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\begin{document} |
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\title[Notes on the Self-Interaction] |
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{Notes on the Self-Interaction} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu.} |
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\affiliation{Department of Chemistry and Biochemistry, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\maketitle |
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\section{The Self-Interaction} |
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The Wolf summation~\cite{Wolf99} and the later damped shifted force |
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(DSF) extension~\cite{Fennell06} included self-interactions that are |
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handled separately from the pairwise interactions between sites. The |
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self-term is normally calculated via a single loop over all sites in |
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the system, and is relatively cheap to evaluate. The self-interaction |
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has contributions from two sources: |
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\begin{itemize} |
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\item The neutralization procedure within the cutoff radius requires a |
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contribution from a charge opposite in sign, but equal in magnitude, |
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to the central charge, which has been spread out over the surface of |
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the cutoff sphere. For a system of undamped charges, the total |
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self-term is |
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\begin{equation} |
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V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
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\label{eq:selfTerm} |
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\end{equation} |
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Note that in this potential and in all electrostatic quantities that |
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follow, the standard $4 \pi \epsilon_{0}$ has been omitted for |
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clarity. |
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\item Charge damping with the complementary error function is a |
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partial analogy to the Ewald procedure which splits the interaction |
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into real and reciprocal space sums. The real space sum is retained |
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in the Wolf and DSF methods. The reciprocal space sum is first |
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minimized by folding the largest contribution (the self-interaction) |
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into the self-interaction from charge neutralization of the damped |
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potential. The remainder of the reciprocal space portion is then |
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discarded (as this contributes the largest computational cost and |
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complexity to the Ewald sum). For a system containing only damped |
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charges, the complete self-interaction can be written as |
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\begin{equation} |
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V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
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\frac{2 \alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
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C_{\bf a}^{2}. |
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\label{eq:dampSelfTerm} |
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\end{equation} |
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\end{itemize} |
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|
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The extension of DSF electrostatics to point multipoles requires |
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treatment of {\it both} the self-neutralization and reciprocal |
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contributions to the self-interaction for higher order multipoles. In |
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this section we give formulae for these interactions up to quadrupolar |
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order. |
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The self-neutralization term is computed by taking the {\it |
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non-shifted} kernel for each interaction, placing a multipole of |
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equal magnitude (but opposite in polarization) on the surface of the |
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cutoff sphere, and averaging over the surface of the cutoff sphere. |
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Because the self term is carried out as a single sum over sites, the |
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reciprocal-space portion is identical to half of the self-term |
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obtained by Smith and Aguado and Madden for the application of the |
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Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
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site which posesses a charge, dipole, and multipole, both types of |
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contribution are given in table \ref{tab:tableSelf}. |
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|
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\begin{table*} |
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\caption{\label{tab:tableSelf} Self-interaction contributions for |
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site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
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$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
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\begin{ruledtabular} |
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\begin{tabular}{lccc} |
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Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
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Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
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Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
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\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
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Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
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$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
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$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
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Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
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h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\ |
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\end{tabular} |
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\end{ruledtabular} |
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\end{table*} |
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|
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For sites which simultaneously contain charges and quadrupoles, the |
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self-interaction includes a cross-interaction between these two |
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multipole orders. Symmetry prevents the charge-dipole and |
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dipole-quadrupole interactions from contributing to the |
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self-interaction. The functions that go into the self-neutralization |
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terms, $f(r), g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
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derivatives of the electrostatic kernel (either the undamped $1/r$ or |
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the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that are |
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evaluated at the cutoff distance. For undamped interactions, $f(r_c) |
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= 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For damped interactions, |
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$f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so on. Appendix XX |
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contains recursion relations that allow rapid evaluation of these |
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derivatives. |
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The self interaction also gives rise to a contribution to the torque |
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on the site |
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\newpage |
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\bibliography{multipole} |
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\end{document} |
