350 Free energy calculations
p ̇η 2 =∑^3 N
α=np^2 α
2 m′α
−(3N−n)kT. (8.10.6)In practice, it is recommended that a a more robust thermostatting scheme be em-
ployed such as Nos ́e–Hoover chains (see Section 4.10), the method in Problem 4.2 of
Chapter 4, or Langevin thermostats to be discussed in Chapter 15.
The time evolution of the system is generated by the Liouville operator. In order
to keep the discussion general, we write this operator as
iL=∑^3 N
α=1[
pα
m′α∂
∂qα+Fα(q)∂
∂pα]
+iLtherm, 1 (Tq) +iLtherm, 2 (T), (8.10.7)whereFα(q) =−∂V /∂q ̃ αandiLtherm, 1 (Tq) andiLtherm, 2 (T) are the Liouville opera-
tors for the two thermostats. If xtdenotes the full phase space vector, including all
variables related to the thermostats, then the time evolution of the system is formally
given by
xt= eiLtx 0. (8.10.8)The key to analyzing this unusual dynamics is to factorize the propagator exp(iLt) in
a way consistent with the adiabatic decoupling. To this end, we definethe following
combinations of terms in eqn. (8.10.7):
iLref, 1 =∑nα=1pα
m′α∂
∂qα+iLtherm, 1 (Tq)iLref, 2 =∑^3 N
α=n+1pα
m′α∂
∂qα
+iLtherm, 2 (T)iL 2 =iLref, 2 +∑^3 N
α=1Fα(q)∂
∂pα. (8.10.9)
We next express the total Liouville operator as
iL=iLref, 1 +iL 2. (8.10.10)Let ∆tbe a time interval characteristic of the motion of the hot, heavy, and slow-
moving reaction coordinatesq 1 ,...,qn. Then, a Trotter decomposition of the propagator
appropriate for the adiabatically decoupled motion is
eiL∆t= eiL^2 ∆t/^2 eiLref,^1 ∆teiL^2 ∆t/^2 +O(
∆t^3)
. (8.10.11)
Note that the operator exp(iL 2 ∆t/2) has terms that vary on a time scale much faster
than ∆tand must be further decomposed. Using the ideas underlying the multiple