Free energy perturbation theory 317ZB=
∫
dNre−βUB(r^1 ,...,rN)=
∫
dNre−βUB(r^1 ,...,rN)e−βUA(r^1 ,...,rN)eβUA(r^1 ,...,rN)=
∫
dNre−βUA(r^1 ,...,rN)e−β(UB(r^1 ,...,rN)−UA(r^1 ,...,rN)). (8.1.4)If we now take the ratioZB/ZA, we find
ZB
ZA=
1
ZA
∫
dNre−βUA(r^1 ,...,rN)e−β(UB(r^1 ,...,rN)−UA(r^1 ,...,rN))=
〈
e−β(UB(r^1 ,...,rN)−UA(r^1 ,...,rN))〉
A, (8.1.5)
where the notation〈···〉Aindicates an average taken with respect to the canonical
configurational distribution of the stateA. Substituting eqn. (8.1.5) into eqn. (8.1.2)
gives
∆AAB=−kTln〈
e−β(UB−UA)〉
A. (8.1.6)
Eqn. (8.1.6) is known as thefree energy perturbationformula (Zwanzig, 1954); it
should be reminiscent of the thermodynamic perturbation formula ineqn. (4.7.8). A
result known as the potential distribution theorem (Widom, 1963; Becket al., 2006) for
calculating the excess chemical potential of a solute molecule in solution is essentially
a variant of eqn. (8.1.6) applied to the Gibbs free energyG=μN.
Eqn. (8.1.6) can be interpreted as follows: We sample a set of configurations
{r 1 ,...,rN}from the canonical distribution of stateAand simply use these config-
urations, unchanged, to sample the canonical distribution of stateBwith potential
UB. However, because these configurations are not sampled from exp(−βUB)/ZBdi-
rectly, we need to “unbias” our sampling by removing the factor exp(−βUA) and
reweighting with exp(−βUB), which leads to eqn. (8.1.6). The difficulty with this ap-
proach is that the configuration spaces of statesAandBmight not have significant
overlap. By this, we mean that configurations sampled from the canonical distribution
of stateAmay not be states of high probability in the canonical distribution of state
B. When this is the case, the potential energy differenceUB−UAbecomes large, and
hence the exponential factor exp[−β(UB−UA)] becomes negligibly small. Thus, most
of the configurations have very low weight in the ensemble average,and the free energy
difference converges slowly. For this reason, it is clear that the free energy perturbation
formula is only useful when the two statesAandBdo not differ significantly. In other
words, the stateBmust be a small perturbation to the stateA.
Even ifBis not a small perturbation toA, the free energy perturbation idea can
still be salvaged by introducing a set ofM−2 intermediate states with potentials
Uα(r 1 ,...,rN), whereα= 1,...,M,α= 1 corresponds to the stateA, andα=M
corresponds to the stateB. Let ∆Uα,α+1=Uα+1−Uα. We now transform the system
from stateAto stateBalong a path through each of the intermediate states and