| 152 |
|
An efficient real-space electrostatic method involves the use of a |
| 153 |
|
pair-wise functional form, |
| 154 |
|
\begin{equation} |
| 155 |
< |
V = \sum_i \sum_{j>i} V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) + |
| 155 |
> |
V = \sum_i \sum_{j>i} V_\mathrm{pair}(\mathbf{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\{ |
| 160 |
> |
V_\mathrm{pair}(\mathbf{r}_{ij},\Omega_i, \Omega_j) = \left\{ |
| 161 |
|
\begin{array}{ll} |
| 162 |
< |
V_\mathrm{approx} (r_{ij}, \Omega_i, \Omega_j) & \quad r \le r_c \\ |
| 163 |
< |
0 & \quad r > r_c , |
| 162 |
> |
V_\mathrm{approx} (\mathbf{r}_{ij}, \Omega_i, \Omega_j) & \quad \left| \mathbf{r}_{ij} \right| \le r_c \\ |
| 163 |
> |
0 & \quad \left| \mathbf{r}_{ij} \right| > r_c , |
| 164 |
|
\end{array} |
| 165 |
|
\right. |
| 166 |
|
\end{equation} |
| 167 |
|
along with an easily computed self-interaction term ($\sum_i |
| 168 |
< |
V_i^\mathrm{self}$) which has linear-scaling with the number of |
| 168 |
> |
V_i^\mathrm{self}$) which scales linearly with the number of |
| 169 |
|
particles. Here $\Omega_i$ and $\Omega_j$ represent orientational |
| 170 |
< |
coordinates of the two sites. The computational efficiency, energy |
| 170 |
> |
coordinates of the two sites, and $\mathbf{r}_{ij}$ is the vector |
| 171 |
> |
between 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 |
| 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 $r_c$. 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 |
| 186 |
< |
parameter, is given by |
| 184 |
> |
computed within $r_c$. Damping using a complementary error function is |
| 185 |
> |
applied to the potential to accelerate convergence. The interaction |
| 186 |
> |
for a pair of charges ($C_i$ and $C_j$) in the DSF method, |
| 187 |
|
\begin{equation*} |
| 188 |
|
V_\mathrm{DSF}(r) = C_i C_j \Biggr{[} |
| 189 |
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
| 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. |
| 196 |
> |
where $\alpha$ is the adjustable damping parameter. Note that in this |
| 197 |
> |
potential and in all electrostatic quantities that follow, the |
| 198 |
> |
standard $1/4 \pi \epsilon_{0}$ has been omitted for clarity. |
| 199 |
|
|
| 200 |
|
To insure net charge neutrality within each cutoff sphere, an |
| 201 |
|
additional ``self'' term is added to the potential. This term is |
| 250 |
|
The Taylor expansion in $r$ can be written using an implied summation |
| 251 |
|
notation. Here Greek indices are used to indicate space coordinates |
| 252 |
|
($x$, $y$, $z$) and the subscripts $k$ and $j$ are reserved for |
| 253 |
< |
labelling specific charges in $\bf a$ and $\bf b$ respectively. The |
| 253 |
> |
labeling specific charges in $\bf a$ and $\bf b$ respectively. The |
| 254 |
|
Taylor expansion, |
| 255 |
|
\begin{equation} |
| 256 |
|
\frac{1}{\lvert \mathbf{r} - \mathbf{r}_k \rvert} = |
| 432 |
|
In general, we can write |
| 433 |
|
% |
| 434 |
|
\begin{equation} |
| 435 |
< |
U= (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 435 |
> |
U^{\text{TSF}}= (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 436 |
|
\label{generic} |
| 437 |
|
\end{equation} |
| 438 |
|
% |
| 614 |
|
|
| 615 |
|
In addition to cutoff-sphere neutralization, the Wolf |
| 616 |
|
summation~\cite{Wolf99} and the damped shifted force (DSF) |
| 617 |
< |
extension~\cite{Fennell:2006zl} also included self-interactions that |
| 617 |
> |
extension~\cite{Fennell:2006zl} also include self-interactions that |
| 618 |
|
are handled separately from the pairwise interactions between |
| 619 |
|
sites. The self-term is normally calculated via a single loop over all |
| 620 |
|
sites in the system, and is relatively cheap to evaluate. The |
| 661 |
|
reciprocal-space portion is identical to half of the self-term |
| 662 |
|
obtained by Smith and Aguado and Madden for the application of the |
| 663 |
|
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
| 664 |
< |
site which posesses a charge, dipole, and multipole, both types of |
| 664 |
> |
site which posesses a charge, dipole, and quadrupole, both types of |
| 665 |
|
contribution are given in table \ref{tab:tableSelf}. |
| 666 |
|
|
| 667 |
|
\begin{table*} |
| 1357 |
|
above. |
| 1358 |
|
|
| 1359 |
|
\section{Related real-space methods} |
| 1360 |
< |
One can also formulate a shifted potential, |
| 1360 |
> |
One can also formulate an extension of the Wolf approach for point |
| 1361 |
> |
multipoles by simply projecting the image multipole onto the surface |
| 1362 |
> |
of the cutoff sphere, and including the interactions with the central |
| 1363 |
> |
multipole and the image. This effectively shifts the total potential |
| 1364 |
> |
to zero at the cutoff radius, |
| 1365 |
|
\begin{equation} |
| 1366 |
< |
U^{\text{SP}} = U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 1367 |
< |
U(\mathbf{r}_c, \hat{\mathbf{a}}, \hat{\mathbf{b}}), |
| 1366 |
> |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 1367 |
> |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 1368 |
|
\label{eq:SP} |
| 1369 |
|
\end{equation} |
| 1370 |
< |
obtained by projecting the image multipole onto the surface of the |
| 1371 |
< |
cutoff sphere. The shifted potential (SP) can be thought of as a |
| 1372 |
< |
simple extension to the original Wolf method. The energies and |
| 1373 |
< |
torques for the SP can be easily obtained by zeroing out the $(r-r_c)$ |
| 1374 |
< |
terms in the final column of table \ref{tab:tableenergy}. SP forces |
| 1375 |
< |
(which retain discontinuities at the cutoff sphere) can be obtained by |
| 1370 |
> |
where the sum describes separate potential shifting that is applied to |
| 1371 |
> |
each orientational contribution to the energy. |
| 1372 |
> |
|
| 1373 |
> |
The energies and torques for the shifted potential (SP) can be easily |
| 1374 |
> |
obtained by zeroing out the $(r-r_c)$ terms in the final column of |
| 1375 |
> |
table \ref{tab:tableenergy}. Forces for the SP method retain |
| 1376 |
> |
discontinuities at the cutoff sphere, and can be obtained by |
| 1377 |
|
eliminating all functions that depend on $r_c$ in the last column of |
| 1378 |
< |
table \ref{tab:tableFORCE}. The self-energy contributions to the SP |
| 1378 |
> |
table \ref{tab:tableFORCE}. The self-energy contributions for the SP |
| 1379 |
|
potential are identical to both the GSF and TSF methods. |
| 1380 |
|
|
| 1381 |
|
\section{Comparison to known multipolar energies} |
| 1469 |
|
where $a$ is the lattice parameter, and $\bar{Q}$ is the effective |
| 1470 |
|
quadrupole moment, |
| 1471 |
|
\begin{equation} |
| 1472 |
< |
\bar{Q}^2 = 4 \left(3 Q : Q - (\text{Tr} Q)^2 \right) |
| 1472 |
> |
\bar{Q}^2 = 2 \left(3 Q : Q - (\text{Tr} Q)^2 \right) |
| 1473 |
|
\end{equation} |
| 1474 |
|
for the primitive quadrupole as defined in Eq. \ref{eq:quadrupole}. |
| 1475 |
|
(For the traceless quadrupole tensor, $\Theta = 3 Q - \text{Tr} Q$, |
| 1476 |
< |
the effective moment, $\bar{Q}^2 = \frac{4}{3} \Theta : \Theta$.) |
| 1476 |
> |
the effective moment, $\bar{Q}^2 = \frac{2}{3} \Theta : \Theta$.) |
| 1477 |
|
|
| 1478 |
|
\section{Conclusion} |
| 1479 |
|
We have presented two efficient real-space methods for computing the |
