Multiple time-scale integration 117p ̇=Ffast(x) and has the associated single-time-step propagator exp(iLfast∆t). The
full propagator is then factorized by applying the Trotter schemeas follows:
exp(iL∆t) = exp(
iLslow∆t
2)
exp(iLfast∆t) exp(
iLslow∆t
2)
. (3.11.6)
This factorization leads to thereference system propagator algorithm(RESPA) (Tuck-
ermanet al., 1992). The idea behind the RESPA algorithm is that the time step ∆t
appearing in eqn. (3.11.6) is chosen according to the time scale of theslow forces.
There are two ways to achieve this: Either the propagator exp(iLfast∆t) is applied
exactly analytically, or exp(iLfast∆t) is further factorized with a smaller time stepδt
that is appropriate for the fast motion. We will discuss these two possibilities below.
First, suppose an analytical solution for the reference system is available. As a
concrete example, consider a harmonic fast forceFfast(x) =−mω^2 x. Acting with the
operators directly yields an algorithm of the form
x(∆t) =x(0) cosω∆t+1
ω[
p(0)
m+
∆t
2 m
Fslow(x(0))]
sinω∆tp(∆t) =[
p(0) +∆t
2Fslow(x(0))]
cosω∆t−mωx(0) sinω∆t+
∆t
2Fslow(x(∆t)), (3.11.7)which can also be written as a step-wise set of instructions using thedirect translation
technique:
p=p+ 0. 5 ∗dt∗Fslow
xtemp =x∗cos(arg) +p/(m∗ω)∗sin(arg)
ptemp =p∗cos(arg)−m∗ω∗x∗sin(arg)
x=xtemp
p=ptemp
Recalculate slow force
p=p+ 0. 5 ∗dt∗Fslow, (3.11.8)where arg =ω∗dt. Generalizations of eqn. (3.11.8) for complex molecular systems
described by potential energy models like eqn. (3.11.1) were recently presented by
Janeˇziˇc and coworkers (Janeˇziˇcet al., 2005). Similar schemes can be worked out for
other analytically solvable systems.
When the reference system cannot be solved analytically, the RESPA concept can
still be applied by introducing a second time stepδt= ∆t/nand writing
exp(iLfast∆t) =[
exp(
δt
2
Ffast∂
∂p)
exp(
δtp
m∂
∂x)
exp(
δt
2
Ffast∂
∂p)]n. (3.11.9)
Substitution of eqn. (3.11.9) into eqn. (3.11.6) yields a purely numerical RESPA prop-
agator given by