Chemistry Reference
In-Depth Information
This modification amounts to the replacement of the middle step of Eq. [6]
with an inner loop over the
steps between slow force updates.
It can be shown that impulse multiple time-step algorithms, such as the
one described here, can be formulated so as to preserve time reversibility. As a
result these methods can, for suitable choices of time-step sizes, avoid systema-
tic energy drift along computed trajectories. In the next section we discuss the
question of feasible time-step size. To consume less computing resources per
unit of simulation time, successful multiple time-step methods must combine
force-splitting approaches and time-stepping algorithms that allow signifi-
cantly lengthened time steps for the most computationally costly force compo-
nents. This issue has been the focus of intense work over the past decade.
t
Fundamental Limitation on Size of MTS Methods
Impulse MTS methods began to show considerable success in the mid-
1990s, with published results reporting computational speed-up by a factor
of 5 compared to traditional MD simulation.
29,30
Two features with regard
to the practical time-step sizes for MTS methods emerged. The first, which
was consistent with results reported by Streett et al.
24
20 years earlier,
involved the size of the small time step used to resolve the highest frequency
motion in the system. That time step needed to be somewhat smaller than
the typical MD time step in order to maintain energy stability. This is of little
practical concern in terms of overall computational efficiency because the
forces being evaluated at each small step are assumed to be very inexpensive in
CPU time. The second feature was more significant: Computed trajectories
demonstrated systematic energy instability whenever the larger steps used to
resolve slower force components exceeded 5 fs.
31
This is important because
the possibility of achieving further efficiency gains with MTS methods require
that the slowest forces be updated much less frequently. The 5 fs barrier, which
for a time seemed to have put a ceiling on further developments, came to be
understood as a resonance artifact
32-34
coinciding with the half-period of
bond vibrations such as O-H. The impulses introduced into the dynamics at
each large step excite the bonds and lead to catastrophic energy growth. This
energy growth is seen initially in the highest frequency (fastest) bonded energy.
As a practical matter, growth in these energy components can be used as an
early diagnosis of trouble in an MTS simulation.
One obvious remedy for this problem is to choose time-step lengths so as
to avoid small integer multiples of half-periods of any oscillatory motion.
However, it has been demonstrated
35
that the molecular dynamics potential
gives rise to motion with a continuum of periods greater than or equal to
10 fs. Furthermore, the energy instability of impulse MTS methods becomes
exponentially worse at larger multiples of the half-periods. This rules out
the possibility that a fortuitously chosen assortment of impulse multiple
time steps longer than 5 fs could yield stable trajectories.
