316 Free energy calculations
acterized by potential energy functionsUA(r 1 ,...,rN) andUB(r 1 ,...,rN). For example,
in a drug-binding study, the stateAmight correspond to the unbound ligand and en-
zyme whileBwould correspond to the bound complex. In this case, the potentialUA
would exclude all interactions between the ligand and the enzyme, but the potential
UBwould include them.
The Helmholtz free energy difference between the statesAandBis simply ∆AAB=
AB−AA. The two free energiesAAandABare given in terms of their respective
canonical partition functionsQAandQB, respectively byAA=−kTlnQAandAB=
−kTlnQB, where
QA(N,V,T) =CN
∫
dNpdNrexp{
−β[N
∑
i=1p^2 i
2 mi
+UA(r 1 ,...,rN)]}
=
ZA(N,V,T)
N!λ^3 NQB(N,V,T) =CN
∫
dNpdNrexp{
−β[N
∑
i=1p^2 i
2 mi+UB(r 1 ,...,rN)]}
=
ZB(N,V,T)
N!λ^3 N. (8.1.1)
The free energy difference is, therefore,
∆AAB=AB−AA=−kTln(
QB
QA
)
=−kTln(
ZB
ZA
)
, (8.1.2)
whereZA andZB are the configurational partition functions for statesAandB,
respectively:
ZA=
∫
dNre−βUA(r^1 ,...,rN)ZB=
∫
dNre−βUB(r^1 ,...,rN). (8.1.3)The ratio of full partition functionsQB/QAreduces to the ratio of configurational
partition functionsZB/ZAbecause the momentum integrations in the former cancel
out of the ratio.
Eqn. (8.1.2) is difficult to implement in practice because in any numericalcalcu-
lation based either on molecular dynamics or Monte Carlo, we can compute averages
of phase space functions, but we do not have direct access to thepartition function.^1
Thus, eqn. (8.1.2) can be computed directly if it can be expressed in terms of a phase
space average. To this end, consider inserting unity into the expression forZBas
follows:
(^1) The Wang–Landau method of Section 7.6 is an exception.