Integrating Nos ́e–Hoover chains 201Consider the action of the operator exp(cx∂/∂x) onx. We can work this out using
a Taylor series:
exp[
cx∂
∂x]
x=[∞
∑
k=0ck
k!(
x∂
∂x)k]
x=x∑∞
k=0ck
k!=xec. (4.11.18)We see that the operator scalesxby the constant ec. Similarly, the action of the
operator exp(cx∂/∂x) on a functionf(x) isf(xec). Using this general result, each of
the operators in eqn. (4.11.17) can be turned into a simple instruction in code (either
translation or scaling) via the direct translation technique from Section 3.10.
At this point, several comments are in order. First, the separation of the non-
Hamiltonian component of the equations of motion from the Hamiltonian component
in eqn. (4.11.8) makes implementation of RESPA integration with Nos ́e–Hoover chains
relatively straightforward. For example, suppose a system has fast and slow forces as
discussed in Section 3.11. Instead of decomposing the Liouville operator as was done
in eqn. (4.11.8), we could expressiLas
iL=iLfast+iLslow+iLNHC (4.11.19)and further decomposeiLfastinto kinetic and force termsiL(1)fast+iL(2)fast, respectively.
Then, the propagator can be factorized according to
eiL∆t= eiLNHC∆t/^2 eiLslow∆t/^2×
[
eiL(2)
fastδt/^2 eiL
(1)
fastδteiL
(2)
fastδt/^2]n×eiLslow∆t/^2 eiLNHC∆t/^2 , (4.11.20)whereδt= ∆t/n. In such a factorization, theτ parameter in eqn. (4.10.2) should
be chosen according to the time scale of the slow forces. On the other hand, if the
thermostats are needed to act on a faster time scale, then they can be pulled into the
reference system by writing the propagator as:
eiL∆t= eiLNHCδt/^2 eiLslow∆t/^2 e−iLNHCδt/^2
×[
eiLNHCδt/^2 eiL(2)
fastδt/^2 eiL
(1)
fastδteiL
(2)
fastδt/^2 eiLNHCδt/^2]n×e−iLNHCδt/^2 eiLslow∆t/^2 eiLNHCδt/^2. (4.11.21)Here, the operator exp(−iLNHCδt/2) is never really applied; its presence in eqn.
(4.11.21) indicates that in the first and last RESPA steps, the Nos ́e–Hoover chain