322 Free energy calculations
8.3 Adiabatic free energy dynamics
Although we cannot entirely eliminate the need to visit the unphysicalstates between
λ= 0 andλ= 1 in adiabatic switching methods, we can substantially alleviate it. If,
instead of preselecting a set ofλvalues, we are willing to allowλto vary continuously
in a molecular dynamics calculation as an additional dynamical degree of freedom,
albeit a fictitious one, then we can exploit the flexibility in our choice of the switching
functionsf(λ) andg(λ) to make the region betweenλ= 0 andλ= 1 energetically
unfavorable. When such a choice is used within a molecular dynamics calculation,λ
will spend most of its time in the physically relevant regions close toλ= 0 andλ= 1.
To understand how to devise such a scheme, let us consider a Hamiltonian that
includes a kinetic energy termp^2 λ/ 2 mλ, wherepλis a “momentum” conjugate toλand
mλis a mass-like parameter needed to define the kinetic energy. The parametermλ
also determines the time scale on whichλevolves dynamically. The total Hamiltonian
is then
Hλ(r,λ,p,pλ) =p^2 λ
2 mλ+
∑N
i=1p^2 i
2 mi
+U(r 1 ,...,rN,λ). (8.3.1)In eqn. (8.3.1),λand its conjugate momentumpλare now part of an extended phase
space (see, for example, the discussion Section 4.8). We can now define a canonical
partition function for the Hamiltonian in eqn. (8.3.1),
Q(N,V,T) =
∫
dpλ∫
dNp∫ 1
0dλ∫
D(V)dNre−βHλ(pλ,λ,p,r), (8.3.2)and therefore compute any ensemble average with respect to thecorresponding canon-
ical distribution. In particular, the probability distributionP(λ′) =〈δ(λ−λ′)〉λ, leads
directly to aλ-dependent free energy functionA(λ) through the relation
A(λ′) =−kTlnP(λ′). (8.3.3)Eqn. (8.3.3) defines an important quantity known as afree energy profile. We will
have more to say about free energy profiles starting in Section 8.6.Note that the free
energy difference
A(1)−A(0) =−kTln[
P(1)
P(0)
]
=−kTln[
QB
QA
]
= ∆AAB, (8.3.4)
sinceP(0) andP(1) are the partition functionsQAandQB, respectively. The distribu-
tion function〈δ(λ−λ′)〉λcan be generated straightforwardly in a molecular dynamics
calculation by accumulating a histogram ofλvalues visited over the course of the
trajectory.
We still need to answer the question of how to maximize the timeλspends near the
endpointsλ= 0 andλ= 1. Eqn. (8.3.3) tells us that free energy is a direct measure of
probability. Thus, consider choosing the functionsf(λ) andg(λ) such thatA(λ) has
a significant barrier separating the regions nearλ= 0 andλ= 1. According to eqn.
(8.3.3), where the free energy is high, the associated phase spaceprobability is low.