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.
Because the self term is carried out as a single sum over sites, the
reciprocal-space portion is identical to half of 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}{lccc}
Multipole order & Summed Quantity & Self-neutralization  & Reciprocal \\ \hline
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) +
  \frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ &
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ &
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left(
  h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\
\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:	3912
Committed:	Thu Jul 11 14:17:40 2013 UTC (11 years, 2 months ago) by gezelter
Content type:	application/x-tex
File size:	5458 byte(s)
Log Message:	Edits for factor of 1/2 in reciprocal space portion.
#	User	Rev	Content
1	gezelter	3908	\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	gezelter	3910	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	gezelter	3908	\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	gezelter	3910	the cutoff sphere. For a system of undamped charges, the total
46			self-term is
47	gezelter	3908	\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	gezelter	3910	complexity to the Ewald sum). For a system containing only damped
63			charges, the complete self-interaction can be written as
64	gezelter	3908	\begin{equation}
65	gezelter	3910	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	gezelter	3908	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	gezelter	3910	this section we give formulae for these interactions up to quadrupolar
76			order.
77	gezelter	3908
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	gezelter	3912	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	gezelter	3908
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	gezelter	3911	\begin{tabular}{lccc}
95	gezelter	3908	Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline
96	gezelter	3912	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	gezelter	3911	Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ &
100	gezelter	3912	$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ &
101			$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\
102	gezelter	3911	Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left(
103	gezelter	3912	h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\
104	gezelter	3908	\end{tabular}
105			\end{ruledtabular}
106			\end{table*}
107
108	gezelter	3910	For sites which simultaneously contain charges and quadrupoles, the
109	gezelter	3908	self-interaction includes a cross-interaction between these two
110	gezelter	3910	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	gezelter	3908
122	gezelter	3910	The self interaction also gives rise to a contribution to the torque
123			on the site
124	gezelter	3908
125			\newpage
126
127			\bibliography{multipole}
128
129			\end{document}