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\begin{document} |
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\preprint{AIP/123-QED} |
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%\preprint{AIP/123-QED} |
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\title[Taylor-shifted and Gradient-shifted electrostatics for multipoles] |
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{Real space alternatives to the Ewald |
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\title{Real space alternatives to the Ewald |
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Sum. I. Taylor-shifted and Gradient-shifted electrostatics for multipoles} |
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\author{Madan Lamichhane} |
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results for ordered arrays of multipoles. |
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\end{abstract} |
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\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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% Classification Scheme. |
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\maketitle |
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|
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The computational efficiency and the accuracy of the DSF method are |
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surprisingly good, particularly for systems with uniform charge |
| 100 |
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density. Additionally, dielectric constants obtained using DSF and |
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similar methods where the force vanishes at $R_\textrm{c}$ are |
| 101 |
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similar methods where the force vanishes at $r_{c}$ are |
| 102 |
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essentially quantitative.\cite{Izvekov:2008wo} The DSF and other |
| 103 |
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related methods have now been widely investigated,\cite{Hansen:2012uq} |
| 104 |
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and DSF is now used routinely in simulations of ionic |
| 105 |
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liquids,\cite{doi:10.1021/la400226g,McCann:2013fk} flow in carbon |
| 106 |
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nanotubes,\cite{kannam:094701} gas sorption in metal-organic framework |
| 107 |
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materials,\cite{Forrest:2012ly} thermal conductivity of methane |
| 108 |
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hydrates,\cite{English:2008kx} condensation coefficients of |
| 109 |
< |
water,\cite{Louden:2013ve} and in tribology at solid-liquid-solid |
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interfaces.\cite{Tokumasu:2013zr} DSF electrostatics provides a |
| 111 |
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compromise between the computational speed of real-space cutoffs and |
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the accuracy of fully-periodic Ewald treatments. |
| 104 |
> |
and DSF is now used routinely in a diverse set of chemical |
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environments.\cite{doi:10.1021/la400226g,McCann:2013fk,kannam:094701,Forrest:2012ly,English:2008kx,Louden:2013ve,Tokumasu:2013zr} |
| 106 |
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DSF electrostatics provides a compromise between the computational |
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speed of real-space cutoffs and the accuracy of fully-periodic Ewald |
| 108 |
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treatments. |
| 109 |
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|
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– |
\subsection{Coarse Graining using Point Multipoles} |
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One common feature of many coarse-graining approaches, which treat |
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entire molecular subsystems as a single rigid body, is simplification |
| 112 |
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of the electrostatic interactions between these bodies so that fewer |
| 113 |
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site-site interactions are required to compute configurational |
| 114 |
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energies. Notably, the force matching approaches of Voth and coworkers |
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are an exciting development in their ability to represent realistic |
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(and {\it reactive}) chemical systems for very large length scales and |
| 122 |
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long times. This approach utilized a coarse-graining in interaction |
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< |
space (CGIS) which fits an effective force for the coarse grained |
| 124 |
< |
system using a variational force-matching method to a fine-grained |
| 125 |
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simulation.\cite{Izvekov:2008wo} |
| 114 |
> |
energies. To do this, the interactions between coarse-grained sites |
| 115 |
> |
are typically taken to be point |
| 116 |
> |
multipoles.\cite{Golubkov06,ISI:000276097500009,ISI:000298664400012} |
| 117 |
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|
| 118 |
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The coarse-graining approaches of Ren \& coworkers,\cite{Golubkov06} |
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and Essex \& |
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coworkers,\cite{ISI:000276097500009,ISI:000298664400012} |
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both utilize Gay-Berne |
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ellipsoids~\cite{Berne72,Gay81,Luckhurst90,Cleaver96,Berardi98,Ravichandran:1999fk,Berardi99,Pasterny00} |
| 123 |
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to model dispersive interactions and point multipoles to model |
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electrostatics for entire molecules or functional groups. |
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|
| 135 |
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Ichiye and coworkers have recently introduced a number of very fast |
| 136 |
< |
water models based on a ``sticky'' multipole model which are |
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< |
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 |
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< |
sticky multipole |
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< |
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 |
| 118 |
> |
Water, in particular, has been modeled recently with point multipoles |
| 119 |
> |
up to octupolar |
| 120 |
> |
order.\cite{Chowdhuri:2006lr,Te:2010rt,Te:2010ys,Te:2010vn} For |
| 121 |
> |
maximum efficiency, these models require the use of an approximate |
| 122 |
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multipole expansion as the exact multipole expansion can become quite |
| 123 |
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expensive (particularly when handled via the Ewald |
| 124 |
> |
sum).\cite{Ichiye:2006qy} Point multipoles and multipole |
| 125 |
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polarizability have also been utilized in the AMOEBA water model and |
| 126 |
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related force fields.\cite{Ponder:2010fk,schnieders:124114,Ren:2011uq} |
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|
| 128 |
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Higher-order multipoles present a peculiar issue for molecular |
| 133 |
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a different orientation can cause energy discontinuities. |
| 134 |
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|
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This paper outlines an extension of the original DSF electrostatic |
| 136 |
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kernel to point multipoles. We have developed two distinct real-space |
| 136 |
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kernel to point multipoles. We describe two distinct real-space |
| 137 |
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interaction models for higher-order multipoles based on two truncated |
| 138 |
|
Taylor expansions that are carried out at the cutoff radius. We are |
| 139 |
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calling these models {\bf Taylor-shifted} and {\bf Gradient-shifted} |
| 140 |
|
electrostatics. Because of differences in the initial assumptions, |
| 141 |
< |
the two methods yield related, but different expressions for energies, |
| 142 |
< |
forces, and torques. |
| 141 |
> |
the two methods yield related, but somewhat different expressions for |
| 142 |
> |
energies, forces, and torques. |
| 143 |
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|
| 144 |
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In this paper we outline the new methodology and give functional forms |
| 145 |
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for the energies, forces, and torques up to quadrupole-quadrupole |
| 146 |
|
order. We also compare the new methods to analytic energy constants |
| 147 |
< |
for periodic arrays of point multipoles. In the following paper, we |
| 148 |
< |
provide extensive numerical comparisons to Ewald-based electrostatics |
| 149 |
< |
in common simulation enviornments. |
| 147 |
> |
for periodic arrays of point multipoles. In the following paper, we |
| 148 |
> |
provide numerical comparisons to Ewald-based electrostatics in common |
| 149 |
> |
simulation enviornments. |
| 150 |
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|
| 151 |
|
\section{Methodology} |
| 152 |
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An efficient real-space electrostatic method involves the use of a |
| 153 |
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pair-wise functional form, |
| 154 |
+ |
\begin{equation} |
| 155 |
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V = \sum_i \sum_{j>i} V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) + |
| 156 |
+ |
\sum_i V_i^\mathrm{self} |
| 157 |
+ |
\end{equation} |
| 158 |
+ |
that is short-ranged and easily truncated at a cutoff radius, |
| 159 |
+ |
\begin{equation} |
| 160 |
+ |
V_\mathrm{pair}(r_{ij},\Omega_i, \Omega_j) = \left\{ |
| 161 |
+ |
\begin{array}{ll} |
| 162 |
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V_\mathrm{approx} (r_{ij}, \Omega_i, \Omega_j) & \quad r \le r_c \\ |
| 163 |
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0 & \quad r > r_c , |
| 164 |
+ |
\end{array} |
| 165 |
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\right. |
| 166 |
+ |
\end{equation} |
| 167 |
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along with an easily computed self-interaction term ($\sum_i |
| 168 |
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V_i^\mathrm{self}$) which has linear-scaling with the number of |
| 169 |
+ |
particles. Here $\Omega_i$ and $\Omega_j$ represent orientational |
| 170 |
+ |
coordinates of the two sites. The computational efficiency, energy |
| 171 |
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conservation, and even some physical properties of a simulation can |
| 172 |
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depend dramatically on how the $V_\mathrm{approx}$ function behaves at |
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the cutoff radius. The goal of any approximation method should be to |
| 174 |
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mimic the real behavior of the electrostatic interactions as closely |
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as possible without sacrificing the near-linear scaling of a cutoff |
| 176 |
+ |
method. |
| 177 |
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|
| 178 |
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\subsection{Self-neutralization, damping, and force-shifting} |
| 179 |
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The DSF and Wolf methods operate by neutralizing the total charge |
| 180 |
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contained within the cutoff sphere surrounding each particle. This is |
| 181 |
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accomplished by shifting the potential functions to generate image |
| 182 |
|
charges on the surface of the cutoff sphere for each pair interaction |
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computed within $R_\textrm{c}$. Damping using a complementary error |
| 183 |
> |
computed within $r_c$. Damping using a complementary error |
| 184 |
|
function is applied to the potential to accelerate convergence. The |
| 185 |
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potential for the DSF method, where $\alpha$ is the adjustable damping |
| 186 |
|
parameter, is given by |
| 187 |
|
\begin{equation*} |
| 188 |
< |
V_\mathrm{DSF}(r) = C_a C_b \Biggr{[} |
| 188 |
> |
V_\mathrm{DSF}(r) = C_i C_j \Biggr{[} |
| 189 |
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
| 190 |
< |
- \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} |
| 190 |
> |
- \frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c} + \left(\frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c^2} |
| 191 |
|
+ \frac{2\alpha}{\pi^{1/2}} |
| 192 |
< |
\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}} |
| 193 |
< |
\right)\left(r_{ij}-R_\mathrm{c}\right)\ \Biggr{]} |
| 192 |
> |
\frac{\exp\left(-\alpha^2r_c^2\right)}{r_c} |
| 193 |
> |
\right)\left(r_{ij}-r_c\right)\ \Biggr{]} |
| 194 |
|
\label{eq:DSFPot} |
| 195 |
|
\end{equation*} |
| 196 |
+ |
Note that in this potential and in all electrostatic quantities that |
| 197 |
+ |
follow, the standard $1/4 \pi \epsilon_{0}$ has been omitted for |
| 198 |
+ |
clarity. |
| 199 |
|
|
| 200 |
|
To insure net charge neutrality within each cutoff sphere, an |
| 201 |
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additional ``self'' term is added to the potential. This term is |
| 206 |
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over the surface of the cutoff sphere. A portion of the self term is |
| 207 |
|
identical to the self term in the Ewald summation, and comes from the |
| 208 |
|
utilization of the complimentary error function for electrostatic |
| 209 |
< |
damping.\cite{deLeeuw80,Wolf99} |
| 209 |
> |
damping.\cite{deLeeuw80,Wolf99} There have also been recent efforts to |
| 210 |
> |
extend the Wolf self-neutralization method to zero out the dipole and |
| 211 |
> |
higher order multipoles contained within the cutoff |
| 212 |
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sphere.\cite{Fukuda:2011jk,Fukuda:2012yu,Fukuda:2013qv} |
| 213 |
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|
| 214 |
< |
There have been recent efforts to extend the Wolf self-neutralization |
| 215 |
< |
method to zero out the dipole and higher order multipoles contained |
| 216 |
< |
within the cutoff |
| 217 |
< |
sphere.\cite{Fukuda:2011jk,Fukuda:2012yu,Fukuda:2013qv} |
| 218 |
< |
|
| 219 |
< |
In this work, we will be extending the idea of self-neutralization for |
| 220 |
< |
the point multipoles in a similar way. In Figure |
| 221 |
< |
\ref{fig:shiftedMultipoles}, the central dipolar site $\mathbf{D}_i$ |
| 222 |
< |
is interacting with point dipole $\mathbf{D}_j$ and point quadrupole, |
| 213 |
< |
$\mathbf{Q}_k$. The self-neutralization scheme for point multipoles |
| 214 |
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involves projecting opposing multipoles for sites $j$ and $k$ on the |
| 215 |
< |
surface of the cutoff sphere. |
| 214 |
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In this work, we extend the idea of self-neutralization for the point |
| 215 |
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multipoles by insuring net charge-neutrality and net-zero moments |
| 216 |
> |
within each cutoff sphere. In Figure \ref{fig:shiftedMultipoles}, the |
| 217 |
> |
central dipolar site $\mathbf{D}_i$ is interacting with point dipole |
| 218 |
> |
$\mathbf{D}_j$ and point quadrupole, $\mathbf{Q}_k$. The |
| 219 |
> |
self-neutralization scheme for point multipoles involves projecting |
| 220 |
> |
opposing multipoles for sites $j$ and $k$ on the surface of the cutoff |
| 221 |
> |
sphere. There are also significant modifications made to make the |
| 222 |
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forces and torques go smoothly to zero at the cutoff distance. |
| 223 |
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|
| 224 |
|
\begin{figure} |
| 225 |
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\includegraphics[width=3in]{SM} |
| 229 |
|
\label{fig:shiftedMultipoles} |
| 230 |
|
\end{figure} |
| 231 |
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|
| 232 |
< |
As in the point-charge approach, there is a contribution from |
| 233 |
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self-neutralization of site $i$. The self term for multipoles is |
| 232 |
> |
As in the point-charge approach, there is an additional contribution |
| 233 |
> |
from self-neutralization of site $i$. The self term for multipoles is |
| 234 |
|
described in section \ref{sec:selfTerm}. |
| 235 |
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|
| 236 |
|
\subsection{The multipole expansion} |
| 244 |
|
a$. Then the electrostatic potential of object $\bf a$ at |
| 245 |
|
$\mathbf{r}$ is given by |
| 246 |
|
\begin{equation} |
| 247 |
< |
V_a(\mathbf r) = \frac{1}{4\pi\epsilon_0} |
| 247 |
> |
V_a(\mathbf r) = |
| 248 |
|
\sum_{k \, \text{in \bf a}} \frac{q_k}{\lvert \mathbf{r} - \mathbf{r}_k \rvert}. |
| 249 |
|
\end{equation} |
| 250 |
|
The Taylor expansion in $r$ can be written using an implied summation |
| 263 |
|
can then be used to express the electrostatic potential on $\bf a$ in |
| 264 |
|
terms of multipole operators, |
| 265 |
|
\begin{equation} |
| 266 |
< |
V_{\bf a}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\hat{M}_{\bf a} \frac{1}{r} |
| 266 |
> |
V_{\bf a}(\mathbf{r}) =\hat{M}_{\bf a} \frac{1}{r} |
| 267 |
|
\end{equation} |
| 268 |
|
where |
| 269 |
|
\begin{equation} |
| 288 |
|
$\bf a$ to $\bf b$ ($\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $), the interaction energy is given by |
| 289 |
|
\begin{equation} |
| 290 |
|
U_{\bf{ab}}(r) |
| 291 |
< |
= \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} . |
| 291 |
> |
= \hat{M}_a \sum_{j \, \text{in \bf b}} \frac {q_j}{\vert \bf{r}+\bf{r}_j \vert} . |
| 292 |
|
\end{equation} |
| 293 |
|
This can also be expanded as a Taylor series in $r$. Using a notation |
| 294 |
|
similar to before to define the multipoles on object {\bf b}, |
| 299 |
|
\end{equation} |
| 300 |
|
we arrive at the multipole expression for the total interaction energy. |
| 301 |
|
\begin{equation} |
| 302 |
< |
U_{\bf{ab}}(r)=\frac{\hat{M}_{\bf a} \hat{M}_{\bf b}}{4\pi \epsilon_0} \frac{1}{r} \label{kernel}. |
| 302 |
> |
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
| 303 |
|
\end{equation} |
| 304 |
|
This form has the benefit of separating out the energies of |
| 305 |
|
interaction into contributions from the charge, dipole, and quadrupole |
| 306 |
< |
of $\bf a$ interacting with the same multipoles $\bf b$. |
| 306 |
> |
of $\bf a$ interacting with the same multipoles on $\bf b$. |
| 307 |
|
|
| 308 |
|
\subsection{Damped Coulomb interactions} |
| 309 |
|
In the standard multipole expansion, one typically uses the bare |
| 318 |
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
| 319 |
|
\end{equation} |
| 320 |
|
We develop equations below using the function $f(r)$ to represent |
| 321 |
< |
either $1/r$ or $B_0(r)$, and all of the techniques can be applied |
| 322 |
< |
either to bare or damped Coulomb kernels as long as derivatives of |
| 323 |
< |
these functions are known. Smith's convenient functions $B_l(r)$ are |
| 324 |
< |
summarized in Appendix A. |
| 321 |
> |
either $1/r$ or $B_0(r)$, and all of the techniques can be applied to |
| 322 |
> |
bare or damped Coulomb kernels (or any other function) as long as |
| 323 |
> |
derivatives of these functions are known. Smith's convenient |
| 324 |
> |
functions $B_l(r)$ are summarized in Appendix A. |
| 325 |
|
|
| 319 |
– |
\subsection{Taylor-shifted force (TSF) electrostatics} |
| 320 |
– |
|
| 326 |
|
The main goal of this work is to smoothly cut off the interaction |
| 327 |
|
energy as well as forces and torques as $r\rightarrow r_c$. To |
| 328 |
|
describe how this goal may be met, we use two examples, charge-charge |
| 329 |
< |
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain |
| 330 |
< |
the idea. |
| 329 |
> |
and charge-dipole, using the bare Coulomb kernel, $f(r)=1/r$, to |
| 330 |
> |
explain the idea. |
| 331 |
|
|
| 332 |
+ |
\subsection{Shifted-force methods} |
| 333 |
|
In the shifted-force approximation, the interaction energy for two |
| 334 |
|
charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is |
| 335 |
|
written: |
| 336 |
|
\begin{equation} |
| 337 |
< |
U_{C_{\bf a}C_{\bf b}}(r)=\frac{1}{4\pi \epsilon_0} C_{\bf a} C_{\bf b} |
| 337 |
> |
U_{C_{\bf a}C_{\bf b}}(r)= C_{\bf a} C_{\bf b} |
| 338 |
|
\left({ \frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
| 339 |
|
\right) . |
| 340 |
|
\end{equation} |
| 341 |
|
Two shifting terms appear in this equations, one from the |
| 342 |
< |
neutralization procedure ($-1/r_c$), and one that will make the first |
| 343 |
< |
derivative also vanish at the cutoff radius. |
| 342 |
> |
neutralization procedure ($-1/r_c$), and one that causes the first |
| 343 |
> |
derivative to vanish at the cutoff radius. |
| 344 |
|
|
| 345 |
|
Since one derivative of the interaction energy is needed for the |
| 346 |
|
force, the minimal perturbation is a term linear in $(r-r_c)$ in the |
| 351 |
|
\right) = \left(- \frac{1}{r^2} + \frac{1}{r_c^2} |
| 352 |
|
\right) . |
| 353 |
|
\end{equation} |
| 354 |
< |
There are a number of ways to generalize this derivative shift for |
| 355 |
< |
higher-order multipoles. |
| 354 |
> |
which clearly vanishes as the $r$ approaches the cutoff radius. There |
| 355 |
> |
are a number of ways to generalize this derivative shift for |
| 356 |
> |
higher-order multipoles. Below, we present two methods, one based on |
| 357 |
> |
higher-order Taylor series for $r$ near $r_c$, and the other based on |
| 358 |
> |
linear shift of the kernel gradients at the cutoff itself. |
| 359 |
|
|
| 360 |
+ |
\subsection{Taylor-shifted force (TSF) electrostatics} |
| 361 |
|
In the Taylor-shifted force (TSF) method, the procedure that we follow |
| 362 |
|
is based on a Taylor expansion containing the same number of |
| 363 |
|
derivatives required for each force term to vanish at the cutoff. For |
| 364 |
|
example, the quadrupole-quadrupole interaction energy requires four |
| 365 |
|
derivatives of the kernel, and the force requires one additional |
| 366 |
< |
derivative. We therefore require shifted energy expressions that |
| 367 |
< |
include enough terms so that all energies, forces, and torques are |
| 368 |
< |
zero as $r \rightarrow r_c$. In each case, we will subtract off a |
| 369 |
< |
function $f_n^{\text{shift}}(r)$ from the kernel $f(r)=1/r$. The |
| 370 |
< |
index $n$ indicates the number of derivatives to be taken when |
| 371 |
< |
deriving a given multipole energy. We choose a function with |
| 372 |
< |
guaranteed smooth derivatives --- a truncated Taylor series of the |
| 373 |
< |
function $f(r)$, e.g., |
| 366 |
> |
derivative. For quadrupole-quadrupole interactions, we therefore |
| 367 |
> |
require shifted energy expressions that include up to $(r-r_c)^5$ so |
| 368 |
> |
that all energies, forces, and torques are zero as $r \rightarrow |
| 369 |
> |
r_c$. In each case, we subtract off a function $f_n^{\text{shift}}(r)$ |
| 370 |
> |
from the kernel $f(r)=1/r$. The subscript $n$ indicates the number of |
| 371 |
> |
derivatives to be taken when deriving a given multipole energy. We |
| 372 |
> |
choose a function with guaranteed smooth derivatives -- a truncated |
| 373 |
> |
Taylor series of the function $f(r)$, e.g., |
| 374 |
|
% |
| 375 |
|
\begin{equation} |
| 376 |
< |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
| 376 |
> |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)}(r_c) . |
| 377 |
|
\end{equation} |
| 378 |
|
% |
| 379 |
|
The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
| 388 |
|
% |
| 389 |
|
\begin{equation} |
| 390 |
|
U_{C_{\bf a}D_{\bf b}}(r)= |
| 391 |
< |
\frac{C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} \frac {\partial f_1(r) }{\partial r_\alpha} |
| 392 |
< |
=\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
| 391 |
> |
C_{\bf a} D_{{\bf b}\alpha} \frac {\partial f_1(r) }{\partial r_\alpha} |
| 392 |
> |
= C_{\bf a} D_{{\bf b}\alpha} |
| 393 |
|
\frac {r_\alpha}{r} \frac {\partial f_1(r)}{\partial r} . |
| 394 |
|
\end{equation} |
| 395 |
|
% |
| 396 |
< |
The force that dipole $\bf b$ puts on charge $\bf a$ is |
| 396 |
> |
The force that dipole $\bf b$ exerts on charge $\bf a$ is |
| 397 |
|
% |
| 398 |
|
\begin{equation} |
| 399 |
< |
F_{C_{\bf a}D_{\bf b}\beta} =\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
| 399 |
> |
F_{C_{\bf a}D_{\bf b}\beta} = C_{\bf a} D_{{\bf b}\alpha} |
| 400 |
|
\left[ \frac{\delta_{\alpha\beta}}{r} \frac {\partial}{\partial r} + |
| 401 |
|
\frac{r_\alpha r_\beta}{r^2} |
| 402 |
|
\left( -\frac{1}{r} \frac {\partial} {\partial r} |
| 403 |
|
+ \frac {\partial ^2} {\partial r^2} \right) \right] f_1(r) . |
| 404 |
|
\end{equation} |
| 405 |
|
% |
| 406 |
< |
For $f(r)=1/r$, we find |
| 406 |
> |
For undamped coulombic interactions, $f(r)=1/r$, we find |
| 407 |
|
% |
| 408 |
|
\begin{equation} |
| 409 |
|
F_{C_{\bf a}D_{\bf b}\beta} = |
| 410 |
< |
\frac{C_{\bf a} D_{{\bf b}\beta} }{4\pi \epsilon_0r} |
| 410 |
> |
\frac{C_{\bf a} D_{{\bf b}\beta}}{r} |
| 411 |
|
\left[ -\frac{1}{r^2}+\frac{1}{r_c^2}-\frac{2(r-r_c)}{r_c^3} \right] |
| 412 |
< |
+\frac{C_{\bf a} D_{{\bf b}\alpha}r_\alpha r_\beta }{4\pi \epsilon_0} |
| 412 |
> |
+C_{\bf a} D_{{\bf b}\alpha}r_\alpha r_\beta |
| 413 |
|
\left[ \frac{3}{r^5}-\frac{3}{r^3r_c^2} \right] . |
| 414 |
|
\end{equation} |
| 415 |
|
% |
| 416 |
|
This expansion shows the expected $1/r^3$ dependence of the force. |
| 417 |
|
|
| 418 |
< |
In general, we write |
| 418 |
> |
In general, we can write |
| 419 |
|
% |
| 420 |
|
\begin{equation} |
| 421 |
< |
U=\frac{1}{4\pi \epsilon_0} (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 421 |
> |
U= (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 422 |
|
\label{generic} |
| 423 |
|
\end{equation} |
| 424 |
|
% |
| 425 |
< |
where $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole |
| 426 |
< |
and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for quadrupole-quadrupole. |
| 427 |
< |
An example is the case of quadrupole-quadrupole for which the $\text{prefactor}$ is |
| 428 |
< |
$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$ and the derivatives are |
| 429 |
< |
$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, with |
| 430 |
< |
implied summation combining the space indices. |
| 425 |
> |
with $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for |
| 426 |
> |
charge-quadrupole and dipole-dipole, $n=3$ for dipole-quadrupole, and |
| 427 |
> |
$n=4$ for quadrupole-quadrupole. For example, in |
| 428 |
> |
quadrupole-quadrupole interactions for which the $\text{prefactor}$ is |
| 429 |
> |
$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$, the derivatives are |
| 430 |
> |
$\partial^4/\partial r_\alpha \partial r_\beta \partial |
| 431 |
> |
r_\gamma \partial r_\delta$, with implied summation combining the |
| 432 |
> |
space indices. |
| 433 |
|
|
| 434 |
< |
To apply this method to the smeared-charge approach, |
| 435 |
< |
we write $f(r)=\text{erfc}(\alpha r)/r$. By using one function $f(r)$ for both |
| 436 |
< |
approaches, we simplify the tabulation of equations used. Because |
| 437 |
< |
of the many derivatives that are taken, the algebra is tedious and are summarized |
| 438 |
< |
in Appendices A and B. |
| 434 |
> |
In the formulas presented in the tables below, the placeholder |
| 435 |
> |
function $f(r)$ is used to represent the electrostatic kernel (either |
| 436 |
> |
damped or undamped). The main functions that go into the force and |
| 437 |
> |
torque terms, $g_n(r), h_n(r), s_n(r), \mathrm{~and~} t_n(r)$ are |
| 438 |
> |
successive derivatives of the shifted electrostatic kernel, $f_n(r)$ |
| 439 |
> |
of the same index $n$. The algebra required to evaluate energies, |
| 440 |
> |
forces and torques is somewhat tedious, so only the final forms are |
| 441 |
> |
presented in tables \ref{tab:tableenergy} and \ref{tab:tableFORCE}. |
| 442 |
|
|
| 443 |
|
\subsection{Gradient-shifted force (GSF) electrostatics} |
| 444 |
< |
|
| 445 |
< |
Note the method used in the previous subsection to shift a force is basically that of using |
| 446 |
< |
a truncated Taylor Series in the radius $r$. An alternate method exists, best explained by |
| 447 |
< |
writing one shifted formula for all interaction energies $U(r)$: |
| 444 |
> |
The second, and conceptually simpler approach to force-shifting |
| 445 |
> |
maintains only the linear $(r-r_c)$ term in the truncated Taylor |
| 446 |
> |
expansion, and has a similar interaction energy for all multipole |
| 447 |
> |
orders: |
| 448 |
|
\begin{equation} |
| 449 |
< |
U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big \lvert _{r_c} . |
| 449 |
> |
U^{\text{GSF}} = |
| 450 |
> |
U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 451 |
> |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r} |
| 452 |
> |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} . |
| 453 |
> |
\label{generic2} |
| 454 |
|
\end{equation} |
| 455 |
< |
Note that this method uses only the linear term, $(r-r_c)$ in the Taylor series, no higher order terms |
| 456 |
< |
$(r-r_c)^n$ appear. The primary difference between methods 1 and 2 originates |
| 457 |
< |
with the stage in the derivation where the Taylor Series is applied; in method 1, it is applied to the |
| 458 |
< |
kernel. In method 2, it is applied to individual interaction energies of the multipole expansion. |
| 459 |
< |
Terms from this method thus have the general form: |
| 455 |
> |
Both the potential and the gradient for force shifting are evaluated |
| 456 |
> |
for an image multipole projected onto the surface of the cutoff sphere |
| 457 |
> |
(see fig \ref{fig:shiftedMultipoles}). The image multipole retains |
| 458 |
> |
the orientation ($\hat{\mathbf{b}}$) of the interacting multipole. No |
| 459 |
> |
higher order terms $(r-r_c)^n$ appear. The primary difference between |
| 460 |
> |
the TSF and GSF methods is the stage at which the Taylor Series is |
| 461 |
> |
applied; in the Taylor-shifted approach, it is applied to the kernel |
| 462 |
> |
itself. In the Gradient-shifted approach, it is applied to individual |
| 463 |
> |
radial interactions terms in the multipole expansion. Energies from |
| 464 |
> |
this method thus have the general form: |
| 465 |
|
\begin{equation} |
| 466 |
< |
U=\frac{1}{4\pi \epsilon_0}\sum (\text{angular factor}) (\text{radial factor}). |
| 467 |
< |
\label{generic2} |
| 466 |
> |
U= \sum (\text{angular factor}) (\text{radial factor}). |
| 467 |
> |
\label{generic3} |
| 468 |
|
\end{equation} |
| 469 |
|
|
| 470 |
< |
Results for both methods can be summarized using the form of Eq.~(\ref{generic2}) |
| 471 |
< |
and are listed in Table I below. |
| 470 |
> |
Functional forms for both methods (TSF and GSF) can both be summarized |
| 471 |
> |
using the form of Eq.~(\ref{generic3}). The basic forms for the |
| 472 |
> |
energy, force, and torque expressions are tabulated for both shifting |
| 473 |
> |
approaches below -- for each separate orientational contribution, only |
| 474 |
> |
the radial factors differ between the two methods. |
| 475 |
|
|
| 476 |
|
\subsection{\label{sec:level2}Body and space axes} |
| 477 |
+ |
Although objects $\bf a$ and $\bf b$ rotate during a molecular |
| 478 |
+ |
dynamics (MD) simulation, their multipole tensors remain fixed in |
| 479 |
+ |
body-frame coordinates. While deriving force and torque expressions, |
| 480 |
+ |
it is therefore convenient to write the energies, forces, and torques |
| 481 |
+ |
in intermediate forms involving the vectors of the rotation matrices. |
| 482 |
+ |
We denote body axes for objects $\bf a$ and $\bf b$ using unit vectors |
| 483 |
+ |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$. |
| 484 |
+ |
In a typical simulation , the initial axes are obtained by |
| 485 |
+ |
diagonalizing the moment of inertia tensors for the objects. (N.B., |
| 486 |
+ |
the body axes are generally {\it not} the same as those for which the |
| 487 |
+ |
quadrupole moment is diagonal.) The rotation matrices are then |
| 488 |
+ |
propagated during the simulation. |
| 489 |
|
|
| 490 |
< |
Up to this point, all energies and forces have been written in terms of fixed space |
| 491 |
< |
coordinates $x$, $y$, $z$. Interaction energies are computed from the generic formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which |
| 453 |
< |
combine prefactors with radial functions. But because objects |
| 454 |
< |
$\bf a$ and $\bf b$ both translate and rotate as part of a MD simulation, |
| 455 |
< |
it is desirable to contract all $r$-dependent terms with dipole and quadrupole |
| 456 |
< |
moments expressed in terms of their body axes. |
| 457 |
< |
Since the interaction energy expressions will be used to derive both forces and torques, |
| 458 |
< |
we follow here the development of Allen and Germano, which was itself based on an |
| 459 |
< |
earlier paper by Price \em et al.\em |
| 460 |
< |
|
| 461 |
< |
Denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
| 462 |
< |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ referring to a convenient |
| 463 |
< |
set of inertial body axes. (Note, these body axes are generally not the same as those for which the |
| 464 |
< |
quadrupole moment is diagonal.) Then, |
| 465 |
< |
% |
| 490 |
> |
The rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
| 491 |
> |
expressed using these unit vectors: |
| 492 |
|
\begin{eqnarray} |
| 467 |
– |
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
| 468 |
– |
\hat{b}_m= b_{mx}\hat{x} + b_{my}\hat{y} + b_{mz}\hat{z} . |
| 469 |
– |
\end{eqnarray} |
| 470 |
– |
Allen and Germano define matrices $\hat{\mathbf {a}}$ |
| 471 |
– |
and $\hat{\mathbf {b}}$ using these unit vectors: |
| 472 |
– |
\begin{eqnarray} |
| 493 |
|
\hat{\mathbf {a}} = |
| 494 |
|
\begin{pmatrix} |
| 495 |
|
\hat{a}_1 \\ |
| 496 |
|
\hat{a}_2 \\ |
| 497 |
|
\hat{a}_3 |
| 498 |
< |
\end{pmatrix} |
| 479 |
< |
= |
| 480 |
< |
\begin{pmatrix} |
| 481 |
< |
a_{1x} \quad a_{1y} \quad a_{1z} \\ |
| 482 |
< |
a_{2x} \quad a_{2y} \quad a_{2z} \\ |
| 483 |
< |
a_{3x} \quad a_{3y} \quad a_{3z} |
| 484 |
< |
\end{pmatrix}\\ |
| 498 |
> |
\end{pmatrix}, \qquad |
| 499 |
|
\hat{\mathbf {b}} = |
| 500 |
|
\begin{pmatrix} |
| 501 |
|
\hat{b}_1 \\ |
| 502 |
|
\hat{b}_2 \\ |
| 503 |
|
\hat{b}_3 |
| 504 |
|
\end{pmatrix} |
| 491 |
– |
= |
| 492 |
– |
\begin{pmatrix} |
| 493 |
– |
b_{1x}\quad b_{1y} \quad b_{1z} \\ |
| 494 |
– |
b_{2x} \quad b_{2y} \quad b_{2z} \\ |
| 495 |
– |
b_{3x} \quad b_{3y} \quad b_{3z} |
| 496 |
– |
\end{pmatrix} . |
| 505 |
|
\end{eqnarray} |
| 506 |
|
% |
| 507 |
< |
These matrices convert from space-fixed $(xyz)$ to object-fixed $(123)$ coordinates. |
| 508 |
< |
All contractions of prefactors with derivatives of functions can be written in terms of these matrices. |
| 509 |
< |
It proves to be equally convenient to just write any contraction in terms of unit vectors |
| 510 |
< |
$\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. |
| 511 |
< |
We have found it useful to write angular-dependent terms in three different fashions, |
| 512 |
< |
illustrated by the following |
| 513 |
< |
three examples from the interaction-energy expressions: |
| 507 |
> |
These matrices convert from space-fixed $(xyz)$ to body-fixed $(123)$ |
| 508 |
> |
coordinates. |
| 509 |
> |
|
| 510 |
> |
Allen and Germano,\cite{Allen:2006fk} following earlier work by Price |
| 511 |
> |
{\em et al.},\cite{Price:1984fk} showed that if the interaction |
| 512 |
> |
energies are written explicitly in terms of $\hat{r}$ and the body |
| 513 |
> |
axes ($\hat{a}_m$, $\hat{b}_n$) : |
| 514 |
|
% |
| 515 |
+ |
\begin{equation} |
| 516 |
+ |
U(r, \{\hat{a}_m \cdot \hat{r} \}, |
| 517 |
+ |
\{\hat{b}_n\cdot \hat{r} \}, |
| 518 |
+ |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
| 519 |
+ |
\label{ugeneral} |
| 520 |
+ |
\end{equation} |
| 521 |
+ |
% |
| 522 |
+ |
the forces come out relatively cleanly, |
| 523 |
+ |
% |
| 524 |
+ |
\begin{equation} |
| 525 |
+ |
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
| 526 |
+ |
= \frac{\partial U}{\partial r} \hat{r} |
| 527 |
+ |
+ \sum_m \left[ |
| 528 |
+ |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 529 |
+ |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 530 |
+ |
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 531 |
+ |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 532 |
+ |
\right] \label{forceequation}. |
| 533 |
+ |
\end{equation} |
| 534 |
+ |
|
| 535 |
+ |
The torques can also be found in a relatively similar |
| 536 |
+ |
manner, |
| 537 |
+ |
% |
| 538 |
|
\begin{eqnarray} |
| 539 |
< |
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
| 540 |
< |
=D_{\bf {a}\alpha} D_{\bf {b}\alpha}= |
| 541 |
< |
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} \\ |
| 542 |
< |
r^2 \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right)= |
| 543 |
< |
r_\alpha Q_{\bf b \alpha \beta} r_\beta = r^2 |
| 544 |
< |
\sum_{mn}(\hat{r} \cdot \hat{b}_m) Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \\ |
| 545 |
< |
r ( \mathbf{D}_{\mathbf{a}} \cdot |
| 546 |
< |
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r})= |
| 547 |
< |
D_{\bf {a}\alpha} Q_{\bf b \alpha \beta} r_\beta |
| 548 |
< |
=r \sum_{lmn} D_{\mathbf{a}l} (\hat{a}_l \cdot \hat{b}_m ) Q_{\mathbf{b}mn} |
| 549 |
< |
(\hat{b}_n \cdot \hat{r}) . |
| 539 |
> |
\mathbf{\tau}_{\bf a} = |
| 540 |
> |
\sum_m |
| 541 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 542 |
> |
( \hat{r} \times \hat{a}_m ) |
| 543 |
> |
-\sum_{mn} |
| 544 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 545 |
> |
(\hat{a}_m \times \hat{b}_n) \\ |
| 546 |
> |
% |
| 547 |
> |
\mathbf{\tau}_{\bf b} = |
| 548 |
> |
\sum_m |
| 549 |
> |
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 550 |
> |
( \hat{r} \times \hat{b}_m) |
| 551 |
> |
+\sum_{mn} |
| 552 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 553 |
> |
(\hat{a}_m \times \hat{b}_n) . |
| 554 |
|
\end{eqnarray} |
| 555 |
+ |
|
| 556 |
+ |
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
| 557 |
+ |
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
| 558 |
+ |
We also made use of the identities, |
| 559 |
|
% |
| 560 |
< |
[Dan, perhaps there are better examples to show here.] |
| 560 |
> |
\begin{align} |
| 561 |
> |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 562 |
> |
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
| 563 |
> |
\right) \\ |
| 564 |
> |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 565 |
> |
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
| 566 |
> |
\right) . |
| 567 |
> |
\end{align} |
| 568 |
|
|
| 569 |
< |
In each line, the first two terms are written using space coordinates. The first form is standard |
| 570 |
< |
in the chemistry literature, and the second is ``physicist style'' using implied summation notation. The third |
| 571 |
< |
form shows explicitly sums over body indices and the dot products now indicate contractions using space indices. |
| 572 |
< |
We find the first form to be useful in writing equations prior to converting to computer code. The second form is helpful |
| 573 |
< |
in derivations of the interaction energy expressions. The third one is specifically helpful when deriving forces and torques, as will |
| 574 |
< |
be discussed below. |
| 569 |
> |
Many of the multipole contractions required can be written in one of |
| 570 |
> |
three equivalent forms using the unit vectors $\hat{r}$, $\hat{a}_m$, |
| 571 |
> |
and $\hat{b}_n$. In the torque expressions, it is useful to have the |
| 572 |
> |
angular-dependent terms available in all three fashions, e.g. for the |
| 573 |
> |
dipole-dipole contraction: |
| 574 |
> |
% |
| 575 |
> |
\begin{equation} |
| 576 |
> |
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
| 577 |
> |
= D_{\bf {a}\alpha} D_{\bf {b}\alpha} = |
| 578 |
> |
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} |
| 579 |
> |
\end{equation} |
| 580 |
> |
% |
| 581 |
> |
The first two forms are written using space coordinates. The first |
| 582 |
> |
form is standard in the chemistry literature, while the second is |
| 583 |
> |
expressed using implied summation notation. The third form shows |
| 584 |
> |
explicit sums over body indices and the dot products now indicate |
| 585 |
> |
contractions using space indices. |
| 586 |
|
|
| 587 |
+ |
In computing our force and torque expressions, we carried out most of |
| 588 |
+ |
the work in body coordinates, and have transformed the expressions |
| 589 |
+ |
back to space-frame coordinates, which are reported below. Interested |
| 590 |
+ |
readers may consult the supplemental information for this paper for |
| 591 |
+ |
the intermediate body-frame expressions. |
| 592 |
|
|
| 593 |
|
\subsection{The Self-Interaction \label{sec:selfTerm}} |
| 594 |
|
|
| 595 |
< |
The Wolf summation~\cite{Wolf99} and the later damped shifted force |
| 596 |
< |
(DSF) extension~\cite{Fennell06} included self-interactions that are |
| 597 |
< |
handled separately from the pairwise interactions between sites. The |
| 598 |
< |
self-term is normally calculated via a single loop over all sites in |
| 599 |
< |
the system, and is relatively cheap to evaluate. The self-interaction |
| 600 |
< |
has contributions from two sources: |
| 601 |
< |
\begin{itemize} |
| 602 |
< |
\item The neutralization procedure within the cutoff radius requires a |
| 603 |
< |
contribution from a charge opposite in sign, but equal in magnitude, |
| 604 |
< |
to the central charge, which has been spread out over the surface of |
| 605 |
< |
the cutoff sphere. For a system of undamped charges, the total |
| 606 |
< |
self-term is |
| 595 |
> |
In addition to cutoff-sphere neutralization, the Wolf |
| 596 |
> |
summation~\cite{Wolf99} and the damped shifted force (DSF) |
| 597 |
> |
extension~\cite{Fennell:2006zl} also included self-interactions that |
| 598 |
> |
are handled separately from the pairwise interactions between |
| 599 |
> |
sites. The self-term is normally calculated via a single loop over all |
| 600 |
> |
sites in the system, and is relatively cheap to evaluate. The |
| 601 |
> |
self-interaction has contributions from two sources. |
| 602 |
> |
|
| 603 |
> |
First, the neutralization procedure within the cutoff radius requires |
| 604 |
> |
a contribution from a charge opposite in sign, but equal in magnitude, |
| 605 |
> |
to the central charge, which has been spread out over the surface of |
| 606 |
> |
the cutoff sphere. For a system of undamped charges, the total |
| 607 |
> |
self-term is |
| 608 |
|
\begin{equation} |
| 609 |
|
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
| 610 |
|
\label{eq:selfTerm} |
| 611 |
|
\end{equation} |
| 612 |
< |
Note that in this potential and in all electrostatic quantities that |
| 613 |
< |
follow, the standard $4 \pi \epsilon_{0}$ has been omitted for |
| 614 |
< |
clarity. |
| 615 |
< |
\item Charge damping with the complementary error function is a |
| 616 |
< |
partial analogy to the Ewald procedure which splits the interaction |
| 617 |
< |
into real and reciprocal space sums. The real space sum is retained |
| 618 |
< |
in the Wolf and DSF methods. The reciprocal space sum is first |
| 619 |
< |
minimized by folding the largest contribution (the self-interaction) |
| 620 |
< |
into the self-interaction from charge neutralization of the damped |
| 621 |
< |
potential. The remainder of the reciprocal space portion is then |
| 622 |
< |
discarded (as this contributes the largest computational cost and |
| 560 |
< |
complexity to the Ewald sum). For a system containing only damped |
| 561 |
< |
charges, the complete self-interaction can be written as |
| 612 |
> |
|
| 613 |
> |
Second, charge damping with the complementary error function is a |
| 614 |
> |
partial analogy to the Ewald procedure which splits the interaction |
| 615 |
> |
into real and reciprocal space sums. The real space sum is retained |
| 616 |
> |
in the Wolf and DSF methods. The reciprocal space sum is first |
| 617 |
> |
minimized by folding the largest contribution (the self-interaction) |
| 618 |
> |
into the self-interaction from charge neutralization of the damped |
| 619 |
> |
potential. The remainder of the reciprocal space portion is then |
| 620 |
> |
discarded (as this contributes the largest computational cost and |
| 621 |
> |
complexity to the Ewald sum). For a system containing only damped |
| 622 |
> |
charges, the complete self-interaction can be written as |
| 623 |
|
\begin{equation} |
| 624 |
|
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
| 625 |
|
\frac{\alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
| 626 |
|
C_{\bf a}^{2}. |
| 627 |
|
\label{eq:dampSelfTerm} |
| 628 |
|
\end{equation} |
| 568 |
– |
\end{itemize} |
| 629 |
|
|
| 630 |
|
The extension of DSF electrostatics to point multipoles requires |
| 631 |
|
treatment of {\it both} the self-neutralization and reciprocal |
| 654 |
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
| 655 |
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
| 656 |
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
| 657 |
< |
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 657 |
> |
Quadrupole & $2 \mathbf{Q}_{\bf a}:\mathbf{Q}_{\bf a} + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 658 |
|
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
| 659 |
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
| 660 |
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
| 668 |
|
multipole orders. Symmetry prevents the charge-dipole and |
| 669 |
|
dipole-quadrupole interactions from contributing to the |
| 670 |
|
self-interaction. The functions that go into the self-neutralization |
| 671 |
< |
terms, $f(r), g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
| 672 |
< |
derivatives of the electrostatic kernel (either the undamped $1/r$ or |
| 673 |
< |
the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that are |
| 674 |
< |
evaluated at the cutoff distance. For undamped interactions, $f(r_c) |
| 675 |
< |
= 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For damped interactions, |
| 676 |
< |
$f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so on. Appendix XX |
| 677 |
< |
contains recursion relations that allow rapid evaluation of these |
| 678 |
< |
derivatives. |
| 671 |
> |
terms, $g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
| 672 |
> |
derivatives of the electrostatic kernel, $f(r)$ (either the undamped |
| 673 |
> |
$1/r$ or the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that |
| 674 |
> |
have been evaluated at the cutoff distance. For undamped |
| 675 |
> |
interactions, $f(r_c) = 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For |
| 676 |
> |
damped interactions, $f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so |
| 677 |
> |
on. Appendix \ref{SmithFunc} contains recursion relations that allow |
| 678 |
> |
rapid evaluation of these derivatives. |
| 679 |
|
|
| 680 |
< |
\section{Energies, forces, and torques} |
| 681 |
< |
\subsection{Interaction energies} |
| 682 |
< |
|
| 683 |
< |
We now list multipole interaction energies using a set of generic |
| 684 |
< |
radial functions. Table \ref{tab:tableenergy} maps between the |
| 685 |
< |
generic functions and the radial functions derived for both the |
| 686 |
< |
Taylor-shifted and Gradient-shifted methods. This set of equations is |
| 687 |
< |
written in terms of space coordinates: |
| 680 |
> |
\section{Interaction energies, forces, and torques} |
| 681 |
> |
The main result of this paper is a set of expressions for the |
| 682 |
> |
energies, forces and torques (up to quadrupole-quadrupole order) that |
| 683 |
> |
work for both the Taylor-shifted and Gradient-shifted approximations. |
| 684 |
> |
These expressions were derived using a set of generic radial |
| 685 |
> |
functions. Without using the shifting approximations mentioned above, |
| 686 |
> |
some of these radial functions would be identical, and the expressions |
| 687 |
> |
coalesce into the familiar forms for unmodified multipole-multipole |
| 688 |
> |
interactions. Table \ref{tab:tableenergy} maps between the generic |
| 689 |
> |
functions and the radial functions derived for both the Taylor-shifted |
| 690 |
> |
and Gradient-shifted methods. The energy equations are written in |
| 691 |
> |
terms of lab-frame representations of the dipoles, quadrupoles, and |
| 692 |
> |
the unit vector connecting the two objects, |
| 693 |
|
|
| 694 |
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
| 695 |
|
% |
| 698 |
|
% |
| 699 |
|
\begin{align} |
| 700 |
|
U_{C_{\bf a}C_{\bf b}}(r)=& |
| 701 |
< |
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) \label{uchch} |
| 701 |
> |
C_{\bf a} C_{\bf b} v_{01}(r) \label{uchch} |
| 702 |
|
\\ |
| 703 |
|
% |
| 704 |
|
% u ca db |
| 705 |
|
% |
| 706 |
|
U_{C_{\bf a}D_{\bf b}}(r)=& |
| 707 |
< |
\frac{C_{\bf a}}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
| 707 |
> |
C_{\bf a} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
| 708 |
|
\label{uchdip} |
| 709 |
|
\\ |
| 710 |
|
% |
| 711 |
|
% u ca qb |
| 712 |
|
% |
| 713 |
< |
U_{C_{\bf a}Q_{\bf b}}(r)=& |
| 714 |
< |
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigl[ \text{Tr}Q_{\bf b} v_{21}(r) |
| 715 |
< |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
| 713 |
> |
U_{C_{\bf a}Q_{\bf b}}(r)=& C_{\bf a } \Bigl[ \text{Tr}Q_{\bf b} |
| 714 |
> |
v_{21}(r) + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot |
| 715 |
> |
\hat{r} \right) v_{22}(r) \Bigr] |
| 716 |
|
\label{uchquad} |
| 717 |
|
\\ |
| 718 |
|
% |
| 726 |
|
% u da db |
| 727 |
|
% |
| 728 |
|
U_{D_{\bf a}D_{\bf b}}(r)=& |
| 729 |
< |
-\frac{1}{4\pi \epsilon_0} \Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
| 729 |
> |
-\Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
| 730 |
|
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
| 731 |
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
| 732 |
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
| 739 |
|
\begin{split} |
| 740 |
|
% 1 |
| 741 |
|
U_{D_{\bf a}Q_{\bf b}}(r) =& |
| 742 |
< |
-\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 742 |
> |
-\Bigl[ |
| 743 |
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 744 |
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
| 745 |
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
| 746 |
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r} ) \Bigr] v_{31}(r) \\ |
| 747 |
|
% 2 |
| 748 |
< |
&-\frac{1}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
| 748 |
> |
&- \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
| 749 |
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
| 750 |
|
\label{udipquad} |
| 751 |
|
\end{split} |
| 782 |
|
\begin{split} |
| 783 |
|
%1 |
| 784 |
|
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
| 785 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 785 |
> |
\Bigl[ |
| 786 |
|
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
| 787 |
< |
+2 \text{Tr} \left( |
| 788 |
< |
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
| 787 |
> |
+2 |
| 788 |
> |
\mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] v_{41}(r) |
| 789 |
|
\\ |
| 790 |
|
% 2 |
| 791 |
< |
&+ \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 791 |
> |
&+\Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 792 |
|
\left( \hat{r} \cdot |
| 793 |
|
\mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) |
| 794 |
|
+\text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 798 |
|
\Bigr] v_{42}(r) |
| 799 |
|
\\ |
| 800 |
|
% 4 |
| 801 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 801 |
> |
&+ |
| 802 |
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) |
| 803 |
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
| 804 |
|
\label{uquadquad} |
| 805 |
|
\end{split} |
| 806 |
|
\end{align} |
| 807 |
< |
|
| 807 |
> |
% |
| 808 |
|
Note that the energies of multipoles on site $\mathbf{b}$ interacting |
| 809 |
|
with those on site $\mathbf{a}$ can be obtained by swapping indices |
| 810 |
|
along with the sign of the intersite vector, $\hat{r}$. |
| 814 |
|
% TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
| 815 |
|
% |
| 816 |
|
|
| 817 |
< |
\begin{table*} |
| 818 |
< |
\caption{\label{tab:tableenergy}Radial functions used in the energy and torque equations. Functions |
| 819 |
< |
used in this table are defined in Appendices B and C.} |
| 820 |
< |
\begin{ruledtabular} |
| 821 |
< |
\begin{tabular}{|l|c|l|l} |
| 822 |
< |
Generic&Coulomb&Taylor-Shifted&Gradient-Shifted |
| 817 |
> |
\begin{sidewaystable} |
| 818 |
> |
\caption{\label{tab:tableenergy}Radial functions used in the energy |
| 819 |
> |
and torque equations. The $f, g, h, s, t, \mathrm{and} u$ |
| 820 |
> |
functions used in this table are defined in Appendices B and C.} |
| 821 |
> |
\begin{tabular}{|c|c|l|l|} \hline |
| 822 |
> |
Generic&Bare Coulomb&Taylor-Shifted&Gradient-Shifted |
| 823 |
|
\\ \hline |
| 824 |
|
% |
| 825 |
|
% |
| 851 |
|
$\frac{3}{r^3} $ & |
| 852 |
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
| 853 |
|
$\left(-\frac{g(r)}{r}+h(r) \right) |
| 854 |
< |
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right) $ \\ |
| 855 |
< |
&&&$ -(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
| 854 |
> |
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right)$ \\ |
| 855 |
> |
&&& $ ~~~-(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
| 856 |
|
\\ |
| 857 |
|
% |
| 858 |
|
% |
| 863 |
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
| 864 |
|
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
| 865 |
|
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
| 866 |
< |
&&&$ -(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)$ |
| 866 |
> |
&&&$ ~~~ -(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)$ |
| 867 |
|
\\ |
| 868 |
|
% |
| 869 |
|
$v_{32}(r)$ & |
| 871 |
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
| 872 |
|
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
| 873 |
|
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
| 874 |
< |
&&&$ -(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)$ |
| 874 |
> |
&&&$ ~~~ -(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)$ |
| 875 |
|
\\ |
| 876 |
|
% |
| 877 |
|
% |
| 882 |
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
| 883 |
|
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
| 884 |
|
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
| 885 |
< |
&&&$ -(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)$ |
| 885 |
> |
&&&$ ~~~ -(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)$ |
| 886 |
|
\\ |
| 887 |
|
% 2 |
| 888 |
|
$v_{42}(r)$ & |
| 890 |
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
| 891 |
|
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
| 892 |
|
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
| 893 |
< |
&&&$ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
| 893 |
> |
&&&$ ~~~ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
| 894 |
|
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
| 895 |
|
\\ |
| 896 |
|
% 3 |
| 898 |
|
$ \frac{105}{r^5} $ & |
| 899 |
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
| 900 |
|
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
| 901 |
< |
&&&$ -\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)$ \\ |
| 902 |
< |
&&&$ -(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} |
| 903 |
< |
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ |
| 901 |
> |
&&&$~~~ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
| 902 |
> |
&&&$~~~ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
| 903 |
> |
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ \hline |
| 904 |
|
\end{tabular} |
| 905 |
< |
\end{ruledtabular} |
| 841 |
< |
\end{table*} |
| 905 |
> |
\end{sidewaystable} |
| 906 |
|
% |
| 907 |
|
% |
| 908 |
|
% FORCE TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
| 909 |
|
% |
| 910 |
|
|
| 911 |
< |
\begin{table} |
| 911 |
> |
\begin{sidewaystable} |
| 912 |
|
\caption{\label{tab:tableFORCE}Radial functions used in the force equations.} |
| 913 |
< |
\begin{ruledtabular} |
| 914 |
< |
\begin{tabular}{cc} |
| 851 |
< |
Generic&Method 1 or Method 2 |
| 913 |
> |
\begin{tabular}{|c|c|l|l|} \hline |
| 914 |
> |
Function&Definition&Taylor-Shifted&Gradient-Shifted |
| 915 |
|
\\ \hline |
| 916 |
|
% |
| 917 |
|
% |
| 918 |
|
% |
| 919 |
|
$w_a(r)$& |
| 920 |
< |
$\frac{d v_{01}}{dr}$ \\ |
| 920 |
> |
$\frac{d v_{01}}{dr}$& |
| 921 |
> |
$g_0(r)$& |
| 922 |
> |
$g(r)-g(r_c)$ \\ |
| 923 |
|
% |
| 924 |
|
% |
| 925 |
|
$w_b(r)$ & |
| 926 |
< |
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $ \\ |
| 926 |
> |
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $& |
| 927 |
> |
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
| 928 |
> |
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
| 929 |
|
% |
| 930 |
|
$w_c(r)$ & |
| 931 |
< |
$\frac{v_{11}(r)}{r}$ \\ |
| 931 |
> |
$\frac{v_{11}(r)}{r}$ & |
| 932 |
> |
$\frac{g_1(r)}{r} $ & |
| 933 |
> |
$\frac{v_{11}(r)}{r}$\\ |
| 934 |
|
% |
| 935 |
|
% |
| 936 |
|
$w_d(r)$& |
| 937 |
< |
$\frac{d v_{21}}{dr}$ \\ |
| 937 |
> |
$\frac{d v_{21}}{dr}$& |
| 938 |
> |
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
| 939 |
> |
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
| 940 |
> |
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $ \\ |
| 941 |
|
% |
| 942 |
|
$w_e(r)$ & |
| 943 |
+ |
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
| 944 |
+ |
$\frac{v_{22}(r)}{r}$ & |
| 945 |
|
$\frac{v_{22}(r)}{r}$ \\ |
| 946 |
|
% |
| 947 |
|
% |
| 948 |
|
$w_f(r)$& |
| 949 |
< |
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$\\ |
| 949 |
> |
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$& |
| 950 |
> |
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
| 951 |
> |
$ \left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) $ \\ |
| 952 |
> |
&&& $ ~~~- \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) |
| 953 |
> |
\right)-\frac{2v_{22}(r)}{r}$\\ |
| 954 |
|
% |
| 955 |
|
$w_g(r)$& |
| 956 |
+ |
$\frac{v_{31}(r)}{r}$& |
| 957 |
+ |
$ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
| 958 |
|
$\frac{v_{31}(r)}{r}$\\ |
| 959 |
|
% |
| 960 |
|
$w_h(r)$ & |
| 961 |
< |
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$\\ |
| 961 |
> |
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$& |
| 962 |
> |
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 963 |
> |
$ \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) $ \\ |
| 964 |
> |
&&& $ ~~~ -\frac{v_{31}(r)}{r}$ \\ |
| 965 |
|
% 2 |
| 966 |
|
$w_i(r)$ & |
| 967 |
< |
$\frac{v_{32}(r)}{r}$ \\ |
| 967 |
> |
$\frac{v_{32}(r)}{r}$ & |
| 968 |
> |
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 969 |
> |
$\frac{v_{32}(r)}{r}$\\ |
| 970 |
|
% |
| 971 |
|
$w_j(r)$ & |
| 972 |
< |
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$ \\ |
| 972 |
> |
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$& |
| 973 |
> |
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
| 974 |
> |
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right)$ \\ |
| 975 |
> |
&&& $~~~-\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} |
| 976 |
> |
-\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
| 977 |
|
% |
| 978 |
|
$w_k(r)$ & |
| 979 |
< |
$\frac{d v_{41}}{dr} $ \\ |
| 979 |
> |
$\frac{d v_{41}}{dr} $ & |
| 980 |
> |
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 981 |
> |
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right) |
| 982 |
> |
-\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
| 983 |
|
% |
| 984 |
|
$w_l(r)$ & |
| 985 |
< |
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ \\ |
| 985 |
> |
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ & |
| 986 |
> |
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 987 |
> |
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
| 988 |
> |
&&& $~~~ -\left(-\frac{9g(r_c)}{r_c^4} +\frac{9h(r_c)}{r_c^3} -\frac{4s(r_c)}{r_c^2} +\frac{t(r_c)}{r_c} \right) |
| 989 |
> |
-\frac{2v_{42}(r)}{r}$\\ |
| 990 |
|
% |
| 991 |
|
$w_m(r)$ & |
| 992 |
< |
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$ \\ |
| 992 |
> |
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$& |
| 993 |
> |
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
| 994 |
> |
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$\\ |
| 995 |
> |
&&& $~~~- \left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
| 996 |
> |
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $\\ |
| 997 |
> |
&&& $~~~-\frac{4v_{43}(r)}{r}$ \\ |
| 998 |
|
% |
| 999 |
|
$w_n(r)$ & |
| 1000 |
< |
$\frac{v_{42}(r)}{r}$ \\ |
| 1000 |
> |
$\frac{v_{42}(r)}{r}$ & |
| 1001 |
> |
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 1002 |
> |
$\frac{v_{42}(r)}{r}$\\ |
| 1003 |
|
% |
| 1004 |
|
$w_o(r)$ & |
| 1005 |
< |
$\frac{v_{43}(r)}{r}$ \\ |
| 1005 |
> |
$\frac{v_{43}(r)}{r}$& |
| 1006 |
> |
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 1007 |
> |
$\frac{v_{43}(r)}{r}$ \\ \hline |
| 1008 |
|
% |
| 1009 |
|
|
| 1010 |
|
\end{tabular} |
| 1011 |
< |
\end{ruledtabular} |
| 907 |
< |
\end{table} |
| 1011 |
> |
\end{sidewaystable} |
| 1012 |
|
% |
| 1013 |
|
% |
| 1014 |
|
% |
| 1015 |
|
|
| 1016 |
|
\subsection{Forces} |
| 1017 |
< |
|
| 1018 |
< |
The force $\mathbf{F}_{\bf a}$ on $\bf{a}$ due to $\bf{b}$ is the negative of |
| 1019 |
< |
the force $\mathbf{F}_{\bf b}$ on $\bf{b}$ due to $\bf{a}$. For a simple charge-charge |
| 1020 |
< |
interaction, these forces will point along the $\pm \hat{r}$ directions, where |
| 1021 |
< |
$\mathbf{r}=\mathbf{r}_b - \mathbf{r}_a $. Thus |
| 1017 |
> |
The force on object $\bf{a}$, $\mathbf{F}_{\bf a}$, due to object |
| 1018 |
> |
$\bf{b}$ is the negative of the force on $\bf{b}$ due to $\bf{a}$. For |
| 1019 |
> |
a simple charge-charge interaction, these forces will point along the |
| 1020 |
> |
$\pm \hat{r}$ directions, where $\mathbf{r}=\mathbf{r}_b - |
| 1021 |
> |
\mathbf{r}_a $. Thus |
| 1022 |
|
% |
| 1023 |
|
\begin{equation} |
| 1024 |
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
| 1026 |
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
| 1027 |
|
\end{equation} |
| 1028 |
|
% |
| 1029 |
< |
The concept of obtaining a force from an energy by taking a gradient is the same for |
| 1030 |
< |
higher-order multipole interactions, the trick is to make sure that all |
| 1031 |
< |
$r$-dependent derivatives are considered. |
| 928 |
< |
As is pointed out by Allen and Germano, this is straightforward if the |
| 929 |
< |
interaction energies are written recognizing explicit |
| 930 |
< |
$\hat{r}$ and body axes ($\hat{a}_m$, $\hat{b}_n$) dependences: |
| 1029 |
> |
We list below the force equations written in terms of lab-frame |
| 1030 |
> |
coordinates. The radial functions used in the two methods are listed |
| 1031 |
> |
in Table \ref{tab:tableFORCE} |
| 1032 |
|
% |
| 1033 |
< |
\begin{equation} |
| 933 |
< |
U(r,\{\hat{a}_m \cdot \hat{r} \}, |
| 934 |
< |
\{\hat{b}_n\cdot \hat{r} \} |
| 935 |
< |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
| 936 |
< |
\label{ugeneral} |
| 937 |
< |
\end{equation} |
| 1033 |
> |
%SPACE COORDINATES FORCE EQUATIONS |
| 1034 |
|
% |
| 939 |
– |
Then, |
| 940 |
– |
% |
| 941 |
– |
\begin{equation} |
| 942 |
– |
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
| 943 |
– |
= \frac{\partial U}{\partial r} \hat{r} |
| 944 |
– |
+ \sum_m \left[ |
| 945 |
– |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 946 |
– |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 947 |
– |
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 948 |
– |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 949 |
– |
\right] \label{forceequation}. |
| 950 |
– |
\end{equation} |
| 951 |
– |
% |
| 952 |
– |
Note our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ is opposite |
| 953 |
– |
that of Allen and Germano. In simplifying the algebra, we also use: |
| 954 |
– |
% |
| 955 |
– |
\begin{eqnarray} |
| 956 |
– |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 957 |
– |
= \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
| 958 |
– |
\right) \\ |
| 959 |
– |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 960 |
– |
= \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
| 961 |
– |
\right) . |
| 962 |
– |
\end{eqnarray} |
| 963 |
– |
% |
| 964 |
– |
We list below the force equations written in terms of space coordinates. The |
| 965 |
– |
radial functions used in the two methods are listed in Table II. |
| 966 |
– |
% |
| 967 |
– |
%SPACE COORDINATES FORCE EQUTIONS |
| 968 |
– |
% |
| 1035 |
|
% ************************************************************************** |
| 1036 |
|
% f ca cb |
| 1037 |
|
% |
| 1038 |
< |
\begin{equation} |
| 1039 |
< |
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
| 1040 |
< |
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
| 975 |
< |
\end{equation} |
| 1038 |
> |
\begin{align} |
| 1039 |
> |
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} =& |
| 1040 |
> |
C_{\bf a} C_{\bf b} w_a(r) \hat{r} \\ |
| 1041 |
|
% |
| 1042 |
|
% |
| 1043 |
|
% |
| 1044 |
< |
\begin{equation} |
| 1045 |
< |
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
| 981 |
< |
\frac{C_{\bf a}}{4\pi \epsilon_0} \Bigl[ |
| 1044 |
> |
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} =& |
| 1045 |
> |
C_{\bf a} \Bigl[ |
| 1046 |
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{b}} \right) |
| 1047 |
|
w_b(r) \hat{r} |
| 1048 |
< |
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] |
| 985 |
< |
\end{equation} |
| 1048 |
> |
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] \\ |
| 1049 |
|
% |
| 1050 |
|
% |
| 1051 |
|
% |
| 1052 |
< |
\begin{equation} |
| 1053 |
< |
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
| 991 |
< |
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigr[ |
| 1052 |
> |
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} =& |
| 1053 |
> |
C_{\bf a } \Bigr[ |
| 1054 |
|
\text{Tr}\mathbf{Q}_{\bf b} w_d(r) \hat{r} |
| 1055 |
|
+ 2 \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} w_e(r) |
| 1056 |
< |
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
| 1057 |
< |
\end{equation} |
| 1056 |
> |
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} |
| 1057 |
> |
\right) w_f(r) \hat{r} \Bigr] \\ |
| 1058 |
|
% |
| 1059 |
|
% |
| 1060 |
|
% |
| 1061 |
< |
\begin{equation} |
| 1062 |
< |
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1063 |
< |
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} \Bigl[ |
| 1064 |
< |
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
| 1065 |
< |
+ \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
| 1066 |
< |
\end{equation} |
| 1061 |
> |
% \begin{equation} |
| 1062 |
> |
% \mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1063 |
> |
% -C_{\bf{b}} \Bigl[ |
| 1064 |
> |
% \left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
| 1065 |
> |
% + \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
| 1066 |
> |
% \end{equation} |
| 1067 |
|
% |
| 1068 |
|
% |
| 1069 |
|
% |
| 1070 |
< |
\begin{equation} |
| 1071 |
< |
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} = |
| 1010 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1070 |
> |
\begin{split} |
| 1071 |
> |
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
| 1072 |
|
- \mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} w_d(r) \hat{r} |
| 1073 |
|
+ \left( \mathbf{D}_{\mathbf {a}} |
| 1074 |
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
| 1075 |
< |
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r) |
| 1075 |
> |
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r)\\ |
| 1076 |
|
% 2 |
| 1077 |
< |
- \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
| 1078 |
< |
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} \Bigr] |
| 1079 |
< |
\end{equation} |
| 1077 |
> |
& - \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
| 1078 |
> |
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} |
| 1079 |
> |
\end{split}\\ |
| 1080 |
|
% |
| 1081 |
|
% |
| 1082 |
|
% |
| 1022 |
– |
\begin{equation} |
| 1083 |
|
\begin{split} |
| 1084 |
< |
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
| 1025 |
< |
& - \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1084 |
> |
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} =& - \Bigl[ |
| 1085 |
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} \mathbf{ D}_{\mathbf{a}} |
| 1086 |
|
+2 \mathbf{D}_{\mathbf{a}} \cdot |
| 1087 |
|
\mathbf{Q}_{\mathbf{b}} \Bigr] w_g(r) |
| 1088 |
< |
- \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1088 |
> |
- \Bigl[ |
| 1089 |
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1090 |
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) |
| 1091 |
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
| 1092 |
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1093 |
|
% 3 |
| 1094 |
< |
& - \frac{1}{4\pi \epsilon_0} \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
| 1094 |
> |
& - \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
| 1095 |
|
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \Bigr] |
| 1096 |
|
w_i(r) |
| 1097 |
|
% 4 |
| 1098 |
< |
-\frac{1}{4\pi \epsilon_0} |
| 1098 |
> |
- |
| 1099 |
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) |
| 1100 |
< |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} |
| 1042 |
< |
\end{split} |
| 1043 |
< |
\end{equation} |
| 1100 |
> |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} \end{split} \\ |
| 1101 |
|
% |
| 1102 |
|
% |
| 1103 |
< |
\begin{equation} |
| 1104 |
< |
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
| 1105 |
< |
\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
| 1106 |
< |
\text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
| 1107 |
< |
+ 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
| 1108 |
< |
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
| 1109 |
< |
\end{equation} |
| 1103 |
> |
% \begin{equation} |
| 1104 |
> |
% \mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
| 1105 |
> |
% \frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
| 1106 |
> |
% \text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
| 1107 |
> |
% + 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
| 1108 |
> |
% + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
| 1109 |
> |
% \end{equation} |
| 1110 |
> |
% % |
| 1111 |
> |
% \begin{equation} |
| 1112 |
> |
% \begin{split} |
| 1113 |
> |
% \mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
| 1114 |
> |
% &\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1115 |
> |
% \text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
| 1116 |
> |
% +2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
| 1117 |
> |
% % 2 |
| 1118 |
> |
% + \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1119 |
> |
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
| 1120 |
> |
% +2 (\mathbf{D}_{\mathbf{b}} \cdot |
| 1121 |
> |
% \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1122 |
> |
% % 3 |
| 1123 |
> |
% &+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
| 1124 |
> |
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
| 1125 |
> |
% +2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
| 1126 |
> |
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
| 1127 |
> |
% % 4 |
| 1128 |
> |
% +\frac{1}{4\pi \epsilon_0} |
| 1129 |
> |
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
| 1130 |
> |
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
| 1131 |
> |
% \end{split} |
| 1132 |
> |
% \end{equation} |
| 1133 |
|
% |
| 1054 |
– |
\begin{equation} |
| 1055 |
– |
\begin{split} |
| 1056 |
– |
\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
| 1057 |
– |
&\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1058 |
– |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
| 1059 |
– |
+2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
| 1060 |
– |
% 2 |
| 1061 |
– |
+ \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1062 |
– |
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
| 1063 |
– |
+2 (\mathbf{D}_{\mathbf{b}} \cdot |
| 1064 |
– |
\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1065 |
– |
% 3 |
| 1066 |
– |
&+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
| 1067 |
– |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
| 1068 |
– |
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
| 1069 |
– |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
| 1070 |
– |
% 4 |
| 1071 |
– |
+\frac{1}{4\pi \epsilon_0} |
| 1072 |
– |
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
| 1073 |
– |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
| 1074 |
– |
\end{split} |
| 1075 |
– |
\end{equation} |
| 1134 |
|
% |
| 1135 |
|
% |
| 1078 |
– |
% |
| 1079 |
– |
\begin{equation} |
| 1136 |
|
\begin{split} |
| 1137 |
< |
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
| 1138 |
< |
+\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1139 |
< |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
| 1140 |
< |
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
| 1137 |
> |
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
| 1138 |
> |
\Bigl[ |
| 1139 |
> |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1140 |
> |
+ 2 \mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] w_k(r) \hat{r} \\ |
| 1141 |
|
% 2 |
| 1142 |
< |
+\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1142 |
> |
&+ \Bigl[ |
| 1143 |
|
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
| 1144 |
|
+ 2\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) |
| 1145 |
|
% 3 |
| 1146 |
|
+4 (\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
| 1147 |
|
+ 4(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}}) \Bigr] w_n(r) \\ |
| 1148 |
|
% 4 |
| 1149 |
< |
+ \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1149 |
> |
&+ \Bigl[ |
| 1150 |
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
| 1151 |
|
+ \text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1152 |
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
| 1154 |
|
+4 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot |
| 1155 |
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
| 1156 |
|
% |
| 1157 |
< |
+ \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1157 |
> |
&+ \Bigl[ |
| 1158 |
|
+ 2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
| 1159 |
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
| 1160 |
|
%6 |
| 1161 |
|
+2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
| 1162 |
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_o(r) \\ |
| 1163 |
|
% 7 |
| 1164 |
< |
+ \frac{1}{4\pi \epsilon_0} |
| 1164 |
> |
&+ |
| 1165 |
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
| 1166 |
< |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} |
| 1167 |
< |
\end{split} |
| 1168 |
< |
\end{equation} |
| 1166 |
> |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} \end{split} |
| 1167 |
> |
\end{align} |
| 1168 |
> |
Note that the forces for higher multipoles on site $\mathbf{a}$ |
| 1169 |
> |
interacting with those of lower order on site $\mathbf{b}$ can be |
| 1170 |
> |
obtained by swapping indices in the expressions above. |
| 1171 |
> |
|
| 1172 |
|
% |
| 1173 |
+ |
% Torques SECTION ----------------------------------------------------------------------------------------- |
| 1174 |
|
% |
| 1115 |
– |
% TORQUES SECTION ----------------------------------------------------------------------------------------- |
| 1116 |
– |
% |
| 1175 |
|
\subsection{Torques} |
| 1176 |
|
|
| 1119 |
– |
Following again Allen and Germano, when energies are written in the form |
| 1120 |
– |
of Eq.~({\ref{ugeneral}), then torques can be expressed as: |
| 1177 |
|
% |
| 1178 |
< |
\begin{eqnarray} |
| 1179 |
< |
\mathbf{\tau}_{\bf a} = |
| 1124 |
< |
\sum_m |
| 1125 |
< |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 1126 |
< |
( \hat{r} \times \hat{a}_m ) |
| 1127 |
< |
-\sum_{mn} |
| 1128 |
< |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 1129 |
< |
(\hat{a}_m \times \hat{b}_n) \\ |
| 1178 |
> |
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
| 1179 |
> |
methods are given in space-frame coordinates: |
| 1180 |
|
% |
| 1131 |
– |
\mathbf{\tau}_{\bf b} = |
| 1132 |
– |
\sum_m |
| 1133 |
– |
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 1134 |
– |
( \hat{r} \times \hat{b}_m) |
| 1135 |
– |
+\sum_{mn} |
| 1136 |
– |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 1137 |
– |
(\hat{a}_m \times \hat{b}_n) . |
| 1138 |
– |
\end{eqnarray} |
| 1181 |
|
% |
| 1182 |
+ |
\begin{align} |
| 1183 |
+ |
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} =& |
| 1184 |
+ |
C_{\bf a} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) \\ |
| 1185 |
|
% |
| 1141 |
– |
Here we list the torque equations written in terms of space coordinates. |
| 1186 |
|
% |
| 1187 |
|
% |
| 1188 |
+ |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} =& |
| 1189 |
+ |
2C_{\bf a} |
| 1190 |
+ |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) \\ |
| 1191 |
|
% |
| 1145 |
– |
\begin{equation} |
| 1146 |
– |
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
| 1147 |
– |
\frac{C_{\bf a}}{4\pi \epsilon_0} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) |
| 1148 |
– |
\end{equation} |
| 1192 |
|
% |
| 1193 |
|
% |
| 1194 |
+ |
% \begin{equation} |
| 1195 |
+ |
% \mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1196 |
+ |
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1197 |
+ |
% (\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
| 1198 |
+ |
% \end{equation} |
| 1199 |
|
% |
| 1152 |
– |
\begin{equation} |
| 1153 |
– |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
| 1154 |
– |
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
| 1155 |
– |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) |
| 1156 |
– |
\end{equation} |
| 1200 |
|
% |
| 1201 |
|
% |
| 1202 |
< |
% |
| 1203 |
< |
\begin{equation} |
| 1161 |
< |
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1162 |
< |
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1163 |
< |
(\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
| 1164 |
< |
\end{equation} |
| 1165 |
< |
% |
| 1166 |
< |
% |
| 1167 |
< |
% |
| 1168 |
< |
\begin{equation} |
| 1169 |
< |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} = |
| 1170 |
< |
\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
| 1202 |
> |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
| 1203 |
> |
\mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
| 1204 |
|
% 2 |
| 1205 |
< |
-\frac{1}{4\pi \epsilon_0} |
| 1205 |
> |
- |
| 1206 |
|
(\hat{r} \times \mathbf{D}_{\mathbf {a}} ) |
| 1207 |
< |
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
| 1175 |
< |
\end{equation} |
| 1207 |
> |
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r)\\ |
| 1208 |
|
% |
| 1209 |
|
% |
| 1210 |
|
% |
| 1211 |
< |
\begin{equation} |
| 1212 |
< |
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
| 1213 |
< |
-\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
| 1214 |
< |
% 2 |
| 1215 |
< |
+\frac{1}{4\pi \epsilon_0} |
| 1216 |
< |
(\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
| 1217 |
< |
(\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
| 1218 |
< |
\end{equation} |
| 1211 |
> |
% \begin{equation} |
| 1212 |
> |
% \mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
| 1213 |
> |
% -\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
| 1214 |
> |
% % 2 |
| 1215 |
> |
% +\frac{1}{4\pi \epsilon_0} |
| 1216 |
> |
% (\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
| 1217 |
> |
% (\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
| 1218 |
> |
% \end{equation} |
| 1219 |
|
% |
| 1220 |
|
% |
| 1221 |
|
% |
| 1222 |
< |
\begin{equation} |
| 1223 |
< |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
| 1192 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1222 |
> |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} =& |
| 1223 |
> |
\Bigl[ |
| 1224 |
|
-\text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1225 |
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
| 1226 |
|
+2 \mathbf{D}_{\mathbf{a}} \times |
| 1227 |
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
| 1228 |
|
\Bigr] v_{31}(r) |
| 1229 |
|
% 3 |
| 1230 |
< |
-\frac{1}{4\pi \epsilon_0} |
| 1231 |
< |
\ (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
| 1201 |
< |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r) |
| 1202 |
< |
\end{equation} |
| 1230 |
> |
- (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
| 1231 |
> |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r)\\ |
| 1232 |
|
% |
| 1233 |
|
% |
| 1234 |
|
% |
| 1235 |
< |
\begin{equation} |
| 1236 |
< |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} = |
| 1208 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1235 |
> |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} =& |
| 1236 |
> |
\Bigl[ |
| 1237 |
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \times |
| 1238 |
|
\hat{r} |
| 1239 |
|
-2 \mathbf{D}_{\mathbf{a}} \times |
| 1240 |
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
| 1241 |
|
\Bigr] v_{31}(r) |
| 1242 |
|
% 2 |
| 1243 |
< |
+\frac{2}{4\pi \epsilon_0} |
| 1243 |
> |
+ |
| 1244 |
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}}) |
| 1245 |
< |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r) |
| 1218 |
< |
\end{equation} |
| 1245 |
> |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r)\\ |
| 1246 |
|
% |
| 1247 |
|
% |
| 1248 |
|
% |
| 1249 |
< |
\begin{equation} |
| 1250 |
< |
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
| 1251 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1252 |
< |
-2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
| 1253 |
< |
+2 \mathbf{D}_{\mathbf{b}} \times |
| 1254 |
< |
(\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
| 1255 |
< |
\Bigr] v_{31}(r) |
| 1256 |
< |
% 3 |
| 1257 |
< |
- \frac{2}{4\pi \epsilon_0} |
| 1258 |
< |
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
| 1259 |
< |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
| 1260 |
< |
\end{equation} |
| 1249 |
> |
% \begin{equation} |
| 1250 |
> |
% \mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
| 1251 |
> |
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1252 |
> |
% -2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
| 1253 |
> |
% +2 \mathbf{D}_{\mathbf{b}} \times |
| 1254 |
> |
% (\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
| 1255 |
> |
% \Bigr] v_{31}(r) |
| 1256 |
> |
% % 3 |
| 1257 |
> |
% - \frac{2}{4\pi \epsilon_0} |
| 1258 |
> |
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
| 1259 |
> |
% (\hat{r} \cdot \mathbf |
| 1260 |
> |
% {Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
| 1261 |
> |
% \end{equation} |
| 1262 |
|
% |
| 1263 |
|
% |
| 1264 |
|
% |
| 1265 |
< |
\begin{equation} |
| 1266 |
< |
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
| 1267 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1268 |
< |
\text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1269 |
< |
(\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
| 1270 |
< |
+2 \mathbf{D}_{\mathbf{b}} \times |
| 1271 |
< |
( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
| 1272 |
< |
% 2 |
| 1273 |
< |
+\frac{1}{4\pi \epsilon_0} |
| 1274 |
< |
(\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
| 1275 |
< |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
| 1276 |
< |
\end{equation} |
| 1265 |
> |
% \begin{equation} |
| 1266 |
> |
% \mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
| 1267 |
> |
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1268 |
> |
% \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1269 |
> |
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
| 1270 |
> |
% +2 \mathbf{D}_{\mathbf{b}} \times |
| 1271 |
> |
% ( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
| 1272 |
> |
% % 2 |
| 1273 |
> |
% +\frac{1}{4\pi \epsilon_0} |
| 1274 |
> |
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
| 1275 |
> |
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
| 1276 |
> |
% \end{equation} |
| 1277 |
|
% |
| 1278 |
|
% |
| 1279 |
|
% |
| 1252 |
– |
\begin{equation} |
| 1280 |
|
\begin{split} |
| 1281 |
< |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
| 1282 |
< |
&-\frac{4}{4\pi \epsilon_0} |
| 1281 |
> |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
| 1282 |
> |
-4 |
| 1283 |
|
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} |
| 1284 |
|
v_{41}(r) \\ |
| 1285 |
|
% 2 |
| 1286 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 1286 |
> |
&+ |
| 1287 |
|
\Bigl[-2\text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1288 |
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times \hat{r} |
| 1289 |
|
+4 \hat{r} \times |
| 1292 |
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} )\times |
| 1293 |
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} ) \Bigr] v_{42}(r) \\ |
| 1294 |
|
% 4 |
| 1295 |
< |
&+ \frac{2}{4\pi \epsilon_0} |
| 1295 |
> |
&+ 2 |
| 1296 |
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
| 1297 |
< |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) |
| 1271 |
< |
\end{split} |
| 1272 |
< |
\end{equation} |
| 1297 |
> |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) \end{split}\\ |
| 1298 |
|
% |
| 1299 |
|
% |
| 1300 |
|
% |
| 1276 |
– |
\begin{equation} |
| 1301 |
|
\begin{split} |
| 1302 |
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} = |
| 1303 |
< |
&\frac{4}{4\pi \epsilon_0} |
| 1303 |
> |
&4 |
| 1304 |
|
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} v_{41}(r) \\ |
| 1305 |
|
% 2 |
| 1306 |
< |
&+ \frac{1}{4\pi \epsilon_0} \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1306 |
> |
&+ \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1307 |
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \times \hat{r} |
| 1308 |
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot |
| 1309 |
|
\mathbf{Q}_{{\mathbf b}} ) \times |
| 1312 |
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
| 1313 |
|
\Bigr] v_{42}(r) \\ |
| 1314 |
|
% 4 |
| 1315 |
< |
&+ \frac{2}{4\pi \epsilon_0} |
| 1315 |
> |
&+2 |
| 1316 |
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
| 1317 |
< |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) |
| 1318 |
< |
\end{split} |
| 1295 |
< |
\end{equation} |
| 1317 |
> |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r)\end{split} |
| 1318 |
> |
\end{align} |
| 1319 |
|
% |
| 1320 |
+ |
Here, we have defined the matrix cross product in an identical form |
| 1321 |
+ |
as in Ref. \onlinecite{Smith98}: |
| 1322 |
+ |
\begin{equation} |
| 1323 |
+ |
\left[\mathbf{A} \times \mathbf{B}\right]_\alpha = \sum_\beta |
| 1324 |
+ |
\left[\mathbf{A}_{\alpha+1,\beta} \mathbf{B}_{\alpha+2,\beta} |
| 1325 |
+ |
-\mathbf{A}_{\alpha+2,\beta} \mathbf{B}_{\alpha+2,\beta} |
| 1326 |
+ |
\right] |
| 1327 |
+ |
\end{equation} |
| 1328 |
+ |
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic |
| 1329 |
+ |
permuations of the matrix indices. |
| 1330 |
|
|
| 1331 |
+ |
All of the radial functions required for torques are identical with |
| 1332 |
+ |
the radial functions previously computed for the interaction energies. |
| 1333 |
+ |
These are tabulated for both shifted force methods in table |
| 1334 |
+ |
\ref{tab:tableenergy}. The torques for higher multipoles on site |
| 1335 |
+ |
$\mathbf{a}$ interacting with those of lower order on site |
| 1336 |
+ |
$\mathbf{b}$ can be obtained by swapping indices in the expressions |
| 1337 |
+ |
above. |
| 1338 |
+ |
|
| 1339 |
+ |
\section{Related real-space methods} |
| 1340 |
+ |
One can also formulate a shifted potential, |
| 1341 |
+ |
\begin{equation} |
| 1342 |
+ |
U^{\text{SP}} = U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 1343 |
+ |
U(\mathbf{r}_c, \hat{\mathbf{a}}, \hat{\mathbf{b}}), |
| 1344 |
+ |
\label{eq:SP} |
| 1345 |
+ |
\end{equation} |
| 1346 |
+ |
obtained by projecting the image multipole onto the surface of the |
| 1347 |
+ |
cutoff sphere. The shifted potential (SP) can be thought of as a |
| 1348 |
+ |
simple extension to the original Wolf method. The energies and |
| 1349 |
+ |
torques for the SP can be easily obtained by zeroing out the $(r-r_c)$ |
| 1350 |
+ |
terms in the final column of table \ref{tab:tableenergy}. SP forces |
| 1351 |
+ |
(which retain discontinuities at the cutoff sphere) can be obtained by |
| 1352 |
+ |
eliminating all functions that depend on $r_c$ in the last column of |
| 1353 |
+ |
table \ref{tab:tableFORCE}. The self-energy contributions to the SP |
| 1354 |
+ |
potential are identical to both the GSF and TSF methods. |
| 1355 |
+ |
|
| 1356 |
|
\section{Comparison to known multipolar energies} |
| 1357 |
|
|
| 1358 |
|
To understand how these new real-space multipole methods behave in |
| 1359 |
|
computer simulations, it is vital to test against established methods |
| 1360 |
|
for computing electrostatic interactions in periodic systems, and to |
| 1361 |
|
evaluate the size and sources of any errors that arise from the |
| 1362 |
< |
real-space cutoffs. In this paper we test Taylor-shifted and |
| 1363 |
< |
Gradient-shifted electrostatics against analytical methods for |
| 1364 |
< |
computing the energies of ordered multipolar arrays. In the following |
| 1365 |
< |
paper, we test the new methods against the multipolar Ewald sum for |
| 1366 |
< |
computing the energies, forces and torques for a wide range of typical |
| 1367 |
< |
condensed-phase (disordered) systems. |
| 1362 |
> |
real-space cutoffs. In this paper we test both TSF and GSF |
| 1363 |
> |
electrostatics against analytical methods for computing the energies |
| 1364 |
> |
of ordered multipolar arrays. In the following paper, we test the new |
| 1365 |
> |
methods against the multipolar Ewald sum for computing the energies, |
| 1366 |
> |
forces and torques for a wide range of typical condensed-phase |
| 1367 |
> |
(disordered) systems. |
| 1368 |
|
|
| 1369 |
|
Because long-range electrostatic effects can be significant in |
| 1370 |
|
crystalline materials, ordered multipolar arrays present one of the |
| 1373 |
|
who computed the energies of ordered dipole arrays of zero |
| 1374 |
|
magnetization and obtained a number of these constants.\cite{Sauer} |
| 1375 |
|
This theory was developed more completely by Luttinger and |
| 1376 |
< |
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays and |
| 1377 |
< |
other periodic structures. We have repeated the Luttinger \& Tisza |
| 1320 |
< |
series summations to much higher order and obtained the following |
| 1321 |
< |
energy constants (converged to one part in $10^9$): |
| 1322 |
< |
\begin{table*} |
| 1323 |
< |
\centering{ |
| 1324 |
< |
\caption{Luttinger \& Tisza arrays and their associated |
| 1325 |
< |
energy constants. Type "A" arrays have nearest neighbor strings of |
| 1326 |
< |
antiparallel dipoles. Type "B" arrays have nearest neighbor |
| 1327 |
< |
strings of antiparallel dipoles if the dipoles are contained in a |
| 1328 |
< |
plane perpendicular to the dipole direction that passes through |
| 1329 |
< |
the dipole.} |
| 1330 |
< |
} |
| 1331 |
< |
\label{tab:LT} |
| 1332 |
< |
\begin{ruledtabular} |
| 1333 |
< |
\begin{tabular}{cccc} |
| 1334 |
< |
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
| 1335 |
< |
A & SC & 001 & -2.676788684 \\ |
| 1336 |
< |
A & BCC & 001 & 0 \\ |
| 1337 |
< |
A & BCC & 111 & -1.770078733 \\ |
| 1338 |
< |
A & FCC & 001 & 2.166932835 \\ |
| 1339 |
< |
A & FCC & 011 & -1.083466417 \\ |
| 1376 |
> |
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays |
| 1377 |
> |
and other periodic structures. |
| 1378 |
|
|
| 1379 |
< |
* & BCC & minimum & -1.985920929 \\ |
| 1380 |
< |
|
| 1381 |
< |
B & SC & 001 & -2.676788684 \\ |
| 1382 |
< |
B & BCC & 001 & -1.338394342 \\ |
| 1383 |
< |
B & BCC & 111 & -1.770078733 \\ |
| 1384 |
< |
B & FCC & 001 & -1.083466417 \\ |
| 1385 |
< |
B & FCC & 011 & -1.807573634 |
| 1348 |
< |
\end{tabular} |
| 1349 |
< |
\end{ruledtabular} |
| 1350 |
< |
\end{table*} |
| 1351 |
< |
|
| 1352 |
< |
In addition to the A and B arrays, there is an additional minimum |
| 1379 |
> |
To test the new electrostatic methods, we have constructed very large, |
| 1380 |
> |
$N=$ 16,000~(bcc) arrays of dipoles in the orientations described in |
| 1381 |
> |
Ref. \onlinecite{LT}. These structures include ``A'' lattices with |
| 1382 |
> |
nearest neighbor chains of antiparallel dipoles, as well as ``B'' |
| 1383 |
> |
lattices with nearest neighbor strings of antiparallel dipoles if the |
| 1384 |
> |
dipoles are contained in a plane perpendicular to the dipole direction |
| 1385 |
> |
that passes through the dipole. We have also studied the minimum |
| 1386 |
|
energy structure for the BCC lattice that was found by Luttinger \& |
| 1387 |
< |
Tisza. The total electrostatic energy for an array is given by: |
| 1387 |
> |
Tisza. The total electrostatic energy for any of the arrays is given |
| 1388 |
> |
by: |
| 1389 |
|
\begin{equation} |
| 1390 |
|
E = C N^2 \mu^2 |
| 1391 |
|
\end{equation} |
| 1392 |
< |
where $C$ is the energy constant given above, $N$ is the number of |
| 1393 |
< |
dipoles per unit volume, and $\mu$ is the strength of the dipole. |
| 1392 |
> |
where $C$ is the energy constant (equivalent to the Madelung |
| 1393 |
> |
constant), $N$ is the number of dipoles per unit volume, and $\mu$ is |
| 1394 |
> |
the strength of the dipole. Energy constants (converged to 1 part in |
| 1395 |
> |
$10^9$) are given in the supplemental information. |
| 1396 |
|
|
| 1397 |
< |
{\it Quadrupolar} analogues to the Madelung constants were first worked out by Nagai and Nakamura who |
| 1398 |
< |
computed the energies of selected quadrupole arrays based on |
| 1399 |
< |
extensions to the Luttinger and Tisza |
| 1400 |
< |
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
| 1401 |
< |
energy constants for the lowest energy configurations for linear |
| 1402 |
< |
quadrupoles shown in table \ref{tab:NNQ} |
| 1397 |
> |
For the purposes of testing the energy expressions and the |
| 1398 |
> |
self-neutralization schemes, the primary quantity of interest is the |
| 1399 |
> |
analytic energy constant for the perfect arrays. Convergence to these |
| 1400 |
> |
constants are shown as a function of both the cutoff radius, $r_c$, |
| 1401 |
> |
and the damping parameter, $\alpha$ in Figs. |
| 1402 |
> |
\ref{fig:energyConstVsCutoff} and XXX. We have simultaneously tested a |
| 1403 |
> |
hard cutoff (where the kernel is simply truncated at the cutoff |
| 1404 |
> |
radius), as well as a shifted potential (SP) form which includes a |
| 1405 |
> |
potential-shifting and self-interaction term, but does not shift the |
| 1406 |
> |
forces and torques smoothly at the cutoff radius. The SP method is |
| 1407 |
> |
essentially an extension of the original Wolf method for multipoles. |
| 1408 |
|
|
| 1409 |
< |
\begin{table*} |
| 1410 |
< |
\centering{ |
| 1411 |
< |
\caption{Nagai and Nakamura Quadurpolar arrays}} |
| 1412 |
< |
\label{tab:NNQ} |
| 1413 |
< |
\begin{ruledtabular} |
| 1414 |
< |
\begin{tabular}{ccc} |
| 1415 |
< |
Lattice & Quadrupole Direction & Energy constants \\ \hline |
| 1416 |
< |
SC & 111 & -8.3 \\ |
| 1417 |
< |
BCC & 011 & -21.7 \\ |
| 1418 |
< |
FCC & 111 & -80.5 |
| 1419 |
< |
\end{tabular} |
| 1420 |
< |
\end{ruledtabular} |
| 1380 |
< |
\end{table*} |
| 1409 |
> |
\begin{figure}[!htbp] |
| 1410 |
> |
\includegraphics[width=4.5in]{energyConstVsCutoff} |
| 1411 |
> |
\caption{Convergence to the analytic energy constants as a function of |
| 1412 |
> |
cutoff radius (normalized by the lattice constant) for the different |
| 1413 |
> |
real-space methods. The two crystals shown here are the ``B'' array |
| 1414 |
> |
for bcc crystals with the dipoles along the 001 direction (upper), |
| 1415 |
> |
as well as the minimum energy bcc lattice (lower). The analytic |
| 1416 |
> |
energy constants are shown as a grey dashed line. The left panel |
| 1417 |
> |
shows results for the undamped kernel ($1/r$), while the damped |
| 1418 |
> |
error function kernel, $B_0(r)$ was used in the right panel. } |
| 1419 |
> |
\label{fig:energyConstVsCutoff} |
| 1420 |
> |
\end{figure} |
| 1421 |
|
|
| 1422 |
+ |
The Hard cutoff exhibits oscillations around the analytic energy |
| 1423 |
+ |
constants, and converges to incorrect energies when the complementary |
| 1424 |
+ |
error function damping kernel is used. The shifted potential (SP) and |
| 1425 |
+ |
gradient-shifted force (GSF) approximations converge to the correct |
| 1426 |
+ |
energy smoothly by $r_c / 6 a$ even for the undamped case. This |
| 1427 |
+ |
indicates that the correction provided by the self term is required |
| 1428 |
+ |
for obtaining accurate energies. The Taylor-shifted force (TSF) |
| 1429 |
+ |
approximation appears to perturb the potential too much inside the |
| 1430 |
+ |
cutoff region to provide accurate measures of the energy constants. |
| 1431 |
+ |
|
| 1432 |
+ |
{\it Quadrupolar} analogues to the Madelung constants were first |
| 1433 |
+ |
worked out by Nagai and Nakamura who computed the energies of selected |
| 1434 |
+ |
quadrupole arrays based on extensions to the Luttinger and Tisza |
| 1435 |
+ |
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
| 1436 |
+ |
energy constants for the lowest energy configurations for linear |
| 1437 |
+ |
quadrupoles. |
| 1438 |
+ |
|
| 1439 |
|
In analogy to the dipolar arrays, the total electrostatic energy for |
| 1440 |
|
the quadrupolar arrays is: |
| 1441 |
|
\begin{equation} |
| 1442 |
|
E = C \frac{3}{4} N^2 Q^2 |
| 1443 |
|
\end{equation} |
| 1444 |
< |
where $Q$ is the quadrupole moment. |
| 1444 |
> |
where $Q$ is the quadrupole moment. The lowest energy |
| 1445 |
|
|
| 1446 |
+ |
\section{Conclusion} |
| 1447 |
+ |
We have presented two efficient real-space methods for computing the |
| 1448 |
+ |
interactions between point multipoles. These methods have the benefit |
| 1449 |
+ |
of smoothly truncating the energies, forces, and torques at the cutoff |
| 1450 |
+ |
radius, making them attractive for both molecular dynamics (MD) and |
| 1451 |
+ |
Monte Carlo (MC) simulations. We find that the Gradient-Shifted Force |
| 1452 |
+ |
(GSF) and the Shifted-Potential (SP) methods converge rapidly to the |
| 1453 |
+ |
correct lattice energies for ordered dipolar and quadrupolar arrays, |
| 1454 |
+ |
while the Taylor-Shifted Force (TSF) is too severe an approximation to |
| 1455 |
+ |
provide accurate convergence to lattice energies. |
| 1456 |
|
|
| 1457 |
+ |
In most cases, GSF can obtain nearly quantitative agreement with the |
| 1458 |
+ |
lattice energy constants with reasonably small cutoff radii. The only |
| 1459 |
+ |
exception we have observed is for crystals which exhibit a bulk |
| 1460 |
+ |
macroscopic dipole moment (e.g. Luttinger \& Tisza's $Z_1$ lattice). |
| 1461 |
+ |
In this particular case, the multipole neutralization scheme can |
| 1462 |
+ |
interfere with the correct computation of the energies. We note that |
| 1463 |
+ |
the energies for these arrangements are typically much larger than for |
| 1464 |
+ |
crystals with net-zero moments, so this is not expected to be an issue |
| 1465 |
+ |
in most simulations. |
| 1466 |
|
|
| 1467 |
+ |
In large systems, these new methods can be made to scale approximately |
| 1468 |
+ |
linearly with system size, and detailed comparisons with the Ewald sum |
| 1469 |
+ |
for a wide range of chemical environments follows in the second paper. |
| 1470 |
|
|
| 1392 |
– |
|
| 1393 |
– |
|
| 1394 |
– |
|
| 1395 |
– |
|
| 1471 |
|
\begin{acknowledgments} |
| 1472 |
< |
Support for this project was provided by the National Science |
| 1473 |
< |
Foundation under grant CHE-0848243. Computational time was provided by |
| 1474 |
< |
the Center for Research Computing (CRC) at the University of Notre |
| 1475 |
< |
Dame. |
| 1472 |
> |
JDG acknowledges helpful discussions with Christopher |
| 1473 |
> |
Fennell. Support for this project was provided by the National |
| 1474 |
> |
Science Foundation under grant CHE-0848243. Computational time was |
| 1475 |
> |
provided by the Center for Research Computing (CRC) at the |
| 1476 |
> |
University of Notre Dame. |
| 1477 |
|
\end{acknowledgments} |
| 1478 |
|
|
| 1479 |
+ |
\newpage |
| 1480 |
|
\appendix |
| 1481 |
|
|
| 1482 |
< |
\section{Smith's $B_l(r)$ functions for smeared-charge distributions} |
| 1483 |
< |
|
| 1484 |
< |
The following summarizes Smith's $B_l(r)$ functions and |
| 1485 |
< |
includes formulas given in his appendix. |
| 1486 |
< |
|
| 1410 |
< |
The first function $B_0(r)$ is defined by |
| 1482 |
> |
\section{Smith's $B_l(r)$ functions for damped-charge distributions} |
| 1483 |
> |
\label{SmithFunc} |
| 1484 |
> |
The following summarizes Smith's $B_l(r)$ functions and includes |
| 1485 |
> |
formulas given in his appendix.\cite{Smith98} The first function |
| 1486 |
> |
$B_0(r)$ is defined by |
| 1487 |
|
% |
| 1488 |
|
\begin{equation} |
| 1489 |
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
| 1497 |
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
| 1498 |
|
\end{equation} |
| 1499 |
|
% |
| 1500 |
< |
and can be rewritten in terms of a function $B_1(r)$: |
| 1500 |
> |
which can be used to define a function $B_1(r)$: |
| 1501 |
|
% |
| 1502 |
|
\begin{equation} |
| 1503 |
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
| 1504 |
|
\end{equation} |
| 1505 |
|
% |
| 1506 |
< |
In general, |
| 1506 |
> |
In general, the recurrence relation, |
| 1507 |
|
\begin{equation} |
| 1508 |
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
| 1509 |
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
| 1510 |
|
\text{e}^{-{\alpha}^2r^2} |
| 1511 |
< |
\right] . |
| 1511 |
> |
\right] , |
| 1512 |
|
\end{equation} |
| 1513 |
+ |
is very useful for building up higher derivatives. Using these |
| 1514 |
+ |
formulas, we find: |
| 1515 |
|
% |
| 1516 |
< |
Using these formulas, we find |
| 1516 |
> |
\begin{align} |
| 1517 |
> |
\frac{dB_0}{dr}=&-rB_1(r) \\ |
| 1518 |
> |
\frac{d^2B_0}{dr^2}=& - B_1(r) + r^2 B_2(r) \\ |
| 1519 |
> |
\frac{d^3B_0}{dr^3}=& 3 r B_2(r) - r^3 B_3(r) \\ |
| 1520 |
> |
\frac{d^4B_0}{dr^4}=& 3 B_2(r) - 6 r^2 B_3(r) + r^4 B_4(r) \\ |
| 1521 |
> |
\frac{d^5B_0}{dr^5}=& - 15 r B_3(r) + 10 r^3 B_4(r) - r^5 B_5(r) . |
| 1522 |
> |
\end{align} |
| 1523 |
|
% |
| 1524 |
< |
\begin{eqnarray} |
| 1525 |
< |
\frac{dB_0}{dr}=-rB_1(r) \\ |
| 1442 |
< |
\frac{d^2B_0}{dr^2}=-B_1(r) + r^2B_2(r) \\ |
| 1443 |
< |
\frac{d^3B_0}{dr^3}=3rB_2(r) - r^3B_3(r) \\ |
| 1444 |
< |
\frac{d^4B_0}{dr^4}=3B_2(r) - 6r^2B_3(r)+r^4B_4(r) \\ |
| 1445 |
< |
\frac{d^5B_0}{dr^5}=-15rB_3(r) + 10r^3B_4(r) -r^5B_5(r) . |
| 1446 |
< |
\end{eqnarray} |
| 1524 |
> |
As noted by Smith, it is possible to approximate the $B_l(r)$ |
| 1525 |
> |
functions, |
| 1526 |
|
% |
| 1448 |
– |
As noted by Smith, |
| 1449 |
– |
% |
| 1527 |
|
\begin{equation} |
| 1528 |
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
| 1529 |
|
+\text{O}(r) . |
| 1530 |
|
\end{equation} |
| 1531 |
+ |
\newpage |
| 1532 |
+ |
\section{The $r$-dependent factors for TSF electrostatics} |
| 1533 |
|
|
| 1455 |
– |
\section{Method 1, the $r$-dependent factors} |
| 1456 |
– |
|
| 1534 |
|
Using the shifted damped functions $f_n(r)$ defined by: |
| 1535 |
|
% |
| 1536 |
|
\begin{equation} |
| 1537 |
< |
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} , |
| 1537 |
> |
f_n(r)= B_0(r) -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)}(r_c) , |
| 1538 |
|
\end{equation} |
| 1539 |
|
% |
| 1540 |
< |
we first provide formulas for successive derivatives of this function. (If there is |
| 1541 |
< |
no damping, then $B_0(r)$ is replaced by $1/r$, as discussed in Section~\ref{damped???}.) First, we find: |
| 1540 |
> |
where the superscript $(m)$ denotes the $m^\mathrm{th}$ derivative. In |
| 1541 |
> |
this Appendix, we provide formulas for successive derivatives of this |
| 1542 |
> |
function. (If there is no damping, then $B_0(r)$ is replaced by |
| 1543 |
> |
$1/r$.) First, we find: |
| 1544 |
|
% |
| 1545 |
|
\begin{equation} |
| 1546 |
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
| 1547 |
|
\end{equation} |
| 1548 |
|
% |
| 1549 |
< |
This formula clearly brings in derivatives of Smith's $B_0(r)$ function, motivating us to |
| 1550 |
< |
define higher-order derivatives as follows: |
| 1549 |
> |
This formula clearly brings in derivatives of Smith's $B_0(r)$ |
| 1550 |
> |
function, and we define higher-order derivatives as follows: |
| 1551 |
|
% |
| 1552 |
< |
\begin{eqnarray} |
| 1553 |
< |
g_n(r)= \frac{d f_n}{d r} = |
| 1554 |
< |
B_0^{(1)} \Big \lvert _r -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)} \Big \lvert _{r_c} \\ |
| 1555 |
< |
h_n(r)= \frac{d^2f_n}{d r^2} = |
| 1556 |
< |
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} \\ |
| 1557 |
< |
s_n(r)= \frac{d^3f_n}{d r^3} = |
| 1558 |
< |
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} \\ |
| 1559 |
< |
t_n(r)= \frac{d^4f_n}{d r^4} = |
| 1560 |
< |
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} \\ |
| 1561 |
< |
u_n(r)= \frac{d^5f_n}{d r^5} = |
| 1562 |
< |
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} . |
| 1563 |
< |
\end{eqnarray} |
| 1552 |
> |
\begin{align} |
| 1553 |
> |
g_n(r)=& \frac{d f_n}{d r} = |
| 1554 |
> |
B_0^{(1)}(r) -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)}(r_c) \\ |
| 1555 |
> |
h_n(r)=& \frac{d^2f_n}{d r^2} = |
| 1556 |
> |
B_0^{(2)}(r) -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)}(r_c) \\ |
| 1557 |
> |
s_n(r)=& \frac{d^3f_n}{d r^3} = |
| 1558 |
> |
B_0^{(3)}(r) -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)}(r_c) \\ |
| 1559 |
> |
t_n(r)=& \frac{d^4f_n}{d r^4} = |
| 1560 |
> |
B_0^{(4)}(r) -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)}(r_c) \\ |
| 1561 |
> |
u_n(r)=& \frac{d^5f_n}{d r^5} = |
| 1562 |
> |
B_0^{(5)}(r) -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)}(r_c) . |
| 1563 |
> |
\end{align} |
| 1564 |
|
% |
| 1565 |
< |
We note that the last function needed (for quadrupole-quadrupole) is |
| 1565 |
> |
We note that the last function needed (for quadrupole-quadrupole interactions) is |
| 1566 |
|
% |
| 1567 |
|
\begin{equation} |
| 1568 |
< |
u_4(r)=B_0^{(5)} \Big \lvert _r - B_0^{(5)} \Big \lvert _{r_c} . |
| 1568 |
> |
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
| 1569 |
|
\end{equation} |
| 1570 |
< |
|
| 1571 |
< |
The functions $f_n(r)$ to $u_n(r)$ are recursively computed and stored for values of $r$ |
| 1572 |
< |
from $0$ to $r_c$. The functions needed are listed schematically below: |
| 1570 |
> |
% The functions |
| 1571 |
> |
% needed are listed schematically below: |
| 1572 |
> |
% % |
| 1573 |
> |
% \begin{eqnarray} |
| 1574 |
> |
% f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
| 1575 |
> |
% g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
| 1576 |
> |
% h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
| 1577 |
> |
% s_2 \quad s_3 \quad &s_4 \nonumber \\ |
| 1578 |
> |
% t_3 \quad &t_4 \nonumber \\ |
| 1579 |
> |
% &u_4 \nonumber . |
| 1580 |
> |
% \end{eqnarray} |
| 1581 |
> |
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
| 1582 |
> |
stored on a grid for values of $r$ from $0$ to $r_c$. Using these |
| 1583 |
> |
functions, we find |
| 1584 |
|
% |
| 1585 |
< |
\begin{eqnarray} |
| 1586 |
< |
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
| 1587 |
< |
g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
| 1588 |
< |
h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
| 1589 |
< |
s_2 \quad s_3 \quad &s_4 \nonumber \\ |
| 1500 |
< |
t_3 \quad &t_4 \nonumber \\ |
| 1501 |
< |
&u_4 \nonumber . |
| 1502 |
< |
\end{eqnarray} |
| 1503 |
< |
|
| 1504 |
< |
Using these functions, we find |
| 1505 |
< |
% |
| 1506 |
< |
\begin{equation} |
| 1507 |
< |
\frac{\partial f_n}{\partial r_\alpha} =r_\alpha \frac {g_n}{r} |
| 1508 |
< |
\end{equation} |
| 1509 |
< |
% |
| 1510 |
< |
\begin{equation} |
| 1511 |
< |
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =\delta_{\alpha \beta}\frac {g_n}{r} |
| 1512 |
< |
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) |
| 1513 |
< |
\end{equation} |
| 1514 |
< |
% |
| 1515 |
< |
\begin{equation} |
| 1516 |
< |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} = |
| 1585 |
> |
\begin{align} |
| 1586 |
> |
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
| 1587 |
> |
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
| 1588 |
> |
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
| 1589 |
> |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta \partial r_\gamma} =& |
| 1590 |
|
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
| 1591 |
|
\delta_{ \beta \gamma} r_\alpha \right) |
| 1592 |
< |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
| 1593 |
< |
+ r_\alpha r_\beta r_\gamma |
| 1594 |
< |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) |
| 1595 |
< |
\end{equation} |
| 1596 |
< |
% |
| 1524 |
< |
\begin{eqnarray} |
| 1525 |
< |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} = |
| 1592 |
> |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) \nonumber \\ |
| 1593 |
> |
& + r_\alpha r_\beta r_\gamma |
| 1594 |
> |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
| 1595 |
> |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta \partial |
| 1596 |
> |
r_\gamma \partial r_\delta} =& |
| 1597 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
| 1598 |
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
| 1599 |
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
| 1600 |
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
| 1601 |
< |
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
| 1602 |
< |
+ \text{5 perm} |
| 1601 |
> |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
| 1602 |
> |
+ \text{5 permutations} |
| 1603 |
|
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
| 1604 |
|
\right) \nonumber \\ |
| 1605 |
< |
+ r_\alpha r_\beta r_\gamma r_\delta |
| 1605 |
> |
&+ r_\alpha r_\beta r_\gamma r_\delta |
| 1606 |
|
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
| 1607 |
< |
+ \frac{t_n}{r^4} \right) |
| 1537 |
< |
\end{eqnarray} |
| 1538 |
< |
% |
| 1539 |
< |
\begin{eqnarray} |
| 1607 |
> |
+ \frac{t_n}{r^4} \right)\\ |
| 1608 |
|
\frac{\partial^5 f_n} |
| 1609 |
< |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} = |
| 1609 |
> |
{\partial r_\alpha \partial r_\beta \partial r_\gamma \partial |
| 1610 |
> |
r_\delta \partial r_\epsilon} =& |
| 1611 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
| 1612 |
< |
+ \text{14 perm} \right) |
| 1612 |
> |
+ \text{14 permutations} \right) |
| 1613 |
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
| 1614 |
< |
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
| 1615 |
< |
+ \text{9 perm} |
| 1614 |
> |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
| 1615 |
> |
+ \text{9 permutations} |
| 1616 |
|
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
| 1617 |
|
\right) \nonumber \\ |
| 1618 |
< |
+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
| 1618 |
> |
&+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
| 1619 |
|
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
| 1620 |
< |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) |
| 1621 |
< |
\end{eqnarray} |
| 1620 |
> |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) \label{eq:b13} |
| 1621 |
> |
\end{align} |
| 1622 |
|
% |
| 1623 |
|
% |
| 1624 |
|
% |
| 1625 |
< |
\section{Method 2, the $r$-dependent factors} |
| 1625 |
> |
\newpage |
| 1626 |
> |
\section{The $r$-dependent factors for GSF electrostatics} |
| 1627 |
|
|
| 1628 |
< |
In method 2, the kernel is not expanded, rather the individual terms in the multipole interaction energies, |
| 1629 |
< |
see Eq. (20?). For a smeared-charge distribution, this still brings into the algebra multiple derivatives |
| 1630 |
< |
of the kernel $B_0(r)$. To denote these terms, we generalize the notation of the previous appendix. |
| 1631 |
< |
For $f(r)=1/r$ (bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
| 1628 |
> |
In Gradient-shifted force electrostatics, the kernel is not expanded, |
| 1629 |
> |
rather the individual terms in the multipole interaction energies. |
| 1630 |
> |
For damped charges , this still brings into the algebra multiple |
| 1631 |
> |
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
| 1632 |
> |
we generalize the notation of the previous appendix. For either |
| 1633 |
> |
$f(r)=1/r$ (undamped) or $f(r)=B_0(r)$ (damped), |
| 1634 |
|
% |
| 1635 |
< |
\begin{eqnarray} |
| 1636 |
< |
g(r)= \frac{df}{d r}\\ |
| 1637 |
< |
h(r)= \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
| 1638 |
< |
s(r)= \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
| 1639 |
< |
t(r)= \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
| 1640 |
< |
u(r)= \frac{dt}{d r} =\frac{d^5f}{d r^5} . |
| 1641 |
< |
\end{eqnarray} |
| 1635 |
> |
\begin{align} |
| 1636 |
> |
g(r)=& \frac{df}{d r}\\ |
| 1637 |
> |
h(r)=& \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
| 1638 |
> |
s(r)=& \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
| 1639 |
> |
t(r)=& \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
| 1640 |
> |
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
| 1641 |
> |
\end{align} |
| 1642 |
|
% |
| 1643 |
< |
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 |
| 1644 |
< |
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) |
| 1645 |
< |
are correct for method 2 if one just eliminates the subscript $n$. |
| 1574 |
< |
|
| 1575 |
< |
\section{Extra Material} |
| 1576 |
< |
% |
| 1577 |
< |
% |
| 1578 |
< |
%Energy in body coordinate form --------------------------------------------------------------- |
| 1579 |
< |
% |
| 1580 |
< |
Here are the interaction energies written in terms of the body coordinates: |
| 1581 |
< |
|
| 1582 |
< |
% |
| 1583 |
< |
% u ca cb |
| 1584 |
< |
% |
| 1585 |
< |
\begin{equation} |
| 1586 |
< |
U_{C_{\bf a}C_{\bf b}}(r)= |
| 1587 |
< |
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) |
| 1588 |
< |
\end{equation} |
| 1589 |
< |
% |
| 1590 |
< |
% u ca db |
| 1591 |
< |
% |
| 1592 |
< |
\begin{equation} |
| 1593 |
< |
U_{C_{\bf a}D_{\bf b}}(r)= |
| 1594 |
< |
\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1595 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
| 1596 |
< |
\end{equation} |
| 1597 |
< |
% |
| 1598 |
< |
% u ca qb |
| 1599 |
< |
% |
| 1600 |
< |
\begin{equation} |
| 1601 |
< |
U_{C_{\bf a}Q_{\bf b}}(r)= |
| 1602 |
< |
\frac{C_{\bf a }\text{Tr}Q_{\bf b}}{4\pi \epsilon_0} |
| 1603 |
< |
v_{21}(r) \nonumber \\ |
| 1604 |
< |
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1605 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) |
| 1606 |
< |
v_{22}(r) |
| 1607 |
< |
\end{equation} |
| 1608 |
< |
% |
| 1609 |
< |
% u da cb |
| 1610 |
< |
% |
| 1611 |
< |
\begin{equation} |
| 1612 |
< |
U_{D_{\bf a}C_{\bf b}}(r)= |
| 1613 |
< |
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1614 |
< |
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
| 1615 |
< |
\end{equation} |
| 1616 |
< |
% |
| 1617 |
< |
% u da db |
| 1618 |
< |
% |
| 1619 |
< |
\begin{equation} |
| 1620 |
< |
\begin{split} |
| 1621 |
< |
% 1 |
| 1622 |
< |
U_{D_{\bf a}D_{\bf b}}(r)&= |
| 1623 |
< |
-\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
| 1624 |
< |
(\hat{a}_m \cdot \hat{b}_n) |
| 1625 |
< |
D_{\mathbf{b}n} v_{21}(r) \\ |
| 1626 |
< |
% 2 |
| 1627 |
< |
&-\frac{1}{4\pi \epsilon_0} |
| 1628 |
< |
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 1629 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} |
| 1630 |
< |
v_{22}(r) |
| 1631 |
< |
\end{split} |
| 1632 |
< |
\end{equation} |
| 1633 |
< |
% |
| 1634 |
< |
% u da qb |
| 1635 |
< |
% |
| 1636 |
< |
\begin{equation} |
| 1637 |
< |
\begin{split} |
| 1638 |
< |
% 1 |
| 1639 |
< |
U_{D_{\bf a}Q_{\bf b}}(r)&= |
| 1640 |
< |
-\frac{1}{4\pi \epsilon_0} \left( |
| 1641 |
< |
\text{Tr}Q_{\mathbf{b}} |
| 1642 |
< |
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
| 1643 |
< |
+2\sum_{lmn}D_{\mathbf{a}l} |
| 1644 |
< |
(\hat{a}_l \cdot \hat{b}_m) |
| 1645 |
< |
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 1646 |
< |
\right) v_{31}(r) \\ |
| 1647 |
< |
% 2 |
| 1648 |
< |
&-\frac{1}{4\pi \epsilon_0} |
| 1649 |
< |
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 1650 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1651 |
< |
Q_{{\mathbf b}mn} |
| 1652 |
< |
(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
| 1653 |
< |
\end{split} |
| 1654 |
< |
\end{equation} |
| 1655 |
< |
% |
| 1656 |
< |
% u qa cb |
| 1657 |
< |
% |
| 1658 |
< |
\begin{equation} |
| 1659 |
< |
U_{Q_{\bf a}C_{\bf b}}(r)= |
| 1660 |
< |
\frac{C_{\bf b }\text{Tr}Q_{\bf a}}{4\pi \epsilon_0} v_{21}(r) |
| 1661 |
< |
+\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1662 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r) |
| 1663 |
< |
\end{equation} |
| 1664 |
< |
% |
| 1665 |
< |
% u qa db |
| 1666 |
< |
% |
| 1667 |
< |
\begin{equation} |
| 1668 |
< |
\begin{split} |
| 1669 |
< |
%1 |
| 1670 |
< |
U_{Q_{\bf a}D_{\bf b}}(r)&= |
| 1671 |
< |
\frac{1}{4\pi \epsilon_0} \left( |
| 1672 |
< |
\text{Tr}Q_{\mathbf{a}} |
| 1673 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
| 1674 |
< |
+2\sum_{lmn}D_{\mathbf{b}l} |
| 1675 |
< |
(\hat{b}_l \cdot \hat{a}_m) |
| 1676 |
< |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 1677 |
< |
\right) v_{31}(r) \\ |
| 1678 |
< |
% 2 |
| 1679 |
< |
&+\frac{1}{4\pi \epsilon_0} |
| 1680 |
< |
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 1681 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1682 |
< |
Q_{{\mathbf a}mn} |
| 1683 |
< |
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
| 1684 |
< |
\end{split} |
| 1685 |
< |
\end{equation} |
| 1686 |
< |
% |
| 1687 |
< |
% u qa qb |
| 1688 |
< |
% |
| 1689 |
< |
\begin{equation} |
| 1690 |
< |
\begin{split} |
| 1691 |
< |
%1 |
| 1692 |
< |
U_{Q_{\bf a}Q_{\bf b}}(r)&= |
| 1693 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1694 |
< |
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
| 1695 |
< |
+2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 1696 |
< |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 1697 |
< |
(\hat{a}_m \cdot \hat{b}_n) \Bigr] |
| 1698 |
< |
v_{41}(r) \\ |
| 1699 |
< |
% 2 |
| 1700 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 1701 |
< |
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
| 1702 |
< |
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
| 1703 |
< |
Q_{{\mathbf b}lm} |
| 1704 |
< |
(\hat{b}_m \cdot \hat{r}) |
| 1705 |
< |
+\text{Tr}Q_{\mathbf{b}} |
| 1706 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
| 1707 |
< |
Q_{{\mathbf a}lm} |
| 1708 |
< |
(\hat{a}_m \cdot \hat{r}) \\ |
| 1709 |
< |
% 3 |
| 1710 |
< |
&+4 \sum_{lmnp} |
| 1711 |
< |
(\hat{r} \cdot \hat{a}_l ) |
| 1712 |
< |
Q_{{\mathbf a}lm} |
| 1713 |
< |
(\hat{a}_m \cdot \hat{b}_n) |
| 1714 |
< |
Q_{{\mathbf b}np} |
| 1715 |
< |
(\hat{b}_p \cdot \hat{r}) |
| 1716 |
< |
\Bigr] v_{42}(r) \\ |
| 1717 |
< |
% 4 |
| 1718 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 1719 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
| 1720 |
< |
Q_{{\mathbf a}lm} |
| 1721 |
< |
(\hat{a}_m \cdot \hat{r}) |
| 1722 |
< |
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
| 1723 |
< |
Q_{{\mathbf b}np} |
| 1724 |
< |
(\hat{b}_p \cdot \hat{r}) v_{43}(r). |
| 1725 |
< |
\end{split} |
| 1726 |
< |
\end{equation} |
| 1727 |
< |
% |
| 1728 |
< |
|
| 1729 |
< |
|
| 1730 |
< |
% BODY coordinates force equations -------------------------------------------- |
| 1731 |
< |
% |
| 1732 |
< |
% |
| 1733 |
< |
Here are the force equations written in terms of body coordinates. |
| 1734 |
< |
% |
| 1735 |
< |
% f ca cb |
| 1736 |
< |
% |
| 1737 |
< |
\begin{equation} |
| 1738 |
< |
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
| 1739 |
< |
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
| 1740 |
< |
\end{equation} |
| 1741 |
< |
% |
| 1742 |
< |
% f ca db |
| 1743 |
< |
% |
| 1744 |
< |
\begin{equation} |
| 1745 |
< |
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
| 1746 |
< |
\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1747 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r} |
| 1748 |
< |
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1749 |
< |
\sum_n D_{\mathbf{b}n} \hat{b}_n w_c(r) |
| 1750 |
< |
\end{equation} |
| 1751 |
< |
% |
| 1752 |
< |
% f ca qb |
| 1753 |
< |
% |
| 1754 |
< |
\begin{equation} |
| 1755 |
< |
\begin{split} |
| 1756 |
< |
% 1 |
| 1757 |
< |
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
| 1758 |
< |
\frac{1}{4\pi \epsilon_0} |
| 1759 |
< |
C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r} |
| 1760 |
< |
+ 2C_{\bf a } \sum_l \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot \hat{r}) w_e(r) \\ |
| 1761 |
< |
% 2 |
| 1762 |
< |
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1763 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} |
| 1764 |
< |
\end{split} |
| 1765 |
< |
\end{equation} |
| 1766 |
< |
% |
| 1767 |
< |
% f da cb |
| 1768 |
< |
% |
| 1769 |
< |
\begin{equation} |
| 1770 |
< |
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1771 |
< |
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
| 1772 |
< |
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r) \hat{r} |
| 1773 |
< |
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
| 1774 |
< |
\sum_n D_{\mathbf{a}n} \hat{a}_n w_c(r) |
| 1775 |
< |
\end{equation} |
| 1776 |
< |
% |
| 1777 |
< |
% f da db |
| 1778 |
< |
% |
| 1779 |
< |
\begin{equation} |
| 1780 |
< |
\begin{split} |
| 1781 |
< |
% 1 |
| 1782 |
< |
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
| 1783 |
< |
-\frac{1}{4\pi \epsilon_0} |
| 1784 |
< |
\sum_{mn} D_{\mathbf {a}m} |
| 1785 |
< |
(\hat{a}_m \cdot \hat{b}_n) |
| 1786 |
< |
D_{\mathbf{b}n} w_d(r) \hat{r} |
| 1787 |
< |
-\frac{1}{4\pi \epsilon_0} |
| 1788 |
< |
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 1789 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r} \\ |
| 1790 |
< |
% 2 |
| 1791 |
< |
& \quad + \frac{1}{4\pi \epsilon_0} |
| 1792 |
< |
\Bigl[ \sum_m D_{\mathbf {a}m} |
| 1793 |
< |
\hat{a}_m \sum_n D_{\mathbf{b}n} |
| 1794 |
< |
(\hat{b}_n \cdot \hat{r}) |
| 1795 |
< |
+ \sum_m D_{\mathbf {b}m} |
| 1796 |
< |
\hat{b}_m \sum_n D_{\mathbf{a}n} |
| 1797 |
< |
(\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r) \\ |
| 1798 |
< |
\end{split} |
| 1799 |
< |
\end{equation} |
| 1800 |
< |
% |
| 1801 |
< |
% f da qb |
| 1802 |
< |
% |
| 1803 |
< |
\begin{equation} |
| 1804 |
< |
\begin{split} |
| 1805 |
< |
% 1 |
| 1806 |
< |
&\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
| 1807 |
< |
- \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1808 |
< |
\text{Tr}Q_{\mathbf{b}} |
| 1809 |
< |
\sum_l D_{\mathbf{a}l} \hat{a}_l |
| 1810 |
< |
+2\sum_{lmn} D_{\mathbf{a}l} |
| 1811 |
< |
(\hat{a}_l \cdot \hat{b}_m) |
| 1812 |
< |
Q_{\mathbf{b}mn} \hat{b}_n \Bigr] w_g(r) \\ |
| 1813 |
< |
% 3 |
| 1814 |
< |
& - \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1815 |
< |
\text{Tr}Q_{\mathbf{b}} |
| 1816 |
< |
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
| 1817 |
< |
+2\sum_{lmn}D_{\mathbf{a}l} |
| 1818 |
< |
(\hat{a}_l \cdot \hat{b}_m) |
| 1819 |
< |
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1820 |
< |
% 4 |
| 1821 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 1822 |
< |
\Bigl[\sum_l D_{\mathbf{a}l} \hat{a}_l |
| 1823 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1824 |
< |
Q_{{\mathbf b}mn} |
| 1825 |
< |
(\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l) |
| 1826 |
< |
D_{\mathbf{a}l} |
| 1827 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1828 |
< |
Q_{{\mathbf b}mn} \hat{b}_n \Bigr] w_i(r)\\ |
| 1829 |
< |
% 6 |
| 1830 |
< |
& -\frac{1}{4\pi \epsilon_0} |
| 1831 |
< |
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 1832 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1833 |
< |
Q_{{\mathbf b}mn} |
| 1834 |
< |
(\hat{b}_n \cdot \hat{r}) w_j(r) \hat{r} |
| 1835 |
< |
\end{split} |
| 1836 |
< |
\end{equation} |
| 1837 |
< |
% |
| 1838 |
< |
% force qa cb |
| 1839 |
< |
% |
| 1840 |
< |
\begin{equation} |
| 1841 |
< |
\begin{split} |
| 1842 |
< |
% 1 |
| 1843 |
< |
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} &= |
| 1844 |
< |
\frac{1}{4\pi \epsilon_0} |
| 1845 |
< |
C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r) |
| 1846 |
< |
+ \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) \\ |
| 1847 |
< |
% 2 |
| 1848 |
< |
& +\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1849 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} |
| 1850 |
< |
\end{split} |
| 1851 |
< |
\end{equation} |
| 1852 |
< |
% |
| 1853 |
< |
% f qa db |
| 1854 |
< |
% |
| 1855 |
< |
\begin{equation} |
| 1856 |
< |
\begin{split} |
| 1857 |
< |
% 1 |
| 1858 |
< |
&\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
| 1859 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1860 |
< |
\text{Tr}Q_{\mathbf{a}} |
| 1861 |
< |
\sum_l D_{\mathbf{b}l} \hat{b}_l |
| 1862 |
< |
+2\sum_{lmn} D_{\mathbf{b}l} |
| 1863 |
< |
(\hat{b}_l \cdot \hat{a}_m) |
| 1864 |
< |
Q_{\mathbf{a}mn} \hat{a}_n \Bigr] |
| 1865 |
< |
w_g(r)\\ |
| 1866 |
< |
% 3 |
| 1867 |
< |
& + \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1868 |
< |
\text{Tr}Q_{\mathbf{a}} |
| 1869 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
| 1870 |
< |
+2\sum_{lmn}D_{\mathbf{b}l} |
| 1871 |
< |
(\hat{b}_l \cdot \hat{a}_m) |
| 1872 |
< |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1873 |
< |
% 4 |
| 1874 |
< |
& + \frac{1}{4\pi \epsilon_0} \Bigl[ \sum_l D_{\mathbf{b}l} \hat{b}_l |
| 1875 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1876 |
< |
Q_{{\mathbf a}mn} |
| 1877 |
< |
(\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l) |
| 1878 |
< |
D_{\mathbf{b}l} |
| 1879 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1880 |
< |
Q_{{\mathbf a}mn} \hat{a}_n \Bigr] w_i(r) \\ |
| 1881 |
< |
% 6 |
| 1882 |
< |
& +\frac{1}{4\pi \epsilon_0} |
| 1883 |
< |
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 1884 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1885 |
< |
Q_{{\mathbf a}mn} |
| 1886 |
< |
(\hat{a}_n \cdot \hat{r}) w_j(r) \hat{r} |
| 1887 |
< |
\end{split} |
| 1888 |
< |
\end{equation} |
| 1889 |
< |
% |
| 1890 |
< |
% f qa qb |
| 1891 |
< |
% |
| 1892 |
< |
\begin{equation} |
| 1893 |
< |
\begin{split} |
| 1894 |
< |
&\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
| 1895 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1896 |
< |
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
| 1897 |
< |
+ 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 1898 |
< |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 1899 |
< |
(\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r}\\ |
| 1900 |
< |
&+\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1901 |
< |
2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
| 1902 |
< |
+ 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} \hat{b}_m \\ |
| 1903 |
< |
&+ 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}) |
| 1904 |
< |
+ 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 |
| 1905 |
< |
\Bigr] w_n(r) \\ |
| 1906 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 1907 |
< |
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
| 1908 |
< |
\sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) |
| 1909 |
< |
+ \text{Tr}Q_{\mathbf{b}} |
| 1910 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) \\ |
| 1911 |
< |
&+4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) |
| 1912 |
< |
Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
| 1913 |
< |
% |
| 1914 |
< |
&+\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1915 |
< |
2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
| 1916 |
< |
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot \hat{r}) \\ |
| 1917 |
< |
&+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
| 1918 |
< |
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr] w_o(r) \hat{r} \\ |
| 1919 |
< |
& + \frac{1}{4\pi \epsilon_0} |
| 1920 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
| 1921 |
< |
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} |
| 1922 |
< |
\end{split} |
| 1923 |
< |
\end{equation} |
| 1924 |
< |
% |
| 1925 |
< |
Here we list the form of the non-zero damped shifted multipole torques showing |
| 1926 |
< |
explicitly dependences on body axes: |
| 1927 |
< |
% |
| 1928 |
< |
% t ca db |
| 1929 |
< |
% |
| 1930 |
< |
\begin{equation} |
| 1931 |
< |
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
| 1932 |
< |
\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1933 |
< |
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
| 1934 |
< |
\end{equation} |
| 1935 |
< |
% |
| 1936 |
< |
% t ca qb |
| 1937 |
< |
% |
| 1938 |
< |
\begin{equation} |
| 1939 |
< |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
| 1940 |
< |
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
| 1941 |
< |
\sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m \cdot \hat{r}) v_{22}(r) |
| 1942 |
< |
\end{equation} |
| 1943 |
< |
% |
| 1944 |
< |
% t da cb |
| 1945 |
< |
% |
| 1946 |
< |
\begin{equation} |
| 1947 |
< |
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1948 |
< |
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1949 |
< |
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
| 1950 |
< |
\end{equation}% |
| 1951 |
< |
% |
| 1952 |
< |
% |
| 1953 |
< |
% ta da db |
| 1954 |
< |
% |
| 1955 |
< |
\begin{equation} |
| 1956 |
< |
\begin{split} |
| 1957 |
< |
% 1 |
| 1958 |
< |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
| 1959 |
< |
\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
| 1960 |
< |
(\hat{a}_m \times \hat{b}_n) |
| 1961 |
< |
D_{\mathbf{b}n} v_{21}(r) \\ |
| 1962 |
< |
% 2 |
| 1963 |
< |
&-\frac{1}{4\pi \epsilon_0} |
| 1964 |
< |
\sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} |
| 1965 |
< |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
| 1966 |
< |
\end{split} |
| 1967 |
< |
\end{equation} |
| 1968 |
< |
% |
| 1969 |
< |
% tb da db |
| 1970 |
< |
% |
| 1971 |
< |
\begin{equation} |
| 1972 |
< |
\begin{split} |
| 1973 |
< |
% 1 |
| 1974 |
< |
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} &= |
| 1975 |
< |
-\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
| 1976 |
< |
(\hat{a}_m \times \hat{b}_n) |
| 1977 |
< |
D_{\mathbf{b}n} v_{21}(r) \\ |
| 1978 |
< |
% 2 |
| 1979 |
< |
&+\frac{1}{4\pi \epsilon_0} |
| 1980 |
< |
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 1981 |
< |
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
| 1982 |
< |
\end{split} |
| 1983 |
< |
\end{equation} |
| 1984 |
< |
% |
| 1985 |
< |
% ta da qb |
| 1986 |
< |
% |
| 1987 |
< |
\begin{equation} |
| 1988 |
< |
\begin{split} |
| 1989 |
< |
% 1 |
| 1990 |
< |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} &= |
| 1991 |
< |
\frac{1}{4\pi \epsilon_0} \left( |
| 1992 |
< |
-\text{Tr}Q_{\mathbf{b}} |
| 1993 |
< |
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} |
| 1994 |
< |
+2\sum_{lmn}D_{\mathbf{a}l} |
| 1995 |
< |
(\hat{a}_l \times \hat{b}_m) |
| 1996 |
< |
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 1997 |
< |
\right) v_{31}(r)\\ |
| 1998 |
< |
% 2 |
| 1999 |
< |
&-\frac{1}{4\pi \epsilon_0} |
| 2000 |
< |
\sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l} |
| 2001 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 2002 |
< |
Q_{{\mathbf b}mn} |
| 2003 |
< |
(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
| 2004 |
< |
\end{split} |
| 2005 |
< |
\end{equation} |
| 2006 |
< |
% |
| 2007 |
< |
% tb da qb |
| 2008 |
< |
% |
| 2009 |
< |
\begin{equation} |
| 2010 |
< |
\begin{split} |
| 2011 |
< |
% 1 |
| 2012 |
< |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} &= |
| 2013 |
< |
\frac{1}{4\pi \epsilon_0} \left( |
| 2014 |
< |
-2\sum_{lmn}D_{\mathbf{a}l} |
| 2015 |
< |
(\hat{a}_l \cdot \hat{b}_m) |
| 2016 |
< |
Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n) |
| 2017 |
< |
-2\sum_{lmn}D_{\mathbf{a}l} |
| 2018 |
< |
(\hat{a}_l \times \hat{b}_m) |
| 2019 |
< |
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 2020 |
< |
\right) v_{31}(r) \\ |
| 2021 |
< |
% 2 |
| 2022 |
< |
&-\frac{2}{4\pi \epsilon_0} |
| 2023 |
< |
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 2024 |
< |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 2025 |
< |
Q_{{\mathbf b}mn} |
| 2026 |
< |
(\hat{r}\times \hat{b}_n) v_{32}(r) |
| 2027 |
< |
\end{split} |
| 2028 |
< |
\end{equation} |
| 2029 |
< |
% |
| 2030 |
< |
% ta qa cb |
| 2031 |
< |
% |
| 2032 |
< |
\begin{equation} |
| 2033 |
< |
\mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
| 2034 |
< |
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
| 2035 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r) |
| 2036 |
< |
\end{equation} |
| 2037 |
< |
% |
| 2038 |
< |
% ta qa db |
| 2039 |
< |
% |
| 2040 |
< |
\begin{equation} |
| 2041 |
< |
\begin{split} |
| 2042 |
< |
% 1 |
| 2043 |
< |
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} &= |
| 2044 |
< |
\frac{1}{4\pi \epsilon_0} \left( |
| 2045 |
< |
2\sum_{lmn}D_{\mathbf{b}l} |
| 2046 |
< |
(\hat{b}_l \cdot \hat{a}_m) |
| 2047 |
< |
Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n) |
| 2048 |
< |
+2\sum_{lmn}D_{\mathbf{b}l} |
| 2049 |
< |
(\hat{a}_l \times \hat{b}_m) |
| 2050 |
< |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 2051 |
< |
\right) v_{31}(r) \\ |
| 2052 |
< |
% 2 |
| 2053 |
< |
&+\frac{2}{4\pi \epsilon_0} |
| 2054 |
< |
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 2055 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 2056 |
< |
Q_{{\mathbf a}mn} |
| 2057 |
< |
(\hat{r}\times \hat{a}_n) v_{32}(r) |
| 2058 |
< |
\end{split} |
| 2059 |
< |
\end{equation} |
| 2060 |
< |
% |
| 2061 |
< |
% tb qa db |
| 2062 |
< |
% |
| 2063 |
< |
\begin{equation} |
| 2064 |
< |
\begin{split} |
| 2065 |
< |
% 1 |
| 2066 |
< |
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} &= |
| 2067 |
< |
\frac{1}{4\pi \epsilon_0} \left( |
| 2068 |
< |
\text{Tr}Q_{\mathbf{a}} |
| 2069 |
< |
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} |
| 2070 |
< |
+2\sum_{lmn}D_{\mathbf{b}l} |
| 2071 |
< |
(\hat{a}_l \times \hat{b}_m) |
| 2072 |
< |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 2073 |
< |
\right) v_{31}(r)\\ |
| 2074 |
< |
% 2 |
| 2075 |
< |
&\frac{1}{4\pi \epsilon_0} |
| 2076 |
< |
\sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l} |
| 2077 |
< |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 2078 |
< |
Q_{{\mathbf a}mn} |
| 2079 |
< |
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
| 2080 |
< |
\end{split} |
| 2081 |
< |
\end{equation} |
| 2082 |
< |
% |
| 2083 |
< |
% ta qa qb |
| 2084 |
< |
% |
| 2085 |
< |
\begin{equation} |
| 2086 |
< |
\begin{split} |
| 2087 |
< |
% 1 |
| 2088 |
< |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} &= |
| 2089 |
< |
-\frac{4}{4\pi \epsilon_0} |
| 2090 |
< |
\sum_{lmnp} (\hat{a}_l \times \hat{b}_p) |
| 2091 |
< |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 2092 |
< |
(\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \\ |
| 2093 |
< |
% 2 |
| 2094 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 2095 |
< |
\Bigl[ |
| 2096 |
< |
2\text{Tr}Q_{\mathbf{b}} |
| 2097 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
| 2098 |
< |
Q_{{\mathbf a}lm} |
| 2099 |
< |
(\hat{r} \times \hat{a}_m) |
| 2100 |
< |
+4 \sum_{lmnp} |
| 2101 |
< |
(\hat{r} \times \hat{a}_l ) |
| 2102 |
< |
Q_{{\mathbf a}lm} |
| 2103 |
< |
(\hat{a}_m \cdot \hat{b}_n) |
| 2104 |
< |
Q_{{\mathbf b}np} |
| 2105 |
< |
(\hat{b}_p \cdot \hat{r}) \\ |
| 2106 |
< |
% 3 |
| 2107 |
< |
&-4 \sum_{lmnp} |
| 2108 |
< |
(\hat{r} \cdot \hat{a}_l ) |
| 2109 |
< |
Q_{{\mathbf a}lm} |
| 2110 |
< |
(\hat{a}_m \times \hat{b}_n) |
| 2111 |
< |
Q_{{\mathbf b}np} |
| 2112 |
< |
(\hat{b}_p \cdot \hat{r}) |
| 2113 |
< |
\Bigr] v_{42}(r) \\ |
| 2114 |
< |
% 4 |
| 2115 |
< |
&+ \frac{2}{4\pi \epsilon_0} |
| 2116 |
< |
\sum_{lm} (\hat{r} \times \hat{a}_l) |
| 2117 |
< |
Q_{{\mathbf a}lm} |
| 2118 |
< |
(\hat{a}_m \cdot \hat{r}) |
| 2119 |
< |
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
| 2120 |
< |
Q_{{\mathbf b}np} |
| 2121 |
< |
(\hat{b}_p \cdot \hat{r}) v_{43}(r)\\ |
| 2122 |
< |
\end{split} |
| 2123 |
< |
\end{equation} |
| 2124 |
< |
% |
| 2125 |
< |
% tb qa qb |
| 2126 |
< |
% |
| 2127 |
< |
\begin{equation} |
| 2128 |
< |
\begin{split} |
| 2129 |
< |
% 1 |
| 2130 |
< |
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} &= |
| 2131 |
< |
\frac{4}{4\pi \epsilon_0} |
| 2132 |
< |
\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 2133 |
< |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 2134 |
< |
(\hat{a}_m \times \hat{b}_n) v_{41}(r) \\ |
| 2135 |
< |
% 2 |
| 2136 |
< |
&+ \frac{1}{4\pi \epsilon_0} |
| 2137 |
< |
\Bigl[ |
| 2138 |
< |
2\text{Tr}Q_{\mathbf{a}} |
| 2139 |
< |
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
| 2140 |
< |
Q_{{\mathbf b}lm} |
| 2141 |
< |
(\hat{r} \times \hat{b}_m) |
| 2142 |
< |
+4 \sum_{lmnp} |
| 2143 |
< |
(\hat{r} \cdot \hat{a}_l ) |
| 2144 |
< |
Q_{{\mathbf a}lm} |
| 2145 |
< |
(\hat{a}_m \cdot \hat{b}_n) |
| 2146 |
< |
Q_{{\mathbf b}np} |
| 2147 |
< |
(\hat{r} \times \hat{b}_p) \\ |
| 2148 |
< |
% 3 |
| 2149 |
< |
&+4 \sum_{lmnp} |
| 2150 |
< |
(\hat{r} \cdot \hat{a}_l ) |
| 2151 |
< |
Q_{{\mathbf a}lm} |
| 2152 |
< |
(\hat{a}_m \times \hat{b}_n) |
| 2153 |
< |
Q_{{\mathbf b}np} |
| 2154 |
< |
(\hat{b}_p \cdot \hat{r}) |
| 2155 |
< |
\Bigr] v_{42}(r) \\ |
| 2156 |
< |
% 4 |
| 2157 |
< |
&+ \frac{2}{4\pi \epsilon_0} |
| 2158 |
< |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
| 2159 |
< |
Q_{{\mathbf a}lm} |
| 2160 |
< |
(\hat{a}_m \cdot \hat{r}) |
| 2161 |
< |
\sum_{np} (\hat{r} \times \hat{b}_n) |
| 2162 |
< |
Q_{{\mathbf b}np} |
| 2163 |
< |
(\hat{b}_p \cdot \hat{r}) v_{43}(r). \\ |
| 2164 |
< |
\end{split} |
| 2165 |
< |
\end{equation} |
| 2166 |
< |
% |
| 2167 |
< |
\begin{table*} |
| 2168 |
< |
\caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
| 2169 |
< |
\begin{ruledtabular} |
| 2170 |
< |
\begin{tabular}{ccc} |
| 2171 |
< |
Generic&Method 1&Method 2 |
| 2172 |
< |
\\ \hline |
| 2173 |
< |
% |
| 2174 |
< |
% |
| 2175 |
< |
% |
| 2176 |
< |
$w_a(r)$& |
| 2177 |
< |
$g_0(r)$& |
| 2178 |
< |
$g(r)-g(r_c)$ \\ |
| 2179 |
< |
% |
| 2180 |
< |
% |
| 2181 |
< |
$w_b(r)$ & |
| 2182 |
< |
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
| 2183 |
< |
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
| 2184 |
< |
% |
| 2185 |
< |
$w_c(r)$ & |
| 2186 |
< |
$\frac{g_1(r)}{r} $ & |
| 2187 |
< |
$\frac{v_{11}(r)}{r}$ \\ |
| 2188 |
< |
% |
| 2189 |
< |
% |
| 2190 |
< |
$w_d(r)$& |
| 2191 |
< |
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
| 2192 |
< |
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
| 2193 |
< |
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\ |
| 2194 |
< |
% |
| 2195 |
< |
$w_e(r)$ & |
| 2196 |
< |
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
| 2197 |
< |
$\frac{v_{22}(r)}{r}$ \\ |
| 2198 |
< |
% |
| 2199 |
< |
% |
| 2200 |
< |
$w_f(r)$& |
| 2201 |
< |
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
| 2202 |
< |
$\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\ |
| 2203 |
< |
&&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\ |
| 2204 |
< |
% |
| 2205 |
< |
$w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
| 2206 |
< |
$\frac{v_{31}(r)}{r}$\\ |
| 2207 |
< |
% |
| 2208 |
< |
$w_h(r)$ & |
| 2209 |
< |
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 2210 |
< |
$\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\ |
| 2211 |
< |
&&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
| 2212 |
< |
&&$-\frac{v_{31}(r)}{r}$\\ |
| 2213 |
< |
% 2 |
| 2214 |
< |
$w_i(r)$ & |
| 2215 |
< |
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 2216 |
< |
$\frac{v_{32}(r)}{r}$ \\ |
| 2217 |
< |
% |
| 2218 |
< |
$w_j(r)$ & |
| 2219 |
< |
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
| 2220 |
< |
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $ \\ |
| 2221 |
< |
&&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
| 2222 |
< |
% |
| 2223 |
< |
$w_k(r)$ & |
| 2224 |
< |
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 2225 |
< |
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right)$ \\ |
| 2226 |
< |
&&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
| 2227 |
< |
% |
| 2228 |
< |
$w_l(r)$ & |
| 2229 |
< |
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 2230 |
< |
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
| 2231 |
< |
&&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right) |
| 2232 |
< |
-\frac{2v_{42}(r)}{r}$ \\ |
| 2233 |
< |
% |
| 2234 |
< |
$w_m(r)$ & |
| 2235 |
< |
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
| 2236 |
< |
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\ |
| 2237 |
< |
&&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
| 2238 |
< |
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ |
| 2239 |
< |
&&$-\frac{4v_{43}(r)}{r}$ \\ |
| 2240 |
< |
% |
| 2241 |
< |
$w_n(r)$ & |
| 2242 |
< |
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 2243 |
< |
$\frac{v_{42}(r)}{r}$ \\ |
| 2244 |
< |
% |
| 2245 |
< |
$w_o(r)$ & |
| 2246 |
< |
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 2247 |
< |
$\frac{v_{43}(r)}{r}$ \\ |
| 2248 |
< |
% |
| 2249 |
< |
\end{tabular} |
| 2250 |
< |
\end{ruledtabular} |
| 2251 |
< |
\end{table*} |
| 1643 |
> |
For undamped charges Table I lists these derivatives under the column |
| 1644 |
> |
``Bare Coulomb.'' Equations \ref{eq:b9} to \ref{eq:b13} are still |
| 1645 |
> |
correct for GSF electrostatics if the subscript $n$ is eliminated. |
| 1646 |
|
|
| 1647 |
|
\newpage |
| 1648 |
|
|
