| 499 |
|
\end{equation} |
| 500 |
|
|
| 501 |
|
Functional forms for both methods (TSF and GSF) can both be summarized |
| 502 |
< |
using the form of Eq.~(\ref{generic3}). The basic forms for the |
| 502 |
> |
using the form of Eq.~\ref{generic3}). The basic forms for the |
| 503 |
|
energy, force, and torque expressions are tabulated for both shifting |
| 504 |
|
approaches below -- for each separate orientational contribution, only |
| 505 |
|
the radial factors differ between the two methods. |
| 523 |
|
method for each orientational contribution, leaving out the $(r-r_c)$ |
| 524 |
|
terms that multiply the gradient. Functional forms for the |
| 525 |
|
shifted-potential (SP) method can also be summarized using the form of |
| 526 |
< |
Eq.~(\ref{eq:generic}). The energy, force, and torque expressions are |
| 526 |
> |
Eq.~\ref{generic3}. The energy, force, and torque expressions are |
| 527 |
|
tabulated below for all three methods. As in the GSF and TSF methods, |
| 528 |
|
for each separate orientational contribution, only the radial factors |
| 529 |
|
differ between the SP, GSF, and TSF methods. |
| 1579 |
|
|
| 1580 |
|
\section{Conclusion} |
| 1581 |
|
We have presented three efficient real-space methods for computing the |
| 1582 |
< |
interactions between point multipoles. These methods have the benefit |
| 1583 |
< |
of smoothly truncating the energies, forces, and torques at the cutoff |
| 1584 |
< |
radius, making them attractive for both molecular dynamics (MD) and |
| 1585 |
< |
Monte Carlo (MC) simulations. We find that the Gradient-Shifted Force |
| 1586 |
< |
(GSF) and the Shifted-Potential (SP) methods converge rapidly to the |
| 1587 |
< |
correct lattice energies for ordered dipolar and quadrupolar arrays, |
| 1588 |
< |
while the Taylor-Shifted Force (TSF) is too severe an approximation to |
| 1589 |
< |
provide accurate convergence to lattice energies. |
| 1582 |
> |
interactions between point multipoles. One of these (SP) is a |
| 1583 |
> |
multipolar generalization of Wolf's method that smoothly shifts |
| 1584 |
> |
electrostatic energies to zero at the cutoff radius. Two of these |
| 1585 |
> |
methods (GSF and TSF) also smoothly truncate the forces and torques |
| 1586 |
> |
(in addition to the energies) at the cutoff radius, making them |
| 1587 |
> |
attractive for both molecular dynamics and Monte Carlo simulations. We |
| 1588 |
> |
find that the Gradient-Shifted Force (GSF) and the Shifted-Potential |
| 1589 |
> |
(SP) methods converge rapidly to the correct lattice energies for |
| 1590 |
> |
ordered dipolar and quadrupolar arrays, while the Taylor-Shifted Force |
| 1591 |
> |
(TSF) is too severe an approximation to provide accurate convergence |
| 1592 |
> |
to lattice energies. |
| 1593 |
|
|
| 1594 |
|
In most cases, GSF can obtain nearly quantitative agreement with the |
| 1595 |
|
lattice energy constants with reasonably small cutoff radii. The only |
