| 276 |
|
a}\alpha\beta}$, respectively. These are the primitive multipoles |
| 277 |
|
which can be expressed as a distribution of charges, |
| 278 |
|
\begin{align} |
| 279 |
< |
C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \\ |
| 280 |
< |
D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha} ,\\ |
| 281 |
< |
Q_{{\bf a}\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} r_{k\beta} . |
| 279 |
> |
C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \label{eq:charge} \\ |
| 280 |
> |
D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha}, \label{eq:dipole}\\ |
| 281 |
> |
Q_{{\bf a}\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k |
| 282 |
> |
r_{k\alpha} r_{k\beta} . \label{eq:quadrupole} |
| 283 |
|
\end{align} |
| 284 |
|
Note that the definition of the primitive quadrupole here differs from |
| 285 |
|
the standard traceless form, and contains an additional Taylor-series |
| 286 |
< |
based factor of $1/2$. |
| 286 |
> |
based factor of $1/2$. We are essentially treating the mass |
| 287 |
> |
distribution with higher priority; the moment of inertia tensor, |
| 288 |
> |
$\overleftrightarrow{\mathsf I}$, is diagonalized to obtain body-fixed |
| 289 |
> |
axes, and the charge distribution may result in a quadrupole tensor |
| 290 |
> |
that is not necessarily diagonal in the body frame. Additional |
| 291 |
> |
reasons for utilizing the primitive quadrupole are discussed in |
| 292 |
> |
section \ref{sec:damped}. |
| 293 |
|
|
| 294 |
|
It is convenient to locate charges $q_j$ relative to the center of mass of $\bf b$. Then with $\bf{r}$ pointing from |
| 295 |
|
$\bf a$ to $\bf b$ ($\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $), the interaction energy is given by |
| 313 |
|
of $\bf a$ interacting with the same multipoles on $\bf b$. |
| 314 |
|
|
| 315 |
|
\subsection{Damped Coulomb interactions} |
| 316 |
+ |
\label{sec:damped} |
| 317 |
|
In the standard multipole expansion, one typically uses the bare |
| 318 |
|
Coulomb potential, with radial dependence $1/r$, as shown in |
| 319 |
|
Eq.~(\ref{kernel}). It is also quite common to use a damped Coulomb |
| 329 |
|
either $1/r$ or $B_0(r)$, and all of the techniques can be applied to |
| 330 |
|
bare or damped Coulomb kernels (or any other function) as long as |
| 331 |
|
derivatives of these functions are known. Smith's convenient |
| 332 |
< |
functions $B_l(r)$ are summarized in Appendix A. |
| 332 |
> |
functions $B_l(r)$ are summarized in Appendix A. (N.B. there is one |
| 333 |
> |
important distinction between the two kernels, which is the behavior |
| 334 |
> |
of $\nabla^2 \frac{1}{r}$ compared with $\nabla^2 B_0(r)$. The former |
| 335 |
> |
is zero everywhere except for a delta function evaluated at the |
| 336 |
> |
origin. The latter also has delta function behavior, but is non-zero |
| 337 |
> |
for $r \neq 0$. Thus the standard justification for using a traceless |
| 338 |
> |
quadrupole tensor fails for the damped case.) |
| 339 |
|
|
| 340 |
|
The main goal of this work is to smoothly cut off the interaction |
| 341 |
|
energy as well as forces and torques as $r\rightarrow r_c$. To |
| 453 |
|
of the same index $n$. The algebra required to evaluate energies, |
| 454 |
|
forces and torques is somewhat tedious, so only the final forms are |
| 455 |
|
presented in tables \ref{tab:tableenergy} and \ref{tab:tableFORCE}. |
| 456 |
+ |
One of the principal findings of our work is that the individual |
| 457 |
+ |
orientational contributions to the various multipole-multipole |
| 458 |
+ |
interactions must be treated with distinct radial functions, but each |
| 459 |
+ |
of these contributions is independently force shifted at the cutoff |
| 460 |
+ |
radius. |
| 461 |
|
|
| 462 |
|
\subsection{Gradient-shifted force (GSF) electrostatics} |
| 463 |
|
The second, and conceptually simpler approach to force-shifting |
| 465 |
|
expansion, and has a similar interaction energy for all multipole |
| 466 |
|
orders: |
| 467 |
|
\begin{equation} |
| 468 |
< |
U^{\text{GSF}} = |
| 450 |
< |
U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 468 |
> |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 469 |
|
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r} |
| 470 |
< |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} . |
| 470 |
> |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} \right] |
| 471 |
|
\label{generic2} |
| 472 |
|
\end{equation} |
| 473 |
< |
Both the potential and the gradient for force shifting are evaluated |
| 474 |
< |
for an image multipole projected onto the surface of the cutoff sphere |
| 475 |
< |
(see fig \ref{fig:shiftedMultipoles}). The image multipole retains |
| 476 |
< |
the orientation ($\hat{\mathbf{b}}$) of the interacting multipole. No |
| 473 |
> |
where the sum describes a separate force-shifting that is applied to |
| 474 |
> |
each orientational contribution to the energy. Both the potential and |
| 475 |
> |
the gradient for force shifting are evaluated for an image multipole |
| 476 |
> |
projected onto the surface of the cutoff sphere (see fig |
| 477 |
> |
\ref{fig:shiftedMultipoles}). The image multipole retains the |
| 478 |
> |
orientation ($\hat{\mathbf{b}}$) of the interacting multipole. No |
| 479 |
|
higher order terms $(r-r_c)^n$ appear. The primary difference between |
| 480 |
|
the TSF and GSF methods is the stage at which the Taylor Series is |
| 481 |
|
applied; in the Taylor-shifted approach, it is applied to the kernel |
| 1459 |
|
In analogy to the dipolar arrays, the total electrostatic energy for |
| 1460 |
|
the quadrupolar arrays is: |
| 1461 |
|
\begin{equation} |
| 1462 |
< |
E = C \frac{3}{4} N^2 Q^2 |
| 1462 |
> |
E = C N \frac{3\bar{Q}^2}{4a^5} |
| 1463 |
|
\end{equation} |
| 1464 |
< |
where $Q$ is the quadrupole moment. The lowest energy |
| 1464 |
> |
where $a$ is the lattice parameter, and $\bar{Q}$ is the effective |
| 1465 |
> |
quadrupole moment, |
| 1466 |
> |
\begin{equation} |
| 1467 |
> |
\bar{Q}^2 = 4 \left(3 Q : Q - (\text{Tr} Q)^2 \right) |
| 1468 |
> |
\end{equation} |
| 1469 |
> |
for the primitive quadrupole as defined in Eq. \ref{eq:quadrupole}. |
| 1470 |
> |
(For the traceless quadrupole tensor, $\Theta = 3 Q - \text{Tr} Q$, |
| 1471 |
> |
the effective moment, $\bar{Q}^2 = \frac{4}{3} \Theta : \Theta$.) |
| 1472 |
|
|
| 1473 |
|
\section{Conclusion} |
| 1474 |
|
We have presented two efficient real-space methods for computing the |
