| 34 |
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\section{The Self-Interaction} |
| 35 |
|
The Wolf summation~\cite{Wolf99} and the later damped shifted force |
| 36 |
|
(DSF) extension~\cite{Fennell06} included self-interactions that are |
| 37 |
< |
handled outside the main pairwise interactions between sites. The |
| 38 |
< |
self-interaction has contributions from two sources: |
| 37 |
> |
handled separately from the pairwise interactions between sites. The |
| 38 |
> |
self-term is normally calculated via a single loop over all sites in |
| 39 |
> |
the system, and is relatively cheap to evaluate. The self-interaction |
| 40 |
> |
has contributions from two sources: |
| 41 |
|
\begin{itemize} |
| 42 |
|
\item The neutralization procedure within the cutoff radius requires a |
| 43 |
|
contribution from a charge opposite in sign, but equal in magnitude, |
| 44 |
|
to the central charge, which has been spread out over the surface of |
| 45 |
< |
the cutoff sphere. This term is calculated via a single loop over |
| 46 |
< |
all charges in the system. For a system of undamped charges, the |
| 45 |
< |
total self-term is |
| 45 |
> |
the cutoff sphere. For a system of undamped charges, the total |
| 46 |
> |
self-term is |
| 47 |
|
\begin{equation} |
| 48 |
|
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
| 49 |
|
\label{eq:selfTerm} |
| 59 |
|
into the self-interaction from charge neutralization of the damped |
| 60 |
|
potential. The remainder of the reciprocal space portion is then |
| 61 |
|
discarded (as this contributes the largest computational cost and |
| 62 |
< |
complexity to the Ewald sum). For the damped charge case the |
| 63 |
< |
complete self-interaction can be written as |
| 62 |
> |
complexity to the Ewald sum). For a system containing only damped |
| 63 |
> |
charges, the complete self-interaction can be written as |
| 64 |
|
\begin{equation} |
| 65 |
< |
V_\textrm{self} = - \left(\frac{2 \alpha}{\sqrt{\pi}} |
| 66 |
< |
+ \frac{\textrm{erfc}(\alpha r_c)}{r_c}\right) \sum_{{\bf a}=1}^N |
| 65 |
> |
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
| 66 |
> |
\frac{2 \alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
| 67 |
|
C_{\bf a}^{2}. |
| 68 |
|
\label{eq:dampSelfTerm} |
| 69 |
|
\end{equation} |
| 72 |
|
The extension of DSF electrostatics to point multipoles requires |
| 73 |
|
treatment of {\it both} the self-neutralization and reciprocal |
| 74 |
|
contributions to the self-interaction for higher order multipoles. In |
| 75 |
< |
this section we give formulae for these interactions and discuss the |
| 76 |
< |
relative sizes of these contributions. |
| 75 |
> |
this section we give formulae for these interactions up to quadrupolar |
| 76 |
> |
order. |
| 77 |
|
|
| 78 |
|
The self-neutralization term is computed by taking the {\it |
| 79 |
|
non-shifted} kernel for each interaction, placing a multipole of |
| 80 |
|
equal magnitude (but opposite in polarization) on the surface of the |
| 81 |
|
cutoff sphere, and averaging over the surface of the cutoff sphere. |
| 82 |
< |
The reciprocal-space portion is identical to theself-term obtained by |
| 83 |
< |
Smith and Aguado and Madden for the application of the Ewald sum to |
| 84 |
< |
multipoles.\cite{Smith82,Smith98,Aguado03} For a given site which |
| 85 |
< |
posesses a charge, dipole, and multipole, both types of contribution |
| 86 |
< |
are given in table \ref{tab:tableSelf}. |
| 82 |
> |
Because the self term is carried out as a single sum over sites, the |
| 83 |
> |
reciprocal-space portion is identical to half of the self-term |
| 84 |
> |
obtained by Smith and Aguado and Madden for the application of the |
| 85 |
> |
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
| 86 |
> |
site which posesses a charge, dipole, and multipole, both types of |
| 87 |
> |
contribution are given in table \ref{tab:tableSelf}. |
| 88 |
|
|
| 89 |
|
\begin{table*} |
| 90 |
|
\caption{\label{tab:tableSelf} Self-interaction contributions for |
| 91 |
|
site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
| 92 |
|
$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
| 93 |
|
\begin{ruledtabular} |
| 94 |
< |
\begin{tabular}{llll} |
| 94 |
> |
\begin{tabular}{lccc} |
| 95 |
|
Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
| 96 |
< |
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{2 \alpha}{\sqrt{\pi}}$ \\ |
| 97 |
< |
Dipole & $|D_{\bf a}|^2$ & $\left( \frac{h(r_c)}{3} + \frac{2 |
| 98 |
< |
g(r_c)}{3 r_c} |
| 99 |
< |
\right)$ & $-\frac{4 \alpha^3}{3 \sqrt{\pi}}$\\ |
| 100 |
< |
Quadrupole & $2 \text{Tr}[Q_{\bf a}^2] + \left(\text{Tr}[Q_{\bf a}]\right)^2$ & |
| 101 |
< |
$\frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & $-\frac{8 |
| 102 |
< |
\alpha^5}{5 \sqrt{\pi}}$ \\ |
| 103 |
< |
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}[Q_{\bf a}]$ & $\left( |
| 102 |
< |
\frac{h(r_c)}{3} + \frac{2 g(r_c)}{3 r_c} \right)$& $-\frac{4 |
| 103 |
< |
\alpha^3}{3 \sqrt{\pi}}$ \\ |
| 96 |
> |
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
| 97 |
> |
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
| 98 |
> |
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
| 99 |
> |
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 100 |
> |
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
| 101 |
> |
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
| 102 |
> |
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
| 103 |
> |
h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\ |
| 104 |
|
\end{tabular} |
| 105 |
|
\end{ruledtabular} |
| 106 |
|
\end{table*} |
| 107 |
|
|
| 108 |
< |
For sites which contain both charges and quadrupoles, the |
| 108 |
> |
For sites which simultaneously contain charges and quadrupoles, the |
| 109 |
|
self-interaction includes a cross-interaction between these two |
| 110 |
< |
multipoles. Symmetry prevents the charge-dipole and dipole-quadrupole |
| 111 |
< |
interactions from contributing to the self-interaction. The functions |
| 112 |
< |
that go into the self-neutralization terms are derivatives of Smith's |
| 113 |
< |
$B_0(r)$ function that have been evaluated at the cutoff distance. |
| 114 |
< |
For example, $f(r_c) = \left(\frac{d B_0}{dr}\right)_{r_c}$, $g(r_c) = |
| 115 |
< |
\left(\frac{d^2 B_0}{dr^2}\right)_{r_c}$, and so on. |
| 110 |
> |
multipole orders. Symmetry prevents the charge-dipole and |
| 111 |
> |
dipole-quadrupole interactions from contributing to the |
| 112 |
> |
self-interaction. The functions that go into the self-neutralization |
| 113 |
> |
terms, $f(r), g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
| 114 |
> |
derivatives of the electrostatic kernel (either the undamped $1/r$ or |
| 115 |
> |
the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that are |
| 116 |
> |
evaluated at the cutoff distance. For undamped interactions, $f(r_c) |
| 117 |
> |
= 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For damped interactions, |
| 118 |
> |
$f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so on. Appendix XX |
| 119 |
> |
contains recursion relations that allow rapid evaluation of these |
| 120 |
> |
derivatives. |
| 121 |
|
|
| 122 |
< |
As the order of multipoles increases, the reciprocal portion is |
| 123 |
< |
expected to shrink rapidly. This is expected as the range of the |
| 119 |
< |
interaction is also decreasing dramatically. |
| 122 |
> |
The self interaction also gives rise to a contribution to the torque |
| 123 |
> |
on the site |
| 124 |
|
|
| 121 |
– |
One final question: Are there torques that arise from the self |
| 122 |
– |
interactions? |
| 123 |
– |
|
| 124 |
– |
|
| 125 |
|
\newpage |
| 126 |
|
|
| 127 |
|
\bibliography{multipole} |
