| 104 |
|
and a long-ranged reciprocal-space summation, |
| 105 |
|
\begin{equation} |
| 106 |
|
\begin{split} |
| 107 |
< |
V_\textrm{elec} = \frac{1}{2}& \sum_{i=1}^N\sum_{j=1}^N \Biggr[ q_i q_j\Biggr( {\sum_{\mathbf{n}}}^\prime \frac{\textrm{erfc}\left( \alpha|\mathbf{r}_{ij}+\mathbf{n}|\right)}{|\mathbf{r}_{ij}+\mathbf{n}|} \\ &+ \frac{1}{\pi L^3}\sum_{\mathbf{k}\ne 0}\frac{4\pi^2}{|\mathbf{k}|^2} \exp{\left(-\frac{\pi^2|\mathbf{k}|^2}{\alpha^2}\right) \cos\left(\mathbf{k}\cdot\mathbf{r}_{ij}\right)}\Biggr)\Biggr] \\ &- \frac{\alpha}{\pi^{1/2}}\sum_{i=1}^N q_i^2 + \frac{2\pi}{3L^3}\left|\sum_{i=1}^N q_i\mathbf{r}_i\right|^2, |
| 107 |
> |
V_\textrm{elec} = \frac{1}{2}& \sum_{i=1}^N\sum_{j=1}^N \Biggr[ q_i q_j\Biggr( {\sum_{\mathbf{n}}}^\prime \frac{\textrm{erfc}\left( \alpha|\mathbf{r}_{ij}+\mathbf{n}|\right)}{|\mathbf{r}_{ij}+\mathbf{n}|} \\ &+ \frac{1}{\pi L^3}\sum_{\mathbf{k}\ne 0}\frac{4\pi^2}{|\mathbf{k}|^2} \exp{\left(-\frac{\pi^2|\mathbf{k}|^2}{\alpha^2}\right) \cos\left(\mathbf{k}\cdot\mathbf{r}_{ij}\right)}\Biggr)\Biggr] \\ &- \frac{\alpha}{\pi^{1/2}}\sum_{i=1}^N q_i^2 + \frac{2\pi}{(2\epsilon_\textrm{S}+1)L^3}\left|\sum_{i=1}^N q_i\mathbf{r}_i\right|^2, |
| 108 |
|
\end{split} |
| 109 |
|
\label{eq:EwaldSum} |
| 110 |
|
\end{equation} |
| 111 |
|
where $\alpha$ is a damping parameter, or separation constant, with |
| 112 |
< |
units of \AA$^{-1}$, and $\mathbf{k}$ are the reciprocal vectors and |
| 113 |
< |
equal $2\pi\mathbf{n}/L^2$. The final two terms of |
| 112 |
> |
units of \AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and equal |
| 113 |
> |
$2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the dielectric |
| 114 |
> |
constant of the encompassing medium. The final two terms of |
| 115 |
|
eq. (\ref{eq:EwaldSum}) are a particle-self term and a dipolar term |
| 116 |
|
for interacting with a surrounding dielectric.\cite{Allen87} This |
| 117 |
|
dipolar term was neglected in early applications in molecular |
| 118 |
|
simulations,\cite{Brush66,Woodcock71} until it was introduced by de |
| 119 |
|
Leeuw {\it et al.} to address situations where the unit cell has a |
| 120 |
|
dipole moment and this dipole moment gets magnified through |
| 121 |
< |
replication of the periodic images.\cite{deLeeuw80} This term is zero |
| 122 |
< |
for systems where $\epsilon_{\rm S} = \infty$. Figure |
| 121 |
> |
replication of the periodic images.\cite{deLeeuw80,Smith81} If this |
| 122 |
> |
term is taken to be zero, the system is using conducting boundary |
| 123 |
> |
conditions, $\epsilon_{\rm S} = \infty$. Figure |
| 124 |
|
\ref{fig:ewaldTime} shows how the Ewald sum has been applied over |
| 125 |
|
time. Initially, due to the small size of systems, the entire |
| 126 |
|
simulation box was replicated to convergence. Currently, we balance a |
| 143 |
|
$\mathscr{O}(N^2)$ algorithm. The separation constant $(\alpha)$ |
| 144 |
|
plays an important role in the computational cost balance between the |
| 145 |
|
direct and reciprocal-space portions of the summation. The choice of |
| 146 |
< |
the magnitude of this value allows one to whether the real-space or |
| 147 |
< |
reciprocal space portion of the summation is an $\mathscr{O}(N^2)$ |
| 148 |
< |
calcualtion, with the other being $\mathscr{O}(N)$.\cite{Sagui99} With |
| 149 |
< |
appropriate choice of $\alpha$ and thoughtful algorithm development, |
| 150 |
< |
this cost can be brought down to |
| 146 |
> |
the magnitude of this value allows one to select whether the |
| 147 |
> |
real-space or reciprocal space portion of the summation is an |
| 148 |
> |
$\mathscr{O}(N^2)$ calcualtion (with the other being |
| 149 |
> |
$\mathscr{O}(N)$).\cite{Sagui99} With appropriate choice of $\alpha$ |
| 150 |
> |
and thoughtful algorithm development, this cost can be brought down to |
| 151 |
|
$\mathscr{O}(N^{3/2})$.\cite{Perram88} The typical route taken to |
| 152 |
< |
accelerate the Ewald summation is to se |
| 152 |
> |
reduce the cost of the Ewald summation further is to set $\alpha$ such |
| 153 |
> |
that the real-space interactions decay rapidly, allowing for a short |
| 154 |
> |
spherical cutoff, and then optimize the reciprocal space summation. |
| 155 |
> |
These optimizations usually involve the utilization of the fast |
| 156 |
> |
Fourier transform (FFT),\cite{Hockney81} leading to the |
| 157 |
> |
particle-particle particle-mesh (P3M) and particle mesh Ewald (PME) |
| 158 |
> |
methods.\cite{Shimada93,Luty94,Luty95,Darden93,Essmann95} In these |
| 159 |
> |
methods, the cost of the reciprocal-space portion of the Ewald |
| 160 |
> |
summation is from $\mathscr{O}(N^2)$ down to $\mathscr{O}(N \log N)$. |
| 161 |
|
|
| 162 |
+ |
These developments and optimizations have led the use of the Ewald |
| 163 |
+ |
summation to become routine in simulations with periodic boundary |
| 164 |
+ |
conditions. However, in certain systems the intrinsic three |
| 165 |
+ |
dimensional periodicity can prove to be problematic, such as two |
| 166 |
+ |
dimensional surfaces and membranes. The Ewald sum has been |
| 167 |
+ |
reformulated to handle 2D |
| 168 |
+ |
systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89}, but the new |
| 169 |
+ |
methods have been found to be computationally |
| 170 |
+ |
expensive.\cite{Spohr97,Yeh99} Inclusion of a correction term in the |
| 171 |
+ |
full Ewald summation is a possible direction for enabling the handling |
| 172 |
+ |
of 2D systems and the inclusion of the optimizations described |
| 173 |
+ |
previously.\cite{Yeh99} |
| 174 |
+ |
|
| 175 |
+ |
Several studies have recognized that the inherent periodicity in the |
| 176 |
+ |
Ewald sum can also have an effect on systems that have the same |
| 177 |
+ |
dimensionality.\cite{Roberts94,Roberts95,Luty96,Hunenberger99a,Hunenberger99b,Weber00} |
| 178 |
+ |
Good examples are solvated proteins kept at high relative |
| 179 |
+ |
concentration due to the periodicity of the electrostatics. In these |
| 180 |
+ |
systems, the more compact folded states of a protein can be |
| 181 |
+ |
artificially stabilized by the periodic replicas introduced by the |
| 182 |
+ |
Ewald summation.\cite{Weber00} Thus, care ought to be taken when |
| 183 |
+ |
considering the use of the Ewald summation where the intrinsic |
| 184 |
+ |
perodicity may negatively affect the system dynamics. |
| 185 |
+ |
|
| 186 |
+ |
|
| 187 |
|
\subsection{The Wolf and Zahn Methods} |
| 188 |
|
In a recent paper by Wolf \textit{et al.}, a procedure was outlined |
| 189 |
|
for the accurate accumulation of electrostatic interactions in an |
| 190 |
< |
efficient pairwise fashion.\cite{Wolf99} Wolf \textit{et al.} observed |
| 191 |
< |
that the electrostatic interaction is effectively short-ranged in |
| 192 |
< |
condensed phase systems and that neutralization of the charge |
| 193 |
< |
contained within the cutoff radius is crucial for potential |
| 194 |
< |
stability. They devised a pairwise summation method that ensures |
| 195 |
< |
charge neutrality and gives results similar to those obtained with |
| 196 |
< |
the Ewald summation. The resulting shifted Coulomb potential |
| 197 |
< |
(Eq. \ref{eq:WolfPot}) includes image-charges subtracted out through |
| 198 |
< |
placement on the cutoff sphere and a distance-dependent damping |
| 199 |
< |
function (identical to that seen in the real-space portion of the |
| 200 |
< |
Ewald sum) to aid convergence |
| 190 |
> |
efficient pairwise fashion and lacks the inherent periodicity of the |
| 191 |
> |
Ewald summation.\cite{Wolf99} Wolf \textit{et al.} observed that the |
| 192 |
> |
electrostatic interaction is effectively short-ranged in condensed |
| 193 |
> |
phase systems and that neutralization of the charge contained within |
| 194 |
> |
the cutoff radius is crucial for potential stability. They devised a |
| 195 |
> |
pairwise summation method that ensures charge neutrality and gives |
| 196 |
> |
results similar to those obtained with the Ewald summation. The |
| 197 |
> |
resulting shifted Coulomb potential (Eq. \ref{eq:WolfPot}) includes |
| 198 |
> |
image-charges subtracted out through placement on the cutoff sphere |
| 199 |
> |
and a distance-dependent damping function (identical to that seen in |
| 200 |
> |
the real-space portion of the Ewald sum) to aid convergence |
| 201 |
|
\begin{equation} |
| 202 |
|
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\}. |
| 203 |
|
\label{eq:WolfPot} |
