trunk/multipole/Supplemental.tex

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\documentclass[%
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preprint,%
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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}
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%\usepackage[mathlines]{lineno}% Enable numbering of text and display math
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\begin{document}

\title{Supplemental Material for: Real space electrostatics for
  multipoles. I. Development of Methods}

\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_a C_b}(r)=&
C_a C_b  v_{01}(r) 
\\
%
% u ca db
%
U_{C_a \mathbf{D}_b}(r)=&
C_a
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn} \,  v_{11}(r) 
\\
%
% u ca qb
%
U_{C_a \mathsf{Q}_b}(r)=&
C_a \text{Tr} \mathsf{Q}_b
v_{21}(r) +C_a
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) Q_{bmn} (\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}})
v_{22}(r) \\
%
% u da cb
%
U_{\mathbf{D}_a C_b}(r)=&
-C_b
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_n) D_{an} \,  v_{11}(r) 
\\
%
% u da db
%
% 1
U_{\mathbf{D}_a \mathbf{D}_b}(r)=&
-  \sum_{mn} D_{am} 
(\hat{\mathbf{A}}_m \cdot   \hat{\mathbf{B}}_n)
D_{bn} v_{21}(r) \nonumber \\
% 2
&-
\sum_m (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) D_{am} 
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn} 
v_{22}(r)
\\
%
% u da qb
%
% 1
U_{\mathbf{D}_a \mathsf{Q}_b}(r)=&
-\left(
\text{Tr}\mathsf{Q}_b
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_n) D_{an}
+2\sum_{lmn}D_{al} 
(\hat{\mathbf{A}}_l \cdot \hat{\mathbf{B}}_m)
Q_{bmn} (\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}})
\right)  v_{31}(r) \nonumber \\
% 2
&-
\sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) D_{al}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) 
Q_{bmn}
(\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) v_{32}(r)
\\
%
% u qa cb
%
U_{\mathsf{Q}_a C_b}(r)=&
C_b \text{Tr}\mathsf{Q}_a v_{21}(r)
+C_b
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) Q_{amn} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) v_{22}(r)
\\
%
% u qa db
%
%1
U_{\mathsf{Q}_a \mathbf{D}_b}(r)=&
\left(
\text{Tr}\mathsf{Q}_a
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn}
+2\sum_{lmn}D_{bl} 
(\hat{\mathbf{B}}_l \cdot \hat{\mathbf{A}}_m)
Q_{amn} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}})
\right) v_{31}(r) \nonumber \\
% 2
&+
\sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l) D_{bl}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
Q_{amn}
(\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) v_{32}(r)
\end{align}

\begin{align}
%
% u qa qb
%
%1
U_{\mathsf{Q}_a \mathsf{Q}_b}(r)=&
 \Bigl[
\text{Tr}\mathsf{Q}_a \text{Tr}\mathsf{Q}_b
+2\sum_{lmnp} (\hat{\mathbf{A}}_l \cdot \hat{\mathbf{B}}_p) 
Q_{alm}  Q_{bnp} 
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{B}}_n) \Bigr] 
v_{41}(r) \nonumber \\
% 2
&+ 
\Bigl[ \text{Tr}\mathsf{Q}_a
\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l )
Q_{blm}
(\hat{\mathbf{B}}_m \cdot \hat{\mathbf{r}})
+\text{Tr}\mathsf{Q}_b
\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l )
Q_{alm}
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{r}}) \nonumber \\
% 3
&+4 \sum_{lmnp}
(\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l ) 
Q_{alm}
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{B}}_n)
Q_{bnp}
(\hat{\mathbf{B}}_p \cdot \hat{\mathbf{r}})
\Bigr] v_{42}(r)  \nonumber \\
% 4
&+
\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) 
Q_{alm}
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{r}})
\sum_{np}  (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) 
Q_{bnp}
(\hat{\mathbf{B}}_p \cdot \hat{\mathbf{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}_{a C_a C_b} =&
C_a C_b  w_a(r) \hat{\mathbf{r}}
\\
%
% f ca db
%
\mathbf{F}_{a C_a \mathbf{D}_b} =& 
C_a
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn} w_b(r) \hat{\mathbf{r}}
+C_a
\sum_n  D_{bn} \hat{\mathbf{B}}_n w_c(r)
\\
%
% f ca qb
%
% 1
\mathbf{F}_{a C_a \mathsf{Q}_b} =&
C_a \text{Tr}\mathsf{Q}_b w_d(r) \hat{\mathbf{r}}
+ 2C_a \sum_l  \hat{\mathbf{B}}_l Q_{bln} (\hat{\mathbf{B}}_n \cdot
\hat{\mathbf{r}}) w_e(r) \nonumber \\
% 2
&+C_a
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) Q_{bmn} (\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) w_f(r) \hat{\mathbf{r}} \\
%
% f da cb
%
\mathbf{F}_{a \mathbf{D}_a C_b} =&
-C_b
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_n) D_{an} w_b(r)  \hat{\mathbf{r}}
-C_b
\sum_n  D_{an} \hat{\mathbf{A}}_n w_c(r)
\\
% 
% f da db
%
% 1
\mathbf{F}_{a \mathbf{D}_a \mathbf{D}_b} =& 
-\sum_{mn} D_{am} 
(\hat{\mathbf{A}}_m \cdot   \hat{\mathbf{B}}_n)
D_{bn}  w_d(r) \hat{\mathbf{r}}
-\sum_m (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) D_{am} 
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn} w_f(r) \hat{\mathbf{r}}
\nonumber \\
% 2
&  + 
\Bigl[ \sum_m D_{am} 
\hat{\mathbf{A}}_m \sum_n D_{bn} 
(\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) 
+ \sum_m D_{bm} 
\hat{\mathbf{B}}_m \sum_n D_{an} 
(\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) \Bigr] w_e(r)  \\
%
% f da qb
%
% 1
\mathbf{F}_{a \mathbf{D}_a \mathsf{Q}_b} =&
 -  \Bigl[
\text{Tr}\mathsf{Q}_b
\sum_l  D_{al} \hat{\mathbf{A}}_l
+2\sum_{lmn} D_{al} 
(\hat{\mathbf{A}}_l \cdot \hat{\mathbf{B}}_m)
Q_{bmn} \hat{\mathbf{B}}_n  \Bigr] w_g(r) \nonumber \\
% 3
&  - \Bigl[
\text{Tr}\mathsf{Q}_b
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_n) D_{an}
+2\sum_{lmn}D_{al} 
(\hat{\mathbf{A}}_l \cdot \hat{\mathbf{B}}_m)
Q_{bmn} (\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) \Bigr] w_h(r) \hat{\mathbf{r}}
\nonumber \\
% 4
&+ 
\Bigl[\sum_l  D_{al} \hat{\mathbf{A}}_l
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) 
Q_{bmn}
(\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) +2 \sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l)
 D_{al}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) 
Q_{bmn} \hat{\mathbf{B}}_n \Bigr]   w_i(r) \nonumber \\
% 6
&  -
\sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) D_{al} 
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) 
Q_{bmn}
(\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}})  w_j(r)  \hat{\mathbf{r}}
\\
%
% force qa cb
%
% 1
\mathbf{F}_{a \mathsf{Q}_a C_b} =& 
C_b \text{Tr} \mathsf{Q}_a \hat{\mathbf{r}} w_d(r)
+ 2C_b  \sum_l  \hat{\mathbf{A}}_l 
Q_{aln} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) w_e(r) \nonumber \\
% 2
&  +C_b
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) Q_{amn} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) w_f(r) \hat{\mathbf{r}} 
\end{align}

\begin{align}
%
% f qa db 
%
% 1
\mathbf{F}_{a \mathsf{Q}_a \mathbf{D}_b} =&
 \Bigl[
\text{Tr}\mathsf{Q}_a
\sum_l  D_{bl} \hat{\mathbf{B}}_l
+2\sum_{lmn} D_{bl} 
(\hat{\mathbf{B}}_l \cdot \hat{\mathbf{A}}_m)
Q_{amn} \hat{\mathbf{A}}_n  \Bigr] 
w_g(r) \nonumber \\
% 3
&  +  \Bigl[
\text{Tr}\mathsf{Q}_a
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn} 
+2\sum_{lmn}D_{bl} 
(\hat{\mathbf{B}}_l \cdot \hat{\mathbf{A}}_m)
Q_{amn} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) \Bigr] w_h(r) \hat{\mathbf{r}}
\nonumber \\
% 4
&  + \Bigl[ \sum_l  D_{bl} \hat{\mathbf{B}}_l
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
Q_{amn}
(\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) +2 \sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l)
 D_{bl}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
Q_{amn} \hat{\mathbf{A}}_n \Bigr]   w_i(r) \nonumber \\
% 6
&  +\sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l) D_{bl} 
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
Q_{amn}
(\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}})  w_j(r)  \hat{\mathbf{r}}
\\
%
% f qa qb
%
\mathbf{F}_{a \mathsf{Q}_a \mathsf{Q}_b} =&
 \Bigl[
\text{Tr}\mathsf{Q}_a \text{Tr} \mathsf{Q}_b
+ 2 \sum_{lmnp} (\hat{\mathbf{A}}_l \cdot \hat{\mathbf{B}}_p) 
Q_{alm}  Q_{bnp} 
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{B}}_n) \Bigr] w_k(r) \hat{\mathbf{r}} \nonumber \\
&+ \Bigl[
2\text{Tr}\mathsf{Q}_b \sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) Q_{alm}  \hat{\mathbf{A}}_m  
+ 2\text{Tr} \mathsf{Q}_a \sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l)
Q_{blm}  \hat{\mathbf{B}}_m \nonumber \\
&+ 4\sum_{lmnp} \hat{\mathbf{A}}_l Q_{alm} (\hat{\mathbf{A}}_m \cdot \hat{\mathbf{B}}_n) 
Q_{bnp} (\hat{\mathbf{B}}_p \cdot \hat{\mathbf{r}})   
+ 4\sum_{lmnp} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) Q_{alm} 
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{B}}_n) Q_{bnp} \hat{\mathbf{B}}_p
\Bigr] w_n(r) \nonumber \\
&+ 
\Bigl[ \text{Tr} \mathsf{Q}_a
\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l) Q_{blm} 
(\hat{\mathbf{B}}_m \cdot \hat{\mathbf{r}}) 
+ \text{Tr} \mathsf{Q}_b
\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) Q_{alm}  
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{r}}) \nonumber \\
&+4\sum_{lmnp} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) 
Q_{alm} (\hat{\mathbf{A}}_m \cdot \hat{\mathbf{B}}_n) 
Q_{bnp}  (\hat{\mathbf{B}}_p \cdot \hat{\mathbf{r}}) \Bigr] w_l(r) \hat{\mathbf{r}} \nonumber \\
%
&+ \Bigl[
2\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) Q_{alm} \hat{\mathbf{A}}_m
\sum_{np} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) Q_{bnp} 
(\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) \nonumber \\
&+2 \sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) Q_{alm} 
(\hat{\mathbf{A}}_m \cdot \hat{\mathbf{r}})
\sum_{np} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) Q_{bnp} \hat{\mathbf{B}}_n \Bigr]
w_o(r) \hat{\mathbf{r}} \nonumber \\
&  + 
\sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) 
Q_{alm} (\hat{\mathbf{A}}_m \cdot \hat{\mathbf{r}})
\sum_{np} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) 
Q_{bnp} (\hat{\mathbf{B}}_p \cdot \hat{\mathbf{r}}) w_m(r) \hat{\mathbf{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}_{b C_a \mathbf{D}_b} =&
C_a
\sum_n  (\hat{\mathbf{r}} \times \hat{\mathbf{B}}_n)  D_{bn} \,  v_{11}(r) 
\\
%
% t ca qb
%
\mathbf{\tau}_{b C_a \mathsf{Q}_b} =&
2C_a
\sum_{lm} (\hat{\mathbf{r}} \times \hat{\mathbf{B}}_l) 
Q_{blm} (\hat{\mathbf{B}}_m \cdot \hat{\mathbf{r}}) v_{22}(r)
\\
%
%  t da cb
%
\mathbf{\tau}_{a \mathbf{D}_a C_b} =&
-C_b
\sum_n  (\hat{\mathbf{r}} \times \hat{\mathbf{A}}_n)  D_{an} \,  v_{11}(r) 
\\
%
%
%  ta da db
%
% 1
\mathbf{\tau}_{a \mathbf{D}_a \mathbf{D}_b} =& 
 \sum_{mn} D_{am} 
(\hat{\mathbf{A}}_m \times  \hat{\mathbf{B}}_n)
D_{bn} v_{21}(r) \nonumber \\
% 2
&-
\sum_m (\hat{\mathbf{r}} \times \hat{\mathbf{A}}_m) D_{am} 
\sum_n (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_n) D_{bn} v_{22}(r)
\\
%
%  tb da db
%
% 1
\mathbf{\tau}_{b \mathbf{D}_a \mathbf{D}_b} =& 
- \sum_{mn} D_{am} 
(\hat{\mathbf{A}}_m \times  \hat{\mathbf{B}}_n)
D_{bn} v_{21}(r) \nonumber \\
% 2
&+
\sum_m (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
D_{am} \sum_n (\hat{\mathbf{r}} \times \hat{\mathbf{B}}_n) D_{bn} v_{22}(r) 
\\
% ta da qb
%
% 1
\mathbf{\tau}_{a \mathbf{D}_a \mathsf{Q}_b} =& 
 \left(
-\text{Tr} \mathsf{Q}_b
\sum_n (\hat{\mathbf{r}} \times \hat{\mathbf{A}}_n) D_{an}
+2\sum_{lmn} D_{al} 
(\hat{\mathbf{A}}_l \times \hat{\mathbf{B}}_m)
Q_{bmn} (\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}})
\right) v_{31}(r) \nonumber \\
% 2
&-
\sum_l (\hat{\mathbf{r}} \times \hat{\mathbf{A}}_l) D_{al}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) 
Q_{bmn}
(\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}}) v_{32}(r) \\
%
% tb da qb
%
% 1
\mathbf{\tau}_{b \mathbf{D}_a \mathsf{Q}_b} =& 
 \left(
-2\sum_{lmn}D_{al} 
(\hat{\mathbf{A}}_l \cdot \hat{\mathbf{B}}_m)
Q_{bmn} (\hat{\mathbf{r}} \times \hat{\mathbf{B}}_n)
-2\sum_{lmn}D_{al} 
(\hat{\mathbf{A}}_l \times \hat{\mathbf{B}}_m)
Q_{bmn} (\hat{\mathbf{B}}_n \cdot \hat{\mathbf{r}})
\right) v_{31}(r) \nonumber \\
% 2
&-2
\sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) 
D_{al} \sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_m) 
Q_{bmn} (\hat{\mathbf{r}}\times \hat{\mathbf{B}}_n) v_{32}(r)
\\
%
% ta qa cb
%
\mathbf{\tau}_{a \mathsf{Q}_a C_b} =&
2C_b \sum_{lm} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_l) 
Q_{alm} (\hat{\mathbf{r}} \times \hat{\mathbf{A}}_m) v_{22}(r)
\\
%
% ta qa db
%
% 1
\mathbf{\tau}_{a \mathsf{Q}_a \mathbf{D}_b} = &
 \left(
2\sum_{lmn}D_{bl} 
(\hat{\mathbf{B}}_l \cdot \hat{\mathbf{A}}_m)
Q_{amn} (\hat{\mathbf{r}} \times \hat{\mathbf{A}}_n)
+2\sum_{lmn}D_{bl} 
(\hat{\mathbf{A}}_l \times \hat{\mathbf{B}}_m)
Q_{amn} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}})
\right) v_{31}(r) \nonumber \\
% 2
&+2
\sum_l (\hat{\mathbf{r}} \cdot \hat{\mathbf{B}}_l) D_{bl}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
Q_{amn}
(\hat{\mathbf{r}}\times \hat{\mathbf{A}}_n) v_{32}(r)
\\
%
% tb qa db
%
% 1
\mathbf{\tau}_{b \mathsf{Q}_a \mathbf{D}_b} =& 
 \left(
\text{Tr} \mathsf{Q}_a
\sum_n (\hat{\mathbf{r}} \times \hat{\mathbf{B}}_n) D_{bn}
+2\sum_{lmn}D_{bl} 
(\hat{\mathbf{A}}_l \times \hat{\mathbf{B}}_m)
Q_{amn} (\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}})
\right) v_{31}(r) \nonumber \\
% 2
& \sum_l (\hat{\mathbf{r}} \times \hat{\mathbf{B}}_l) D_{bl}
\sum_{mn} (\hat{\mathbf{r}} \cdot \hat{\mathbf{A}}_m) 
Q_{amn}
(\hat{\mathbf{A}}_n \cdot \hat{\mathbf{r}}) v_{32}(r)
\end{align}

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

\section{}

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}.


\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 Quadrupolar arrays.  Note that these
    take into account the factor of two corrections in
    Ref. \onlinecite{Nagai01091963}}} 
\label{tab:NNQ}
\begin{ruledtabular}
\begin{tabular}{ccc}
 Lattice &   Quadrupole Direction &    Energy constants \\ \hline
  SC       &      111         &      -16.6 \\
  BCC      &      011         &      -43.4 \\
  FCC      &      111         &      -161 
\end{tabular}
\end{ruledtabular}
\end{table*}

\newpage
\bibliography{multipole}
\end{document}
%
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