Extending the Time Scale in Atomically Detailed Simulations - Reviews in Computational Chemistry - page 371

with R is a ''healing length'' over which the

switching function S varies smoothly between one and zero. The form of the
switching function is somewhat arbitrary, although sufficient smoothness is
required.
The forces due to bonded interaction potentials V b , V a , V d , and V i (see
Eq. [10]) are included in the fast component along with the nonbonded forces
that change the fastest. It is not uncommon for the two-scale splitting
described here to be generalized to a hierarchy of more than two classes.
We should mention that efficiency gains cannot typically be realized from
the simple splitting in which bonded forces comprise one class and nonbonded
forces the other. The reason is that atomic collisions cause the short-range
nonbonded forces to vary over roughly the same (short) time scale as the
bonded forces. The details of force splitting can be rather complicated and sys-
tem dependent. Later in this work, we will address this important issue influ-
encing efficient implementation of force splitting in MTS methods.

¼½ r ð r split l Þ= l . Here l

Numerical Integration with Force Splitting:
Extrapolation vs. Impulse
The general plan for a multiple time-step numerical method is that F fast
i
will be evaluated at every step of the integration at time increments h , while
F slow
i

will be evaluated less frequently, typically at time increments t h where

1 is an integer. The key question is: How should F slo i be incorporated
into the numerical dynamics? In the original work of Streett et al., 24 the
slow force on particle i was approximated by a truncated Taylor series at
each step j ,0

t >

< j < t , between updates at steps t n and t n þ t h :

slow
i

slow
i

jh F

2

F

F slow
i

Þ¼ F slow
i

1

ð t n þ jh ð t n Þþ ð t n Þþ

2 ð

jh Þ ð t n Þþ ½ 16

A natural simplification is to truncate the Taylor series after the constant term,
resulting in a constant extrapolation of the slow force. The velocity Verlet
method [6] can be easily modified to implement this constant extrapolation
multiple time-step method:

h
2 M 1

v n þ1=2
i

¼ v i þ ð F fast

i

þ F slow
i

Þ ½ 17a

hv n þ1=2
i

r n þ1
i

¼ r i þ ½ 17b

update F fast
i

0 update F slow
i

if ð n þ 1 Þ mod t ¼

h
2 M 1

¼ v n þ1=2
i

v n þ1
i

ð F fast
i

þ F slow
i

þ Þ ½ 17c