348 Free energy calculations
toh(s)→h(s) +α, whereα= lnf. This is equivalent to scalingg(s)→fg(s). As
the simulation proceeds,g(s) approaches the true probabilityP(s), and the histogram
H(s) will become flat. Typically,fis initially chosen large, e.g.f= e^1 , and is gradually
reduced to 1, which means thatαis initially chosen to near 1 and is reduced to 0. Note
that the Metropolis acceptance rule in eqn. (8.9.2) is equivalent to the usual acceptance
rule using a modified potentialU ̃(r 1 ,...,rN) =U(r 1 ,...,rN)−kTh(f 1 (r 1 ,...,rN)).
8.10 Adiabatic dynamics
In this section, we show how the AFED approach introduced in Section 8.3 can be
extended to treat reaction coordinates, and we will provide a detailed analysis of the
adiabatic dynamics, demonstrating how it leads to the free energy profile directly from
the adiabatic probability distribution function.
Suppose there aren < 3 Ngeneralized coordinatesqαthat describe a certain pro-
cess and for which we are interested in the free energy hypersurfaceA(q 1 ,...,qn).
Consider the full canonical partition function
Q(N,V,T) =CN
∫
dNpdNrexp{
−β[N
∑
i=1p^2 i
2 mi+U(r 1 ,...,rN)]}
. (8.10.1)
We now introduce the transformation to generalized coordinatesqα=fα(r 1 ,...,rN)
in the configurational part of the partition function, leaving the momenta unchanged,
which yields
Q(N,V,T) =CN
∫
dNpd^3 Nqexp{
−β[N
∑
i=1p^2 i
2 mi+V ̃(q 1 ,...,q 3 N,β)]}
, (8.10.2)
whereV ̃(q 1 ,...,q 3 N,β) =U ̃(q 1 ,...,q 3 N)−kTlnJ(q 1 ,...,q 3 N) andJ(q 1 ,...,q 3 N) is the
Jacobian of the transformation. Of course, the partition functions in eqns. (8.10.1)
and (8.10.2) are equal and, therefore, yield the same thermodynamic properties of
the system. However, consider using the argument of eqn. (8.10.2) as a Hamiltonian
with the 3NCartesian components of the momenta as “conjugate” to the generalized
coordinatesq 1 ,...,q 3 N. This Hamiltonian takes the form
H ̃(q,p,β) =∑N
i=1p^2 i
2 mi+V ̃(q 1 ,...,q 3 N,β)=
∑N
i=1p^2 i
2 mi+U ̃(q 1 ,...,q 3 N)−kTlnJ(q 1 ,...,q 3 N) (8.10.3)and is not equivalent to the physical Hamiltonian in the argument of the exponential in
eqn. (8.10.1). Therefore,H ̃does not yield the same dynamics. Nevertheless, trajectories
of either Hamiltonian, if used in conjunction with a thermostat, yield configurations
consistent with the canonical ensemble for the system, underscoring the fact that
the integrals in the configurational partition function can be performed in any set of
coordinates.