trunk/multipole/Supplemental.tex

% ****** Start of file aipsamp.tex ******
%
%   This file is part of the AIP files in the AIP distribution for REVTeX 4.
%   Version 4.1 of REVTeX, October 2009
%
%   Copyright (c) 2009 American Institute of Physics.
%
%   See the AIP README file for restrictions and more information.
%
% TeX'ing this file requires that you have AMS-LaTeX 2.0 installed
% as well as the rest of the prerequisites for REVTeX 4.1
%
% It also requires running BibTeX. The commands are as follows:
%
%  1)  latex  aipsamp
%  2)  bibtex aipsamp
%  3)  latex  aipsamp
%  4)  latex  aipsamp
%
% Use this file as a source of example code for your aip document.
% Use the file aiptemplate.tex as a template for your document.
\documentclass[%
 aip,jcp,
 amsmath,amssymb,
preprint,%
% reprint,%
%author-year,%
%author-numerical,%
jcp]{revtex4-1}

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

%\usepackage[mathlines]{lineno}% Enable numbering of text and display math
%\linenumbers\relax % Commence numbering lines

\begin{document}

\title[Taylor-shifted and Gradient-shifted electrostatics for multipoles]
{Supplemental Material for: Real space alternatives to the Ewald
Sum. I. Taylor-shifted and Gradient-shifted electrostatics for multipoles}

\author{Madan Lamichhane}
\affiliation{Department of Physics, University
of Notre Dame, Notre Dame, IN 46556}

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

\author{Kathie E. Newman}
\affiliation{Department of Physics, University
of Notre Dame, Notre Dame, IN 46556}

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

\maketitle

\section{Interaction Energies in body-frame coordiantes}
%
%
%Energy in body coordinate form ---------------------------------------------------------------
%
Although they are not as widely used as space-frame coordinates, the
body-frame versions may occasionally prove useful.  In this section,
we list the interaction energies, forces, and torques in terms of the
body coordinates for both the Taylor-Shifted and Gradient-Shifted
approximations. The radial functions ($v_{ij}(r)$ and $w_{\alpha}(r)$)
are given in the Tables I and II in the paper.  These functions depend
on the choice of electrostatic kernel as well as the approximation
method being utilized.  Again, all energy, force, and torque equations
have an an implied factor of $1/4\pi \epsilon_0$:
%
% u ca cb
%
\begin{align}
U_{C_{\bf a}C_{\bf b}}(r)=&
C_{\bf a} C_{\bf b}  v_{01}(r) 
\\
%
% u ca db
%
U_{C_{\bf a}D_{\bf b}}(r)=&
C_{\bf a}
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \,  v_{11}(r) 
\\
%
% u ca qb
%
U_{C_{\bf a}Q_{\bf b}}(r)=&
C_{\bf a }\text{Tr}Q_{\bf b}  
v_{21}(r) +C_{\bf a}
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r})
v_{22}(r) \\
%
% u da cb
%
U_{D_{\bf a}C_{\bf b}}(r)=&
-C_{\bf b}
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \,  v_{11}(r) 
\\
%
% u da db
%
% 1
U_{D_{\bf a}D_{\bf b}}(r)=&
-  \sum_{mn} D_{\mathbf {a}m} 
(\hat{a}_m \cdot   \hat{b}_n)
D_{\mathbf{b}n} v_{21}(r) \nonumber \\
% 2
&-
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} 
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} 
v_{22}(r)
\\
%
% u da qb
%
% 1
U_{D_{\bf a}Q_{\bf b}}(r)=&
-\left(
\text{Tr}Q_{\mathbf{b}}
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n}
+2\sum_{lmn}D_{\mathbf{a}l} 
(\hat{a}_l \cdot \hat{b}_m)
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r})
\right)  v_{31}(r) \nonumber \\
% 2
&-
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l}
\sum_{mn} (\hat{r} \cdot \hat{b}_m) 
Q_{{\mathbf b}mn}
(\hat{b}_n \cdot \hat{r}) v_{32}(r)
\\
%
% u qa cb
%
U_{Q_{\bf a}C_{\bf b}}(r)=&
C_{\bf b }\text{Tr}Q_{\bf a} v_{21}(r)
+C_{\bf b}
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r)
\\
%
% u qa db
%
%1
U_{Q_{\bf a}D_{\bf b}}(r)=&
\left(
\text{Tr}Q_{\mathbf{a}}
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n}
+2\sum_{lmn}D_{\mathbf{b}l} 
(\hat{b}_l \cdot \hat{a}_m)
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r})
\right) v_{31}(r) \nonumber \\
% 2
&+
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l}
\sum_{mn} (\hat{r} \cdot \hat{a}_m) 
Q_{{\mathbf a}mn}
(\hat{a}_n \cdot \hat{r}) v_{32}(r)
\end{align}

\begin{align}
%
% u qa qb
%
%1
U_{Q_{\bf a}Q_{\bf b}}(r)=&
 \Bigl[
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}}
+2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) 
Q_{\mathbf{a}lm}  Q_{\mathbf{b}np} 
(\hat{a}_m \cdot \hat{b}_n) \Bigr] 
v_{41}(r) \nonumber \\
% 2
&+ 
\Bigl[ \text{Tr}Q_{\mathbf{a}}
\sum_{lm} (\hat{r} \cdot \hat{b}_l )
Q_{{\mathbf b}lm}
(\hat{b}_m \cdot \hat{r})
+\text{Tr}Q_{\mathbf{b}}
\sum_{lm} (\hat{r} \cdot \hat{a}_l )
Q_{{\mathbf a}lm}
(\hat{a}_m \cdot \hat{r}) \nonumber \\
% 3
&+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 \cdot \hat{r})
\Bigr] v_{42}(r)  \nonumber \\
% 4
&+
\sum_{lm} (\hat{r} \cdot \hat{a}_l) 
Q_{{\mathbf a}lm}
(\hat{a}_m \cdot \hat{r})
\sum_{np}  (\hat{r} \cdot \hat{b}_n) 
Q_{{\mathbf b}np}
(\hat{b}_p \cdot \hat{r})  v_{43}(r).
\end{align}


% BODY coordinates force equations --------------------------------------------
%
%
Here are the force equations written in terms of body coordinates.
%
% f ca cb
%
\begin{align}
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} =&
C_{\bf a} C_{\bf b}  w_a(r) \hat{r}
\\
%
% f ca db
%
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} =& 
C_{\bf a}
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r}
+C_{\bf a}
\sum_n  D_{\mathbf{b}n} \hat{b}_n w_c(r)
\\
%
% f ca qb
%
% 1
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} =&
C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r}
+ 2C_{\bf a } \sum_l  \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot
\hat{r}) w_e(r) \nonumber \\
% 2
&+C_{\bf a}
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} \\
%
% f da cb
%
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} =&
-C_{\bf{b}}
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r)  \hat{r}
-C_{\bf{b}}
\sum_n  D_{\mathbf{a}n} \hat{a}_n w_c(r)
\\
% 
% f da db
%
% 1
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} =& 
-\sum_{mn} D_{\mathbf {a}m} 
(\hat{a}_m \cdot   \hat{b}_n)
D_{\mathbf{b}n}  w_d(r) \hat{r}
-\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} 
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r}
\nonumber \\
% 2
&  + 
\Bigl[ \sum_m D_{\mathbf {a}m} 
\hat{a}_m \sum_n D_{\mathbf{b}n} 
(\hat{b}_n \cdot \hat{r}) 
+ \sum_m D_{\mathbf {b}m} 
\hat{b}_m \sum_n D_{\mathbf{a}n} 
(\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r)  \\
%
% f da qb
%
% 1
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} =&
 -  \Bigl[
\text{Tr}Q_{\mathbf{b}}
\sum_l  D_{\mathbf{a}l} \hat{a}_l
+2\sum_{lmn} D_{\mathbf{a}l} 
(\hat{a}_l \cdot \hat{b}_m)
Q_{\mathbf{b}mn} \hat{b}_n  \Bigr] w_g(r) \nonumber \\
% 3
&  - \Bigl[
\text{Tr}Q_{\mathbf{b}}
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n}
+2\sum_{lmn}D_{\mathbf{a}l} 
(\hat{a}_l \cdot \hat{b}_m)
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r}
\nonumber \\
% 4
&+ 
\Bigl[\sum_l  D_{\mathbf{a}l} \hat{a}_l
\sum_{mn} (\hat{r} \cdot \hat{b}_m) 
Q_{{\mathbf b}mn}
(\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l)
 D_{\mathbf{a}l}
\sum_{mn} (\hat{r} \cdot \hat{b}_m) 
Q_{{\mathbf b}mn} \hat{b}_n \Bigr]   w_i(r) \nonumber \\
% 6
&  -
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} 
\sum_{mn} (\hat{r} \cdot \hat{b}_m) 
Q_{{\mathbf b}mn}
(\hat{b}_n \cdot \hat{r})  w_j(r)  \hat{r}
\\
%
% force qa cb
%
% 1
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} =& 
C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r)
+ 2C_{\bf b }  \sum_l  \hat{a}_l Q_{{\mathbf
    a}ln} (\hat{a}_n \cdot \hat{r}) w_e(r) \nonumber \\
% 2
&  +C_{\bf b}
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} 
\end{align}

\begin{align}
%
% f qa db 
%
% 1
\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} =&
 \Bigl[
\text{Tr}Q_{\mathbf{a}}
\sum_l  D_{\mathbf{b}l} \hat{b}_l
+2\sum_{lmn} D_{\mathbf{b}l} 
(\hat{b}_l \cdot \hat{a}_m)
Q_{\mathbf{a}mn} \hat{a}_n  \Bigr] 
w_g(r) \nonumber \\
% 3
&  +  \Bigl[
\text{Tr}Q_{\mathbf{a}}
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} 
+2\sum_{lmn}D_{\mathbf{b}l} 
(\hat{b}_l \cdot \hat{a}_m)
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r}
\nonumber \\
% 4
&  + \Bigl[ \sum_l  D_{\mathbf{b}l} \hat{b}_l
\sum_{mn} (\hat{r} \cdot \hat{a}_m) 
Q_{{\mathbf a}mn}
(\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l)
 D_{\mathbf{b}l}
\sum_{mn} (\hat{r} \cdot \hat{a}_m) 
Q_{{\mathbf a}mn} \hat{a}_n \Bigr]   w_i(r) \nonumber \\
% 6
&  +\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} 
\sum_{mn} (\hat{r} \cdot \hat{a}_m) 
Q_{{\mathbf a}mn}
(\hat{a}_n \cdot \hat{r})  w_j(r)  \hat{r}
\\
%
% f qa qb
%
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =&
 \Bigl[
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} 
+ 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) 
Q_{\mathbf{a}lm}  Q_{\mathbf{b}np} 
(\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r} \nonumber \\
&+ \Bigl[
2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm}  \hat{a}_m  
+ 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l)
Q_{\mathbf{b}lm}  \hat{b}_m \nonumber \\
&+ 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})   
+ 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
\Bigr] w_n(r) \nonumber \\
&+ 
\Bigl[ \text{Tr}Q_{\mathbf{a}}
\sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) 
+ \text{Tr}Q_{\mathbf{b}}
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm}  (\hat{a}_m \cdot \hat{r}) \\
&+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 \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \nonumber \\
%
&+ \Bigl[
2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot
\hat{r}) \nonumber \\
&+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r})
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr]
w_o(r) \hat{r} \nonumber \\
&  + 
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r})
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} 
\end{align}
%
Here we list the form of the non-zero damped shifted multipole torques showing
explicitly dependences on body axes:
%
%  t ca db
%
\begin{align}
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} =&
C_{\bf a}
\sum_n  (\hat{r} \times \hat{b}_n)  D_{\mathbf{b}n} \,  v_{11}(r) 
\\
%
% t ca qb
%
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} =&
2C_{\bf a}
\sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m
\cdot \hat{r}) v_{22}(r)
\\
%
%  t da cb
%
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} =&
-C_{\bf b}
\sum_n  (\hat{r} \times \hat{a}_n)  D_{\mathbf{a}n} \,  v_{11}(r) 
\\
%
%
%  ta da db
%
% 1
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} =& 
 \sum_{mn} D_{\mathbf {a}m} 
(\hat{a}_m \times  \hat{b}_n)
D_{\mathbf{b}n} v_{21}(r) \nonumber \\
% 2
&-
\sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} 
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r)
\\
%
%  tb da db
%
% 1
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} =& 
- \sum_{mn} D_{\mathbf {a}m} 
(\hat{a}_m \times  \hat{b}_n)
D_{\mathbf{b}n} v_{21}(r) \nonumber \\
% 2
&+
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} 
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) 
\\
% ta da qb
%
% 1
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} =& 
 \left(
-\text{Tr}Q_{\mathbf{b}}
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n}
+2\sum_{lmn}D_{\mathbf{a}l} 
(\hat{a}_l \times \hat{b}_m)
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r})
\right) v_{31}(r) \nonumber \\
% 2
&-
\sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l}
\sum_{mn} (\hat{r} \cdot \hat{b}_m) 
Q_{{\mathbf b}mn}
(\hat{b}_n \cdot \hat{r}) v_{32}(r) \\
%
% tb da qb
%
% 1
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} =& 
 \left(
-2\sum_{lmn}D_{\mathbf{a}l} 
(\hat{a}_l \cdot \hat{b}_m)
Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n)
-2\sum_{lmn}D_{\mathbf{a}l} 
(\hat{a}_l \times \hat{b}_m)
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r})
\right) v_{31}(r) \nonumber \\
% 2
&-2
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l}
\sum_{mn} (\hat{r} \cdot \hat{b}_m) 
Q_{{\mathbf b}mn}
(\hat{r}\times \hat{b}_n) v_{32}(r)
\\
%
% ta qa cb
%
\mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} =&
2C_{\bf a}
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r)
\\
%
% ta qa db
%
% 1
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = &
 \left(
2\sum_{lmn}D_{\mathbf{b}l} 
(\hat{b}_l \cdot \hat{a}_m)
Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n)
+2\sum_{lmn}D_{\mathbf{b}l} 
(\hat{a}_l \times \hat{b}_m)
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r})
\right) v_{31}(r) \nonumber \\
% 2
&+2
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l}
\sum_{mn} (\hat{r} \cdot \hat{a}_m) 
Q_{{\mathbf a}mn}
(\hat{r}\times \hat{a}_n) v_{32}(r)
\\
%
% tb qa db
%
% 1
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} =& 
 \left(
\text{Tr}Q_{\mathbf{a}}
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n}
+2\sum_{lmn}D_{\mathbf{b}l} 
(\hat{a}_l \times \hat{b}_m)
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r})
\right) v_{31}(r) \nonumber \\
% 2
& \sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l}
\sum_{mn} (\hat{r} \cdot \hat{a}_m) 
Q_{{\mathbf a}mn}
(\hat{a}_n \cdot \hat{r}) v_{32}(r)
\end{align}

%
% ta qa qb
%
\begin{align}
% 1
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& 
-4
\sum_{lmnp} (\hat{a}_l \times \hat{b}_p)
Q_{\mathbf{a}lm}  Q_{\mathbf{b}np} 
(\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \nonumber \\
% 2
&+ 
\Bigl[
2\text{Tr}Q_{\mathbf{b}}
\sum_{lm} (\hat{r} \cdot \hat{a}_l )
Q_{{\mathbf a}lm}
(\hat{r} \times \hat{a}_m)
+4 \sum_{lmnp}
(\hat{r} \times \hat{a}_l ) 
Q_{{\mathbf a}lm}
(\hat{a}_m \cdot \hat{b}_n)
Q_{{\mathbf b}np}
(\hat{b}_p \cdot \hat{r}) \nonumber \\
% 3
&-4 \sum_{lmnp}
(\hat{r} \cdot \hat{a}_l ) 
Q_{{\mathbf a}lm}
(\hat{a}_m \times \hat{b}_n)
Q_{{\mathbf b}np}
(\hat{b}_p \cdot \hat{r})
\Bigr] v_{42}(r) \nonumber \\
% 4
&+2
\sum_{lm} (\hat{r} \times \hat{a}_l) 
Q_{{\mathbf a}lm}
(\hat{a}_m \cdot \hat{r})
\sum_{np}  (\hat{r} \cdot \hat{b}_n) 
Q_{{\mathbf b}np}
(\hat{b}_p \cdot \hat{r})  v_{43}(r)\\
%
% tb qa qb
%
% 1
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} =& 
4 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p)
Q_{\mathbf{a}lm}  Q_{\mathbf{b}np} 
(\hat{a}_m \times \hat{b}_n) v_{41}(r) \nonumber \\
% 2
&+ 
\Bigl[
2\text{Tr}Q_{\mathbf{a}}
\sum_{lm} (\hat{r} \cdot \hat{b}_l )
Q_{{\mathbf b}lm}
(\hat{r} \times \hat{b}_m)
+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{r} \times \hat{b}_p) \nonumber \\
% 3
&+4 \sum_{lmnp}
(\hat{r} \cdot \hat{a}_l ) 
Q_{{\mathbf a}lm}
(\hat{a}_m \times \hat{b}_n)
Q_{{\mathbf b}np}
(\hat{b}_p \cdot \hat{r})
\Bigr] v_{42}(r)  \nonumber \\
% 4
&+2
\sum_{lm} (\hat{r} \cdot \hat{a}_l) 
Q_{{\mathbf a}lm}
(\hat{a}_m \cdot \hat{r})
\sum_{np}  (\hat{r} \times \hat{b}_n) 
Q_{{\mathbf b}np}
(\hat{b}_p \cdot \hat{r}) v_{43}(r).
\end{align}


% \begin{table*}
% \caption{\label{tab:tableFORCE2}Radial functions used in the force equations.}
% \begin{ruledtabular}
% \begin{tabular}{|l|l|l|}
% Generic&Taylor-shifted Force&Gradient-shifted Force
% \\ \hline
% %
% %
% %
% $w_a(r)$&
% $g_0(r)$&
% $g(r)-g(r_c)$ \\
% %
% %
% $w_b(r)$ &
% $\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ &
% $h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\
% %
% $w_c(r)$ &
% $\frac{g_1(r)}{r} $ &
% $\frac{v_{11}(r)}{r}$ \\
% %
% %
% $w_d(r)$&
% $\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ &
% $\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right)
% -\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\
% %
% $w_e(r)$ &
% $\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ &
% $\frac{v_{22}(r)}{r}$ \\
% % 
% %
% $w_f(r)$&
% $\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ &
% $\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\
% &&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\
% %
% $w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& 
% $\frac{v_{31}(r)}{r}$\\
% %
% $w_h(r)$ &
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $  &
% $\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\
% &&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\
% &&$-\frac{v_{31}(r)}{r}$\\
% % 2
% $w_i(r)$ &
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $  &
% $\frac{v_{32}(r)}{r}$ \\
% % 
% $w_j(r)$ &
% $\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right)  $ &
% $\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $  \\
% &&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\
% %
% $w_k(r)$ &
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2}  \right)$ &
% $\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2}  \right)$  \\
% &&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2}  \right)$ \\
% %
% $w_l(r)$ &
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ &
% $\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\
% &&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)
% -\frac{2v_{42}(r)}{r}$ \\
% %
% $w_m(r)$ &
% $\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ &
% $\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\
% &&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} 
% +\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ 
% &&$-\frac{4v_{43}(r)}{r}$ \\
% %
% $w_n(r)$ &
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2}  \right)$ &
% $\frac{v_{42}(r)}{r}$ \\
% %
% $w_o(r)$ &
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ &
% $\frac{v_{43}(r)}{r}$ \\
% %
% \end{tabular}
% \end{ruledtabular}
% \end{table*}
%
% \newpage
%
% \bibliography{multipole}
% 

To test the gradient-shifted force (GSF) and Taylor-shifted force
(TSF) methods against known energies for multipolar crystals, we
repeated the Luttinger \& Tisza series summations and have obtained
the energy constants (converged to one part in $10^9$) in table
\ref{tab:LT}.

\section{}
\begin{table*}[h]
\centering{
  \caption{Luttinger \& Tisza arrays and their associated
    energy constants. Type ``A'' arrays have nearest neighbor strings of
    antiparallel dipoles.  Type ``B'' arrays have nearest neighbor
    strings of antiparallel dipoles if the dipoles are contained in a
    plane perpendicular to the dipole direction that passes through
    the dipole.}
}
\label{tab:LT}
\begin{ruledtabular}
\begin{tabular}{cccc}
Array Type &  Lattice &   Dipole Direction &    Energy constants \\ \hline
   A     &      SC       &      001         &      -2.676788684 \\
   A     &      BCC      &      001         &       0 \\
   A     &      BCC      &      111         &      -1.770078733 \\
   A     &      FCC      &      001         &       2.166932835 \\
   A     &      FCC      &      011         &      -1.083466417 \\
   B     &      SC       &      001         &      -2.676788684 \\
   B     &      BCC      &      001         &      -1.338394342 \\
   B     &      BCC      &      111         &      -1.770078733 \\
   B     &      FCC      &      001         &      -1.083466417 \\
   B     &      FCC      &      011         &      -1.807573634 \\
  --     &      BCC      &    minimum       &      -1.985920929 \\
\end{tabular}
\end{ruledtabular}
\end{table*}

We have also tested agains the energy constants for Quadrupolar
arrays.  Nagai and Nakamura computed the energies of selected
quadrupole arrays based on extensions to the Luttinger and Tisza
approach.  These energy constants are given in table \ref{tab:NNQ}.

\begin{table*}
\centering{
  \caption{Nagai and Nakamura Quadurpolar arrays}}
\label{tab:NNQ}
\begin{ruledtabular}
\begin{tabular}{ccc}
 Lattice &   Quadrupole Direction &    Energy constants \\ \hline
  SC       &      111         &      -8.3 \\
  BCC      &      011         &      -21.7 \\
  FCC      &      111         &      -80.5 
\end{tabular}
\end{ruledtabular}
\end{table*}


\end{document}
%
% ****** End of file multipole.tex ******
Revision:	3990
Committed:	Fri Jan 3 18:28:01 2014 UTC (11 years, 9 months ago) by gezelter
Content type:	application/x-tex
File size:	20861 byte(s)
Log Message:	most recent edits