1 |
gezelter |
3906 |
% ****** Start of file aipsamp.tex ****** |
2 |
|
|
% |
3 |
|
|
% This file is part of the AIP files in the AIP distribution for REVTeX 4. |
4 |
|
|
% Version 4.1 of REVTeX, October 2009 |
5 |
|
|
% |
6 |
|
|
% Copyright (c) 2009 American Institute of Physics. |
7 |
|
|
% |
8 |
|
|
% See the AIP README file for restrictions and more information. |
9 |
|
|
% |
10 |
|
|
% TeX'ing this file requires that you have AMS-LaTeX 2.0 installed |
11 |
|
|
% as well as the rest of the prerequisites for REVTeX 4.1 |
12 |
|
|
% |
13 |
|
|
% It also requires running BibTeX. The commands are as follows: |
14 |
|
|
% |
15 |
|
|
% 1) latex aipsamp |
16 |
|
|
% 2) bibtex aipsamp |
17 |
|
|
% 3) latex aipsamp |
18 |
|
|
% 4) latex aipsamp |
19 |
|
|
% |
20 |
|
|
% Use this file as a source of example code for your aip document. |
21 |
|
|
% Use the file aiptemplate.tex as a template for your document. |
22 |
|
|
\documentclass[% |
23 |
gezelter |
3980 |
aip,jcp, |
24 |
gezelter |
3906 |
amsmath,amssymb, |
25 |
|
|
preprint,% |
26 |
|
|
% reprint,% |
27 |
|
|
%author-year,% |
28 |
|
|
%author-numerical,% |
29 |
|
|
]{revtex4-1} |
30 |
|
|
|
31 |
|
|
\usepackage{graphicx}% Include figure files |
32 |
|
|
\usepackage{dcolumn}% Align table columns on decimal point |
33 |
gezelter |
3980 |
%\usepackage{bm}% bold math |
34 |
|
|
\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
35 |
|
|
\usepackage{url} |
36 |
|
|
|
37 |
gezelter |
3906 |
%\usepackage[mathlines]{lineno}% Enable numbering of text and display math |
38 |
|
|
%\linenumbers\relax % Commence numbering lines |
39 |
|
|
|
40 |
|
|
\begin{document} |
41 |
|
|
|
42 |
|
|
\preprint{AIP/123-QED} |
43 |
|
|
|
44 |
gezelter |
3980 |
\title[Taylor-shifted and Gradient-shifted electrostatics for multipoles] |
45 |
|
|
{Real space alternatives to the Ewald |
46 |
|
|
Sum. I. Taylor-shifted and Gradient-shifted electrostatics for multipoles} |
47 |
gezelter |
3906 |
|
48 |
|
|
\author{Madan Lamichhane} |
49 |
|
|
\affiliation{Department of Physics, University |
50 |
|
|
of Notre Dame, Notre Dame, IN 46556} |
51 |
|
|
|
52 |
|
|
\author{J. Daniel Gezelter} |
53 |
|
|
\email{gezelter@nd.edu.} |
54 |
|
|
\affiliation{Department of Chemistry and Biochemistry, University |
55 |
|
|
of Notre Dame, Notre Dame, IN 46556} |
56 |
|
|
|
57 |
|
|
\author{Kathie E. Newman} |
58 |
|
|
\affiliation{Department of Physics, University |
59 |
|
|
of Notre Dame, Notre Dame, IN 46556} |
60 |
|
|
|
61 |
|
|
|
62 |
|
|
\date{\today}% It is always \today, today, |
63 |
|
|
% but any date may be explicitly specified |
64 |
|
|
|
65 |
|
|
\begin{abstract} |
66 |
gezelter |
3980 |
We have extended the original damped-shifted force (DSF) |
67 |
|
|
electrostatic kernel and have been able to derive two new |
68 |
|
|
electrostatic potentials for higher-order multipoles that are based |
69 |
|
|
on truncated Taylor expansions around the cutoff radius. For |
70 |
|
|
multipole-multipole interactions, we find that each of the distinct |
71 |
|
|
orientational contributions has a separate radial function to ensure |
72 |
|
|
that the overall forces and torques vanish at the cutoff radius. In |
73 |
|
|
this paper, we present energy, force, and torque expressions for the |
74 |
|
|
new models, and compare these real-space interaction models to exact |
75 |
|
|
results for ordered arrays of multipoles. |
76 |
gezelter |
3906 |
\end{abstract} |
77 |
|
|
|
78 |
|
|
\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
79 |
|
|
% Classification Scheme. |
80 |
|
|
\keywords{Suggested keywords}%Use showkeys class option if keyword |
81 |
|
|
%display desired |
82 |
|
|
\maketitle |
83 |
|
|
|
84 |
|
|
\section{Introduction} |
85 |
|
|
|
86 |
gezelter |
3980 |
There is increasing interest in real-space methods for calculating |
87 |
|
|
electrostatic interactions in computer simulations of condensed |
88 |
|
|
molecular |
89 |
|
|
systems.\cite{Wolf99,Zahn02,Kast03,BeckD.A.C._bi0486381,Ma05,Fennell:2006zl,Chen:2004du,Chen:2006ii,Rodgers:2006nw,Denesyuk:2008ez,Izvekov:2008wo} |
90 |
|
|
The simplest of these techniques was developed by Wolf {\it et al.} |
91 |
|
|
in their work towards an $\mathcal{O}(N)$ Coulombic sum.\cite{Wolf99} |
92 |
|
|
Fennell and Gezelter showed that a simple damped shifted force (DSF) |
93 |
|
|
modification to Wolf's method could give nearly quantitative agreement |
94 |
|
|
with smooth particle mesh Ewald (SPME)\cite{Essmann95} for quantities |
95 |
|
|
of interest in Monte Carlo (i.e., configurational energy differences) |
96 |
|
|
and Molecular Dynamics (i.e., atomic force and molecular torque |
97 |
|
|
vectors).\cite{Fennell:2006zl} |
98 |
gezelter |
3906 |
|
99 |
gezelter |
3980 |
The computational efficiency and the accuracy of the DSF method are |
100 |
|
|
surprisingly good, particularly for systems with uniform charge |
101 |
|
|
density. Additionally, dielectric constants obtained using DSF and |
102 |
|
|
similar methods where the force vanishes at $R_\textrm{c}$ are |
103 |
|
|
essentially quantitative.\cite{Izvekov:2008wo} The DSF and other |
104 |
|
|
related methods have now been widely investigated,\cite{Hansen:2012uq} |
105 |
|
|
and DSF is now used routinely in simulations of ionic |
106 |
|
|
liquids,\cite{doi:10.1021/la400226g,McCann:2013fk} flow in carbon |
107 |
|
|
nanotubes,\cite{kannam:094701} gas sorption in metal-organic framework |
108 |
|
|
materials,\cite{Forrest:2012ly} thermal conductivity of methane |
109 |
|
|
hydrates,\cite{English:2008kx} condensation coefficients of |
110 |
|
|
water,\cite{Louden:2013ve} and in tribology at solid-liquid-solid |
111 |
|
|
interfaces.\cite{Tokumasu:2013zr} DSF electrostatics provides a |
112 |
|
|
compromise between the computational speed of real-space cutoffs and |
113 |
|
|
the accuracy of fully-periodic Ewald treatments. |
114 |
gezelter |
3906 |
|
115 |
gezelter |
3980 |
One common feature of many coarse-graining approaches, which treat |
116 |
|
|
entire molecular subsystems as a single rigid body, is simplification |
117 |
|
|
of the electrostatic interactions between these bodies so that fewer |
118 |
|
|
site-site interactions are required to compute configurational |
119 |
|
|
energies. Notably, the force matching approaches of Voth and coworkers |
120 |
|
|
are an exciting development in their ability to represent realistic |
121 |
|
|
(and {\it reactive}) chemical systems for very large length scales and |
122 |
|
|
long times. This approach utilized a coarse-graining in interaction |
123 |
|
|
space (CGIS) which fits an effective force for the coarse grained |
124 |
|
|
system using a variational force-matching method to a fine-grained |
125 |
|
|
simulation.\cite{Izvekov:2008wo} |
126 |
gezelter |
3906 |
|
127 |
gezelter |
3980 |
The coarse-graining approaches of Ren \& coworkers,\cite{Golubkov06} |
128 |
|
|
and Essex \& |
129 |
|
|
coworkers,\cite{ISI:000276097500009,ISI:000298664400012} |
130 |
|
|
both utilize Gay-Berne |
131 |
|
|
ellipsoids~\cite{Berne72,Gay81,Luckhurst90,Cleaver96,Berardi98,Ravichandran:1999fk,Berardi99,Pasterny00} |
132 |
|
|
to model dispersive interactions and point multipoles to model |
133 |
|
|
electrostatics for entire molecules or functional groups. |
134 |
gezelter |
3906 |
|
135 |
gezelter |
3980 |
Ichiye and coworkers have recently introduced a number of very fast |
136 |
|
|
water models based on a ``sticky'' multipole model which are |
137 |
|
|
qualitatively better at reproducing the behavior of real water than |
138 |
|
|
the more common point-charge models (SPC/E, TIPnP). The point charge |
139 |
|
|
models are also substantially more computationally demanding than the |
140 |
|
|
sticky multipole |
141 |
|
|
approach.\cite{Chowdhuri:2006lr,Te:2010rt,Te:2010ys,Te:2010vn} The |
142 |
|
|
SSDQO model requires the use of an approximate multipole expansion |
143 |
|
|
(AME) as the exact multipole expansion is quite expensive |
144 |
|
|
(particularly when handled via the Ewald sum).\cite{Ichiye:2006qy} |
145 |
|
|
Another particularly important use of point multipoles (and multipole |
146 |
|
|
polarizability) is in the very high-quality AMOEBA water model and |
147 |
|
|
related force fields.\cite{Ponder:2010fk,schnieders:124114,Ren:2011uq} |
148 |
gezelter |
3906 |
|
149 |
gezelter |
3980 |
Higher-order multipoles present a peculiar issue for molecular |
150 |
|
|
dynamics. Multipolar interactions are inherently short-ranged, and |
151 |
|
|
should not need the relatively expensive Ewald treatment. However, |
152 |
|
|
real-space cutoff methods are normally applied in an orientation-blind |
153 |
|
|
fashion so multipoles which leave and then re-enter a cutoff sphere in |
154 |
|
|
a different orientation can cause energy discontinuities. |
155 |
gezelter |
3906 |
|
156 |
gezelter |
3980 |
This paper outlines an extension of the original DSF electrostatic |
157 |
|
|
kernel to point multipoles. We present in this paper two distinct |
158 |
|
|
real-space interaction models for higher-order multipoles based on two |
159 |
|
|
truncated Taylor expansions that are carried out at the cutoff radius. |
160 |
|
|
We are calling these models {\bf Taylor-shifted} and {\bf |
161 |
|
|
Gradient-shifted} electrostatics. Because of differences in the |
162 |
|
|
initial assumptions, the two methods yield related, but different |
163 |
|
|
expressions for energies, forces, and torques. |
164 |
gezelter |
3906 |
|
165 |
gezelter |
3980 |
\section{Methodology} |
166 |
gezelter |
3906 |
|
167 |
gezelter |
3980 |
\subsection{Self-neutralization, damping, and force-shifting} |
168 |
gezelter |
3906 |
|
169 |
gezelter |
3980 |
The DSF and Wolf methods operate by neutralizing the total charge |
170 |
|
|
contained within the cutoff sphere surrounding each particle. This is |
171 |
|
|
accomplished by shifting the potential functions to generate image |
172 |
|
|
charges on the surface of the cutoff sphere for each pair interaction |
173 |
|
|
computed within $R_\textrm{c}$. An Ewald-like complementary error |
174 |
|
|
function damping is applied to the potential to accelerate |
175 |
|
|
convergence. The potential for the DSF method, where $\alpha$ is the |
176 |
|
|
adjustable damping parameter, is given by |
177 |
|
|
\begin{equation*} |
178 |
|
|
V_\mathrm{DSF}(r) = C_a C_b \Biggr{[} |
179 |
|
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
180 |
|
|
- \frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}} + \left(\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}^2} |
181 |
|
|
+ \frac{2\alpha}{\pi^{1/2}} |
182 |
|
|
\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}} |
183 |
|
|
\right)\left(r_{ij}-R_\mathrm{c}\right)\ \Biggr{]} |
184 |
|
|
\label{eq:DSFPot} |
185 |
|
|
\end{equation*} |
186 |
|
|
|
187 |
|
|
To insure net charge neutrality within each cutoff sphere, an |
188 |
|
|
additional ``self'' term is added to the potential. This term is |
189 |
|
|
constant (as long as the charges and cutoff radius do not change), and |
190 |
|
|
exists outside the normal pair-loop for molecular simulations. It can |
191 |
|
|
be thought of as a contribution from a charge opposite in sign, but |
192 |
|
|
equal in magnitude, to the central charge, which has been spread out |
193 |
|
|
over the surface of the cutoff sphere. A portion of the self term is |
194 |
|
|
identical to the self term in the Ewald summation, and comes from the |
195 |
|
|
utilization of the complimentary error function for electrostatic |
196 |
|
|
damping.\cite{deLeeuw80,Wolf99} |
197 |
|
|
|
198 |
|
|
\begin{figure} |
199 |
|
|
\includegraphics[width=\linewidth]{SM} |
200 |
|
|
\caption{Reversed multipoles are projected onto the surface of the |
201 |
|
|
cutoff sphere. The forces, torques, and potential are then smoothly |
202 |
|
|
shifted to zero as the sites leave the cutoff region.} |
203 |
|
|
\label{fig:shiftedMultipoles} |
204 |
|
|
\end{figure} |
205 |
|
|
|
206 |
gezelter |
3906 |
\subsection{Multipole Expansion} |
207 |
|
|
|
208 |
gezelter |
3980 |
Consider two discrete rigid collections of point charges, denoted as |
209 |
|
|
objects $\bf a$ and $\bf b$. In the following, we assume that the two |
210 |
|
|
objects only interact via electrostatics and describe those |
211 |
|
|
interactions in terms of a multipole expansion. Putting the origin of |
212 |
|
|
the coordinate system at the center of mass of $\bf a$, we use vectors |
213 |
|
|
$\mathbf{r}_k$ to denote the positions of all charges $q_k$ in $\bf |
214 |
|
|
a$. Then the electrostatic potential of object $\bf a$ at |
215 |
|
|
$\mathbf{r}$ is given by |
216 |
gezelter |
3906 |
\begin{equation} |
217 |
|
|
V_a(\mathbf r) = \frac{1}{4\pi\epsilon_0} |
218 |
|
|
\sum_{k \, \text{in \bf a}} \frac{q_k}{\lvert \mathbf{r} - \mathbf{r}_k \rvert}. |
219 |
|
|
\end{equation} |
220 |
gezelter |
3980 |
We write the Taylor series expansion in $r$ using an implied summation |
221 |
|
|
notation, Greek indices are used to indicate space coordinates ($x$, |
222 |
|
|
$y$, $z$) and the subscripts $k$ and $j$ are reserved for labelling |
223 |
|
|
specific charges in $\bf a$ and $\bf b$ respectively, and find: |
224 |
gezelter |
3906 |
\begin{equation} |
225 |
|
|
\frac{1}{\lvert \mathbf{r} - \mathbf{r}_k \rvert} = |
226 |
|
|
\left( 1 |
227 |
|
|
- r_{k\alpha} \frac{\partial}{\partial r_{\alpha}} |
228 |
|
|
+ \frac{1}{2} r_{k\alpha} r_{k\beta} \frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} +\dots |
229 |
|
|
\right) |
230 |
|
|
\frac{1}{r} . |
231 |
|
|
\end{equation} |
232 |
gezelter |
3980 |
The electrostatic potential on $\bf a$ can then be expressed in terms |
233 |
|
|
of multipole operators, |
234 |
gezelter |
3906 |
\begin{equation} |
235 |
|
|
V_{\bf a}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\hat{M}_{\bf a} \frac{1}{r} |
236 |
|
|
\end{equation} |
237 |
|
|
where |
238 |
|
|
\begin{equation} |
239 |
|
|
\hat{M}_{\bf a} = C_{\bf a} - D_{{\bf a}\alpha} \frac{\partial}{\partial r_{\alpha}} |
240 |
|
|
+ Q_{{\bf a}\alpha\beta} |
241 |
|
|
\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
242 |
|
|
\end{equation} |
243 |
gezelter |
3980 |
Here, the point charge, dipole, and quadrupole for object $\bf a$ are |
244 |
|
|
given by $C_{\bf a}$, $D_{{\bf a}\alpha}$, and $Q_{{\bf |
245 |
|
|
a}\alpha\beta}$, respectively. These can be tied to distribution |
246 |
|
|
of charges |
247 |
gezelter |
3906 |
\begin{equation} |
248 |
|
|
C_{\bf a}=\sum_{k \, \text{in \bf a}} q_k , |
249 |
|
|
\end{equation} |
250 |
|
|
\begin{equation} |
251 |
|
|
D_{{\bf a}\alpha} = \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} , |
252 |
|
|
\end{equation} |
253 |
|
|
\begin{equation} |
254 |
|
|
Q_{{\bf a}\alpha\beta} = \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} r_{k\beta} . |
255 |
|
|
\end{equation} |
256 |
|
|
|
257 |
|
|
It is convenient to locate charges $q_j$ relative to the center of mass of $\bf b$. Then with $\bf{r}$ pointing from |
258 |
|
|
$\bf a$ to $\bf b$ ($\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $), the interaction energy is given by |
259 |
|
|
\begin{eqnarray} |
260 |
|
|
U_{\bf{ab}}(r) =&& \frac{1}{4\pi \epsilon_0} |
261 |
|
|
\sum_{k \, \text{in \bf a}} \sum_{j \, \text{in \bf b}} |
262 |
|
|
\frac{q_k q_j}{\vert \bf{r}_k - (\bf{r}+\bf{r}_j) \vert} \nonumber\\ |
263 |
|
|
=&& \frac{1}{4\pi \epsilon_0} |
264 |
|
|
\sum_{k \, \text{in \bf a}} \sum_{j \, \text{in \bf b}} |
265 |
|
|
\frac{q_k q_j}{\vert \bf{r}+ (\bf{r}_j-\bf{r}_k) \vert} \nonumber\\ |
266 |
|
|
=&&\frac{1}{4\pi \epsilon_0} \sum_{j \, \text{in \bf b}} q_j V_a(\bf{r}+\bf{r}_j) \nonumber\\ |
267 |
|
|
=&&\frac{1}{4\pi \epsilon_0} \hat{M}_a \sum_{j \, \text{in \bf b}} \frac {q_j}{\vert \bf{r}+\bf{r}_j \vert} . |
268 |
|
|
\end{eqnarray} |
269 |
|
|
The last expression can also be expanded as a Taylor series in $r$. Using a notation similar to before, we define |
270 |
|
|
\begin{equation} |
271 |
|
|
\hat{M}_{\bf b} = C_{\bf b} + D_{{\bf b}\alpha} \frac{\partial}{\partial r_{\alpha}} |
272 |
|
|
+ Q_{{\bf b}\alpha\beta} |
273 |
|
|
\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
274 |
|
|
\end{equation} |
275 |
|
|
and |
276 |
|
|
\begin{equation} |
277 |
|
|
U_{\bf{ab}}(r)=\frac{\hat{M}_{\bf a} \hat{M}_{\bf b}}{4\pi \epsilon_0} \frac{1}{r} \label{kernel}. |
278 |
|
|
\end{equation} |
279 |
|
|
Note the ease of separting out the respective energies of interaction of the charge, dipole, and quadrupole of $\bf a$ from those of $\bf b$. |
280 |
|
|
|
281 |
|
|
\subsection{Bare Coulomb versus smeared charge} |
282 |
|
|
|
283 |
|
|
With the four types of methods being considered here, it is desirable to list the approximations in as transparent a form |
284 |
|
|
as possible. First, one may use the bare Coulomb potential, with radial dependence $1/r$, |
285 |
|
|
as shown in Eq.~(\ref{kernel}). Alternatively, one may use |
286 |
|
|
a smeared charge distribution, then the``kernel'' $1/r$ of the expansion is replaced with a function: |
287 |
|
|
\begin{equation} |
288 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r} |
289 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
290 |
|
|
\end{equation} |
291 |
|
|
We develop equations using a function $f(r)$ to represent either $1/r$ or $B_0(r)$, dependent on the the type of approach being considered. |
292 |
|
|
Smith's convenient functions $B_l(r)$ are summarized in Appendix A. |
293 |
|
|
|
294 |
|
|
\subsection{Shifting the force, method 1} |
295 |
|
|
|
296 |
|
|
As discussed in the introduction, it is desirable to cutoff the electrostatic energy at a radius |
297 |
|
|
$r_c$. For consistency in approximation, we want the interaction energy as well as the force and |
298 |
|
|
torque to go to zero at $r=r_c$. |
299 |
|
|
To describe how this goal may be met using a radial approximation, we use two examples, charge-charge |
300 |
|
|
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain the idea. |
301 |
|
|
|
302 |
|
|
In the shifted-force approximation, the interaction energy $U_{\bf{ab}}(r_c)=0$ |
303 |
|
|
for two charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is written: |
304 |
|
|
\begin{equation} |
305 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)=\frac{1}{4\pi \epsilon_0} C_{\bf a} C_{\bf b} |
306 |
|
|
\left({ \frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
307 |
|
|
\right) . |
308 |
|
|
\end{equation} |
309 |
|
|
Two shifting terms appear in this equations because we want the force to |
310 |
|
|
also be shifted due to an image charge located at a distance $r_c$. |
311 |
|
|
Since one derivative of the interaction energy is needed for the force, we want a term |
312 |
|
|
linear in $(r-r_c)$ in the interaction energy, that is: |
313 |
|
|
\begin{equation} |
314 |
|
|
\frac{d\,}{dr} |
315 |
|
|
\left( {\frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
316 |
|
|
\right) = \left(- \frac{1}{r^2} + \frac{1}{r_c^2} |
317 |
|
|
\right) . |
318 |
|
|
\end{equation} |
319 |
|
|
This demonstrates the need of the third term in the brackets of the energy expression, but leads to the question, how does this idea generalize for higher-order multipole energies? |
320 |
|
|
|
321 |
|
|
In method 1, the procedure that we follow is based on the number of derivatives need for each energy, force, or torque. That is, |
322 |
|
|
a quadrupole-quadrupole interaction energy will have four derivatives, |
323 |
|
|
$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, |
324 |
|
|
and the force or torque will bring in yet another derivative. |
325 |
|
|
We thus want shifted energy expressions to include terms so that all energies, forces, and torques |
326 |
|
|
are zero at $r=r_c$. In each case, we will subtract off a function $f_n^{\text{shift}}(r)$ from the |
327 |
|
|
kernel $f(r)=1/r$. The index $n$ indicates the number of derivatives to be taken when |
328 |
|
|
deriving a given multipole energy. |
329 |
|
|
We choose a function with guaranteed smooth derivatives --- a truncated Taylor series of the function |
330 |
|
|
$f(r)$, e.g., |
331 |
|
|
% |
332 |
|
|
\begin{equation} |
333 |
|
|
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
334 |
|
|
\end{equation} |
335 |
|
|
% |
336 |
|
|
The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
337 |
|
|
Thus, for $f(r)=1/r$, we find |
338 |
|
|
% |
339 |
|
|
\begin{equation} |
340 |
|
|
f_1(r)=\frac{1}{r}- \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} - \frac{(r-r_c)^2}{r_c^3} . |
341 |
|
|
\end{equation} |
342 |
|
|
% |
343 |
|
|
Continuing with the example of a charge $\bf a$ interacting with a dipole $\bf b$, we write |
344 |
|
|
% |
345 |
|
|
\begin{equation} |
346 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
347 |
|
|
\frac{C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} \frac {\partial f_1(r) }{\partial r_\alpha} |
348 |
|
|
=\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
349 |
|
|
\frac {r_\alpha}{r} \frac {\partial f_1(r)}{\partial r} . |
350 |
|
|
\end{equation} |
351 |
|
|
% |
352 |
|
|
The force that dipole $\bf b$ puts on charge $\bf a$ is |
353 |
|
|
% |
354 |
|
|
\begin{equation} |
355 |
|
|
F_{C_{\bf a}D_{\bf b}\beta} =\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
356 |
|
|
\left[ \frac{\delta_{\alpha\beta}}{r} \frac {\partial}{\partial r} + |
357 |
|
|
\frac{r_\alpha r_\beta}{r^2} |
358 |
|
|
\left( -\frac{1}{r} \frac {\partial} {\partial r} |
359 |
|
|
+ \frac {\partial ^2} {\partial r^2} \right) \right] f_1(r) . |
360 |
|
|
\end{equation} |
361 |
|
|
% |
362 |
|
|
For $f(r)=1/r$, we find |
363 |
|
|
% |
364 |
|
|
\begin{equation} |
365 |
|
|
F_{C_{\bf a}D_{\bf b}\beta} = |
366 |
|
|
\frac{C_{\bf a} D_{{\bf b}\beta} }{4\pi \epsilon_0r} |
367 |
|
|
\left[ -\frac{1}{r^2}+\frac{1}{r_c^2}-\frac{2(r-r_c)}{r_c^3} \right] |
368 |
|
|
+\frac{C_{\bf a} D_{{\bf b}\alpha}r_\alpha r_\beta }{4\pi \epsilon_0} |
369 |
|
|
\left[ \frac{3}{r^5}-\frac{3}{r^3r_c^2} \right] . |
370 |
|
|
\end{equation} |
371 |
|
|
% |
372 |
|
|
This expansion shows the expected $1/r^3$ dependence of the force. |
373 |
|
|
|
374 |
|
|
In general, we write |
375 |
|
|
% |
376 |
|
|
\begin{equation} |
377 |
|
|
U=\frac{1}{4\pi \epsilon_0} (\text{prefactor}) (\text{derivatives}) f_n(r) |
378 |
|
|
\label{generic} |
379 |
|
|
\end{equation} |
380 |
|
|
% |
381 |
|
|
where $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole |
382 |
|
|
and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for quadrupole-quadrupole. |
383 |
|
|
An example is the case of quadrupole-quadrupole for which the $\text{prefactor}$ is |
384 |
|
|
$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$ and the derivatives are |
385 |
|
|
$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, with |
386 |
|
|
implied summation combining the space indices. |
387 |
|
|
|
388 |
|
|
To apply this method to the smeared-charge approach, |
389 |
|
|
we write $f(r)=\text{erfc}(\alpha r)/r$. By using one function $f(r)$ for both |
390 |
|
|
approaches, we simplify the tabulation of equations used. Because |
391 |
|
|
of the many derivatives that are taken, the algebra is tedious and are summarized |
392 |
|
|
in Appendices A and B. |
393 |
|
|
|
394 |
|
|
\subsection{Shifting the force, method 2} |
395 |
|
|
|
396 |
|
|
Note the method used in the previous subsection to shift a force is basically that of using |
397 |
|
|
a truncated Taylor Series in the radius $r$. An alternate method exists, best explained by |
398 |
|
|
writing one shifted formula for all interaction energies $U(r)$: |
399 |
|
|
\begin{equation} |
400 |
|
|
U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big \lvert _{r_c} . |
401 |
|
|
\end{equation} |
402 |
|
|
Note that this method uses only the linear term, $(r-r_c)$ in the Taylor series, no higher order terms |
403 |
|
|
$(r-r_c)^n$ appear. The primary difference between methods 1 and 2 originates |
404 |
|
|
with the stage in the derivation where the Taylor Series is applied; in method 1, it is applied to the |
405 |
|
|
kernel. In method 2, it is applied to individual interaction energies of the multipole expansion. |
406 |
|
|
Terms from this method thus have the general form: |
407 |
|
|
\begin{equation} |
408 |
|
|
U=\frac{1}{4\pi \epsilon_0}\sum (\text{angular factor}) (\text{radial factor}). |
409 |
|
|
\label{generic2} |
410 |
|
|
\end{equation} |
411 |
|
|
|
412 |
|
|
Results for both methods can be summarized using the form of Eq.~(\ref{generic2}) |
413 |
|
|
and are listed in Table I below. |
414 |
|
|
|
415 |
|
|
\subsection{\label{sec:level2}Body and space axes} |
416 |
|
|
|
417 |
|
|
Up to this point, all energies and forces have been written in terms of fixed space |
418 |
|
|
coordinates $x$, $y$, $z$. Interaction energies are computed from the generic formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which |
419 |
|
|
combine prefactors with radial functions. But because objects |
420 |
|
|
$\bf a$ and $\bf b$ both translate and rotate as part of a MD simulation, |
421 |
|
|
it is desirable to contract all $r$-dependent terms with dipole and quadrupole |
422 |
|
|
moments expressed in terms of their body axes. |
423 |
|
|
Since the interaction energy expressions will be used to derive both forces and torques, |
424 |
|
|
we follow here the development of Allen and Germano, which was itself based on an |
425 |
|
|
earlier paper by Price \em et al.\em |
426 |
|
|
|
427 |
|
|
Denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
428 |
|
|
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ referring to a convenient |
429 |
|
|
set of inertial body axes. (Note, these body axes are generally not the same as those for which the |
430 |
|
|
quadrupole moment is diagonal.) Then, |
431 |
|
|
% |
432 |
|
|
\begin{eqnarray} |
433 |
|
|
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
434 |
|
|
\hat{b}_m= b_{mx}\hat{x} + b_{my}\hat{y} + b_{mz}\hat{z} . |
435 |
|
|
\end{eqnarray} |
436 |
|
|
Allen and Germano define matrices $\hat{\mathbf {a}}$ |
437 |
|
|
and $\hat{\mathbf {b}}$ using these unit vectors: |
438 |
|
|
\begin{eqnarray} |
439 |
|
|
\hat{\mathbf {a}} = |
440 |
|
|
\begin{pmatrix} |
441 |
|
|
\hat{a}_1 \\ |
442 |
|
|
\hat{a}_2 \\ |
443 |
|
|
\hat{a}_3 |
444 |
|
|
\end{pmatrix} |
445 |
|
|
= |
446 |
|
|
\begin{pmatrix} |
447 |
|
|
a_{1x} \quad a_{1y} \quad a_{1z} \\ |
448 |
|
|
a_{2x} \quad a_{2y} \quad a_{2z} \\ |
449 |
|
|
a_{3x} \quad a_{3y} \quad a_{3z} |
450 |
|
|
\end{pmatrix}\\ |
451 |
|
|
\hat{\mathbf {b}} = |
452 |
|
|
\begin{pmatrix} |
453 |
|
|
\hat{b}_1 \\ |
454 |
|
|
\hat{b}_2 \\ |
455 |
|
|
\hat{b}_3 |
456 |
|
|
\end{pmatrix} |
457 |
|
|
= |
458 |
|
|
\begin{pmatrix} |
459 |
|
|
b_{1x}\quad b_{1y} \quad b_{1z} \\ |
460 |
|
|
b_{2x} \quad b_{2y} \quad b_{2z} \\ |
461 |
|
|
b_{3x} \quad b_{3y} \quad b_{3z} |
462 |
|
|
\end{pmatrix} . |
463 |
|
|
\end{eqnarray} |
464 |
|
|
% |
465 |
|
|
These matrices convert from space-fixed $(xyz)$ to object-fixed $(123)$ coordinates. |
466 |
|
|
All contractions of prefactors with derivatives of functions can be written in terms of these matrices. |
467 |
|
|
It proves to be equally convenient to just write any contraction in terms of unit vectors |
468 |
|
|
$\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. |
469 |
|
|
We have found it useful to write angular-dependent terms in three different fashions, |
470 |
|
|
illustrated by the following |
471 |
|
|
three examples from the interaction-energy expressions: |
472 |
|
|
% |
473 |
|
|
\begin{eqnarray} |
474 |
|
|
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
475 |
|
|
=D_{\bf {a}\alpha} D_{\bf {b}\alpha}= |
476 |
|
|
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} \\ |
477 |
|
|
r^2 \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right)= |
478 |
|
|
r_\alpha Q_{\bf b \alpha \beta} r_\beta = r^2 |
479 |
|
|
\sum_{mn}(\hat{r} \cdot \hat{b}_m) Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \\ |
480 |
|
|
r ( \mathbf{D}_{\mathbf{a}} \cdot |
481 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r})= |
482 |
|
|
D_{\bf {a}\alpha} Q_{\bf b \alpha \beta} r_\beta |
483 |
|
|
=r \sum_{lmn} D_{\mathbf{a}l} (\hat{a}_l \cdot \hat{b}_m ) Q_{\mathbf{b}mn} |
484 |
|
|
(\hat{b}_n \cdot \hat{r}) . |
485 |
|
|
\end{eqnarray} |
486 |
|
|
% |
487 |
|
|
[Dan, perhaps there are better examples to show here.] |
488 |
|
|
|
489 |
|
|
In each line, the first two terms are written using space coordinates. The first form is standard |
490 |
|
|
in the chemistry literature, and the second is ``physicist style'' using implied summation notation. The third |
491 |
|
|
form shows explicitly sums over body indices and the dot products now indicate contractions using space indices. |
492 |
|
|
We find the first form to be useful in writing equations prior to converting to computer code. The second form is helpful |
493 |
|
|
in derivations of the interaction energy expressions. The third one is specifically helpful when deriving forces and torques, as will |
494 |
|
|
be discussed below. |
495 |
|
|
|
496 |
gezelter |
3980 |
|
497 |
|
|
\section{The Self-Interaction} |
498 |
|
|
The Wolf summation~\cite{Wolf99} and the later damped shifted force |
499 |
|
|
(DSF) extension~\cite{Fennell06} included self-interactions that are |
500 |
|
|
handled separately from the pairwise interactions between sites. The |
501 |
|
|
self-term is normally calculated via a single loop over all sites in |
502 |
|
|
the system, and is relatively cheap to evaluate. The self-interaction |
503 |
|
|
has contributions from two sources: |
504 |
|
|
\begin{itemize} |
505 |
|
|
\item The neutralization procedure within the cutoff radius requires a |
506 |
|
|
contribution from a charge opposite in sign, but equal in magnitude, |
507 |
|
|
to the central charge, which has been spread out over the surface of |
508 |
|
|
the cutoff sphere. For a system of undamped charges, the total |
509 |
|
|
self-term is |
510 |
|
|
\begin{equation} |
511 |
|
|
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
512 |
|
|
\label{eq:selfTerm} |
513 |
|
|
\end{equation} |
514 |
|
|
Note that in this potential and in all electrostatic quantities that |
515 |
|
|
follow, the standard $4 \pi \epsilon_{0}$ has been omitted for |
516 |
|
|
clarity. |
517 |
|
|
\item Charge damping with the complementary error function is a |
518 |
|
|
partial analogy to the Ewald procedure which splits the interaction |
519 |
|
|
into real and reciprocal space sums. The real space sum is retained |
520 |
|
|
in the Wolf and DSF methods. The reciprocal space sum is first |
521 |
|
|
minimized by folding the largest contribution (the self-interaction) |
522 |
|
|
into the self-interaction from charge neutralization of the damped |
523 |
|
|
potential. The remainder of the reciprocal space portion is then |
524 |
|
|
discarded (as this contributes the largest computational cost and |
525 |
|
|
complexity to the Ewald sum). For a system containing only damped |
526 |
|
|
charges, the complete self-interaction can be written as |
527 |
|
|
\begin{equation} |
528 |
|
|
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
529 |
|
|
\frac{\alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
530 |
|
|
C_{\bf a}^{2}. |
531 |
|
|
\label{eq:dampSelfTerm} |
532 |
|
|
\end{equation} |
533 |
|
|
\end{itemize} |
534 |
|
|
|
535 |
|
|
The extension of DSF electrostatics to point multipoles requires |
536 |
|
|
treatment of {\it both} the self-neutralization and reciprocal |
537 |
|
|
contributions to the self-interaction for higher order multipoles. In |
538 |
|
|
this section we give formulae for these interactions up to quadrupolar |
539 |
|
|
order. |
540 |
|
|
|
541 |
|
|
The self-neutralization term is computed by taking the {\it |
542 |
|
|
non-shifted} kernel for each interaction, placing a multipole of |
543 |
|
|
equal magnitude (but opposite in polarization) on the surface of the |
544 |
|
|
cutoff sphere, and averaging over the surface of the cutoff sphere. |
545 |
|
|
Because the self term is carried out as a single sum over sites, the |
546 |
|
|
reciprocal-space portion is identical to half of the self-term |
547 |
|
|
obtained by Smith and Aguado and Madden for the application of the |
548 |
|
|
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
549 |
|
|
site which posesses a charge, dipole, and multipole, both types of |
550 |
|
|
contribution are given in table \ref{tab:tableSelf}. |
551 |
|
|
|
552 |
|
|
\begin{table*} |
553 |
|
|
\caption{\label{tab:tableSelf} Self-interaction contributions for |
554 |
|
|
site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
555 |
|
|
$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
556 |
|
|
\begin{ruledtabular} |
557 |
|
|
\begin{tabular}{lccc} |
558 |
|
|
Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
559 |
|
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
560 |
|
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
561 |
|
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
562 |
|
|
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
563 |
|
|
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
564 |
|
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
565 |
|
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
566 |
|
|
h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\ |
567 |
|
|
\end{tabular} |
568 |
|
|
\end{ruledtabular} |
569 |
|
|
\end{table*} |
570 |
|
|
|
571 |
|
|
For sites which simultaneously contain charges and quadrupoles, the |
572 |
|
|
self-interaction includes a cross-interaction between these two |
573 |
|
|
multipole orders. Symmetry prevents the charge-dipole and |
574 |
|
|
dipole-quadrupole interactions from contributing to the |
575 |
|
|
self-interaction. The functions that go into the self-neutralization |
576 |
|
|
terms, $f(r), g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
577 |
|
|
derivatives of the electrostatic kernel (either the undamped $1/r$ or |
578 |
|
|
the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that are |
579 |
|
|
evaluated at the cutoff distance. For undamped interactions, $f(r_c) |
580 |
|
|
= 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For damped interactions, |
581 |
|
|
$f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so on. Appendix XX |
582 |
|
|
contains recursion relations that allow rapid evaluation of these |
583 |
|
|
derivatives. |
584 |
|
|
|
585 |
gezelter |
3906 |
\section{Energies, forces, and torques} |
586 |
|
|
\subsection{Interaction energies} |
587 |
|
|
|
588 |
|
|
We now list multipole interaction energies for the four types of approximation. |
589 |
|
|
A ``generic'' set of radial functions is introduced so to be able to present the results in Table I. This set of |
590 |
|
|
equations is written in terms of space coordinates: |
591 |
|
|
|
592 |
|
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
593 |
|
|
% |
594 |
|
|
% |
595 |
|
|
% u ca cb |
596 |
|
|
% |
597 |
|
|
\begin{equation} |
598 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)= |
599 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) \label{uchch} |
600 |
|
|
\end{equation} |
601 |
|
|
% |
602 |
|
|
% u ca db |
603 |
|
|
% |
604 |
|
|
\begin{equation} |
605 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
606 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
607 |
|
|
\label{uchdip} |
608 |
|
|
\end{equation} |
609 |
|
|
% |
610 |
|
|
% u ca qb |
611 |
|
|
% |
612 |
|
|
\begin{equation} |
613 |
|
|
U_{C_{\bf a}Q_{\bf b}}(r)= |
614 |
|
|
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigl[ \text{Tr}Q_{\bf b} v_{21}(r) |
615 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
616 |
|
|
\label{uchquad} |
617 |
|
|
\end{equation} |
618 |
|
|
% |
619 |
|
|
% u da cb |
620 |
|
|
% |
621 |
|
|
\begin{equation} |
622 |
|
|
U_{D_{\bf a}C_{\bf b}}(r)= |
623 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
624 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
625 |
|
|
\end{equation} |
626 |
|
|
% |
627 |
|
|
% u da db |
628 |
|
|
% |
629 |
|
|
\begin{equation} |
630 |
|
|
U_{D_{\bf a}D_{\bf b}}(r)= |
631 |
|
|
-\frac{1}{4\pi \epsilon_0} \Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
632 |
|
|
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
633 |
|
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
634 |
|
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
635 |
|
|
v_{22}(r) \Bigr] |
636 |
|
|
\label{udipdip} |
637 |
|
|
\end{equation} |
638 |
|
|
% |
639 |
|
|
% u da qb |
640 |
|
|
% |
641 |
|
|
\begin{equation} |
642 |
|
|
\begin{split} |
643 |
|
|
% 1 |
644 |
|
|
U_{D_{\bf a}Q_{\bf b}}(r)&= |
645 |
|
|
-\frac{1}{4\pi \epsilon_0} \Bigl[ |
646 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
647 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
648 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
649 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r} ) \Bigr] v_{31}(r) \\ |
650 |
|
|
% 2 |
651 |
|
|
&-\frac{1}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
652 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
653 |
|
|
\label{udipquad} |
654 |
|
|
\end{split} |
655 |
|
|
\end{equation} |
656 |
|
|
% |
657 |
|
|
% u qa cb |
658 |
|
|
% |
659 |
|
|
\begin{equation} |
660 |
|
|
U_{Q_{\bf a}C_{\bf b}}(r)= |
661 |
|
|
\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
662 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
663 |
|
|
\label{uquadch} |
664 |
|
|
\end{equation} |
665 |
|
|
% |
666 |
|
|
% u qa db |
667 |
|
|
% |
668 |
|
|
\begin{equation} |
669 |
|
|
\begin{split} |
670 |
|
|
%1 |
671 |
|
|
U_{Q_{\bf a}D_{\bf b}}(r)&= |
672 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
673 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} |
674 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
675 |
|
|
+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
676 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
677 |
|
|
% 2 |
678 |
|
|
+\frac{1}{4\pi \epsilon_0} |
679 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
680 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
681 |
|
|
\label{uquaddip} |
682 |
|
|
\end{split} |
683 |
|
|
\end{equation} |
684 |
|
|
% |
685 |
|
|
% u qa qb |
686 |
|
|
% |
687 |
|
|
\begin{equation} |
688 |
|
|
\begin{split} |
689 |
|
|
%1 |
690 |
|
|
U_{Q_{\bf a}Q_{\bf b}}(r)&= |
691 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
692 |
|
|
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
693 |
|
|
+2 \text{Tr} \left( |
694 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
695 |
|
|
\\ |
696 |
|
|
% 2 |
697 |
|
|
&+ \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
698 |
|
|
\left( \hat{r} \cdot |
699 |
|
|
\mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) |
700 |
|
|
+\text{Tr}\mathbf{Q}_{\mathbf{b}} |
701 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} |
702 |
|
|
\cdot \hat{r} \right) +4 (\hat{r} \cdot |
703 |
|
|
\mathbf{Q}_{{\mathbf a}}\cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
704 |
|
|
\Bigr] v_{42}(r) |
705 |
|
|
\\ |
706 |
|
|
% 4 |
707 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
708 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) |
709 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
710 |
|
|
\label{uquadquad} |
711 |
|
|
\end{split} |
712 |
|
|
\end{equation} |
713 |
|
|
|
714 |
|
|
|
715 |
|
|
% |
716 |
|
|
% |
717 |
|
|
% TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
718 |
|
|
% |
719 |
|
|
|
720 |
|
|
\begin{table*} |
721 |
|
|
\caption{\label{tab:tableenergy}Radial functions used in the energy and torque equations. Functions |
722 |
|
|
used in this table are defined in Appendices B and C.} |
723 |
|
|
\begin{ruledtabular} |
724 |
|
|
\begin{tabular}{cccc} |
725 |
|
|
Generic&Coulomb&Method 1&Method 2 |
726 |
|
|
\\ \hline |
727 |
|
|
% |
728 |
|
|
% |
729 |
|
|
% |
730 |
|
|
%Ch-Ch& |
731 |
|
|
$v_{01}(r)$ & |
732 |
|
|
$\frac{1}{r}$ & |
733 |
|
|
$f_0(r)$ & |
734 |
|
|
$f(r)-f(r_c)-(r-r_c)g(r_c)$ |
735 |
|
|
\\ |
736 |
|
|
% |
737 |
|
|
% |
738 |
|
|
% |
739 |
|
|
%Ch-Di& |
740 |
|
|
$v_{11}(r)$ & |
741 |
|
|
$-\frac{1}{r^2}$ & |
742 |
|
|
$g_1(r)$ & |
743 |
|
|
$g(r)-g(r_c)-(r-r_c)h(r_c)$ \\ |
744 |
|
|
% |
745 |
|
|
% |
746 |
|
|
% |
747 |
|
|
%Ch-Qu/Di-Di& |
748 |
|
|
$v_{21}(r)$ & |
749 |
|
|
$-\frac{1}{r^3} $ & |
750 |
|
|
$\frac{g_2(r)}{r} $ & |
751 |
|
|
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c} -(r-r_c) |
752 |
|
|
\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right)$ \\ |
753 |
|
|
$v_{22}(r)$ & |
754 |
|
|
$\frac{3}{r^3} $ & |
755 |
|
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
756 |
|
|
$\left(-\frac{g(r)}{r}+h(r) \right) |
757 |
|
|
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right) $ \\ |
758 |
|
|
&&&$ -(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
759 |
|
|
\\ |
760 |
|
|
% |
761 |
|
|
% |
762 |
|
|
% |
763 |
|
|
%Di-Qu & |
764 |
|
|
$v_{31}(r)$ & |
765 |
|
|
$\frac{3}{r^4} $ & |
766 |
|
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
767 |
|
|
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
768 |
|
|
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
769 |
|
|
&&&$ -(r-r_c) \left(\frac{2g(r_c)}{r_c^3}-\frac{2h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ |
770 |
|
|
\\ |
771 |
|
|
% |
772 |
|
|
$v_{32}(r)$ & |
773 |
|
|
$-\frac{15}{r^4} $ & |
774 |
|
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
775 |
|
|
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
776 |
|
|
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
777 |
|
|
&&&$ -(r-r_c) \left( \frac{-6g(r_c)}{r_c^3}+\frac{6h(r_c)}{r_c^2}-\frac{3s(r_c)}{r_c}+t(r_c) \right)$ |
778 |
|
|
\\ |
779 |
|
|
% |
780 |
|
|
% |
781 |
|
|
% |
782 |
|
|
%Qu-Qu& |
783 |
|
|
$v_{41}(r)$ & |
784 |
|
|
$\frac{3}{r^5} $ & |
785 |
|
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
786 |
|
|
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
787 |
|
|
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
788 |
|
|
&&&$ -(r-r_c) \left( \frac{3g(r_c)}{r_c^4}-\frac{3h(r_c)}{r_c^3}+\frac{s(r_c)}{r_c^2} \right)$ |
789 |
|
|
\\ |
790 |
|
|
% 2 |
791 |
|
|
$v_{42}(r)$ & |
792 |
|
|
$- \frac{15}{r^5} $ & |
793 |
|
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
794 |
|
|
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
795 |
|
|
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
796 |
|
|
&&&$ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
797 |
|
|
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
798 |
|
|
\\ |
799 |
|
|
% 3 |
800 |
|
|
$v_{43}(r)$ & |
801 |
|
|
$ \frac{105}{r^5} $ & |
802 |
|
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
803 |
|
|
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
804 |
|
|
&&&$ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
805 |
|
|
&&&$ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
806 |
|
|
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ |
807 |
|
|
\end{tabular} |
808 |
|
|
\end{ruledtabular} |
809 |
|
|
\end{table*} |
810 |
|
|
% |
811 |
|
|
% |
812 |
|
|
% FORCE TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
813 |
|
|
% |
814 |
|
|
|
815 |
|
|
\begin{table} |
816 |
|
|
\caption{\label{tab:tableFORCE}Radial functions used in the force equations.} |
817 |
|
|
\begin{ruledtabular} |
818 |
|
|
\begin{tabular}{cc} |
819 |
|
|
Generic&Method 1 or Method 2 |
820 |
|
|
\\ \hline |
821 |
|
|
% |
822 |
|
|
% |
823 |
|
|
% |
824 |
|
|
$w_a(r)$& |
825 |
|
|
$\frac{d v_{01}}{dr}$ \\ |
826 |
|
|
% |
827 |
|
|
% |
828 |
|
|
$w_b(r)$ & |
829 |
|
|
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $ \\ |
830 |
|
|
% |
831 |
|
|
$w_c(r)$ & |
832 |
|
|
$\frac{v_{11}(r)}{r}$ \\ |
833 |
|
|
% |
834 |
|
|
% |
835 |
|
|
$w_d(r)$& |
836 |
|
|
$\frac{d v_{21}}{dr}$ \\ |
837 |
|
|
% |
838 |
|
|
$w_e(r)$ & |
839 |
|
|
$\frac{v_{22}(r)}{r}$ \\ |
840 |
|
|
% |
841 |
|
|
% |
842 |
|
|
$w_f(r)$& |
843 |
|
|
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$\\ |
844 |
|
|
% |
845 |
|
|
$w_g(r)$& |
846 |
|
|
$\frac{v_{31}(r)}{r}$\\ |
847 |
|
|
% |
848 |
|
|
$w_h(r)$ & |
849 |
|
|
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$\\ |
850 |
|
|
% 2 |
851 |
|
|
$w_i(r)$ & |
852 |
|
|
$\frac{v_{32}(r)}{r}$ \\ |
853 |
|
|
% |
854 |
|
|
$w_j(r)$ & |
855 |
|
|
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$ \\ |
856 |
|
|
% |
857 |
|
|
$w_k(r)$ & |
858 |
|
|
$\frac{d v_{41}}{dr} $ \\ |
859 |
|
|
% |
860 |
|
|
$w_l(r)$ & |
861 |
|
|
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ \\ |
862 |
|
|
% |
863 |
|
|
$w_m(r)$ & |
864 |
|
|
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$ \\ |
865 |
|
|
% |
866 |
|
|
$w_n(r)$ & |
867 |
|
|
$\frac{v_{42}(r)}{r}$ \\ |
868 |
|
|
% |
869 |
|
|
$w_o(r)$ & |
870 |
|
|
$\frac{v_{43}(r)}{r}$ \\ |
871 |
|
|
% |
872 |
|
|
|
873 |
|
|
\end{tabular} |
874 |
|
|
\end{ruledtabular} |
875 |
|
|
\end{table} |
876 |
|
|
% |
877 |
|
|
% |
878 |
|
|
% |
879 |
|
|
|
880 |
|
|
\subsection{Forces} |
881 |
|
|
|
882 |
|
|
The force $\mathbf{F}_{\bf a}$ on $\bf{a}$ due to $\bf{b}$ is the negative of |
883 |
|
|
the force $\mathbf{F}_{\bf b}$ on $\bf{b}$ due to $\bf{a}$. For a simple charge-charge |
884 |
|
|
interaction, these forces will point along the $\pm \hat{r}$ directions, where |
885 |
|
|
$\mathbf{r}=\mathbf{r}_b - \mathbf{r}_a $. Thus |
886 |
|
|
% |
887 |
|
|
\begin{equation} |
888 |
|
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
889 |
|
|
\quad \text{and} \quad F_{\bf b \alpha} |
890 |
|
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
891 |
|
|
\end{equation} |
892 |
|
|
% |
893 |
|
|
The concept of obtaining a force from an energy by taking a gradient is the same for |
894 |
|
|
higher-order multipole interactions, the trick is to make sure that all |
895 |
|
|
$r$-dependent derivatives are considered. |
896 |
|
|
As is pointed out by Allen and Germano, this is straightforward if the |
897 |
|
|
interaction energies are written recognizing explicit |
898 |
|
|
$\hat{r}$ and body axes ($\hat{a}_m$, $\hat{b}_n$) dependences: |
899 |
|
|
% |
900 |
|
|
\begin{equation} |
901 |
|
|
U(r,\{\hat{a}_m \cdot \hat{r} \}, |
902 |
|
|
\{\hat{b}_n\cdot \hat{r} \} |
903 |
|
|
\{\hat{a}_m \cdot \hat{b}_n \}) . |
904 |
|
|
\label{ugeneral} |
905 |
|
|
\end{equation} |
906 |
|
|
% |
907 |
|
|
Then, |
908 |
|
|
% |
909 |
|
|
\begin{equation} |
910 |
|
|
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
911 |
|
|
= \frac{\partial U}{\partial r} \hat{r} |
912 |
|
|
+ \sum_m \left[ |
913 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
914 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
915 |
|
|
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
916 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
917 |
|
|
\right] \label{forceequation}. |
918 |
|
|
\end{equation} |
919 |
|
|
% |
920 |
|
|
Note our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ is opposite |
921 |
|
|
that of Allen and Germano. In simplifying the algebra, we also use: |
922 |
|
|
% |
923 |
|
|
\begin{eqnarray} |
924 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
925 |
|
|
= \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
926 |
|
|
\right) \\ |
927 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
928 |
|
|
= \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
929 |
|
|
\right) . |
930 |
|
|
\end{eqnarray} |
931 |
|
|
% |
932 |
|
|
We list below the force equations written in terms of space coordinates. The |
933 |
|
|
radial functions used in the two methods are listed in Table II. |
934 |
|
|
% |
935 |
|
|
%SPACE COORDINATES FORCE EQUTIONS |
936 |
|
|
% |
937 |
|
|
% ************************************************************************** |
938 |
|
|
% f ca cb |
939 |
|
|
% |
940 |
|
|
\begin{equation} |
941 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
942 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
943 |
|
|
\end{equation} |
944 |
|
|
% |
945 |
|
|
% |
946 |
|
|
% |
947 |
|
|
\begin{equation} |
948 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
949 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} \Bigl[ |
950 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{b}} \right) |
951 |
|
|
w_b(r) \hat{r} |
952 |
|
|
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] |
953 |
|
|
\end{equation} |
954 |
|
|
% |
955 |
|
|
% |
956 |
|
|
% |
957 |
|
|
\begin{equation} |
958 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
959 |
|
|
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigr[ |
960 |
|
|
\text{Tr}\mathbf{Q}_{\bf b} w_d(r) \hat{r} |
961 |
|
|
+ 2 \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} w_e(r) |
962 |
|
|
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
963 |
|
|
\end{equation} |
964 |
|
|
% |
965 |
|
|
% |
966 |
|
|
% |
967 |
|
|
\begin{equation} |
968 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
969 |
|
|
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} \Bigl[ |
970 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
971 |
|
|
+ \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
972 |
|
|
\end{equation} |
973 |
|
|
% |
974 |
|
|
% |
975 |
|
|
% |
976 |
|
|
\begin{equation} |
977 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} = |
978 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
979 |
|
|
- \mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} w_d(r) \hat{r} |
980 |
|
|
+ \left( \mathbf{D}_{\mathbf {a}} |
981 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
982 |
|
|
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r) |
983 |
|
|
% 2 |
984 |
|
|
- \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
985 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} \Bigr] |
986 |
|
|
\end{equation} |
987 |
|
|
% |
988 |
|
|
% |
989 |
|
|
% |
990 |
|
|
\begin{equation} |
991 |
|
|
\begin{split} |
992 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
993 |
|
|
& - \frac{1}{4\pi \epsilon_0} \Bigl[ |
994 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} \mathbf{ D}_{\mathbf{a}} |
995 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \cdot |
996 |
|
|
\mathbf{Q}_{\mathbf{b}} \Bigr] w_g(r) |
997 |
|
|
- \frac{1}{4\pi \epsilon_0} \Bigl[ |
998 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
999 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) |
1000 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
1001 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1002 |
|
|
% 3 |
1003 |
|
|
& - \frac{1}{4\pi \epsilon_0} \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1004 |
|
|
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \Bigr] |
1005 |
|
|
w_i(r) |
1006 |
|
|
% 4 |
1007 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1008 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) |
1009 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} |
1010 |
|
|
\end{split} |
1011 |
|
|
\end{equation} |
1012 |
|
|
% |
1013 |
|
|
% |
1014 |
|
|
\begin{equation} |
1015 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
1016 |
|
|
\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
1017 |
|
|
\text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
1018 |
|
|
+ 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
1019 |
|
|
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
1020 |
|
|
\end{equation} |
1021 |
|
|
% |
1022 |
|
|
\begin{equation} |
1023 |
|
|
\begin{split} |
1024 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1025 |
|
|
&\frac{1}{4\pi \epsilon_0} \Bigl[ |
1026 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
1027 |
|
|
+2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
1028 |
|
|
% 2 |
1029 |
|
|
+ \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1030 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1031 |
|
|
+2 (\mathbf{D}_{\mathbf{b}} \cdot |
1032 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1033 |
|
|
% 3 |
1034 |
|
|
&+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
1035 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1036 |
|
|
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1037 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
1038 |
|
|
% 4 |
1039 |
|
|
+\frac{1}{4\pi \epsilon_0} |
1040 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1041 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
1042 |
|
|
\end{split} |
1043 |
|
|
\end{equation} |
1044 |
|
|
% |
1045 |
|
|
% |
1046 |
|
|
% |
1047 |
|
|
\begin{equation} |
1048 |
|
|
\begin{split} |
1049 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
1050 |
|
|
+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1051 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
1052 |
|
|
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
1053 |
|
|
% 2 |
1054 |
|
|
+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1055 |
|
|
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1056 |
|
|
+ 2\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) |
1057 |
|
|
% 3 |
1058 |
|
|
+4 (\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1059 |
|
|
+ 4(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}}) \Bigr] w_n(r) \\ |
1060 |
|
|
% 4 |
1061 |
|
|
+ \frac{1}{4\pi \epsilon_0} \Bigl[ |
1062 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1063 |
|
|
+ \text{Tr}\mathbf{Q}_{\mathbf{b}} |
1064 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1065 |
|
|
% 5 |
1066 |
|
|
+4 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot |
1067 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1068 |
|
|
% |
1069 |
|
|
+ \frac{1}{4\pi \epsilon_0} \Bigl[ |
1070 |
|
|
+ 2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1071 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1072 |
|
|
%6 |
1073 |
|
|
+2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1074 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_o(r) \\ |
1075 |
|
|
% 7 |
1076 |
|
|
+ \frac{1}{4\pi \epsilon_0} |
1077 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1078 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} |
1079 |
|
|
\end{split} |
1080 |
|
|
\end{equation} |
1081 |
|
|
% |
1082 |
|
|
% |
1083 |
|
|
% TORQUES SECTION ----------------------------------------------------------------------------------------- |
1084 |
|
|
% |
1085 |
|
|
\subsection{Torques} |
1086 |
|
|
|
1087 |
|
|
Following again Allen and Germano, when energies are written in the form |
1088 |
|
|
of Eq.~({\ref{ugeneral}), then torques can be expressed as: |
1089 |
|
|
% |
1090 |
|
|
\begin{eqnarray} |
1091 |
|
|
\mathbf{\tau}_{\bf a} = |
1092 |
|
|
\sum_m |
1093 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
1094 |
|
|
( \hat{r} \times \hat{a}_m ) |
1095 |
|
|
-\sum_{mn} |
1096 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
1097 |
|
|
(\hat{a}_m \times \hat{b}_n) \\ |
1098 |
|
|
% |
1099 |
|
|
\mathbf{\tau}_{\bf b} = |
1100 |
|
|
\sum_m |
1101 |
|
|
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
1102 |
|
|
( \hat{r} \times \hat{b}_m) |
1103 |
|
|
+\sum_{mn} |
1104 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
1105 |
|
|
(\hat{a}_m \times \hat{b}_n) . |
1106 |
|
|
\end{eqnarray} |
1107 |
|
|
% |
1108 |
|
|
% |
1109 |
|
|
Here we list the torque equations written in terms of space coordinates. |
1110 |
|
|
% |
1111 |
|
|
% |
1112 |
|
|
% |
1113 |
|
|
\begin{equation} |
1114 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
1115 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) |
1116 |
|
|
\end{equation} |
1117 |
|
|
% |
1118 |
|
|
% |
1119 |
|
|
% |
1120 |
|
|
\begin{equation} |
1121 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
1122 |
|
|
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
1123 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) |
1124 |
|
|
\end{equation} |
1125 |
|
|
% |
1126 |
|
|
% |
1127 |
|
|
% |
1128 |
|
|
\begin{equation} |
1129 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1130 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
1131 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
1132 |
|
|
\end{equation} |
1133 |
|
|
% |
1134 |
|
|
% |
1135 |
|
|
% |
1136 |
|
|
\begin{equation} |
1137 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} = |
1138 |
|
|
\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1139 |
|
|
% 2 |
1140 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1141 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf {a}} ) |
1142 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
1143 |
|
|
\end{equation} |
1144 |
|
|
% |
1145 |
|
|
% |
1146 |
|
|
% |
1147 |
|
|
\begin{equation} |
1148 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
1149 |
|
|
-\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1150 |
|
|
% 2 |
1151 |
|
|
+\frac{1}{4\pi \epsilon_0} |
1152 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
1153 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
1154 |
|
|
\end{equation} |
1155 |
|
|
% |
1156 |
|
|
% |
1157 |
|
|
% |
1158 |
|
|
\begin{equation} |
1159 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
1160 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1161 |
|
|
-\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1162 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1163 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \times |
1164 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1165 |
|
|
\Bigr] v_{31}(r) |
1166 |
|
|
% 3 |
1167 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1168 |
|
|
\ (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1169 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r) |
1170 |
|
|
\end{equation} |
1171 |
|
|
% |
1172 |
|
|
% |
1173 |
|
|
% |
1174 |
|
|
\begin{equation} |
1175 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} = |
1176 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1177 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \times |
1178 |
|
|
\hat{r} |
1179 |
|
|
-2 \mathbf{D}_{\mathbf{a}} \times |
1180 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1181 |
|
|
\Bigr] v_{31}(r) |
1182 |
|
|
% 2 |
1183 |
|
|
+\frac{2}{4\pi \epsilon_0} |
1184 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}}) |
1185 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r) |
1186 |
|
|
\end{equation} |
1187 |
|
|
% |
1188 |
|
|
% |
1189 |
|
|
% |
1190 |
|
|
\begin{equation} |
1191 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1192 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1193 |
|
|
-2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
1194 |
|
|
+2 \mathbf{D}_{\mathbf{b}} \times |
1195 |
|
|
(\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1196 |
|
|
\Bigr] v_{31}(r) |
1197 |
|
|
% 3 |
1198 |
|
|
- \frac{2}{4\pi \epsilon_0} |
1199 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
1200 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
1201 |
|
|
\end{equation} |
1202 |
|
|
% |
1203 |
|
|
% |
1204 |
|
|
% |
1205 |
|
|
\begin{equation} |
1206 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
1207 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1208 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1209 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1210 |
|
|
+2 \mathbf{D}_{\mathbf{b}} \times |
1211 |
|
|
( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
1212 |
|
|
% 2 |
1213 |
|
|
+\frac{1}{4\pi \epsilon_0} |
1214 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1215 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
1216 |
|
|
\end{equation} |
1217 |
|
|
% |
1218 |
|
|
% |
1219 |
|
|
% |
1220 |
|
|
\begin{equation} |
1221 |
|
|
\begin{split} |
1222 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
1223 |
|
|
&-\frac{4}{4\pi \epsilon_0} |
1224 |
|
|
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} |
1225 |
|
|
v_{41}(r) \\ |
1226 |
|
|
% 2 |
1227 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1228 |
|
|
\Bigl[-2\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1229 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times \hat{r} |
1230 |
|
|
+4 \hat{r} \times |
1231 |
|
|
( \mathbf{Q}_{{\mathbf a}} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1232 |
|
|
% 3 |
1233 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} )\times |
1234 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} ) \Bigr] v_{42}(r) \\ |
1235 |
|
|
% 4 |
1236 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
1237 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1238 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) |
1239 |
|
|
\end{split} |
1240 |
|
|
\end{equation} |
1241 |
|
|
% |
1242 |
|
|
% |
1243 |
|
|
% |
1244 |
|
|
\begin{equation} |
1245 |
|
|
\begin{split} |
1246 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} = |
1247 |
|
|
&\frac{4}{4\pi \epsilon_0} |
1248 |
|
|
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} v_{41}(r) \\ |
1249 |
|
|
% 2 |
1250 |
|
|
&+ \frac{1}{4\pi \epsilon_0} \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1251 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \times \hat{r} |
1252 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot |
1253 |
|
|
\mathbf{Q}_{{\mathbf b}} ) \times |
1254 |
|
|
\hat{r} |
1255 |
|
|
+4 ( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times |
1256 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1257 |
|
|
\Bigr] v_{42}(r) \\ |
1258 |
|
|
% 4 |
1259 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
1260 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1261 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) |
1262 |
|
|
\end{split} |
1263 |
|
|
\end{equation} |
1264 |
|
|
% |
1265 |
gezelter |
3980 |
|
1266 |
|
|
\section{Comparison to known multipolar energies} |
1267 |
|
|
|
1268 |
|
|
To understand how these new real-space multipole methods behave in |
1269 |
|
|
computer simulations, it is vital to test against established methods |
1270 |
|
|
for computing electrostatic interactions in periodic systems, and to |
1271 |
|
|
evaluate the size and sources of any errors that arise from the |
1272 |
|
|
real-space cutoffs. In this paper we test Taylor-shifted and |
1273 |
|
|
Gradient-shifted electrostatics against analytical methods for |
1274 |
|
|
computing the energies of ordered multipolar arrays. In the following |
1275 |
|
|
paper, we test the new methods against the multipolar Ewald sum for |
1276 |
|
|
computing the energies, forces and torques for a wide range of typical |
1277 |
|
|
condensed-phase (disordered) systems. |
1278 |
|
|
|
1279 |
|
|
Because long-range electrostatic effects can be significant in |
1280 |
|
|
crystalline materials, ordered multipolar arrays present one of the |
1281 |
|
|
biggest challenges for real-space cutoff methods. The dipolar |
1282 |
|
|
analogues to the Madelung constants were first worked out by Sauer, |
1283 |
|
|
who computed the energies of ordered dipole arrays of zero |
1284 |
|
|
magnetization and obtained a number of these constants.\cite{Sauer} |
1285 |
|
|
This theory was developed more completely by Luttinger and |
1286 |
|
|
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays and |
1287 |
|
|
other periodic structures. We have repeated the Luttinger \& Tisza |
1288 |
|
|
series summations to much higher order and obtained the following |
1289 |
|
|
energy constants (converged to one part in $10^9$): |
1290 |
|
|
\begin{table*} |
1291 |
|
|
\centering{ |
1292 |
|
|
\caption{Luttinger \& Tisza arrays and their associated |
1293 |
|
|
energy constants. Type "A" arrays have nearest neighbor strings of |
1294 |
|
|
antiparallel dipoles. Type "B" arrays have nearest neighbor |
1295 |
|
|
strings of antiparallel dipoles if the dipoles are contained in a |
1296 |
|
|
plane perpendicular to the dipole direction that passes through |
1297 |
|
|
the dipole.} |
1298 |
|
|
} |
1299 |
|
|
\label{tab:LT} |
1300 |
|
|
\begin{ruledtabular} |
1301 |
|
|
\begin{tabular}{cccc} |
1302 |
|
|
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
1303 |
|
|
A & SC & 001 & -2.676788684 \\ |
1304 |
|
|
A & BCC & 001 & 0 \\ |
1305 |
|
|
A & BCC & 111 & -1.770078733 \\ |
1306 |
|
|
A & FCC & 001 & 2.166932835 \\ |
1307 |
|
|
A & FCC & 011 & -1.083466417 \\ |
1308 |
|
|
|
1309 |
|
|
* & BCC & minimum & -1.985920929 \\ |
1310 |
|
|
|
1311 |
|
|
B & SC & 001 & -2.676788684 \\ |
1312 |
|
|
B & BCC & 001 & -1.338394342 \\ |
1313 |
|
|
B & BCC & 111 & -1.770078733 \\ |
1314 |
|
|
B & FCC & 001 & -1.083466417 \\ |
1315 |
|
|
B & FCC & 011 & -1.807573634 |
1316 |
|
|
\end{tabular} |
1317 |
|
|
\end{ruledtabular} |
1318 |
|
|
\end{table*} |
1319 |
|
|
|
1320 |
|
|
In addition to the A and B arrays, there is an additional minimum |
1321 |
|
|
energy structure for the BCC lattice that was found by Luttinger \& |
1322 |
|
|
Tisza. The total electrostatic energy for an array is given by: |
1323 |
|
|
\begin{equation} |
1324 |
|
|
E = C N^2 \mu^2 |
1325 |
|
|
\end{equation} |
1326 |
|
|
where $C$ is the energy constant given above, $N$ is the number of |
1327 |
|
|
dipoles per unit volume, and $\mu$ is the strength of the dipole. |
1328 |
|
|
|
1329 |
|
|
{\it Quadrupolar} analogues to the Madelung constants were first worked out by Nagai and Nakamura who |
1330 |
|
|
computed the energies of selected quadrupole arrays based on |
1331 |
|
|
extensions to the Luttinger and Tisza |
1332 |
|
|
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
1333 |
|
|
energy constants for the lowest energy configurations for linear |
1334 |
|
|
quadrupoles shown in table \ref{tab:NNQ} |
1335 |
|
|
|
1336 |
|
|
\begin{table*} |
1337 |
|
|
\centering{ |
1338 |
|
|
\caption{Nagai and Nakamura Quadurpolar arrays}} |
1339 |
|
|
\label{tab:NNQ} |
1340 |
|
|
\begin{ruledtabular} |
1341 |
|
|
\begin{tabular}{ccc} |
1342 |
|
|
Lattice & Quadrupole Direction & Energy constants \\ \hline |
1343 |
|
|
SC & 111 & -8.3 \\ |
1344 |
|
|
BCC & 011 & -21.7 \\ |
1345 |
|
|
FCC & 111 & -80.5 |
1346 |
|
|
\end{tabular} |
1347 |
|
|
\end{ruledtabular} |
1348 |
|
|
\end{table*} |
1349 |
|
|
|
1350 |
|
|
In analogy to the dipolar arrays, the total electrostatic energy for |
1351 |
|
|
the quadrupolar arrays is: |
1352 |
|
|
\begin{equation} |
1353 |
|
|
E = C \frac{3}{4} N^2 Q^2 |
1354 |
|
|
\end{equation} |
1355 |
|
|
where $Q$ is the quadrupole moment. |
1356 |
|
|
|
1357 |
|
|
|
1358 |
|
|
|
1359 |
|
|
|
1360 |
|
|
|
1361 |
|
|
|
1362 |
|
|
|
1363 |
|
|
|
1364 |
gezelter |
3906 |
\begin{acknowledgments} |
1365 |
gezelter |
3980 |
Support for this project was provided by the National Science |
1366 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
1367 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
1368 |
|
|
Dame. |
1369 |
gezelter |
3906 |
\end{acknowledgments} |
1370 |
|
|
|
1371 |
|
|
\appendix |
1372 |
|
|
|
1373 |
|
|
\section{Smith's $B_l(r)$ functions for smeared-charge distributions} |
1374 |
|
|
|
1375 |
|
|
The following summarizes Smith's $B_l(r)$ functions and |
1376 |
|
|
includes formulas given in his appendix. |
1377 |
|
|
|
1378 |
|
|
The first function $B_0(r)$ is defined by |
1379 |
|
|
% |
1380 |
|
|
\begin{equation} |
1381 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
1382 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
1383 |
|
|
\end{equation} |
1384 |
|
|
% |
1385 |
|
|
The first derivative of this function is |
1386 |
|
|
% |
1387 |
|
|
\begin{equation} |
1388 |
|
|
\frac{dB_0(r)}{dr}=-\frac{1}{r^2}\text{erfc}(\alpha r) |
1389 |
|
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
1390 |
|
|
\end{equation} |
1391 |
|
|
% |
1392 |
|
|
and can be rewritten in terms of a function $B_1(r)$: |
1393 |
|
|
% |
1394 |
|
|
\begin{equation} |
1395 |
|
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
1396 |
|
|
\end{equation} |
1397 |
|
|
% |
1398 |
|
|
In general, |
1399 |
|
|
\begin{equation} |
1400 |
|
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
1401 |
|
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
1402 |
|
|
\text{e}^{-{\alpha}^2r^2} |
1403 |
|
|
\right] . |
1404 |
|
|
\end{equation} |
1405 |
|
|
% |
1406 |
|
|
Using these formulas, we find |
1407 |
|
|
% |
1408 |
|
|
\begin{eqnarray} |
1409 |
|
|
\frac{dB_0}{dr}=-rB_1(r) \\ |
1410 |
|
|
\frac{d^2B_0}{dr^2}=-B_1(r) + r^2B_2(r) \\ |
1411 |
|
|
\frac{d^3B_0}{dr^3}=3rB_2(r) - r^3B_3(r) \\ |
1412 |
|
|
\frac{d^4B_0}{dr^4}=3B_2(r) - 6r^2B_3(r)+r^4B_4(r) \\ |
1413 |
|
|
\frac{d^5B_0}{dr^5}=-15rB_3(r) + 10r^3B_4(r) -r^5B_5(r) . |
1414 |
|
|
\end{eqnarray} |
1415 |
|
|
% |
1416 |
|
|
As noted by Smith, |
1417 |
|
|
% |
1418 |
|
|
\begin{equation} |
1419 |
|
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
1420 |
|
|
+\text{O}(r) . |
1421 |
|
|
\end{equation} |
1422 |
|
|
|
1423 |
|
|
\section{Method 1, the $r$-dependent factors} |
1424 |
|
|
|
1425 |
|
|
Using the shifted damped functions $f_n(r)$ defined by: |
1426 |
|
|
% |
1427 |
|
|
\begin{equation} |
1428 |
|
|
f_n(r)= B_0 \Big \lvert _r -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)} \Big \lvert _{r_c} , |
1429 |
|
|
\end{equation} |
1430 |
|
|
% |
1431 |
|
|
we first provide formulas for successive derivatives of this function. (If there is |
1432 |
|
|
no damping, then $B_0(r)$ is replaced by $1/r$, as discussed in Section~\ref{damped???}.) First, we find: |
1433 |
|
|
% |
1434 |
|
|
\begin{equation} |
1435 |
|
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
1436 |
|
|
\end{equation} |
1437 |
|
|
% |
1438 |
|
|
This formula clearly brings in derivatives of Smith's $B_0(r)$ function, motivating us to |
1439 |
|
|
define higher-order derivatives as follows: |
1440 |
|
|
% |
1441 |
|
|
\begin{eqnarray} |
1442 |
|
|
g_n(r)= \frac{d f_n}{d r} = |
1443 |
|
|
B_0^{(1)} \Big \lvert _r -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)} \Big \lvert _{r_c} \\ |
1444 |
|
|
h_n(r)= \frac{d^2f_n}{d r^2} = |
1445 |
|
|
B_0^{(2)} \Big \lvert _r -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)} \Big \lvert _{r_c} \\ |
1446 |
|
|
s_n(r)= \frac{d^3f_n}{d r^3} = |
1447 |
|
|
B_0^{(3)} \Big \lvert _r -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)} \Big \lvert _{r_c} \\ |
1448 |
|
|
t_n(r)= \frac{d^4f_n}{d r^4} = |
1449 |
|
|
B_0^{(4)} \Big \lvert _r -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)} \Big \lvert _{r_c} \\ |
1450 |
|
|
u_n(r)= \frac{d^5f_n}{d r^5} = |
1451 |
|
|
B_0^{(5)} \Big \lvert _r -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)} \Big \lvert _{r_c} . |
1452 |
|
|
\end{eqnarray} |
1453 |
|
|
% |
1454 |
|
|
We note that the last function needed (for quadrupole-quadrupole) is |
1455 |
|
|
% |
1456 |
|
|
\begin{equation} |
1457 |
|
|
u_4(r)=B_0^{(5)} \Big \lvert _r - B_0^{(5)} \Big \lvert _{r_c} . |
1458 |
|
|
\end{equation} |
1459 |
|
|
|
1460 |
|
|
The functions $f_n(r)$ to $u_n(r)$ are recursively computed and stored for values of $r$ |
1461 |
|
|
from $0$ to $r_c$. The functions needed are listed schematically below: |
1462 |
|
|
% |
1463 |
|
|
\begin{eqnarray} |
1464 |
|
|
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
1465 |
|
|
g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
1466 |
|
|
h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
1467 |
|
|
s_2 \quad s_3 \quad &s_4 \nonumber \\ |
1468 |
|
|
t_3 \quad &t_4 \nonumber \\ |
1469 |
|
|
&u_4 \nonumber . |
1470 |
|
|
\end{eqnarray} |
1471 |
|
|
|
1472 |
|
|
Using these functions, we find |
1473 |
|
|
% |
1474 |
|
|
\begin{equation} |
1475 |
|
|
\frac{\partial f_n}{\partial r_\alpha} =r_\alpha \frac {g_n}{r} |
1476 |
|
|
\end{equation} |
1477 |
|
|
% |
1478 |
|
|
\begin{equation} |
1479 |
|
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =\delta_{\alpha \beta}\frac {g_n}{r} |
1480 |
|
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) |
1481 |
|
|
\end{equation} |
1482 |
|
|
% |
1483 |
|
|
\begin{equation} |
1484 |
|
|
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} = |
1485 |
|
|
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
1486 |
|
|
\delta_{ \beta \gamma} r_\alpha \right) |
1487 |
|
|
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
1488 |
|
|
+ r_\alpha r_\beta r_\gamma |
1489 |
|
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) |
1490 |
|
|
\end{equation} |
1491 |
|
|
% |
1492 |
|
|
\begin{eqnarray} |
1493 |
|
|
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} = |
1494 |
|
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
1495 |
|
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
1496 |
|
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
1497 |
|
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
1498 |
|
|
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
1499 |
|
|
+ \text{5 perm} |
1500 |
|
|
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
1501 |
|
|
\right) \nonumber \\ |
1502 |
|
|
+ r_\alpha r_\beta r_\gamma r_\delta |
1503 |
|
|
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
1504 |
|
|
+ \frac{t_n}{r^4} \right) |
1505 |
|
|
\end{eqnarray} |
1506 |
|
|
% |
1507 |
|
|
\begin{eqnarray} |
1508 |
|
|
\frac{\partial^5 f_n} |
1509 |
|
|
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} = |
1510 |
|
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
1511 |
|
|
+ \text{14 perm} \right) |
1512 |
|
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
1513 |
|
|
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
1514 |
|
|
+ \text{9 perm} |
1515 |
|
|
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
1516 |
|
|
\right) \nonumber \\ |
1517 |
|
|
+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
1518 |
|
|
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
1519 |
|
|
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) |
1520 |
|
|
\end{eqnarray} |
1521 |
|
|
% |
1522 |
|
|
% |
1523 |
|
|
% |
1524 |
|
|
\section{Method 2, the $r$-dependent factors} |
1525 |
|
|
|
1526 |
|
|
In method 2, the kernel is not expanded, rather the individual terms in the multipole interaction energies, |
1527 |
|
|
see Eq. (20?). For a smeared-charge distribution, this still brings into the algebra multiple derivatives |
1528 |
|
|
of the kernel $B_0(r)$. To denote these terms, we generalize the notation of the previous appendix. |
1529 |
|
|
For $f(r)=1/r$ (bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
1530 |
|
|
% |
1531 |
|
|
\begin{eqnarray} |
1532 |
|
|
g(r)= \frac{df}{d r}\\ |
1533 |
|
|
h(r)= \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
1534 |
|
|
s(r)= \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
1535 |
|
|
t(r)= \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
1536 |
|
|
u(r)= \frac{dt}{d r} =\frac{d^5f}{d r^5} . |
1537 |
|
|
\end{eqnarray} |
1538 |
|
|
% |
1539 |
|
|
For $f(r)=1/r$, Table I lists these derivatives under the column ``Bare Coulomb.'' Checks of algebra can be made by using limiting forms |
1540 |
|
|
of equations, e.g., the leading term in the function $g_n(r)$ has $r$ dependence given by $g(r)$. Equations (B9) to B(13) |
1541 |
|
|
are correct for method 2 if one just eliminates the subscript $n$. |
1542 |
|
|
|
1543 |
|
|
\section{Extra Material} |
1544 |
|
|
% |
1545 |
|
|
% |
1546 |
|
|
%Energy in body coordinate form --------------------------------------------------------------- |
1547 |
|
|
% |
1548 |
|
|
Here are the interaction energies written in terms of the body coordinates: |
1549 |
|
|
|
1550 |
|
|
% |
1551 |
|
|
% u ca cb |
1552 |
|
|
% |
1553 |
|
|
\begin{equation} |
1554 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)= |
1555 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) |
1556 |
|
|
\end{equation} |
1557 |
|
|
% |
1558 |
|
|
% u ca db |
1559 |
|
|
% |
1560 |
|
|
\begin{equation} |
1561 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
1562 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} |
1563 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
1564 |
|
|
\end{equation} |
1565 |
|
|
% |
1566 |
|
|
% u ca qb |
1567 |
|
|
% |
1568 |
|
|
\begin{equation} |
1569 |
|
|
U_{C_{\bf a}Q_{\bf b}}(r)= |
1570 |
|
|
\frac{C_{\bf a }\text{Tr}Q_{\bf b}}{4\pi \epsilon_0} |
1571 |
|
|
v_{21}(r) \nonumber \\ |
1572 |
|
|
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
1573 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) |
1574 |
|
|
v_{22}(r) |
1575 |
|
|
\end{equation} |
1576 |
|
|
% |
1577 |
|
|
% u da cb |
1578 |
|
|
% |
1579 |
|
|
\begin{equation} |
1580 |
|
|
U_{D_{\bf a}C_{\bf b}}(r)= |
1581 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
1582 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
1583 |
|
|
\end{equation} |
1584 |
|
|
% |
1585 |
|
|
% u da db |
1586 |
|
|
% |
1587 |
|
|
\begin{equation} |
1588 |
|
|
\begin{split} |
1589 |
|
|
% 1 |
1590 |
|
|
U_{D_{\bf a}D_{\bf b}}(r)&= |
1591 |
|
|
-\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1592 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1593 |
|
|
D_{\mathbf{b}n} v_{21}(r) \\ |
1594 |
|
|
% 2 |
1595 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1596 |
|
|
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1597 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} |
1598 |
|
|
v_{22}(r) |
1599 |
|
|
\end{split} |
1600 |
|
|
\end{equation} |
1601 |
|
|
% |
1602 |
|
|
% u da qb |
1603 |
|
|
% |
1604 |
|
|
\begin{equation} |
1605 |
|
|
\begin{split} |
1606 |
|
|
% 1 |
1607 |
|
|
U_{D_{\bf a}Q_{\bf b}}(r)&= |
1608 |
|
|
-\frac{1}{4\pi \epsilon_0} \left( |
1609 |
|
|
\text{Tr}Q_{\mathbf{b}} |
1610 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
1611 |
|
|
+2\sum_{lmn}D_{\mathbf{a}l} |
1612 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1613 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1614 |
|
|
\right) v_{31}(r) \\ |
1615 |
|
|
% 2 |
1616 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1617 |
|
|
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1618 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1619 |
|
|
Q_{{\mathbf b}mn} |
1620 |
|
|
(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
1621 |
|
|
\end{split} |
1622 |
|
|
\end{equation} |
1623 |
|
|
% |
1624 |
|
|
% u qa cb |
1625 |
|
|
% |
1626 |
|
|
\begin{equation} |
1627 |
|
|
U_{Q_{\bf a}C_{\bf b}}(r)= |
1628 |
|
|
\frac{C_{\bf b }\text{Tr}Q_{\bf a}}{4\pi \epsilon_0} v_{21}(r) |
1629 |
|
|
+\frac{C_{\bf b}}{4\pi \epsilon_0} |
1630 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r) |
1631 |
|
|
\end{equation} |
1632 |
|
|
% |
1633 |
|
|
% u qa db |
1634 |
|
|
% |
1635 |
|
|
\begin{equation} |
1636 |
|
|
\begin{split} |
1637 |
|
|
%1 |
1638 |
|
|
U_{Q_{\bf a}D_{\bf b}}(r)&= |
1639 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1640 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1641 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
1642 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
1643 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1644 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
1645 |
|
|
\right) v_{31}(r) \\ |
1646 |
|
|
% 2 |
1647 |
|
|
&+\frac{1}{4\pi \epsilon_0} |
1648 |
|
|
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1649 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1650 |
|
|
Q_{{\mathbf a}mn} |
1651 |
|
|
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
1652 |
|
|
\end{split} |
1653 |
|
|
\end{equation} |
1654 |
|
|
% |
1655 |
|
|
% u qa qb |
1656 |
|
|
% |
1657 |
|
|
\begin{equation} |
1658 |
|
|
\begin{split} |
1659 |
|
|
%1 |
1660 |
|
|
U_{Q_{\bf a}Q_{\bf b}}(r)&= |
1661 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1662 |
|
|
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
1663 |
|
|
+2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1664 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1665 |
|
|
(\hat{a}_m \cdot \hat{b}_n) \Bigr] |
1666 |
|
|
v_{41}(r) \\ |
1667 |
|
|
% 2 |
1668 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1669 |
|
|
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
1670 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
1671 |
|
|
Q_{{\mathbf b}lm} |
1672 |
|
|
(\hat{b}_m \cdot \hat{r}) |
1673 |
|
|
+\text{Tr}Q_{\mathbf{b}} |
1674 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
1675 |
|
|
Q_{{\mathbf a}lm} |
1676 |
|
|
(\hat{a}_m \cdot \hat{r}) \\ |
1677 |
|
|
% 3 |
1678 |
|
|
&+4 \sum_{lmnp} |
1679 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
1680 |
|
|
Q_{{\mathbf a}lm} |
1681 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1682 |
|
|
Q_{{\mathbf b}np} |
1683 |
|
|
(\hat{b}_p \cdot \hat{r}) |
1684 |
|
|
\Bigr] v_{42}(r) \\ |
1685 |
|
|
% 4 |
1686 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1687 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
1688 |
|
|
Q_{{\mathbf a}lm} |
1689 |
|
|
(\hat{a}_m \cdot \hat{r}) |
1690 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
1691 |
|
|
Q_{{\mathbf b}np} |
1692 |
|
|
(\hat{b}_p \cdot \hat{r}) v_{43}(r). |
1693 |
|
|
\end{split} |
1694 |
|
|
\end{equation} |
1695 |
|
|
% |
1696 |
|
|
|
1697 |
|
|
|
1698 |
|
|
% BODY coordinates force equations -------------------------------------------- |
1699 |
|
|
% |
1700 |
|
|
% |
1701 |
|
|
Here are the force equations written in terms of body coordinates. |
1702 |
|
|
% |
1703 |
|
|
% f ca cb |
1704 |
|
|
% |
1705 |
|
|
\begin{equation} |
1706 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
1707 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
1708 |
|
|
\end{equation} |
1709 |
|
|
% |
1710 |
|
|
% f ca db |
1711 |
|
|
% |
1712 |
|
|
\begin{equation} |
1713 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
1714 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} |
1715 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r} |
1716 |
|
|
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
1717 |
|
|
\sum_n D_{\mathbf{b}n} \hat{b}_n w_c(r) |
1718 |
|
|
\end{equation} |
1719 |
|
|
% |
1720 |
|
|
% f ca qb |
1721 |
|
|
% |
1722 |
|
|
\begin{equation} |
1723 |
|
|
\begin{split} |
1724 |
|
|
% 1 |
1725 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
1726 |
|
|
\frac{1}{4\pi \epsilon_0} |
1727 |
|
|
C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r} |
1728 |
|
|
+ 2C_{\bf a } \sum_l \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot \hat{r}) w_e(r) \\ |
1729 |
|
|
% 2 |
1730 |
|
|
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
1731 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} |
1732 |
|
|
\end{split} |
1733 |
|
|
\end{equation} |
1734 |
|
|
% |
1735 |
|
|
% f da cb |
1736 |
|
|
% |
1737 |
|
|
\begin{equation} |
1738 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1739 |
|
|
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
1740 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r) \hat{r} |
1741 |
|
|
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
1742 |
|
|
\sum_n D_{\mathbf{a}n} \hat{a}_n w_c(r) |
1743 |
|
|
\end{equation} |
1744 |
|
|
% |
1745 |
|
|
% f da db |
1746 |
|
|
% |
1747 |
|
|
\begin{equation} |
1748 |
|
|
\begin{split} |
1749 |
|
|
% 1 |
1750 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
1751 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1752 |
|
|
\sum_{mn} D_{\mathbf {a}m} |
1753 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1754 |
|
|
D_{\mathbf{b}n} w_d(r) \hat{r} |
1755 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1756 |
|
|
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1757 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r} \\ |
1758 |
|
|
% 2 |
1759 |
|
|
& \quad + \frac{1}{4\pi \epsilon_0} |
1760 |
|
|
\Bigl[ \sum_m D_{\mathbf {a}m} |
1761 |
|
|
\hat{a}_m \sum_n D_{\mathbf{b}n} |
1762 |
|
|
(\hat{b}_n \cdot \hat{r}) |
1763 |
|
|
+ \sum_m D_{\mathbf {b}m} |
1764 |
|
|
\hat{b}_m \sum_n D_{\mathbf{a}n} |
1765 |
|
|
(\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r) \\ |
1766 |
|
|
\end{split} |
1767 |
|
|
\end{equation} |
1768 |
|
|
% |
1769 |
|
|
% f da qb |
1770 |
|
|
% |
1771 |
|
|
\begin{equation} |
1772 |
|
|
\begin{split} |
1773 |
|
|
% 1 |
1774 |
|
|
&\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
1775 |
|
|
- \frac{1}{4\pi \epsilon_0} \Bigl[ |
1776 |
|
|
\text{Tr}Q_{\mathbf{b}} |
1777 |
|
|
\sum_l D_{\mathbf{a}l} \hat{a}_l |
1778 |
|
|
+2\sum_{lmn} D_{\mathbf{a}l} |
1779 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1780 |
|
|
Q_{\mathbf{b}mn} \hat{b}_n \Bigr] w_g(r) \\ |
1781 |
|
|
% 3 |
1782 |
|
|
& - \frac{1}{4\pi \epsilon_0} \Bigl[ |
1783 |
|
|
\text{Tr}Q_{\mathbf{b}} |
1784 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
1785 |
|
|
+2\sum_{lmn}D_{\mathbf{a}l} |
1786 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1787 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1788 |
|
|
% 4 |
1789 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1790 |
|
|
\Bigl[\sum_l D_{\mathbf{a}l} \hat{a}_l |
1791 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1792 |
|
|
Q_{{\mathbf b}mn} |
1793 |
|
|
(\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l) |
1794 |
|
|
D_{\mathbf{a}l} |
1795 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1796 |
|
|
Q_{{\mathbf b}mn} \hat{b}_n \Bigr] w_i(r)\\ |
1797 |
|
|
% 6 |
1798 |
|
|
& -\frac{1}{4\pi \epsilon_0} |
1799 |
|
|
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1800 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1801 |
|
|
Q_{{\mathbf b}mn} |
1802 |
|
|
(\hat{b}_n \cdot \hat{r}) w_j(r) \hat{r} |
1803 |
|
|
\end{split} |
1804 |
|
|
\end{equation} |
1805 |
|
|
% |
1806 |
|
|
% force qa cb |
1807 |
|
|
% |
1808 |
|
|
\begin{equation} |
1809 |
|
|
\begin{split} |
1810 |
|
|
% 1 |
1811 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} &= |
1812 |
|
|
\frac{1}{4\pi \epsilon_0} |
1813 |
|
|
C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r) |
1814 |
|
|
+ \frac{2C_{\bf b }}{4\pi \epsilon_0} \sum_l \hat{a}_l Q_{{\mathbf a}ln} (\hat{a}_n \cdot \hat{r}) w_e(r) \\ |
1815 |
|
|
% 2 |
1816 |
|
|
& +\frac{C_{\bf b}}{4\pi \epsilon_0} |
1817 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} |
1818 |
|
|
\end{split} |
1819 |
|
|
\end{equation} |
1820 |
|
|
% |
1821 |
|
|
% f qa db |
1822 |
|
|
% |
1823 |
|
|
\begin{equation} |
1824 |
|
|
\begin{split} |
1825 |
|
|
% 1 |
1826 |
|
|
&\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1827 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1828 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1829 |
|
|
\sum_l D_{\mathbf{b}l} \hat{b}_l |
1830 |
|
|
+2\sum_{lmn} D_{\mathbf{b}l} |
1831 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1832 |
|
|
Q_{\mathbf{a}mn} \hat{a}_n \Bigr] |
1833 |
|
|
w_g(r)\\ |
1834 |
|
|
% 3 |
1835 |
|
|
& + \frac{1}{4\pi \epsilon_0} \Bigl[ |
1836 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1837 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
1838 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
1839 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1840 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1841 |
|
|
% 4 |
1842 |
|
|
& + \frac{1}{4\pi \epsilon_0} \Bigl[ \sum_l D_{\mathbf{b}l} \hat{b}_l |
1843 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1844 |
|
|
Q_{{\mathbf a}mn} |
1845 |
|
|
(\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l) |
1846 |
|
|
D_{\mathbf{b}l} |
1847 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1848 |
|
|
Q_{{\mathbf a}mn} \hat{a}_n \Bigr] w_i(r) \\ |
1849 |
|
|
% 6 |
1850 |
|
|
& +\frac{1}{4\pi \epsilon_0} |
1851 |
|
|
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1852 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1853 |
|
|
Q_{{\mathbf a}mn} |
1854 |
|
|
(\hat{a}_n \cdot \hat{r}) w_j(r) \hat{r} |
1855 |
|
|
\end{split} |
1856 |
|
|
\end{equation} |
1857 |
|
|
% |
1858 |
|
|
% f qa qb |
1859 |
|
|
% |
1860 |
|
|
\begin{equation} |
1861 |
|
|
\begin{split} |
1862 |
|
|
&\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
1863 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1864 |
|
|
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
1865 |
|
|
+ 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1866 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1867 |
|
|
(\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r}\\ |
1868 |
|
|
&+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1869 |
|
|
2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
1870 |
|
|
+ 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} \hat{b}_m \\ |
1871 |
|
|
&+ 4\sum_{lmnp} \hat{a}_l Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) |
1872 |
|
|
+ 4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_p |
1873 |
|
|
\Bigr] w_n(r) \\ |
1874 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1875 |
|
|
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
1876 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) |
1877 |
|
|
+ \text{Tr}Q_{\mathbf{b}} |
1878 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) \\ |
1879 |
|
|
&+4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) |
1880 |
|
|
Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1881 |
|
|
% |
1882 |
|
|
&+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1883 |
|
|
2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
1884 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot \hat{r}) \\ |
1885 |
|
|
&+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
1886 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr] w_o(r) \hat{r} \\ |
1887 |
|
|
& + \frac{1}{4\pi \epsilon_0} |
1888 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
1889 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} |
1890 |
|
|
\end{split} |
1891 |
|
|
\end{equation} |
1892 |
|
|
% |
1893 |
|
|
Here we list the form of the non-zero damped shifted multipole torques showing |
1894 |
|
|
explicitly dependences on body axes: |
1895 |
|
|
% |
1896 |
|
|
% t ca db |
1897 |
|
|
% |
1898 |
|
|
\begin{equation} |
1899 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
1900 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} |
1901 |
|
|
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
1902 |
|
|
\end{equation} |
1903 |
|
|
% |
1904 |
|
|
% t ca qb |
1905 |
|
|
% |
1906 |
|
|
\begin{equation} |
1907 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
1908 |
|
|
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
1909 |
|
|
\sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m \cdot \hat{r}) v_{22}(r) |
1910 |
|
|
\end{equation} |
1911 |
|
|
% |
1912 |
|
|
% t da cb |
1913 |
|
|
% |
1914 |
|
|
\begin{equation} |
1915 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1916 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
1917 |
|
|
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
1918 |
|
|
\end{equation}% |
1919 |
|
|
% |
1920 |
|
|
% |
1921 |
|
|
% ta da db |
1922 |
|
|
% |
1923 |
|
|
\begin{equation} |
1924 |
|
|
\begin{split} |
1925 |
|
|
% 1 |
1926 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
1927 |
|
|
\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1928 |
|
|
(\hat{a}_m \times \hat{b}_n) |
1929 |
|
|
D_{\mathbf{b}n} v_{21}(r) \\ |
1930 |
|
|
% 2 |
1931 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1932 |
|
|
\sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} |
1933 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
1934 |
|
|
\end{split} |
1935 |
|
|
\end{equation} |
1936 |
|
|
% |
1937 |
|
|
% tb da db |
1938 |
|
|
% |
1939 |
|
|
\begin{equation} |
1940 |
|
|
\begin{split} |
1941 |
|
|
% 1 |
1942 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} &= |
1943 |
|
|
-\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1944 |
|
|
(\hat{a}_m \times \hat{b}_n) |
1945 |
|
|
D_{\mathbf{b}n} v_{21}(r) \\ |
1946 |
|
|
% 2 |
1947 |
|
|
&+\frac{1}{4\pi \epsilon_0} |
1948 |
|
|
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1949 |
|
|
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
1950 |
|
|
\end{split} |
1951 |
|
|
\end{equation} |
1952 |
|
|
% |
1953 |
|
|
% ta da qb |
1954 |
|
|
% |
1955 |
|
|
\begin{equation} |
1956 |
|
|
\begin{split} |
1957 |
|
|
% 1 |
1958 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} &= |
1959 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1960 |
|
|
-\text{Tr}Q_{\mathbf{b}} |
1961 |
|
|
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} |
1962 |
|
|
+2\sum_{lmn}D_{\mathbf{a}l} |
1963 |
|
|
(\hat{a}_l \times \hat{b}_m) |
1964 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1965 |
|
|
\right) v_{31}(r)\\ |
1966 |
|
|
% 2 |
1967 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1968 |
|
|
\sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l} |
1969 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1970 |
|
|
Q_{{\mathbf b}mn} |
1971 |
|
|
(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
1972 |
|
|
\end{split} |
1973 |
|
|
\end{equation} |
1974 |
|
|
% |
1975 |
|
|
% tb da qb |
1976 |
|
|
% |
1977 |
|
|
\begin{equation} |
1978 |
|
|
\begin{split} |
1979 |
|
|
% 1 |
1980 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} &= |
1981 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1982 |
|
|
-2\sum_{lmn}D_{\mathbf{a}l} |
1983 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1984 |
|
|
Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n) |
1985 |
|
|
-2\sum_{lmn}D_{\mathbf{a}l} |
1986 |
|
|
(\hat{a}_l \times \hat{b}_m) |
1987 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1988 |
|
|
\right) v_{31}(r) \\ |
1989 |
|
|
% 2 |
1990 |
|
|
&-\frac{2}{4\pi \epsilon_0} |
1991 |
|
|
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1992 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1993 |
|
|
Q_{{\mathbf b}mn} |
1994 |
|
|
(\hat{r}\times \hat{b}_n) v_{32}(r) |
1995 |
|
|
\end{split} |
1996 |
|
|
\end{equation} |
1997 |
|
|
% |
1998 |
|
|
% ta qa cb |
1999 |
|
|
% |
2000 |
|
|
\begin{equation} |
2001 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
2002 |
|
|
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
2003 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r) |
2004 |
|
|
\end{equation} |
2005 |
|
|
% |
2006 |
|
|
% ta qa db |
2007 |
|
|
% |
2008 |
|
|
\begin{equation} |
2009 |
|
|
\begin{split} |
2010 |
|
|
% 1 |
2011 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} &= |
2012 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
2013 |
|
|
2\sum_{lmn}D_{\mathbf{b}l} |
2014 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
2015 |
|
|
Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n) |
2016 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
2017 |
|
|
(\hat{a}_l \times \hat{b}_m) |
2018 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
2019 |
|
|
\right) v_{31}(r) \\ |
2020 |
|
|
% 2 |
2021 |
|
|
&+\frac{2}{4\pi \epsilon_0} |
2022 |
|
|
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
2023 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
2024 |
|
|
Q_{{\mathbf a}mn} |
2025 |
|
|
(\hat{r}\times \hat{a}_n) v_{32}(r) |
2026 |
|
|
\end{split} |
2027 |
|
|
\end{equation} |
2028 |
|
|
% |
2029 |
|
|
% tb qa db |
2030 |
|
|
% |
2031 |
|
|
\begin{equation} |
2032 |
|
|
\begin{split} |
2033 |
|
|
% 1 |
2034 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} &= |
2035 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
2036 |
|
|
\text{Tr}Q_{\mathbf{a}} |
2037 |
|
|
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} |
2038 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
2039 |
|
|
(\hat{a}_l \times \hat{b}_m) |
2040 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
2041 |
|
|
\right) v_{31}(r)\\ |
2042 |
|
|
% 2 |
2043 |
|
|
&\frac{1}{4\pi \epsilon_0} |
2044 |
|
|
\sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l} |
2045 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
2046 |
|
|
Q_{{\mathbf a}mn} |
2047 |
|
|
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
2048 |
|
|
\end{split} |
2049 |
|
|
\end{equation} |
2050 |
|
|
% |
2051 |
|
|
% ta qa qb |
2052 |
|
|
% |
2053 |
|
|
\begin{equation} |
2054 |
|
|
\begin{split} |
2055 |
|
|
% 1 |
2056 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} &= |
2057 |
|
|
-\frac{4}{4\pi \epsilon_0} |
2058 |
|
|
\sum_{lmnp} (\hat{a}_l \times \hat{b}_p) |
2059 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
2060 |
|
|
(\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \\ |
2061 |
|
|
% 2 |
2062 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
2063 |
|
|
\Bigl[ |
2064 |
|
|
2\text{Tr}Q_{\mathbf{b}} |
2065 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
2066 |
|
|
Q_{{\mathbf a}lm} |
2067 |
|
|
(\hat{r} \times \hat{a}_m) |
2068 |
|
|
+4 \sum_{lmnp} |
2069 |
|
|
(\hat{r} \times \hat{a}_l ) |
2070 |
|
|
Q_{{\mathbf a}lm} |
2071 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
2072 |
|
|
Q_{{\mathbf b}np} |
2073 |
|
|
(\hat{b}_p \cdot \hat{r}) \\ |
2074 |
|
|
% 3 |
2075 |
|
|
&-4 \sum_{lmnp} |
2076 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
2077 |
|
|
Q_{{\mathbf a}lm} |
2078 |
|
|
(\hat{a}_m \times \hat{b}_n) |
2079 |
|
|
Q_{{\mathbf b}np} |
2080 |
|
|
(\hat{b}_p \cdot \hat{r}) |
2081 |
|
|
\Bigr] v_{42}(r) \\ |
2082 |
|
|
% 4 |
2083 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
2084 |
|
|
\sum_{lm} (\hat{r} \times \hat{a}_l) |
2085 |
|
|
Q_{{\mathbf a}lm} |
2086 |
|
|
(\hat{a}_m \cdot \hat{r}) |
2087 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
2088 |
|
|
Q_{{\mathbf b}np} |
2089 |
|
|
(\hat{b}_p \cdot \hat{r}) v_{43}(r)\\ |
2090 |
|
|
\end{split} |
2091 |
|
|
\end{equation} |
2092 |
|
|
% |
2093 |
|
|
% tb qa qb |
2094 |
|
|
% |
2095 |
|
|
\begin{equation} |
2096 |
|
|
\begin{split} |
2097 |
|
|
% 1 |
2098 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} &= |
2099 |
|
|
\frac{4}{4\pi \epsilon_0} |
2100 |
|
|
\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
2101 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
2102 |
|
|
(\hat{a}_m \times \hat{b}_n) v_{41}(r) \\ |
2103 |
|
|
% 2 |
2104 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
2105 |
|
|
\Bigl[ |
2106 |
|
|
2\text{Tr}Q_{\mathbf{a}} |
2107 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
2108 |
|
|
Q_{{\mathbf b}lm} |
2109 |
|
|
(\hat{r} \times \hat{b}_m) |
2110 |
|
|
+4 \sum_{lmnp} |
2111 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
2112 |
|
|
Q_{{\mathbf a}lm} |
2113 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
2114 |
|
|
Q_{{\mathbf b}np} |
2115 |
|
|
(\hat{r} \times \hat{b}_p) \\ |
2116 |
|
|
% 3 |
2117 |
|
|
&+4 \sum_{lmnp} |
2118 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
2119 |
|
|
Q_{{\mathbf a}lm} |
2120 |
|
|
(\hat{a}_m \times \hat{b}_n) |
2121 |
|
|
Q_{{\mathbf b}np} |
2122 |
|
|
(\hat{b}_p \cdot \hat{r}) |
2123 |
|
|
\Bigr] v_{42}(r) \\ |
2124 |
|
|
% 4 |
2125 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
2126 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
2127 |
|
|
Q_{{\mathbf a}lm} |
2128 |
|
|
(\hat{a}_m \cdot \hat{r}) |
2129 |
|
|
\sum_{np} (\hat{r} \times \hat{b}_n) |
2130 |
|
|
Q_{{\mathbf b}np} |
2131 |
|
|
(\hat{b}_p \cdot \hat{r}) v_{43}(r). \\ |
2132 |
|
|
\end{split} |
2133 |
|
|
\end{equation} |
2134 |
|
|
% |
2135 |
|
|
\begin{table*} |
2136 |
|
|
\caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
2137 |
|
|
\begin{ruledtabular} |
2138 |
|
|
\begin{tabular}{ccc} |
2139 |
|
|
Generic&Method 1&Method 2 |
2140 |
|
|
\\ \hline |
2141 |
|
|
% |
2142 |
|
|
% |
2143 |
|
|
% |
2144 |
|
|
$w_a(r)$& |
2145 |
|
|
$g_0(r)$& |
2146 |
|
|
$g(r)-g(r_c)$ \\ |
2147 |
|
|
% |
2148 |
|
|
% |
2149 |
|
|
$w_b(r)$ & |
2150 |
|
|
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
2151 |
|
|
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
2152 |
|
|
% |
2153 |
|
|
$w_c(r)$ & |
2154 |
|
|
$\frac{g_1(r)}{r} $ & |
2155 |
|
|
$\frac{v_{11}(r)}{r}$ \\ |
2156 |
|
|
% |
2157 |
|
|
% |
2158 |
|
|
$w_d(r)$& |
2159 |
|
|
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
2160 |
|
|
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
2161 |
|
|
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\ |
2162 |
|
|
% |
2163 |
|
|
$w_e(r)$ & |
2164 |
|
|
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
2165 |
|
|
$\frac{v_{22}(r)}{r}$ \\ |
2166 |
|
|
% |
2167 |
|
|
% |
2168 |
|
|
$w_f(r)$& |
2169 |
|
|
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
2170 |
|
|
$\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\ |
2171 |
|
|
&&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\ |
2172 |
|
|
% |
2173 |
|
|
$w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
2174 |
|
|
$\frac{v_{31}(r)}{r}$\\ |
2175 |
|
|
% |
2176 |
|
|
$w_h(r)$ & |
2177 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
2178 |
|
|
$\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\ |
2179 |
|
|
&&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
2180 |
|
|
&&$-\frac{v_{31}(r)}{r}$\\ |
2181 |
|
|
% 2 |
2182 |
|
|
$w_i(r)$ & |
2183 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
2184 |
|
|
$\frac{v_{32}(r)}{r}$ \\ |
2185 |
|
|
% |
2186 |
|
|
$w_j(r)$ & |
2187 |
|
|
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
2188 |
|
|
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $ \\ |
2189 |
|
|
&&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
2190 |
|
|
% |
2191 |
|
|
$w_k(r)$ & |
2192 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
2193 |
|
|
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right)$ \\ |
2194 |
|
|
&&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
2195 |
|
|
% |
2196 |
|
|
$w_l(r)$ & |
2197 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
2198 |
|
|
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
2199 |
|
|
&&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right) |
2200 |
|
|
-\frac{2v_{42}(r)}{r}$ \\ |
2201 |
|
|
% |
2202 |
|
|
$w_m(r)$ & |
2203 |
|
|
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
2204 |
|
|
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\ |
2205 |
|
|
&&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
2206 |
|
|
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ |
2207 |
|
|
&&$-\frac{4v_{43}(r)}{r}$ \\ |
2208 |
|
|
% |
2209 |
|
|
$w_n(r)$ & |
2210 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
2211 |
|
|
$\frac{v_{42}(r)}{r}$ \\ |
2212 |
|
|
% |
2213 |
|
|
$w_o(r)$ & |
2214 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
2215 |
|
|
$\frac{v_{43}(r)}{r}$ \\ |
2216 |
|
|
% |
2217 |
|
|
\end{tabular} |
2218 |
|
|
\end{ruledtabular} |
2219 |
|
|
\end{table*} |
2220 |
gezelter |
3980 |
|
2221 |
|
|
\newpage |
2222 |
|
|
|
2223 |
|
|
\bibliography{multipole} |
2224 |
|
|
|
2225 |
gezelter |
3906 |
\end{document} |
2226 |
|
|
% |
2227 |
|
|
% ****** End of file multipole.tex ****** |