We begin our treatment of free energy differences by examining the problem of transforming a system from one thermodynamic state to another. Let these states be denoted generically as and . At the microscopic level, these two states are characterized by potential energy functions and . For example, in a drug-binding study, the state might correspond to the unbound ligand and enzyme, while would correspond to the bound complex. In this case, the potential would exclude all interactions between the enzyme and the ligand and the enzyme, whereas they would be included in the potential .

The Helmholtz free energy difference between the states and
is simply
. The two free energies
and are given in terms of their respective
canonical partition functions and ,
respectively by
and
, where

The free energy difference is, therefore,

where and are the configurational partition functions for states and , respectively,

The ratio of full partition functions reduces to the ratio of configurational partition functions because the momentum integrations in the former cancel out of the ratio.

Eqn. (2) is difficult to implement
in practice because in any numerical calculation via either
molecular dynamics or Monte Carlo, we do not have direct access
to the partition function only averages of phase-space functions
corresponding to physical observables. However, if we are willing to
extend the class of phase-space functions whose averages we seek to
functions that do not necessarily correspond to direct observables, then
the ratio of configurational partition functions can be manipulated
to be in the form of such an average. Consider inserting
unity into the expression for as follows:

If we now take the ratio , we find

where the notation indicates an average taken with respect to the canonical configurational distribution of the state . Substituting eqn. (5) into eqn. (2), we find

Eqn. (6) is known as the

If is not a small perturbation to , then
the free-energy perturbation formula can still be salvaged
by introducing a set of intermediate states
with potentials
, where
, corresponds to the
state and corresponds to the state .
Let
.
We can now imagine transforming the system from
state to state by passing through these
intermediate states and computing the average of
in the state .
Applying the free-energy perturbation formula
to this protocol yields the free-energy difference as