Extending the Time Scale in Atomically Detailed Simulations - Reviews in Computational Chemistry - page 372

The important feature to note is that the fast forces are computed at each
step, while the slow forces are computed t times less frequently, with updates

given by

F fast
i

¼ F fast
i

F slow
i

¼ F slow
i

ð r n þ1
i

ð r n þ1
i

Þ Þ ½ 18

However, the simplicity of this modification hides a potentially disastrous
flaw! The Verlet method is popularly employed in virtually every molecular
dynamics simulation done today because its geometric symmetry ensures
that the total energy along computed solutions does not drift but remains
essentially constant, respecting the underlying Newtonian physics of the model.
By modifying the force updates in the multiple time-step method given above,
we have disrupted the symmetry of the original method. The result is that the
energy will drift significantly and systematically. The situation is improved,
but not solved, by using a higher order Taylor approximation for the slow forces
and a higher order integration scheme. Street et al. 24 used a third-order Taylor
approximation for the slow forces along with a high-order Gear predictor-
corrector integration method. In this way, the energy drift can be made small
relative to the time step, so that relatively long simulations can be computed
with less apparent problems from energy growth.
A new era in multiple time-step methods arrived in the early 1990s when
Grubm ¨ ller et al. 27 and Tuckerman et al. 28 independently published multiple
time-step methods that appeared to overcome the energy instability of extra-
polation methods. Their idea is to mimic the ''kick-drift'' nature of the velocity
Verlet method itself. In Eq. [6] the force supplies a ''kick,'' or impulse, in the
first line, and the system then ''drifts'' as the updated half-step velocity contri-
butes to the position at the new step. The velocity Verlet method can be mod-
ified so that the slow force is also applied as an impulse:

v i ¼ v i þ t

h
2 M 1 F slow

ð r i Þ

i

½ 19a

r i ¼ r i

For j ¼ 1 : t ,

h
2 M 1 F fast

v j þ1=2
i

¼ v i þ ð r i Þ ½ 19b

i

hv j þ1=2
i

r j þ1
i

¼ r i þ ½ 19c

h
2 M 1 F fast

v j þ1
i

¼ v j þ1=2
i

ð r j þ1
i

þ Þ ½ 19d

i

end

r n þ1
i

¼ r i

¼ v i þ t

h
2 M 1 F slow

v n þ1
i

ð r n þ1
i

Þ ½ 19e

i