electrostatic energy for |
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the quadrupolar arrays is: |
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(For the traceless quadrupole tensor, $\Theta = 3 Q - \text{Tr} Q$, |
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the effective moment, $\bar{Q}^2 = \frac{2}{3} \Theta : \Theta$.) |
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To test the new electrostatic methods for quadrupoles, we have |
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constructed very large, $N=$ 8,000~(sc), 16,000~(bcc), and |
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32,000~(fcc) arrays of linear quadrupoles in the orientations |
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described in Ref. \onlinecite{Nagai01081960}. We have compared the |
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energy constants for the lowest energy configurations for these linear |
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quadrupoles. Convergence to these constants are shown as a function |
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of both the cutoff radius, $r_c$, and the damping parameter, $\alpha$ |
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in Figs.~\ref{fig:Quadrupoles_rCut} and \ref{fig:Quadrupoles_alpha}. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{Quadrupoles_rcutConvergence.pdf} |
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\caption{Convergence of the lattice energy constants as a function of |
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are indicated with grey dashed lines. The left panel shows results |
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for the undamped kernel ($1/r$), while the damped error function |
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kernel, $B_0(r)$ was used in the right panel.} |
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\label{fig:QuadrupolesrcutCovergence} |
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\label{fig:Quadrupoles_rCut} |
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\end{figure} |
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\begin{figure}[!htbp] |
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\includegraphics[width=3.5in]{Quadrupoles_alphaConvergence-crop.pdf} |
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\caption{Convergence to the analytic energy constants as a function of |
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cutoff damping alpha for the different |
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real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF, and analytic methods are represented by the black sold-circle, red solid-square, green solid-diamond, and grey dashed line respectively. |
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The left panel 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:Quadrupoles_alphaCovergence-crop.pdf} |
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\includegraphics[width=\linewidth]{Quadrupoles_newAlpha.pdf} |
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\caption{Convergence to the lattice energy constants as a function of |
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the reduced damping parameter ($\alpha^* = \alpha a$) for the |
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different real-space methods in the same three quadrupolar crystals |
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in Figure \ref{fig:Quadrupoles_rCut}. The left panel shows |
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results for a relatively small cutoff radius ($r_c = 4.5 a$) while a |
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larger cutoff radius ($r_c = 6 a$) was used in the right panel. } |
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\label{fig:Quadrupoles_alpha} |
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\end{figure} |
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Again, we find that the hard cutoff exhibits oscillations around the |
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analytic energy constants. 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 $r_c = 4 a$ even for the undamped case. The |
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Taylor-shifted force (TSF) approximation again appears to perturb the |
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potential too much inside the cutoff region to provide accurate |
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measures of the energy constants. |
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\section{Conclusion} |
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We have presented two efficient real-space methods for computing the |
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We have presented three efficient real-space methods for computing the |
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interactions between point multipoles. These methods have the benefit |
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of smoothly truncating the energies, forces, and torques at the cutoff |
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radius, making them attractive for both molecular dynamics (MD) and |
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(GSF) and the Shifted-Potential (SP) methods converge rapidly to the |
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correct lattice energies for ordered dipolar and quadrupolar arrays, |
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while the Taylor-Shifted Force (TSF) is too severe an approximation to |
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provide accurate convergence to lattice energies. |
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provide accurate convergence to lattice energies. |
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In most cases, GSF can obtain nearly quantitative agreement with the |
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lattice energy constants with reasonably small cutoff radii. The only |
