| 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}(\mathbf{r}_{ij}, \Omega_i, \Omega_j) + |
| 156 |
< |
\sum_i V_i^\mathrm{self} |
| 155 |
> |
U = \sum_i \sum_{j>i} U_\mathrm{pair}(\mathbf{r}_{ij}, \Omega_i, \Omega_j) + |
| 156 |
> |
\sum_i U_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}(\mathbf{r}_{ij},\Omega_i, \Omega_j) = \left\{ |
| 160 |
> |
U_\mathrm{pair}(\mathbf{r}_{ij},\Omega_i, \Omega_j) = \left\{ |
| 161 |
|
\begin{array}{ll} |
| 162 |
< |
V_\mathrm{approx} (\mathbf{r}_{ij}, \Omega_i, \Omega_j) & \quad \left| \mathbf{r}_{ij} \right| \le r_c \\ |
| 162 |
> |
U_\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 scales linearly with the number of |
| 168 |
> |
U_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, 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 |
| 173 |
> |
depend dramatically on how the $U_\mathrm{approx}$ 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 |
| 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{[} |
| 188 |
> |
U_\mathrm{DSF}(r_{ij}) = C_i C_j \Biggr{[} |
| 189 |
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
| 190 |
|
- \frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c} + \left(\frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c^2} |
| 191 |
|
+ \frac{2\alpha}{\pi^{1/2}} |
| 244 |
|
a$. Then the electrostatic potential of object $\bf a$ at |
| 245 |
|
$\mathbf{r}$ is given by |
| 246 |
|
\begin{equation} |
| 247 |
< |
V_a(\mathbf r) = |
| 247 |
> |
\phi_a(\mathbf r) = |
| 248 |
|
\sum_{k \, \text{in \bf a}} \frac{q_k}{\lvert \mathbf{r} - \mathbf{r}_k \rvert}. |
| 249 |
|
\end{equation} |
| 250 |
|
The Taylor expansion in $r$ can be written using an implied summation |
| 263 |
|
can then be used to express the electrostatic potential on $\bf a$ in |
| 264 |
|
terms of multipole operators, |
| 265 |
|
\begin{equation} |
| 266 |
< |
V_{\bf a}(\mathbf{r}) =\hat{M}_{\bf a} \frac{1}{r} |
| 266 |
> |
\phi_{\bf a}(\mathbf{r}) =\hat{M}_{\bf a} \frac{1}{r} |
| 267 |
|
\end{equation} |
| 268 |
|
where |
| 269 |
|
\begin{equation} |
| 626 |
|
the cutoff sphere. For a system of undamped charges, the total |
| 627 |
|
self-term is |
| 628 |
|
\begin{equation} |
| 629 |
< |
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
| 629 |
> |
U_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
| 630 |
|
\label{eq:selfTerm} |
| 631 |
|
\end{equation} |
| 632 |
|
|
| 641 |
|
complexity to the Ewald sum). For a system containing only damped |
| 642 |
|
charges, the complete self-interaction can be written as |
| 643 |
|
\begin{equation} |
| 644 |
< |
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
| 644 |
> |
U_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
| 645 |
|
\frac{\alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
| 646 |
|
C_{\bf a}^{2}. |
| 647 |
|
\label{eq:dampSelfTerm} |
