| 99 |
|
The computational efficiency and the accuracy of the DSF method are |
| 100 |
|
surprisingly good, particularly for systems with uniform charge |
| 101 |
|
density. Additionally, dielectric constants obtained using DSF and |
| 102 |
< |
similar methods where the force vanishes at $R_\textrm{c}$ are |
| 102 |
> |
similar methods where the force vanishes at $r_{c}$ are |
| 103 |
|
essentially quantitative.\cite{Izvekov:2008wo} The DSF and other |
| 104 |
|
related methods have now been widely investigated,\cite{Hansen:2012uq} |
| 105 |
|
and DSF is now used routinely in a diverse set of chemical |
| 112 |
|
entire molecular subsystems as a single rigid body, is simplification |
| 113 |
|
of the electrostatic interactions between these bodies so that fewer |
| 114 |
|
site-site interactions are required to compute configurational |
| 115 |
< |
energies. The coarse-graining approaches of Ren \& |
| 116 |
< |
coworkers,\cite{Golubkov06} and Essex \& |
| 117 |
< |
coworkers,\cite{ISI:000276097500009,ISI:000298664400012} both utilize |
| 118 |
< |
point multipoles to model electrostatics for entire molecules or |
| 119 |
< |
functional groups. |
| 115 |
> |
energies. To do this, the interactions between coarse-grained sites |
| 116 |
> |
are typically taken to be point |
| 117 |
> |
multipoles.\cite{Golubkov06,ISI:000276097500009,ISI:000298664400012} |
| 118 |
|
|
| 119 |
< |
Ichiye and coworkers have recently introduced a number of very fast |
| 120 |
< |
water models based on a ``sticky'' multipole model which are |
| 121 |
< |
qualitatively better at reproducing the behavior of real water than |
| 122 |
< |
the more common point-charge models (SPC/E, |
| 123 |
< |
TIPnP).\cite{Chowdhuri:2006lr,Te:2010rt,Te:2010ys,Te:2010vn} The SSDQO |
| 124 |
< |
model requires the use of an approximate multipole expansion (AME) as |
| 125 |
< |
the exact multipole expansion is quite expensive (particularly when |
| 126 |
< |
handled via the Ewald sum).\cite{Ichiye:2006qy} |
| 129 |
< |
|
| 130 |
< |
Another particularly important use of point multipoles (and multipole |
| 131 |
< |
polarizability) is in the very high-quality AMOEBA water model and |
| 119 |
> |
Water, in particular, has been modeled recently with point multipoles |
| 120 |
> |
up to octupolar |
| 121 |
> |
order.\cite{Chowdhuri:2006lr,Te:2010rt,Te:2010ys,Te:2010vn} For |
| 122 |
> |
maximum efficiency, these models require the use of an approximate |
| 123 |
> |
multipole expansion as the exact multipole expansion can become quite |
| 124 |
> |
expensive (particularly when handled via the Ewald |
| 125 |
> |
sum).\cite{Ichiye:2006qy} Point multipoles and multipole |
| 126 |
> |
polarizability have also been utilized in the AMOEBA water model and |
| 127 |
|
related force fields.\cite{Ponder:2010fk,schnieders:124114,Ren:2011uq} |
| 128 |
|
|
| 129 |
|
Higher-order multipoles present a peculiar issue for molecular |
| 139 |
|
Taylor expansions that are carried out at the cutoff radius. We are |
| 140 |
|
calling these models {\bf Taylor-shifted} and {\bf Gradient-shifted} |
| 141 |
|
electrostatics. Because of differences in the initial assumptions, |
| 142 |
< |
the two methods yield related, but different expressions for energies, |
| 143 |
< |
forces, and torques. |
| 142 |
> |
the two methods yield related, but somewhat different expressions for |
| 143 |
> |
energies, forces, and torques. |
| 144 |
|
|
| 145 |
|
In this paper we outline the new methodology and give functional forms |
| 146 |
|
for the energies, forces, and torques up to quadrupole-quadrupole |
| 147 |
|
order. We also compare the new methods to analytic energy constants |
| 148 |
< |
for periodic arrays of point multipoles. In the following paper, we |
| 148 |
> |
for periodic arrays of point multipoles. In the following paper, we |
| 149 |
|
provide numerical comparisons to Ewald-based electrostatics in common |
| 150 |
|
simulation enviornments. |
| 151 |
|
|
| 152 |
|
\section{Methodology} |
| 153 |
+ |
An efficient real-space electrostatic method involves the use of a |
| 154 |
+ |
pair-wise functional form, |
| 155 |
+ |
\begin{equation} |
| 156 |
+ |
V = \sum_i \sum_{j>i} V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) + |
| 157 |
+ |
\sum_i V_i^\mathrm{correction} |
| 158 |
+ |
\end{equation} |
| 159 |
+ |
that is short-ranged and easily truncated at a cutoff radius, |
| 160 |
+ |
\begin{equation} |
| 161 |
+ |
V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) = \left\{ |
| 162 |
+ |
\begin{array}{ll} |
| 163 |
+ |
V_\mathrm{approx} (r_{ij}, \Omega_i, \Omega_j) & \quad r \le r_c \\ |
| 164 |
+ |
0 & \quad r > r_c , |
| 165 |
+ |
\end{array} |
| 166 |
+ |
\right. |
| 167 |
+ |
\end{equation} |
| 168 |
+ |
along with an easily computed correction term ($\sum_i |
| 169 |
+ |
V_i^\mathrm{correction}$) which has linear-scaling with the number of |
| 170 |
+ |
particles. Here $\Omega_i$ and $\Omega_j$ represent orientational |
| 171 |
+ |
coordinates of the two sites. The computational efficiency, energy |
| 172 |
+ |
conservation, and even some physical properties of a simulation can |
| 173 |
+ |
depend dramatically on how the $V_\mathrm{approx}$ function behaves at |
| 174 |
+ |
the cutoff radius. The goal of any approximation method should be to |
| 175 |
+ |
mimic the real behavior of the electrostatic interactions as closely |
| 176 |
+ |
as possible without sacrificing the near-linear scaling of a cutoff |
| 177 |
+ |
method. |
| 178 |
|
|
| 179 |
|
\subsection{Self-neutralization, damping, and force-shifting} |
| 180 |
|
The DSF and Wolf methods operate by neutralizing the total charge |
| 181 |
|
contained within the cutoff sphere surrounding each particle. This is |
| 182 |
|
accomplished by shifting the potential functions to generate image |
| 183 |
|
charges on the surface of the cutoff sphere for each pair interaction |
| 184 |
< |
computed within $R_\textrm{c}$. Damping using a complementary error |
| 184 |
> |
computed within $r_c$. Damping using a complementary error |
| 185 |
|
function is applied to the potential to accelerate convergence. The |
| 186 |
|
potential for the DSF method, where $\alpha$ is the adjustable damping |
| 187 |
|
parameter, is given by |
| 188 |
|
\begin{equation*} |
| 189 |
< |
V_\mathrm{DSF}(r) = C_a C_b \Biggr{[} |
| 189 |
> |
V_\mathrm{DSF}(r) = C_i C_j \Biggr{[} |
| 190 |
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
| 191 |
< |
- \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} |
| 191 |
> |
- \frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c} + \left(\frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c^2} |
| 192 |
|
+ \frac{2\alpha}{\pi^{1/2}} |
| 193 |
< |
\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}} |
| 194 |
< |
\right)\left(r_{ij}-R_\mathrm{c}\right)\ \Biggr{]} |
| 193 |
> |
\frac{\exp\left(-\alpha^2r_c^2\right)}{r_c} |
| 194 |
> |
\right)\left(r_{ij}-r_c\right)\ \Biggr{]} |
| 195 |
|
\label{eq:DSFPot} |
| 196 |
|
\end{equation*} |
| 197 |
|
Note that in this potential and in all electrostatic quantities that |
| 198 |
< |
follow, the standard $4 \pi \epsilon_{0}$ has been omitted for |
| 198 |
> |
follow, the standard $1/4 \pi \epsilon_{0}$ has been omitted for |
| 199 |
|
clarity. |
| 200 |
|
|
| 201 |
|
To insure net charge neutrality within each cutoff sphere, an |
| 207 |
|
over the surface of the cutoff sphere. A portion of the self term is |
| 208 |
|
identical to the self term in the Ewald summation, and comes from the |
| 209 |
|
utilization of the complimentary error function for electrostatic |
| 210 |
< |
damping.\cite{deLeeuw80,Wolf99} |
| 211 |
< |
|
| 212 |
< |
There have been recent efforts to extend the Wolf self-neutralization |
| 193 |
< |
method to zero out the dipole and higher order multipoles contained |
| 194 |
< |
within the cutoff |
| 210 |
> |
damping.\cite{deLeeuw80,Wolf99} There have also been recent efforts to |
| 211 |
> |
extend the Wolf self-neutralization method to zero out the dipole and |
| 212 |
> |
higher order multipoles contained within the cutoff |
| 213 |
|
sphere.\cite{Fukuda:2011jk,Fukuda:2012yu,Fukuda:2013qv} |
| 214 |
|
|
| 215 |
|
In this work, we extend the idea of self-neutralization for the point |
| 230 |
|
\label{fig:shiftedMultipoles} |
| 231 |
|
\end{figure} |
| 232 |
|
|
| 233 |
< |
As in the point-charge approach, there is a contribution from |
| 234 |
< |
self-neutralization of site $i$. The self term for multipoles is |
| 233 |
> |
As in the point-charge approach, there is an additional contribution |
| 234 |
> |
from self-neutralization of site $i$. The self term for multipoles is |
| 235 |
|
described in section \ref{sec:selfTerm}. |
| 236 |
|
|
| 237 |
|
\subsection{The multipole expansion} |
| 304 |
|
\end{equation} |
| 305 |
|
This form has the benefit of separating out the energies of |
| 306 |
|
interaction into contributions from the charge, dipole, and quadrupole |
| 307 |
< |
of $\bf a$ interacting with the same multipoles $\bf b$. |
| 307 |
> |
of $\bf a$ interacting with the same multipoles on $\bf b$. |
| 308 |
|
|
| 309 |
|
\subsection{Damped Coulomb interactions} |
| 310 |
|
In the standard multipole expansion, one typically uses the bare |
| 319 |
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
| 320 |
|
\end{equation} |
| 321 |
|
We develop equations below using the function $f(r)$ to represent |
| 322 |
< |
either $1/r$ or $B_0(r)$, and all of the techniques can be applied |
| 323 |
< |
either to bare or damped Coulomb kernels as long as derivatives of |
| 324 |
< |
these functions are known. Smith's convenient functions $B_l(r)$ are |
| 325 |
< |
summarized in Appendix A. |
| 322 |
> |
either $1/r$ or $B_0(r)$, and all of the techniques can be applied to |
| 323 |
> |
bare or damped Coulomb kernels (or any other function) as long as |
| 324 |
> |
derivatives of these functions are known. Smith's convenient |
| 325 |
> |
functions $B_l(r)$ are summarized in Appendix A. |
| 326 |
|
|
| 327 |
|
The main goal of this work is to smoothly cut off the interaction |
| 328 |
|
energy as well as forces and torques as $r\rightarrow r_c$. To |
| 329 |
|
describe how this goal may be met, we use two examples, charge-charge |
| 330 |
< |
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain |
| 331 |
< |
the idea. |
| 330 |
> |
and charge-dipole, using the bare Coulomb kernel, $f(r)=1/r$, to |
| 331 |
> |
explain the idea. |
| 332 |
|
|
| 333 |
|
\subsection{Shifted-force methods} |
| 334 |
|
In the shifted-force approximation, the interaction energy for two |
| 364 |
|
derivatives required for each force term to vanish at the cutoff. For |
| 365 |
|
example, the quadrupole-quadrupole interaction energy requires four |
| 366 |
|
derivatives of the kernel, and the force requires one additional |
| 367 |
< |
derivative. We therefore require shifted energy expressions that |
| 368 |
< |
include enough terms so that all energies, forces, and torques are |
| 369 |
< |
zero as $r \rightarrow r_c$. In each case, we will subtract off a |
| 370 |
< |
function $f_n^{\text{shift}}(r)$ from the kernel $f(r)=1/r$. The |
| 371 |
< |
subscript $n$ indicates the number of derivatives to be taken when |
| 372 |
< |
deriving a given multipole energy. We choose a function with |
| 373 |
< |
guaranteed smooth derivatives --- a truncated Taylor series of the |
| 374 |
< |
function $f(r)$, e.g., |
| 367 |
> |
derivative. For quadrupole-quadrupole interactions, we therefore |
| 368 |
> |
require shifted energy expressions that include up to $(r-r_c)^5$ so |
| 369 |
> |
that all energies, forces, and torques are zero as $r \rightarrow |
| 370 |
> |
r_c$. In each case, we subtract off a function $f_n^{\text{shift}}(r)$ |
| 371 |
> |
from the kernel $f(r)=1/r$. The subscript $n$ indicates the number of |
| 372 |
> |
derivatives to be taken when deriving a given multipole energy. We |
| 373 |
> |
choose a function with guaranteed smooth derivatives -- a truncated |
| 374 |
> |
Taylor series of the function $f(r)$, e.g., |
| 375 |
|
% |
| 376 |
|
\begin{equation} |
| 377 |
|
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)}(r_c) . |
| 439 |
|
successive derivatives of the shifted electrostatic kernel, $f_n(r)$ |
| 440 |
|
of the same index $n$. The algebra required to evaluate energies, |
| 441 |
|
forces and torques is somewhat tedious, so only the final forms are |
| 442 |
< |
presented in tables XX and YY. |
| 442 |
> |
presented in tables \ref{tab:tableenergy} and \ref{tab:tableFORCE}. |
| 443 |
|
|
| 444 |
|
\subsection{Gradient-shifted force (GSF) electrostatics} |
| 445 |
|
The second, and conceptually simpler approach to force-shifting |
| 452 |
|
\label{generic2} |
| 453 |
|
\end{equation} |
| 454 |
|
Here the gradient for force shifting is evaluated for an image |
| 455 |
< |
multipole on the surface of the cutoff sphere (see fig |
| 455 |
> |
multipole projected onto the surface of the cutoff sphere (see fig |
| 456 |
|
\ref{fig:shiftedMultipoles}). No higher order terms $(r-r_c)^n$ |
| 457 |
|
appear. The primary difference between the TSF and GSF methods is the |
| 458 |
|
stage at which the Taylor Series is applied; in the Taylor-shifted |
| 465 |
|
\label{generic3} |
| 466 |
|
\end{equation} |
| 467 |
|
|
| 468 |
< |
Functional forms for both methods (TSF and GSF) can be summarized |
| 468 |
> |
Functional forms for both methods (TSF and GSF) can both be summarized |
| 469 |
|
using the form of Eq.~(\ref{generic3}). The basic forms for the |
| 470 |
|
energy, force, and torque expressions are tabulated for both shifting |
| 471 |
< |
approaches below - for each separate orientational contribution, only |
| 471 |
> |
approaches below -- for each separate orientational contribution, only |
| 472 |
|
the radial factors differ between the two methods. |
| 473 |
|
|
| 474 |
|
\subsection{\label{sec:level2}Body and space axes} |
| 1376 |
|
who computed the energies of ordered dipole arrays of zero |
| 1377 |
|
magnetization and obtained a number of these constants.\cite{Sauer} |
| 1378 |
|
This theory was developed more completely by Luttinger and |
| 1379 |
< |
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays and |
| 1380 |
< |
other periodic structures. We have repeated the Luttinger \& Tisza |
| 1381 |
< |
series summations to much higher order and obtained the following |
| 1382 |
< |
energy constants (converged to one part in $10^9$): |
| 1383 |
< |
\begin{table*} |
| 1379 |
> |
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays |
| 1380 |
> |
and other periodic structures. We have repeated the Luttinger \& |
| 1381 |
> |
Tisza series summations to much higher order and obtained the energy |
| 1382 |
> |
constants (converged to one part in $10^9$) in table \ref{tab:LT}. |
| 1383 |
> |
|
| 1384 |
> |
\begin{table*}[h] |
| 1385 |
|
\centering{ |
| 1386 |
|
\caption{Luttinger \& Tisza arrays and their associated |
| 1387 |
|
energy constants. Type "A" arrays have nearest neighbor strings of |
| 1399 |
|
A & BCC & 111 & -1.770078733 \\ |
| 1400 |
|
A & FCC & 001 & 2.166932835 \\ |
| 1401 |
|
A & FCC & 011 & -1.083466417 \\ |
| 1402 |
< |
|
| 1403 |
< |
* & BCC & minimum & -1.985920929 \\ |
| 1404 |
< |
|
| 1405 |
< |
B & SC & 001 & -2.676788684 \\ |
| 1406 |
< |
B & BCC & 001 & -1.338394342 \\ |
| 1407 |
< |
B & BCC & 111 & -1.770078733 \\ |
| 1389 |
< |
B & FCC & 001 & -1.083466417 \\ |
| 1390 |
< |
B & FCC & 011 & -1.807573634 |
| 1402 |
> |
B & SC & 001 & -2.676788684 \\ |
| 1403 |
> |
B & BCC & 001 & -1.338394342 \\ |
| 1404 |
> |
B & BCC & 111 & -1.770078733 \\ |
| 1405 |
> |
B & FCC & 001 & -1.083466417 \\ |
| 1406 |
> |
B & FCC & 011 & -1.807573634 \\ |
| 1407 |
> |
-- & BCC & minimum & -1.985920929 \\ |
| 1408 |
|
\end{tabular} |
| 1409 |
|
\end{ruledtabular} |
| 1410 |
|
\end{table*} |
| 1411 |
|
|
| 1412 |
|
In addition to the A and B arrays, there is an additional minimum |
| 1413 |
|
energy structure for the BCC lattice that was found by Luttinger \& |
| 1414 |
< |
Tisza. The total electrostatic energy for an array is given by: |
| 1414 |
> |
Tisza. The total electrostatic energy for any of the arrays is given |
| 1415 |
> |
by: |
| 1416 |
|
\begin{equation} |
| 1417 |
|
E = C N^2 \mu^2 |
| 1418 |
|
\end{equation} |
| 1419 |
< |
where $C$ is the energy constant given above, $N$ is the number of |
| 1420 |
< |
dipoles per unit volume, and $\mu$ is the strength of the dipole. |
| 1419 |
> |
where $C$ is the energy constant given in table \ref{tab:LT}, $N$ is |
| 1420 |
> |
the number of dipoles per unit volume, and $\mu$ is the strength of |
| 1421 |
> |
the dipole. |
| 1422 |
|
|
| 1423 |
< |
{\it Quadrupolar} analogues to the Madelung constants were first worked out by Nagai and Nakamura who |
| 1424 |
< |
computed the energies of selected quadrupole arrays based on |
| 1425 |
< |
extensions to the Luttinger and Tisza |
| 1426 |
< |
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
| 1423 |
> |
To test the new electrostatic methods, we have constructed very large |
| 1424 |
> |
($N = 8,000, 16,000, 32,000$) arrays of dipoles in the orientations |
| 1425 |
> |
described in \ref{tab:LT}. For the purposes of this paper, the primary |
| 1426 |
> |
quantity of interest is the analytic energy constant for the perfect |
| 1427 |
> |
arrays. Convergence to these constants are shown as a function of both |
| 1428 |
> |
the cutoff radius, $r_c$, and the damping parameter, $\alpha$ in Fig. |
| 1429 |
> |
XXX. We have simultaneously tested a hard cutoff (where the kernel is |
| 1430 |
> |
simply truncated at the cutoff radius), as well as a shifted potential |
| 1431 |
> |
(SP) form which includes a potential-shifting and self-interaction |
| 1432 |
> |
term, but does not shift the forces and torques smoothly at the cutoff |
| 1433 |
> |
radius. |
| 1434 |
> |
|
| 1435 |
> |
{\it Quadrupolar} analogues to the Madelung constants were first |
| 1436 |
> |
worked out by Nagai and Nakamura who computed the energies of selected |
| 1437 |
> |
quadrupole arrays based on extensions to the Luttinger and Tisza |
| 1438 |
> |
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
| 1439 |
|
energy constants for the lowest energy configurations for linear |
| 1440 |
|
quadrupoles shown in table \ref{tab:NNQ} |
| 1441 |
|
|
| 1660 |
|
under the column ``Bare Coulomb.'' Equations \ref{eq:b9} to |
| 1661 |
|
\ref{eq:b13} are still correct for GSF electrostatics if the subscript |
| 1662 |
|
$n$ is eliminated. |
| 1632 |
– |
|
| 1633 |
– |
% \section{Extra Material} |
| 1634 |
– |
% % |
| 1635 |
– |
% % |
| 1636 |
– |
% %Energy in body coordinate form --------------------------------------------------------------- |
| 1637 |
– |
% % |
| 1638 |
– |
% Here are the interaction energies written in terms of the body coordinates: |
| 1639 |
– |
|
| 1640 |
– |
% % |
| 1641 |
– |
% % u ca cb |
| 1642 |
– |
% % |
| 1643 |
– |
% \begin{equation} |
| 1644 |
– |
% U_{C_{\bf a}C_{\bf b}}(r)= |
| 1645 |
– |
% \frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) |
| 1646 |
– |
% \end{equation} |
| 1647 |
– |
% % |
| 1648 |
– |
% % u ca db |
| 1649 |
– |
% % |
| 1650 |
– |
% \begin{equation} |
| 1651 |
– |
% U_{C_{\bf a}D_{\bf b}}(r)= |
| 1652 |
– |
% \frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1653 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
| 1654 |
– |
% \end{equation} |
| 1655 |
– |
% % |
| 1656 |
– |
% % u ca qb |
| 1657 |
– |
% % |
| 1658 |
– |
% \begin{equation} |
| 1659 |
– |
% U_{C_{\bf a}Q_{\bf b}}(r)= |
| 1660 |
– |
% \frac{C_{\bf a }\text{Tr}Q_{\bf b}}{4\pi \epsilon_0} |
| 1661 |
– |
% v_{21}(r) \nonumber \\ |
| 1662 |
– |
% +\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1663 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) |
| 1664 |
– |
% v_{22}(r) |
| 1665 |
– |
% \end{equation} |
| 1666 |
– |
% % |
| 1667 |
– |
% % u da cb |
| 1668 |
– |
% % |
| 1669 |
– |
% \begin{equation} |
| 1670 |
– |
% U_{D_{\bf a}C_{\bf b}}(r)= |
| 1671 |
– |
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1672 |
– |
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
| 1673 |
– |
% \end{equation} |
| 1674 |
– |
% % |
| 1675 |
– |
% % u da db |
| 1676 |
– |
% % |
| 1677 |
– |
% \begin{equation} |
| 1678 |
– |
% \begin{split} |
| 1679 |
– |
% % 1 |
| 1680 |
– |
% U_{D_{\bf a}D_{\bf b}}(r)&= |
| 1681 |
– |
% -\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
| 1682 |
– |
% (\hat{a}_m \cdot \hat{b}_n) |
| 1683 |
– |
% D_{\mathbf{b}n} v_{21}(r) \\ |
| 1684 |
– |
% % 2 |
| 1685 |
– |
% &-\frac{1}{4\pi \epsilon_0} |
| 1686 |
– |
% \sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 1687 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} |
| 1688 |
– |
% v_{22}(r) |
| 1689 |
– |
% \end{split} |
| 1690 |
– |
% \end{equation} |
| 1691 |
– |
% % |
| 1692 |
– |
% % u da qb |
| 1693 |
– |
% % |
| 1694 |
– |
% \begin{equation} |
| 1695 |
– |
% \begin{split} |
| 1696 |
– |
% % 1 |
| 1697 |
– |
% U_{D_{\bf a}Q_{\bf b}}(r)&= |
| 1698 |
– |
% -\frac{1}{4\pi \epsilon_0} \left( |
| 1699 |
– |
% \text{Tr}Q_{\mathbf{b}} |
| 1700 |
– |
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
| 1701 |
– |
% +2\sum_{lmn}D_{\mathbf{a}l} |
| 1702 |
– |
% (\hat{a}_l \cdot \hat{b}_m) |
| 1703 |
– |
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 1704 |
– |
% \right) v_{31}(r) \\ |
| 1705 |
– |
% % 2 |
| 1706 |
– |
% &-\frac{1}{4\pi \epsilon_0} |
| 1707 |
– |
% \sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 1708 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1709 |
– |
% Q_{{\mathbf b}mn} |
| 1710 |
– |
% (\hat{b}_n \cdot \hat{r}) v_{32}(r) |
| 1711 |
– |
% \end{split} |
| 1712 |
– |
% \end{equation} |
| 1713 |
– |
% % |
| 1714 |
– |
% % u qa cb |
| 1715 |
– |
% % |
| 1716 |
– |
% \begin{equation} |
| 1717 |
– |
% U_{Q_{\bf a}C_{\bf b}}(r)= |
| 1718 |
– |
% \frac{C_{\bf b }\text{Tr}Q_{\bf a}}{4\pi \epsilon_0} v_{21}(r) |
| 1719 |
– |
% +\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1720 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r) |
| 1721 |
– |
% \end{equation} |
| 1722 |
– |
% % |
| 1723 |
– |
% % u qa db |
| 1724 |
– |
% % |
| 1725 |
– |
% \begin{equation} |
| 1726 |
– |
% \begin{split} |
| 1727 |
– |
% %1 |
| 1728 |
– |
% U_{Q_{\bf a}D_{\bf b}}(r)&= |
| 1729 |
– |
% \frac{1}{4\pi \epsilon_0} \left( |
| 1730 |
– |
% \text{Tr}Q_{\mathbf{a}} |
| 1731 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
| 1732 |
– |
% +2\sum_{lmn}D_{\mathbf{b}l} |
| 1733 |
– |
% (\hat{b}_l \cdot \hat{a}_m) |
| 1734 |
– |
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 1735 |
– |
% \right) v_{31}(r) \\ |
| 1736 |
– |
% % 2 |
| 1737 |
– |
% &+\frac{1}{4\pi \epsilon_0} |
| 1738 |
– |
% \sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 1739 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1740 |
– |
% Q_{{\mathbf a}mn} |
| 1741 |
– |
% (\hat{a}_n \cdot \hat{r}) v_{32}(r) |
| 1742 |
– |
% \end{split} |
| 1743 |
– |
% \end{equation} |
| 1744 |
– |
% % |
| 1745 |
– |
% % u qa qb |
| 1746 |
– |
% % |
| 1747 |
– |
% \begin{equation} |
| 1748 |
– |
% \begin{split} |
| 1749 |
– |
% %1 |
| 1750 |
– |
% U_{Q_{\bf a}Q_{\bf b}}(r)&= |
| 1751 |
– |
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1752 |
– |
% \text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
| 1753 |
– |
% +2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 1754 |
– |
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 1755 |
– |
% (\hat{a}_m \cdot \hat{b}_n) \Bigr] |
| 1756 |
– |
% v_{41}(r) \\ |
| 1757 |
– |
% % 2 |
| 1758 |
– |
% &+ \frac{1}{4\pi \epsilon_0} |
| 1759 |
– |
% \Bigl[ \text{Tr}Q_{\mathbf{a}} |
| 1760 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
| 1761 |
– |
% Q_{{\mathbf b}lm} |
| 1762 |
– |
% (\hat{b}_m \cdot \hat{r}) |
| 1763 |
– |
% +\text{Tr}Q_{\mathbf{b}} |
| 1764 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
| 1765 |
– |
% Q_{{\mathbf a}lm} |
| 1766 |
– |
% (\hat{a}_m \cdot \hat{r}) \\ |
| 1767 |
– |
% % 3 |
| 1768 |
– |
% &+4 \sum_{lmnp} |
| 1769 |
– |
% (\hat{r} \cdot \hat{a}_l ) |
| 1770 |
– |
% Q_{{\mathbf a}lm} |
| 1771 |
– |
% (\hat{a}_m \cdot \hat{b}_n) |
| 1772 |
– |
% Q_{{\mathbf b}np} |
| 1773 |
– |
% (\hat{b}_p \cdot \hat{r}) |
| 1774 |
– |
% \Bigr] v_{42}(r) \\ |
| 1775 |
– |
% % 4 |
| 1776 |
– |
% &+ \frac{1}{4\pi \epsilon_0} |
| 1777 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) |
| 1778 |
– |
% Q_{{\mathbf a}lm} |
| 1779 |
– |
% (\hat{a}_m \cdot \hat{r}) |
| 1780 |
– |
% \sum_{np} (\hat{r} \cdot \hat{b}_n) |
| 1781 |
– |
% Q_{{\mathbf b}np} |
| 1782 |
– |
% (\hat{b}_p \cdot \hat{r}) v_{43}(r). |
| 1783 |
– |
% \end{split} |
| 1784 |
– |
% \end{equation} |
| 1785 |
– |
% % |
| 1663 |
|
|
| 1787 |
– |
|
| 1788 |
– |
% % BODY coordinates force equations -------------------------------------------- |
| 1789 |
– |
% % |
| 1790 |
– |
% % |
| 1791 |
– |
% Here are the force equations written in terms of body coordinates. |
| 1792 |
– |
% % |
| 1793 |
– |
% % f ca cb |
| 1794 |
– |
% % |
| 1795 |
– |
% \begin{equation} |
| 1796 |
– |
% \mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
| 1797 |
– |
% \frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
| 1798 |
– |
% \end{equation} |
| 1799 |
– |
% % |
| 1800 |
– |
% % f ca db |
| 1801 |
– |
% % |
| 1802 |
– |
% \begin{equation} |
| 1803 |
– |
% \mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
| 1804 |
– |
% \frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1805 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r} |
| 1806 |
– |
% +\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1807 |
– |
% \sum_n D_{\mathbf{b}n} \hat{b}_n w_c(r) |
| 1808 |
– |
% \end{equation} |
| 1809 |
– |
% % |
| 1810 |
– |
% % f ca qb |
| 1811 |
– |
% % |
| 1812 |
– |
% \begin{equation} |
| 1813 |
– |
% \begin{split} |
| 1814 |
– |
% % 1 |
| 1815 |
– |
% \mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
| 1816 |
– |
% \frac{1}{4\pi \epsilon_0} |
| 1817 |
– |
% C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r} |
| 1818 |
– |
% + 2C_{\bf a } \sum_l \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot \hat{r}) w_e(r) \\ |
| 1819 |
– |
% % 2 |
| 1820 |
– |
% +\frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1821 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} |
| 1822 |
– |
% \end{split} |
| 1823 |
– |
% \end{equation} |
| 1824 |
– |
% % |
| 1825 |
– |
% % f da cb |
| 1826 |
– |
% % |
| 1827 |
– |
% \begin{equation} |
| 1828 |
– |
% \mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 1829 |
– |
% -\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
| 1830 |
– |
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r) \hat{r} |
| 1831 |
– |
% -\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
| 1832 |
– |
% \sum_n D_{\mathbf{a}n} \hat{a}_n w_c(r) |
| 1833 |
– |
% \end{equation} |
| 1834 |
– |
% % |
| 1835 |
– |
% % f da db |
| 1836 |
– |
% % |
| 1837 |
– |
% \begin{equation} |
| 1838 |
– |
% \begin{split} |
| 1839 |
– |
% % 1 |
| 1840 |
– |
% \mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
| 1841 |
– |
% -\frac{1}{4\pi \epsilon_0} |
| 1842 |
– |
% \sum_{mn} D_{\mathbf {a}m} |
| 1843 |
– |
% (\hat{a}_m \cdot \hat{b}_n) |
| 1844 |
– |
% D_{\mathbf{b}n} w_d(r) \hat{r} |
| 1845 |
– |
% -\frac{1}{4\pi \epsilon_0} |
| 1846 |
– |
% \sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 1847 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r} \\ |
| 1848 |
– |
% % 2 |
| 1849 |
– |
% & \quad + \frac{1}{4\pi \epsilon_0} |
| 1850 |
– |
% \Bigl[ \sum_m D_{\mathbf {a}m} |
| 1851 |
– |
% \hat{a}_m \sum_n D_{\mathbf{b}n} |
| 1852 |
– |
% (\hat{b}_n \cdot \hat{r}) |
| 1853 |
– |
% + \sum_m D_{\mathbf {b}m} |
| 1854 |
– |
% \hat{b}_m \sum_n D_{\mathbf{a}n} |
| 1855 |
– |
% (\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r) \\ |
| 1856 |
– |
% \end{split} |
| 1857 |
– |
% \end{equation} |
| 1858 |
– |
% % |
| 1859 |
– |
% % f da qb |
| 1860 |
– |
% % |
| 1861 |
– |
% \begin{equation} |
| 1862 |
– |
% \begin{split} |
| 1863 |
– |
% % 1 |
| 1864 |
– |
% &\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
| 1865 |
– |
% - \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1866 |
– |
% \text{Tr}Q_{\mathbf{b}} |
| 1867 |
– |
% \sum_l D_{\mathbf{a}l} \hat{a}_l |
| 1868 |
– |
% +2\sum_{lmn} D_{\mathbf{a}l} |
| 1869 |
– |
% (\hat{a}_l \cdot \hat{b}_m) |
| 1870 |
– |
% Q_{\mathbf{b}mn} \hat{b}_n \Bigr] w_g(r) \\ |
| 1871 |
– |
% % 3 |
| 1872 |
– |
% & - \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1873 |
– |
% \text{Tr}Q_{\mathbf{b}} |
| 1874 |
– |
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
| 1875 |
– |
% +2\sum_{lmn}D_{\mathbf{a}l} |
| 1876 |
– |
% (\hat{a}_l \cdot \hat{b}_m) |
| 1877 |
– |
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1878 |
– |
% % 4 |
| 1879 |
– |
% &+ \frac{1}{4\pi \epsilon_0} |
| 1880 |
– |
% \Bigl[\sum_l D_{\mathbf{a}l} \hat{a}_l |
| 1881 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1882 |
– |
% Q_{{\mathbf b}mn} |
| 1883 |
– |
% (\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l) |
| 1884 |
– |
% D_{\mathbf{a}l} |
| 1885 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1886 |
– |
% Q_{{\mathbf b}mn} \hat{b}_n \Bigr] w_i(r)\\ |
| 1887 |
– |
% % 6 |
| 1888 |
– |
% & -\frac{1}{4\pi \epsilon_0} |
| 1889 |
– |
% \sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 1890 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 1891 |
– |
% Q_{{\mathbf b}mn} |
| 1892 |
– |
% (\hat{b}_n \cdot \hat{r}) w_j(r) \hat{r} |
| 1893 |
– |
% \end{split} |
| 1894 |
– |
% \end{equation} |
| 1895 |
– |
% % |
| 1896 |
– |
% % force qa cb |
| 1897 |
– |
% % |
| 1898 |
– |
% \begin{equation} |
| 1899 |
– |
% \begin{split} |
| 1900 |
– |
% % 1 |
| 1901 |
– |
% \mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} &= |
| 1902 |
– |
% \frac{1}{4\pi \epsilon_0} |
| 1903 |
– |
% C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r) |
| 1904 |
– |
% + \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) \\ |
| 1905 |
– |
% % 2 |
| 1906 |
– |
% & +\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 1907 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} |
| 1908 |
– |
% \end{split} |
| 1909 |
– |
% \end{equation} |
| 1910 |
– |
% % |
| 1911 |
– |
% % f qa db |
| 1912 |
– |
% % |
| 1913 |
– |
% \begin{equation} |
| 1914 |
– |
% \begin{split} |
| 1915 |
– |
% % 1 |
| 1916 |
– |
% &\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
| 1917 |
– |
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1918 |
– |
% \text{Tr}Q_{\mathbf{a}} |
| 1919 |
– |
% \sum_l D_{\mathbf{b}l} \hat{b}_l |
| 1920 |
– |
% +2\sum_{lmn} D_{\mathbf{b}l} |
| 1921 |
– |
% (\hat{b}_l \cdot \hat{a}_m) |
| 1922 |
– |
% Q_{\mathbf{a}mn} \hat{a}_n \Bigr] |
| 1923 |
– |
% w_g(r)\\ |
| 1924 |
– |
% % 3 |
| 1925 |
– |
% & + \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1926 |
– |
% \text{Tr}Q_{\mathbf{a}} |
| 1927 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
| 1928 |
– |
% +2\sum_{lmn}D_{\mathbf{b}l} |
| 1929 |
– |
% (\hat{b}_l \cdot \hat{a}_m) |
| 1930 |
– |
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
| 1931 |
– |
% % 4 |
| 1932 |
– |
% & + \frac{1}{4\pi \epsilon_0} \Bigl[ \sum_l D_{\mathbf{b}l} \hat{b}_l |
| 1933 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1934 |
– |
% Q_{{\mathbf a}mn} |
| 1935 |
– |
% (\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l) |
| 1936 |
– |
% D_{\mathbf{b}l} |
| 1937 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1938 |
– |
% Q_{{\mathbf a}mn} \hat{a}_n \Bigr] w_i(r) \\ |
| 1939 |
– |
% % 6 |
| 1940 |
– |
% & +\frac{1}{4\pi \epsilon_0} |
| 1941 |
– |
% \sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 1942 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 1943 |
– |
% Q_{{\mathbf a}mn} |
| 1944 |
– |
% (\hat{a}_n \cdot \hat{r}) w_j(r) \hat{r} |
| 1945 |
– |
% \end{split} |
| 1946 |
– |
% \end{equation} |
| 1947 |
– |
% % |
| 1948 |
– |
% % f qa qb |
| 1949 |
– |
% % |
| 1950 |
– |
% \begin{equation} |
| 1951 |
– |
% \begin{split} |
| 1952 |
– |
% &\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
| 1953 |
– |
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1954 |
– |
% \text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
| 1955 |
– |
% + 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 1956 |
– |
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 1957 |
– |
% (\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r}\\ |
| 1958 |
– |
% &+\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1959 |
– |
% 2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
| 1960 |
– |
% + 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} \hat{b}_m \\ |
| 1961 |
– |
% &+ 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}) |
| 1962 |
– |
% + 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 |
| 1963 |
– |
% \Bigr] w_n(r) \\ |
| 1964 |
– |
% &+ \frac{1}{4\pi \epsilon_0} |
| 1965 |
– |
% \Bigl[ \text{Tr}Q_{\mathbf{a}} |
| 1966 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) |
| 1967 |
– |
% + \text{Tr}Q_{\mathbf{b}} |
| 1968 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) \\ |
| 1969 |
– |
% &+4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) |
| 1970 |
– |
% Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
| 1971 |
– |
% % |
| 1972 |
– |
% &+\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 1973 |
– |
% 2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
| 1974 |
– |
% \sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot \hat{r}) \\ |
| 1975 |
– |
% &+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
| 1976 |
– |
% \sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr] w_o(r) \hat{r} \\ |
| 1977 |
– |
% & + \frac{1}{4\pi \epsilon_0} |
| 1978 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
| 1979 |
– |
% \sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} |
| 1980 |
– |
% \end{split} |
| 1981 |
– |
% \end{equation} |
| 1982 |
– |
% % |
| 1983 |
– |
% Here we list the form of the non-zero damped shifted multipole torques showing |
| 1984 |
– |
% explicitly dependences on body axes: |
| 1985 |
– |
% % |
| 1986 |
– |
% % t ca db |
| 1987 |
– |
% % |
| 1988 |
– |
% \begin{equation} |
| 1989 |
– |
% \mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
| 1990 |
– |
% \frac{C_{\bf a}}{4\pi \epsilon_0} |
| 1991 |
– |
% \sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
| 1992 |
– |
% \end{equation} |
| 1993 |
– |
% % |
| 1994 |
– |
% % t ca qb |
| 1995 |
– |
% % |
| 1996 |
– |
% \begin{equation} |
| 1997 |
– |
% \mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
| 1998 |
– |
% \frac{2C_{\bf a}}{4\pi \epsilon_0} |
| 1999 |
– |
% \sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m \cdot \hat{r}) v_{22}(r) |
| 2000 |
– |
% \end{equation} |
| 2001 |
– |
% % |
| 2002 |
– |
% % t da cb |
| 2003 |
– |
% % |
| 2004 |
– |
% \begin{equation} |
| 2005 |
– |
% \mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
| 2006 |
– |
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 2007 |
– |
% \sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
| 2008 |
– |
% \end{equation}% |
| 2009 |
– |
% % |
| 2010 |
– |
% % |
| 2011 |
– |
% % ta da db |
| 2012 |
– |
% % |
| 2013 |
– |
% \begin{equation} |
| 2014 |
– |
% \begin{split} |
| 2015 |
– |
% % 1 |
| 2016 |
– |
% \mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
| 2017 |
– |
% \frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
| 2018 |
– |
% (\hat{a}_m \times \hat{b}_n) |
| 2019 |
– |
% D_{\mathbf{b}n} v_{21}(r) \\ |
| 2020 |
– |
% % 2 |
| 2021 |
– |
% &-\frac{1}{4\pi \epsilon_0} |
| 2022 |
– |
% \sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} |
| 2023 |
– |
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
| 2024 |
– |
% \end{split} |
| 2025 |
– |
% \end{equation} |
| 2026 |
– |
% % |
| 2027 |
– |
% % tb da db |
| 2028 |
– |
% % |
| 2029 |
– |
% \begin{equation} |
| 2030 |
– |
% \begin{split} |
| 2031 |
– |
% % 1 |
| 2032 |
– |
% \mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} &= |
| 2033 |
– |
% -\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
| 2034 |
– |
% (\hat{a}_m \times \hat{b}_n) |
| 2035 |
– |
% D_{\mathbf{b}n} v_{21}(r) \\ |
| 2036 |
– |
% % 2 |
| 2037 |
– |
% &+\frac{1}{4\pi \epsilon_0} |
| 2038 |
– |
% \sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 2039 |
– |
% \sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
| 2040 |
– |
% \end{split} |
| 2041 |
– |
% \end{equation} |
| 2042 |
– |
% % |
| 2043 |
– |
% % ta da qb |
| 2044 |
– |
% % |
| 2045 |
– |
% \begin{equation} |
| 2046 |
– |
% \begin{split} |
| 2047 |
– |
% % 1 |
| 2048 |
– |
% \mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} &= |
| 2049 |
– |
% \frac{1}{4\pi \epsilon_0} \left( |
| 2050 |
– |
% -\text{Tr}Q_{\mathbf{b}} |
| 2051 |
– |
% \sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} |
| 2052 |
– |
% +2\sum_{lmn}D_{\mathbf{a}l} |
| 2053 |
– |
% (\hat{a}_l \times \hat{b}_m) |
| 2054 |
– |
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 2055 |
– |
% \right) v_{31}(r)\\ |
| 2056 |
– |
% % 2 |
| 2057 |
– |
% &-\frac{1}{4\pi \epsilon_0} |
| 2058 |
– |
% \sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l} |
| 2059 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 2060 |
– |
% Q_{{\mathbf b}mn} |
| 2061 |
– |
% (\hat{b}_n \cdot \hat{r}) v_{32}(r) |
| 2062 |
– |
% \end{split} |
| 2063 |
– |
% \end{equation} |
| 2064 |
– |
% % |
| 2065 |
– |
% % tb da qb |
| 2066 |
– |
% % |
| 2067 |
– |
% \begin{equation} |
| 2068 |
– |
% \begin{split} |
| 2069 |
– |
% % 1 |
| 2070 |
– |
% \mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} &= |
| 2071 |
– |
% \frac{1}{4\pi \epsilon_0} \left( |
| 2072 |
– |
% -2\sum_{lmn}D_{\mathbf{a}l} |
| 2073 |
– |
% (\hat{a}_l \cdot \hat{b}_m) |
| 2074 |
– |
% Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n) |
| 2075 |
– |
% -2\sum_{lmn}D_{\mathbf{a}l} |
| 2076 |
– |
% (\hat{a}_l \times \hat{b}_m) |
| 2077 |
– |
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 2078 |
– |
% \right) v_{31}(r) \\ |
| 2079 |
– |
% % 2 |
| 2080 |
– |
% &-\frac{2}{4\pi \epsilon_0} |
| 2081 |
– |
% \sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 2082 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 2083 |
– |
% Q_{{\mathbf b}mn} |
| 2084 |
– |
% (\hat{r}\times \hat{b}_n) v_{32}(r) |
| 2085 |
– |
% \end{split} |
| 2086 |
– |
% \end{equation} |
| 2087 |
– |
% % |
| 2088 |
– |
% % ta qa cb |
| 2089 |
– |
% % |
| 2090 |
– |
% \begin{equation} |
| 2091 |
– |
% \mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
| 2092 |
– |
% \frac{2C_{\bf a}}{4\pi \epsilon_0} |
| 2093 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r) |
| 2094 |
– |
% \end{equation} |
| 2095 |
– |
% % |
| 2096 |
– |
% % ta qa db |
| 2097 |
– |
% % |
| 2098 |
– |
% \begin{equation} |
| 2099 |
– |
% \begin{split} |
| 2100 |
– |
% % 1 |
| 2101 |
– |
% \mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} &= |
| 2102 |
– |
% \frac{1}{4\pi \epsilon_0} \left( |
| 2103 |
– |
% 2\sum_{lmn}D_{\mathbf{b}l} |
| 2104 |
– |
% (\hat{b}_l \cdot \hat{a}_m) |
| 2105 |
– |
% Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n) |
| 2106 |
– |
% +2\sum_{lmn}D_{\mathbf{b}l} |
| 2107 |
– |
% (\hat{a}_l \times \hat{b}_m) |
| 2108 |
– |
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 2109 |
– |
% \right) v_{31}(r) \\ |
| 2110 |
– |
% % 2 |
| 2111 |
– |
% &+\frac{2}{4\pi \epsilon_0} |
| 2112 |
– |
% \sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 2113 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 2114 |
– |
% Q_{{\mathbf a}mn} |
| 2115 |
– |
% (\hat{r}\times \hat{a}_n) v_{32}(r) |
| 2116 |
– |
% \end{split} |
| 2117 |
– |
% \end{equation} |
| 2118 |
– |
% % |
| 2119 |
– |
% % tb qa db |
| 2120 |
– |
% % |
| 2121 |
– |
% \begin{equation} |
| 2122 |
– |
% \begin{split} |
| 2123 |
– |
% % 1 |
| 2124 |
– |
% \mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} &= |
| 2125 |
– |
% \frac{1}{4\pi \epsilon_0} \left( |
| 2126 |
– |
% \text{Tr}Q_{\mathbf{a}} |
| 2127 |
– |
% \sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} |
| 2128 |
– |
% +2\sum_{lmn}D_{\mathbf{b}l} |
| 2129 |
– |
% (\hat{a}_l \times \hat{b}_m) |
| 2130 |
– |
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 2131 |
– |
% \right) v_{31}(r)\\ |
| 2132 |
– |
% % 2 |
| 2133 |
– |
% &\frac{1}{4\pi \epsilon_0} |
| 2134 |
– |
% \sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l} |
| 2135 |
– |
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 2136 |
– |
% Q_{{\mathbf a}mn} |
| 2137 |
– |
% (\hat{a}_n \cdot \hat{r}) v_{32}(r) |
| 2138 |
– |
% \end{split} |
| 2139 |
– |
% \end{equation} |
| 2140 |
– |
% % |
| 2141 |
– |
% % ta qa qb |
| 2142 |
– |
% % |
| 2143 |
– |
% \begin{equation} |
| 2144 |
– |
% \begin{split} |
| 2145 |
– |
% % 1 |
| 2146 |
– |
% \mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} &= |
| 2147 |
– |
% -\frac{4}{4\pi \epsilon_0} |
| 2148 |
– |
% \sum_{lmnp} (\hat{a}_l \times \hat{b}_p) |
| 2149 |
– |
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 2150 |
– |
% (\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \\ |
| 2151 |
– |
% % 2 |
| 2152 |
– |
% &+ \frac{1}{4\pi \epsilon_0} |
| 2153 |
– |
% \Bigl[ |
| 2154 |
– |
% 2\text{Tr}Q_{\mathbf{b}} |
| 2155 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
| 2156 |
– |
% Q_{{\mathbf a}lm} |
| 2157 |
– |
% (\hat{r} \times \hat{a}_m) |
| 2158 |
– |
% +4 \sum_{lmnp} |
| 2159 |
– |
% (\hat{r} \times \hat{a}_l ) |
| 2160 |
– |
% Q_{{\mathbf a}lm} |
| 2161 |
– |
% (\hat{a}_m \cdot \hat{b}_n) |
| 2162 |
– |
% Q_{{\mathbf b}np} |
| 2163 |
– |
% (\hat{b}_p \cdot \hat{r}) \\ |
| 2164 |
– |
% % 3 |
| 2165 |
– |
% &-4 \sum_{lmnp} |
| 2166 |
– |
% (\hat{r} \cdot \hat{a}_l ) |
| 2167 |
– |
% Q_{{\mathbf a}lm} |
| 2168 |
– |
% (\hat{a}_m \times \hat{b}_n) |
| 2169 |
– |
% Q_{{\mathbf b}np} |
| 2170 |
– |
% (\hat{b}_p \cdot \hat{r}) |
| 2171 |
– |
% \Bigr] v_{42}(r) \\ |
| 2172 |
– |
% % 4 |
| 2173 |
– |
% &+ \frac{2}{4\pi \epsilon_0} |
| 2174 |
– |
% \sum_{lm} (\hat{r} \times \hat{a}_l) |
| 2175 |
– |
% Q_{{\mathbf a}lm} |
| 2176 |
– |
% (\hat{a}_m \cdot \hat{r}) |
| 2177 |
– |
% \sum_{np} (\hat{r} \cdot \hat{b}_n) |
| 2178 |
– |
% Q_{{\mathbf b}np} |
| 2179 |
– |
% (\hat{b}_p \cdot \hat{r}) v_{43}(r)\\ |
| 2180 |
– |
% \end{split} |
| 2181 |
– |
% \end{equation} |
| 2182 |
– |
% % |
| 2183 |
– |
% % tb qa qb |
| 2184 |
– |
% % |
| 2185 |
– |
% \begin{equation} |
| 2186 |
– |
% \begin{split} |
| 2187 |
– |
% % 1 |
| 2188 |
– |
% \mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} &= |
| 2189 |
– |
% \frac{4}{4\pi \epsilon_0} |
| 2190 |
– |
% \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 2191 |
– |
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 2192 |
– |
% (\hat{a}_m \times \hat{b}_n) v_{41}(r) \\ |
| 2193 |
– |
% % 2 |
| 2194 |
– |
% &+ \frac{1}{4\pi \epsilon_0} |
| 2195 |
– |
% \Bigl[ |
| 2196 |
– |
% 2\text{Tr}Q_{\mathbf{a}} |
| 2197 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
| 2198 |
– |
% Q_{{\mathbf b}lm} |
| 2199 |
– |
% (\hat{r} \times \hat{b}_m) |
| 2200 |
– |
% +4 \sum_{lmnp} |
| 2201 |
– |
% (\hat{r} \cdot \hat{a}_l ) |
| 2202 |
– |
% Q_{{\mathbf a}lm} |
| 2203 |
– |
% (\hat{a}_m \cdot \hat{b}_n) |
| 2204 |
– |
% Q_{{\mathbf b}np} |
| 2205 |
– |
% (\hat{r} \times \hat{b}_p) \\ |
| 2206 |
– |
% % 3 |
| 2207 |
– |
% &+4 \sum_{lmnp} |
| 2208 |
– |
% (\hat{r} \cdot \hat{a}_l ) |
| 2209 |
– |
% Q_{{\mathbf a}lm} |
| 2210 |
– |
% (\hat{a}_m \times \hat{b}_n) |
| 2211 |
– |
% Q_{{\mathbf b}np} |
| 2212 |
– |
% (\hat{b}_p \cdot \hat{r}) |
| 2213 |
– |
% \Bigr] v_{42}(r) \\ |
| 2214 |
– |
% % 4 |
| 2215 |
– |
% &+ \frac{2}{4\pi \epsilon_0} |
| 2216 |
– |
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) |
| 2217 |
– |
% Q_{{\mathbf a}lm} |
| 2218 |
– |
% (\hat{a}_m \cdot \hat{r}) |
| 2219 |
– |
% \sum_{np} (\hat{r} \times \hat{b}_n) |
| 2220 |
– |
% Q_{{\mathbf b}np} |
| 2221 |
– |
% (\hat{b}_p \cdot \hat{r}) v_{43}(r). \\ |
| 2222 |
– |
% \end{split} |
| 2223 |
– |
% \end{equation} |
| 2224 |
– |
% |
| 2225 |
– |
% \begin{table*} |
| 2226 |
– |
% \caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
| 2227 |
– |
% \begin{ruledtabular} |
| 2228 |
– |
% \begin{tabular}{|l|l|l|} |
| 2229 |
– |
% Generic&Taylor-shifted Force&Gradient-shifted Force |
| 2230 |
– |
% \\ \hline |
| 2231 |
– |
% % |
| 2232 |
– |
% % |
| 2233 |
– |
% % |
| 2234 |
– |
% $w_a(r)$& |
| 2235 |
– |
% $g_0(r)$& |
| 2236 |
– |
% $g(r)-g(r_c)$ \\ |
| 2237 |
– |
% % |
| 2238 |
– |
% % |
| 2239 |
– |
% $w_b(r)$ & |
| 2240 |
– |
% $\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
| 2241 |
– |
% $h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
| 2242 |
– |
% % |
| 2243 |
– |
% $w_c(r)$ & |
| 2244 |
– |
% $\frac{g_1(r)}{r} $ & |
| 2245 |
– |
% $\frac{v_{11}(r)}{r}$ \\ |
| 2246 |
– |
% % |
| 2247 |
– |
% % |
| 2248 |
– |
% $w_d(r)$& |
| 2249 |
– |
% $\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
| 2250 |
– |
% $\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
| 2251 |
– |
% -\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\ |
| 2252 |
– |
% % |
| 2253 |
– |
% $w_e(r)$ & |
| 2254 |
– |
% $\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
| 2255 |
– |
% $\frac{v_{22}(r)}{r}$ \\ |
| 2256 |
– |
% % |
| 2257 |
– |
% % |
| 2258 |
– |
% $w_f(r)$& |
| 2259 |
– |
% $\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
| 2260 |
– |
% $\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\ |
| 2261 |
– |
% &&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\ |
| 2262 |
– |
% % |
| 2263 |
– |
% $w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
| 2264 |
– |
% $\frac{v_{31}(r)}{r}$\\ |
| 2265 |
– |
% % |
| 2266 |
– |
% $w_h(r)$ & |
| 2267 |
– |
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 2268 |
– |
% $\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\ |
| 2269 |
– |
% &&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
| 2270 |
– |
% &&$-\frac{v_{31}(r)}{r}$\\ |
| 2271 |
– |
% % 2 |
| 2272 |
– |
% $w_i(r)$ & |
| 2273 |
– |
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 2274 |
– |
% $\frac{v_{32}(r)}{r}$ \\ |
| 2275 |
– |
% % |
| 2276 |
– |
% $w_j(r)$ & |
| 2277 |
– |
% $\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
| 2278 |
– |
% $\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $ \\ |
| 2279 |
– |
% &&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
| 2280 |
– |
% % |
| 2281 |
– |
% $w_k(r)$ & |
| 2282 |
– |
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 2283 |
– |
% $\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right)$ \\ |
| 2284 |
– |
% &&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
| 2285 |
– |
% % |
| 2286 |
– |
% $w_l(r)$ & |
| 2287 |
– |
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 2288 |
– |
% $\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
| 2289 |
– |
% &&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right) |
| 2290 |
– |
% -\frac{2v_{42}(r)}{r}$ \\ |
| 2291 |
– |
% % |
| 2292 |
– |
% $w_m(r)$ & |
| 2293 |
– |
% $\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
| 2294 |
– |
% $\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\ |
| 2295 |
– |
% &&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
| 2296 |
– |
% +\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ |
| 2297 |
– |
% &&$-\frac{4v_{43}(r)}{r}$ \\ |
| 2298 |
– |
% % |
| 2299 |
– |
% $w_n(r)$ & |
| 2300 |
– |
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 2301 |
– |
% $\frac{v_{42}(r)}{r}$ \\ |
| 2302 |
– |
% % |
| 2303 |
– |
% $w_o(r)$ & |
| 2304 |
– |
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 2305 |
– |
% $\frac{v_{43}(r)}{r}$ \\ |
| 2306 |
– |
% % |
| 2307 |
– |
% \end{tabular} |
| 2308 |
– |
% \end{ruledtabular} |
| 2309 |
– |
% \end{table*} |
| 2310 |
– |
|
| 1664 |
|
\newpage |
| 1665 |
|
|
| 1666 |
|
\bibliography{multipole} |
