| 140 |
|
\end{split} |
| 141 |
|
\label{eq:EwaldSum} |
| 142 |
|
\end{equation} |
| 143 |
< |
where $\alpha$ is a damping parameter, or separation constant, with |
| 144 |
< |
units of \AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and are |
| 145 |
< |
equal to $2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the |
| 146 |
< |
dielectric constant of the surrounding medium. The final two terms of |
| 143 |
> |
where $\alpha$ is the damping or convergence parameter with units of |
| 144 |
> |
\AA$^{-1}$, $\mathbf{k}$ are the reciprocal vectors and are equal to |
| 145 |
> |
$2\pi\mathbf{n}/L^2$, and $\epsilon_\textrm{S}$ is the dielectric |
| 146 |
> |
constant of the surrounding medium. The final two terms of |
| 147 |
|
eq. (\ref{eq:EwaldSum}) are a particle-self term and a dipolar term |
| 148 |
|
for interacting with a surrounding dielectric.\cite{Allen87} This |
| 149 |
|
dipolar term was neglected in early applications in molecular |
| 159 |
|
convergence. In more modern simulations, the simulation boxes have |
| 160 |
|
grown large enough that a real-space cutoff could potentially give |
| 161 |
|
convergent behavior. Indeed, it has often been observed that the |
| 162 |
< |
reciprocal-space portion of the Ewald sum can be vanishingly |
| 163 |
< |
small compared to the real-space portion.\cite{XXX} |
| 162 |
> |
reciprocal-space portion of the Ewald sum can be small and rapidly |
| 163 |
> |
convergent compared to the real-space portion with the choice of small |
| 164 |
> |
$\alpha$.\cite{Karasawa89,Kolafa92} |
| 165 |
|
|
| 166 |
|
\begin{figure} |
| 167 |
|
\centering |
| 177 |
|
\end{figure} |
| 178 |
|
|
| 179 |
|
The original Ewald summation is an $\mathscr{O}(N^2)$ algorithm. The |
| 180 |
< |
separation constant $(\alpha)$ plays an important role in balancing |
| 180 |
> |
convergence parameter $(\alpha)$ plays an important role in balancing |
| 181 |
|
the computational cost between the direct and reciprocal-space |
| 182 |
|
portions of the summation. The choice of this value allows one to |
| 183 |
|
select whether the real-space or reciprocal space portion of the |
| 715 |
|
manner across all systems and configurations. |
| 716 |
|
|
| 717 |
|
The althernative methods were also evaluated with three different |
| 718 |
< |
cutoff radii (9, 12, and 15 \AA). It should be noted that the damping |
| 719 |
< |
parameter chosen in SPME, or so called ``Ewald Coefficient'', has a |
| 720 |
< |
significant effect on the energies and forces calculated. Typical |
| 721 |
< |
molecular mechanics packages set this to a value dependent on the |
| 722 |
< |
cutoff radius and a tolerance (typically less than $1 \times 10^{-4}$ |
| 723 |
< |
kcal/mol). Smaller tolerances are typically associated with increased |
| 724 |
< |
accuracy at the expense of increased time spent calculating the |
| 725 |
< |
reciprocal-space portion of the summation.\cite{Perram88,Essmann95} |
| 726 |
< |
The default TINKER tolerance of $1 \times 10^{-8}$ kcal/mol was used |
| 727 |
< |
in all SPME calculations, resulting in Ewald Coefficients of 0.4200, |
| 728 |
< |
0.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ |
| 729 |
< |
respectively. |
| 718 |
> |
cutoff radii (9, 12, and 15 \AA). As noted previously, the |
| 719 |
> |
convergence parameter ($\alpha$) plays a role in the balance of the |
| 720 |
> |
real-space and reciprocal-space portions of the Ewald calculation. |
| 721 |
> |
Typical molecular mechanics packages set this to a value dependent on |
| 722 |
> |
the cutoff radius and a tolerance (typically less than $1 \times |
| 723 |
> |
10^{-4}$ kcal/mol). Smaller tolerances are typically associated with |
| 724 |
> |
increased accuracy at the expense of increased time spent calculating |
| 725 |
> |
the reciprocal-space portion of the |
| 726 |
> |
summation.\cite{Perram88,Essmann95} The default TINKER tolerance of $1 |
| 727 |
> |
\times 10^{-8}$ kcal/mol was used in all SPME calculations, resulting |
| 728 |
> |
in Ewald Coefficients of 0.4200, 0.3119, and 0.2476 \AA$^{-1}$ for |
| 729 |
> |
cutoff radii of 9, 12, and 15 \AA\ respectively. |
| 730 |
|
|
| 731 |
|
\section{Results and Discussion} |
| 732 |
|
|
