Extending the Time Scale in Atomically Detailed Simulations - Reviews in Computational Chemistry - page 370

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