1549380323-Statistical Mechanics Theory and Molecular Simulation

318 Free energy calculations

compute the average of ∆Uα,α+1in each stateα. The free energy difference ∆AAB
is the sum of contributions obtained using the free energy perturbation formula from
each intermediate state along the path:

∆AAB=−kT

M∑− 1

α=1

ln

〈

e−β∆Uα,α+1

〉

α, (8.1.7)

where〈···〉αrepresents an average taken over the distribution exp(−βUα). The key
to applying eqn. (8.1.7) is to choose the thermodynamic path betweenAandBso as
to achieve sufficient overlap between successive intermediate states without requiring
a large number of them.

8.2 Adiabatic switching and thermodynamic integration

The free energy perturbation approach evokes a physical picture in which configura-
tions sampled from the canonical distribution of stateAare immediately “switched”
to the stateBby simply changing the potential fromUAtoUB. When there is in-
sufficient overlap between the statesAandB, a set of intermediate states can be
employed to define an optimal transformation path. The use of intermediate states
conjures a picture in which the system is slowly switched fromAtoB. In this section,
we will discuss an alternative approach in which the system is continuously,adiabati-
callyswitched fromAtoB. An adiabatic path is one along which the system is fully
relaxed at each point of the path. In order to effect the switching from one state to
the other, we employ a common trick of introducing an “external” switching variable
λin order to parameterize the adiabatic path. This parameter is usedto define a new
potential energy function, sometimes called a “metapotential,” defined as

U(r 1 ,...,rN,λ)≡f(λ)UA(r 1 ,...,rN) +g(λ)UB(r 1 ,...,rN). (8.2.1)

The functionsf(λ) andg(λ) areswitching functionsthat must satisfy the conditions
f(0) = 1,f(1) = 0,g(0) = 0,g(1) = 1. ThusU(r 1 ,...,rN,0) =UA(r 1 ,...,rN) and
U(r 1 ,...,rN,1) =UB(r 1 ,...,rN). Apart from these conditions,f(λ) andg(λ) are com-
pletely arbitrary. The mechanism of eqn. (8.2.1) is one in which an imaginary external
controlling influence (“hand of God” in the form of theλparameter) starts the system
off in stateA(λ= 0) and slowly switches off the potentialUAwhile simultaneously
switching on the potentialUB. The process is complete whenλ= 1. A simple choice
for the functionsf(λ) andg(λ) isf(λ) = 1−λandg(λ) =λ.
In order to see how eqn. (8.2.1) is used to compute the free energydifference ∆AAB,
consider the canonical partition function of a system described bythe potential of eqn.
(8.2.1) for a particular choice ofλ:

Q(N,V,T,λ) =CN

∫

dNpdNrexp

{

−β

[N

∑

i=1

p^2 i

2 mi

+U(r 1 ,...,rN,λ)

]}

. (8.2.2)

This partition function leads to a free energyA(N,V,T,λ) via

A(N,V,T,λ) =−kTlnQ(N,V,T,λ). (8.2.3)