| 149 |
|
calling these models {\bf Taylor-shifted} (TSF), {\bf |
| 150 |
|
gradient-shifted} (GSF) and {\bf shifted potential} (SP) |
| 151 |
|
electrostatics. Because of differences in the initial assumptions, |
| 152 |
< |
the two methods yield related, but distinct expressions for energies, |
| 152 |
> |
the three methods yield related, but distinct expressions for energies, |
| 153 |
|
forces, and torques. |
| 154 |
|
|
| 155 |
|
In this paper we outline the new methodology and give functional forms |
| 321 |
|
\end{equation} |
| 322 |
|
This form has the benefit of separating out the energies of |
| 323 |
|
interaction into contributions from the charge, dipole, and quadrupole |
| 324 |
< |
of $\bf a$ interacting with the same multipoles in $\bf b$. |
| 324 |
> |
of $\bf a$ interacting with the same types of multipoles in $\bf b$. |
| 325 |
|
|
| 326 |
|
\subsection{Damped Coulomb interactions} |
| 327 |
|
\label{sec:damped} |
| 502 |
|
\end{equation} |
| 503 |
|
|
| 504 |
|
Functional forms for both methods (TSF and GSF) can both be summarized |
| 505 |
< |
using the form of Eq.~\ref{generic3}). The basic forms for the |
| 505 |
> |
using the form of Eq.~\ref{generic3}. The basic forms for the |
| 506 |
|
energy, force, and torque expressions are tabulated for both shifting |
| 507 |
|
approaches below -- for each separate orientational contribution, only |
| 508 |
|
the radial factors differ between the two methods. |
| 1566 |
|
It is also useful to understand the convergence to the lattice energy |
| 1567 |
|
constants as a function of the reduced damping parameter ($\alpha^* = |
| 1568 |
|
\alpha a$) for the different real-space methods. |
| 1569 |
< |
Figures. \ref{fig:Dipoles_alpha} and \ref{fig:Quadrupoles_alpha} show |
| 1569 |
> |
Figures \ref{fig:Dipoles_alpha} and \ref{fig:Quadrupoles_alpha} show |
| 1570 |
|
this comparison for the dipolar and quadrupolar lattices, |
| 1571 |
|
respectively. All of the methods (except for TSF) have excellent |
| 1572 |
|
behavior for the undamped or weakly-damped cases. All of the methods |
