Chemistry Reference
In-Depth Information
Success of the multiple time-step methodology depends on computing these
kN
N
2
interactions at each step, i.e., at intervals of
h
Eqs. [3]-[6] and
[8], while computing the remaining
N
pair interactions (which corre-
spond to forces that vary more slowly) at longer time intervals
ð
N
k
Þ
h
.
Street et al.
24
originally presented the MTS method in the context of a
distance truncated Lennard-Jones potential, so that the total number of com-
puted interactions was somewhat less that
N
2
. For biological MD applications
there is evidence that cutoffs can cause undesirable artifacts.
25,26
Before discussing the implementation details, we need to state the general
issues of multiple time-step numerical methods. The central objectives are: (1)
to devise a splitting of the systematic forces into a hierarchy of two or more
force classes based on the time interval over which they vary significantly, and
(2) to incorporate these force classes into a numerical method in a way that
realizes enhanced computational efficiency and maintains stability and accu-
racy of the computed solution.
t
Splitting the Force
Streett and co-workers
24
proposed a splitting of forces based on a dis-
tance parameter
r
split
. In the potential energy formalism we write
X
X
V
ð
r
Þ¼
V
ij
þ
V
ij
½
11
j
r
i
r
j
j<
r
split
j
r
i
r
j
j
r
split
¼
V
fast
ð
r
Þþ
V
slow
ð
r
Þ;
½
12
with
F
fast
ð
r
Þ :¼r
r
V
fast
F
slow
ð
r
Þ :¼r
r
V
slow
ð
r
Þ
ð
r
Þ
½
13
As a practical matter, particles will move in and out of the
r
split
sphere for a
given particle over the course of a simulation. To avoid discontinuities that
result from a particle suddenly changing classification from the slow force
component to the fast force component, the force can be decomposed into
fast and slow components using a switching function
S
ð
r
Þ
,
F
i
¼
F
fast
i
þ
F
slow
i
¼
S
ð
r
Þr
r
i
V
ð
1
S
ð
r
ÞÞr
r
i
V
½
14
where
8
<
1
r
<
r
split
l
R
2
S
ð
r
Þ¼
1
þ
ð
2
R
3
Þ
r
split
l
r
r
split
½
15
:
0
r
split
<
r
![](Reviews-in-Computational-Chemistry_images/img-gen2261.png)