1549380323-Statistical Mechanics Theory and Molecular Simulation

Adiabatic dynamics 349

In order to exploit eqn. (8.10.3) for the development of a free energy method, it
is useful to express both kinetic and potential energies as explicit functions of 3N
variables. To this end, let us reindex theNmassesm 1 ,...,mNusing the notationm′α,
whereα= 1,..., 3 N. Here, of course,m′ 1 =m′ 2 =m′ 3 =m 1 ,m′ 4 =m′ 5 =m′ 6 =m 2 ,
and so forth. Letαindex the 3NCartesian components of theNmomentum vectors,
we can write eqn. (8.10.3) as

H ̃(q,p,β) =

∑^3 N

α=1

p^2 α

2 m′α

+V ̃(q 1 ,...,q 3 N). (8.10.4)

Eqn. (8.10.4) is the starting point for our analysis of adiabatic motion.
We now develop a scheme for computing the free energy surfaceA(q 1 ,...,qn) of
the firstnreaction coordinates when this surface is characterized by high barriers and
direct sampling of the probability distribution function is not possible.As was done
in Section 8.3, we propose to assign these firstncoordinates a temperatureTq≫T
so that the free energy barriers can be easily surmounted. However, we recognize
that introducing two temperatures into the system leads to incorrect thermodynamics
unless we also allow the massesm′ 1 ,...,m′nto be much larger than the remaining 3N−n
masses. In this way, the firstncoordinates will be adiabatically decoupled from the
remaining 3N−ncoordinates, and it can be shown that the adiabatic probability
distribution functionPadb(q 1 ,...,qn) generated by the dynamics of eqn. (8.10.4) under
the adiabatic conditions is

A(q 1 ,...,qn) =−kTqlnPadb(q 1 ,...,qn) + const. (8.10.5)

The remainder of this section will be devoted to a derivation of eqn. (8.10.5) by a
detailed analysis of the adiabatic dynamics followed by several illustrative examples.
Readers wishing to skip the analysis can jump ahead to the examples without loss of
continuity.
In order to maintain the two temperatures, we introduce two separate thermostats,
one atTqfor the firstncoordinates and the second atTfor the remaining 3N−n
coordinates. For notational simplicity, we express the equations of motion in terms of
a simple Nos ́e–Hoover type (see Section 4.8.3),

q ̇α=

pα

m′α

p ̇α=−

∂V ̃

∂qα

−

pη 1

Q 1

pα α= 1,...,n

p ̇α=−

∂V ̃

∂qα

−

pη 2

Q 2

pα α=n+ 1,..., 3 N

η ̇j=

pηj

Qj

p ̇η 1 =

∑n

α=1

p^2 α

2 m′α

−nkTq