| 87 |
|
classified as implicit methods (i.e., continuum dielectrics, static |
| 88 |
|
dipolar fields),\cite{Born20,Grossfield00} explicit methods (i.e., |
| 89 |
|
Ewald summations, interaction shifting or |
| 90 |
< |
trucation),\cite{Ewald21,Steinbach94} or a mixture of the two (i.e., |
| 90 |
> |
truncation),\cite{Ewald21,Steinbach94} or a mixture of the two (i.e., |
| 91 |
|
reaction field type methods, fast multipole |
| 92 |
|
methods).\cite{Onsager36,Rokhlin85} The explicit or mixed methods are |
| 93 |
|
often preferred because they incorporate dynamic solvent molecules in |
| 106 |
|
modification to the direct pairwise sum, and they lack the added |
| 107 |
|
periodicity of the Ewald sum. Below, these methods are evaluated using |
| 108 |
|
a variety of model systems and comparison methodologies to establish |
| 109 |
< |
their useability in molecular simulations. |
| 109 |
> |
their usability in molecular simulations. |
| 110 |
|
|
| 111 |
|
\subsection{The Ewald Sum} |
| 112 |
|
The complete accumulation electrostatic interactions in a system with |
| 126 |
|
$j$, and $\phi$ is Poisson's equation ($\phi(\mathbf{r}_{ij}) = q_i |
| 127 |
|
q_j |\mathbf{r}_{ij}|^{-1}$ for charge-charge interactions). In the |
| 128 |
|
case of monopole electrostatics, eq. (\ref{eq:PBCSum}) is |
| 129 |
< |
conditionally convergent and is discontiuous for non-neutral systems. |
| 129 |
> |
conditionally convergent and is discontinuous for non-neutral systems. |
| 130 |
|
|
| 131 |
|
This electrostatic summation problem was originally studied by Ewald |
| 132 |
|
for the case of an infinite crystal.\cite{Ewald21}. The approach he |
| 176 |
|
direct and reciprocal-space portions of the summation. The choice of |
| 177 |
|
the magnitude of this value allows one to select whether the |
| 178 |
|
real-space or reciprocal space portion of the summation is an |
| 179 |
< |
$\mathscr{O}(N^2)$ calcualtion (with the other being |
| 179 |
> |
$\mathscr{O}(N^2)$ calculation (with the other being |
| 180 |
|
$\mathscr{O}(N)$).\cite{Sagui99} With appropriate choice of $\alpha$ |
| 181 |
|
and thoughtful algorithm development, this cost can be brought down to |
| 182 |
|
$\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route taken to |
| 212 |
|
artificially stabilized by the periodic replicas introduced by the |
| 213 |
|
Ewald summation.\cite{Weber00} Thus, care ought to be taken when |
| 214 |
|
considering the use of the Ewald summation where the intrinsic |
| 215 |
< |
perodicity may negatively affect the system dynamics. |
| 215 |
> |
periodicity may negatively affect the system dynamics. |
| 216 |
|
|
| 217 |
|
|
| 218 |
|
\subsection{The Wolf and Zahn Methods} |
| 230 |
|
and a distance-dependent damping function (identical to that seen in |
| 231 |
|
the real-space portion of the Ewald sum) to aid convergence |
| 232 |
|
\begin{equation} |
| 233 |
< |
V_{\textrm{Wolf}}(r_{ij})= \frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}\right\}. |
| 233 |
> |
V_{\textrm{Wolf}}(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha r_{ij})}{r_{ij}}\right\}. |
| 234 |
|
\label{eq:WolfPot} |
| 235 |
|
\end{equation} |
| 236 |
|
Eq. (\ref{eq:WolfPot}) is essentially the common form of a shifted |
| 738 |
|
Correcting the resulting charged cutoff sphere is one of the purposes |
| 739 |
|
of the damped Coulomb summation proposed by Wolf \textit{et |
| 740 |
|
al.},\cite{Wolf99} and this correction indeed improves the results as |
| 741 |
< |
seen in the Shifted-Potental rows. While the undamped case of this |
| 741 |
> |
seen in the {\sc sp} rows. While the undamped case of this |
| 742 |
|
method is a significant improvement over the pure cutoff, it still |
| 743 |
|
doesn't correlate that well with SPME. Inclusion of potential damping |
| 744 |
|
improves the results, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows |
| 951 |
|
increased, these peaks are smoothed out, and approach the SPME |
| 952 |
|
curve. The damping acts as a distance dependent Gaussian screening of |
| 953 |
|
the point charges for the pairwise summation methods; thus, the |
| 954 |
< |
collisions are more elastic in the undamped {\sc sf} potental, and the |
| 954 |
> |
collisions are more elastic in the undamped {\sc sf} potential, and the |
| 955 |
|
stiffness of the potential is diminished as the electrostatic |
| 956 |
|
interactions are softened by the damping function. With $\alpha$ |
| 957 |
|
values of 0.2 \AA$^{-1}$, the {\sc sf} and {\sc sp} functions are |
