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 outside the main pairwise interactions between sites. The |
38 |
self-interaction has contributions from two sources: |
39 |
\begin{itemize} |
40 |
\item The neutralization procedure within the cutoff radius requires a |
41 |
contribution from a charge opposite in sign, but equal in magnitude, |
42 |
to the central charge, which has been spread out over the surface of |
43 |
the cutoff sphere. This term is calculated via a single loop over |
44 |
all charges in the system. For a system of undamped charges, the |
45 |
total self-term is |
46 |
\begin{equation} |
47 |
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
48 |
\label{eq:selfTerm} |
49 |
\end{equation} |
50 |
Note that in this potential and in all electrostatic quantities that |
51 |
follow, the standard $4 \pi \epsilon_{0}$ has been omitted for |
52 |
clarity. |
53 |
\item Charge damping with the complementary error function is a |
54 |
partial analogy to the Ewald procedure which splits the interaction |
55 |
into real and reciprocal space sums. The real space sum is retained |
56 |
in the Wolf and DSF methods. The reciprocal space sum is first |
57 |
minimized by folding the largest contribution (the self-interaction) |
58 |
into the self-interaction from charge neutralization of the damped |
59 |
potential. The remainder of the reciprocal space portion is then |
60 |
discarded (as this contributes the largest computational cost and |
61 |
complexity to the Ewald sum). For the damped charge case the |
62 |
complete self-interaction can be written as |
63 |
\begin{equation} |
64 |
V_\textrm{self} = - \left(\frac{2 \alpha}{\sqrt{\pi}} |
65 |
+ \frac{\textrm{erfc}(\alpha r_c)}{r_c}\right) \sum_{{\bf a}=1}^N |
66 |
C_{\bf a}^{2}. |
67 |
\label{eq:dampSelfTerm} |
68 |
\end{equation} |
69 |
\end{itemize} |
70 |
|
71 |
The extension of DSF electrostatics to point multipoles requires |
72 |
treatment of {\it both} the self-neutralization and reciprocal |
73 |
contributions to the self-interaction for higher order multipoles. In |
74 |
this section we give formulae for these interactions and discuss the |
75 |
relative sizes of these contributions. |
76 |
|
77 |
The self-neutralization term is computed by taking the {\it |
78 |
non-shifted} kernel for each interaction, placing a multipole of |
79 |
equal magnitude (but opposite in polarization) on the surface of the |
80 |
cutoff sphere, and averaging over the surface of the cutoff sphere. |
81 |
The reciprocal-space portion is identical to theself-term obtained by |
82 |
Smith and Aguado and Madden for the application of the Ewald sum to |
83 |
multipoles.\cite{Smith82,Smith98,Aguado03} For a given site which |
84 |
posesses a charge, dipole, and multipole, both types of contribution |
85 |
are given in table \ref{tab:tableSelf}. |
86 |
|
87 |
\begin{table*} |
88 |
\caption{\label{tab:tableSelf} Self-interaction contributions for |
89 |
site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
90 |
$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
91 |
\begin{ruledtabular} |
92 |
\begin{tabular}{llll} |
93 |
Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
94 |
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{2 \alpha}{\sqrt{\pi}}$ \\ |
95 |
Dipole & $|D_{\bf a}|^2$ & $\left( \frac{h(r_c)}{3} + \frac{2 |
96 |
g(r_c)}{3 r_c} |
97 |
\right)$ & $-\frac{4 \alpha^3}{3 \sqrt{\pi}}$\\ |
98 |
Quadrupole & $2 \text{Tr}[Q_{\bf a}^2] + \left(\text{Tr}[Q_{\bf a}]\right)^2$ & |
99 |
$\frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & $-\frac{8 |
100 |
\alpha^5}{5 \sqrt{\pi}}$ \\ |
101 |
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}[Q_{\bf a}]$ & $\left( |
102 |
\frac{h(r_c)}{3} + \frac{2 g(r_c)}{3 r_c} \right)$& $-\frac{4 |
103 |
\alpha^3}{3 \sqrt{\pi}}$ \\ |
104 |
\end{tabular} |
105 |
\end{ruledtabular} |
106 |
\end{table*} |
107 |
|
108 |
For sites which contain both 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. |
116 |
|
117 |
As the order of multipoles increases, the reciprocal portion is |
118 |
expected to shrink rapidly. This is expected as the range of the |
119 |
interaction is also decreasing dramatically. |
120 |
|
121 |
One final question: Are there torques that arise from the self |
122 |
interactions? |
123 |
|
124 |
|
125 |
\newpage |
126 |
|
127 |
\bibliography{multipole} |
128 |
|
129 |
\end{document} |