trunk/multipole/selfterm.tex

\documentclass[%
 aip,
 jmp,
 amsmath,amssymb,
preprint,%
% reprint,%
%author-year,%
%author-numerical,%
]{revtex4-1}

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{natbib}
\usepackage[version=3]{mhchem}  % this is a great package for formatting chemical reactions
\usepackage{url}
\usepackage{wrapfig,lipsum,booktabs}

\begin{document}

\title[Notes on the Self-Interaction]
{Notes on the Self-Interaction}

\author{J. Daniel Gezelter}
 \email{gezelter@nd.edu.}
\affiliation{Department of Chemistry and Biochemistry, University
of Notre Dame, Notre Dame, IN 46556}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified
\maketitle


\section{The Self-Interaction}
The Wolf summation~\cite{Wolf99} and the later damped shifted force
(DSF) extension~\cite{Fennell06} included self-interactions that are
handled separately from the pairwise interactions between sites. The
self-term is normally calculated via a single loop over all sites in
the system, and is relatively cheap to evaluate. The self-interaction
has contributions from two sources:
\begin{itemize}
\item The neutralization procedure within the cutoff radius requires a
  contribution from a charge opposite in sign, but equal in magnitude,
  to the central charge, which has been spread out over the surface of
  the cutoff sphere.  For a system of undamped charges, the total
  self-term is
\begin{equation}
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2}
\label{eq:selfTerm}
\end{equation}
Note that in this potential and in all electrostatic quantities that
follow, the standard $4 \pi \epsilon_{0}$ has been omitted for
clarity.
\item Charge damping with the complementary error function is a
  partial analogy to the Ewald procedure which splits the interaction
  into real and reciprocal space sums.  The real space sum is retained
  in the Wolf and DSF methods.  The reciprocal space sum is first
  minimized by folding the largest contribution (the self-interaction)
  into the self-interaction from charge neutralization of the damped
  potential.  The remainder of the reciprocal space portion is then
  discarded (as this contributes the largest computational cost and
  complexity to the Ewald sum).  For a system containing only damped
  charges, the complete self-interaction can be written as
\begin{equation}
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + 
  \frac{2 \alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N
      C_{\bf a}^{2}.
\label{eq:dampSelfTerm}
\end{equation}
\end{itemize}

The extension of DSF electrostatics to point multipoles requires
treatment of {\it both} the self-neutralization and reciprocal
contributions to the self-interaction for higher order multipoles.  In
this section we give formulae for these interactions up to quadrupolar
order.

The self-neutralization term is computed by taking the {\it
  non-shifted} kernel for each interaction, placing a multipole of
equal magnitude (but opposite in polarization) on the surface of the
cutoff sphere, and averaging over the surface of the cutoff sphere.
The reciprocal-space term is identical to the self-term obtained by
Smith and Aguado and Madden for the application of the Ewald sum to
multipoles.\cite{Smith82,Smith98,Aguado03} For a given site which
posesses a charge, dipole, and multipole, both types of contribution
are given in table \ref{tab:tableSelf}.

\begin{table*}
  \caption{\label{tab:tableSelf} Self-interaction contributions for 
    site ({\bf a}) that has a charge $(C_{\bf a})$, dipole
    $(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$}
\begin{ruledtabular}
\begin{tabular}{llcc}
Multipole order & Summed Quantity & Self-neutralization  & Reciprocal \\ \hline
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{2 \alpha}{\sqrt{\pi}}$ \\
Dipole & $|D_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + \frac{2
    g(r_c)}{r_c}
\right)$ & $-\frac{4 \alpha^3}{3 \sqrt{\pi}}$\\
Quadrupole & $2 \text{Tr}[Q_{\bf a}^2] + \left(\text{Tr}[Q_{\bf a}]\right)^2$ &
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & $-\frac{8
  \alpha^5}{5 \sqrt{\pi}}$ \\
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}[Q_{\bf a}]$ & $\frac{1}{3} \left(
  h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{4
  \alpha^3}{3 \sqrt{\pi}}$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}

For sites which simultaneously contain charges and quadrupoles, the
self-interaction includes a cross-interaction between these two
multipole orders.  Symmetry prevents the charge-dipole and
dipole-quadrupole interactions from contributing to the
self-interaction.  The functions that go into the self-neutralization
terms, $f(r), g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive
derivatives of the electrostatic kernel (either the undamped $1/r$ or
the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that are
evaluated at the cutoff distance.  For undamped interactions, $f(r_c)
= 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on.  For damped interactions,
$f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so on.  Appendix XX
contains recursion relations that allow rapid evaluation of these
derivatives.

The self interaction also gives rise to a contribution to the torque
on the site 

\newpage

\bibliography{multipole}

\end{document}
Revision:	3910
Committed:	Thu Jul 11 02:57:13 2013 UTC (11 years, 9 months ago) by gezelter
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File size:	5367 byte(s)
Log Message:	Updated text, fixed a sign error
#	Content
1	\documentclass[%
2	aip,
3	jmp,
4	amsmath,amssymb,
5	preprint,%
6	% reprint,%
7	%author-year,%
8	%author-numerical,%
9	]{revtex4-1}
10
11	\usepackage{graphicx}% Include figure files
12	\usepackage{dcolumn}% Align table columns on decimal point
13	\usepackage{bm}% bold math
14	\usepackage{natbib}
15	\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions
16	\usepackage{url}
17	\usepackage{wrapfig,lipsum,booktabs}
18
19	\begin{document}
20
21	\title[Notes on the Self-Interaction]
22	{Notes on the Self-Interaction}
23
24	\author{J. Daniel Gezelter}
25	\email{gezelter@nd.edu.}
26	\affiliation{Department of Chemistry and Biochemistry, University
27	of Notre Dame, Notre Dame, IN 46556}
28
29	\date{\today}% It is always \today, today,
30	% but any date may be explicitly specified
31	\maketitle
32
33
34	\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 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. 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}
50	\end{equation}
51	Note that in this potential and in all electrostatic quantities that
52	follow, the standard $4 \pi \epsilon_{0}$ has been omitted for
53	clarity.
54	\item Charge damping with the complementary error function is a
55	partial analogy to the Ewald procedure which splits the interaction
56	into real and reciprocal space sums. The real space sum is retained
57	in the Wolf and DSF methods. The reciprocal space sum is first
58	minimized by folding the largest contribution (the self-interaction)
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 a system containing only damped
63	charges, the complete self-interaction can be written as
64	\begin{equation}
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}
70	\end{itemize}
71
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 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 term is identical to the self-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}.
87
88	\begin{table*}
89	\caption{\label{tab:tableSelf} Self-interaction contributions for
90	site ({\bf a}) that has a charge $(C_{\bf a})$, dipole
91	$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$}
92	\begin{ruledtabular}
93	\begin{tabular}{llcc}
94	Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline
95	Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{2 \alpha}{\sqrt{\pi}}$ \\
96	Dipole & $\|D_{\bf a}\|^2$ & $\frac{1}{3} \left( h(r_c) + \frac{2
97	g(r_c)}{r_c}
98	\right)$ & $-\frac{4 \alpha^3}{3 \sqrt{\pi}}$\\
99	Quadrupole & $2 \text{Tr}[Q_{\bf a}^2] + \left(\text{Tr}[Q_{\bf a}]\right)^2$ &
100	$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & $-\frac{8
101	\alpha^5}{5 \sqrt{\pi}}$ \\
102	Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}[Q_{\bf a}]$ & $\frac{1}{3} \left(
103	h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{4
104	\alpha^3}{3 \sqrt{\pi}}$ \\
105	\end{tabular}
106	\end{ruledtabular}
107	\end{table*}
108
109	For sites which simultaneously contain charges and quadrupoles, the
110	self-interaction includes a cross-interaction between these two
111	multipole orders. Symmetry prevents the charge-dipole and
112	dipole-quadrupole interactions from contributing to the
113	self-interaction. The functions that go into the self-neutralization
114	terms, $f(r), g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive
115	derivatives of the electrostatic kernel (either the undamped $1/r$ or
116	the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that are
117	evaluated at the cutoff distance. For undamped interactions, $f(r_c)
118	= 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For damped interactions,
119	$f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so on. Appendix XX
120	contains recursion relations that allow rapid evaluation of these
121	derivatives.
122
123	The self interaction also gives rise to a contribution to the torque
124	on the site
125
126	\newpage
127
128	\bibliography{multipole}
129
130	\end{document}