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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 outside the main pairwise interactions between sites. The |
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self-interaction 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. This term is calculated via a single loop over |
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all charges in the system. For a system of undamped charges, the |
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total 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 the damped charge case the |
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complete self-interaction can be written as |
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\begin{equation} |
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V_\textrm{self} = - \left(\frac{2 \alpha}{\sqrt{\pi}} |
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+ \frac{\textrm{erfc}(\alpha r_c)}{r_c}\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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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 and discuss the |
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relative sizes of these contributions. |
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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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The reciprocal-space portion is identical to theself-term obtained by |
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Smith and Aguado and Madden for the application of the Ewald sum to |
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multipoles.\cite{Smith82,Smith98,Aguado03} For a given site which |
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posesses a charge, dipole, and multipole, both types of contribution |
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are given in table \ref{tab:tableSelf}. |
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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}{llll} |
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Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
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Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{2 \alpha}{\sqrt{\pi}}$ \\ |
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Dipole & $|D_{\bf a}|^2$ & $\left( \frac{h(r_c)}{3} + \frac{2 |
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g(r_c)}{3 r_c} |
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\right)$ & $-\frac{4 \alpha^3}{3 \sqrt{\pi}}$\\ |
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Quadrupole & $2 \text{Tr}[Q_{\bf a}^2] + \left(\text{Tr}[Q_{\bf a}]\right)^2$ & |
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$\frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & $-\frac{8 |
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\alpha^5}{5 \sqrt{\pi}}$ \\ |
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Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}[Q_{\bf a}]$ & $\left( |
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\frac{h(r_c)}{3} + \frac{2 g(r_c)}{3 r_c} \right)$& $-\frac{4 |
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\alpha^3}{3 \sqrt{\pi}}$ \\ |
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\end{tabular} |
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\end{ruledtabular} |
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\end{table*} |
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For sites which contain both charges and quadrupoles, the |
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self-interaction includes a cross-interaction between these two |
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multipoles. Symmetry prevents the charge-dipole and dipole-quadrupole |
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interactions from contributing to the self-interaction. The functions |
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that go into the self-neutralization terms are derivatives of Smith's |
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$B_0(r)$ function that have been evaluated at the cutoff distance. |
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For example, $f(r_c) = \left(\frac{d B_0}{dr}\right)_{r_c}$, $g(r_c) = |
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\left(\frac{d^2 B_0}{dr^2}\right)_{r_c}$, and so on. |
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As the order of multipoles increases, the reciprocal portion is |
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expected to shrink rapidly. This is expected as the range of the |
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interaction is also decreasing dramatically. |
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One final question: Are there torques that arise from the self |
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interactions? |
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\newpage |
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\bibliography{multipole} |
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\end{document} |
