| 1022 |
|
evaluation is to apply cutoff where particles farther than a |
| 1023 |
|
predetermined distance, are not included in the calculation |
| 1024 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 1025 |
< |
discontinuity in the potential energy curve |
| 1026 |
< |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
| 1027 |
< |
potential to ensure the potential curve go smoothly to zero at the |
| 1028 |
< |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
| 1029 |
< |
interaction because of its short range nature. However, simply |
| 1030 |
< |
truncating the electrostatic interaction with the use of cutoff has |
| 1031 |
< |
been shown to lead to severe artifacts in simulations. Ewald |
| 1032 |
< |
summation, in which the slowly conditionally convergent Coulomb |
| 1033 |
< |
potential is transformed into direct and reciprocal sums with rapid |
| 1034 |
< |
and absolute convergence, has proved to minimize the periodicity |
| 1035 |
< |
artifacts in liquid simulations. Taking the advantages of the fast |
| 1036 |
< |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
| 1037 |
< |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
| 1038 |
< |
$O(N logN)$. An alternative approach is \emph{fast multipole |
| 1039 |
< |
method}, which treats Coulombic interaction exactly at short range, |
| 1040 |
< |
and approximate the potential at long range through multipolar |
| 1041 |
< |
expansion. In spite of their wide acceptances at the molecular |
| 1042 |
< |
simulation community, these two methods are hard to be implemented |
| 1043 |
< |
correctly and efficiently. Instead, we use a damped and |
| 1044 |
< |
charge-neutralized Coulomb potential method developed by Wolf and |
| 1045 |
< |
his coworkers. The shifted Coulomb potential for particle $i$ and |
| 1046 |
< |
particle $j$ at distance $r_{rj}$ is given by: |
| 1025 |
> |
discontinuity in the potential energy curve. Fortunately, one can |
| 1026 |
> |
shift the potential to ensure the potential curve go smoothly to |
| 1027 |
> |
zero at the cutoff radius. Cutoff strategy works pretty well for |
| 1028 |
> |
Lennard-Jones interaction because of its short range nature. |
| 1029 |
> |
However, simply truncating the electrostatic interaction with the |
| 1030 |
> |
use of cutoff has been shown to lead to severe artifacts in |
| 1031 |
> |
simulations. Ewald summation, in which the slowly conditionally |
| 1032 |
> |
convergent Coulomb potential is transformed into direct and |
| 1033 |
> |
reciprocal sums with rapid and absolute convergence, has proved to |
| 1034 |
> |
minimize the periodicity artifacts in liquid simulations. Taking the |
| 1035 |
> |
advantages of the fast Fourier transform (FFT) for calculating |
| 1036 |
> |
discrete Fourier transforms, the particle mesh-based methods are |
| 1037 |
> |
accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative |
| 1038 |
> |
approach is \emph{fast multipole method}, which treats Coulombic |
| 1039 |
> |
interaction exactly at short range, and approximate the potential at |
| 1040 |
> |
long range through multipolar expansion. In spite of their wide |
| 1041 |
> |
acceptances at the molecular simulation community, these two methods |
| 1042 |
> |
are hard to be implemented correctly and efficiently. Instead, we |
| 1043 |
> |
use a damped and charge-neutralized Coulomb potential method |
| 1044 |
> |
developed by Wolf and his coworkers. The shifted Coulomb potential |
| 1045 |
> |
for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 1046 |
|
\begin{equation} |
| 1047 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 1048 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
