| 831 |
|
error of splitting method in terms of commutator of the |
| 832 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
| 833 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
| 834 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
| 834 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 835 |
|
\begin{equation} |
| 836 |
|
\exp (hX + hY) = \exp (hZ) |
| 837 |
|
\end{equation} |
| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{eqnarray*} |
| 849 |
> |
\begin{equation} |
| 850 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 851 |
|
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 852 |
|
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 853 |
|
\ldots ) |
| 854 |
< |
\end{eqnarray*} |
| 854 |
> |
\end{equation} |
| 855 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 856 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 857 |
|
procedure can be applied to general splitting, of the form |
| 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 |
| 1237 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1238 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1239 |
|
\begin{equation} |
| 1240 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 1240 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1241 |
|
\end{equation} |
| 1242 |
|
which is used to ensure rotation matrix's orthogonality. |
| 1243 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
