8 Free energy calculations
Our treatment of the classical equilibrium ensembles makes clear that the free energy
is a quantity of particular importance in statistical mechanics. Beingrelated to the
logarithm of the partition function, the free energy is the generator through which
other thermodynamic quantities are obtained via differentiation. Often, however, we
are less interested in the absolute free energy than we are in the free energydiffer-
encebetween two thermodynamic states. Free energy differences tellus, for example,
whether a chemical reaction occurs spontaneously or requires input of work. They tell
us whether a given solute is hydrophobic or hydrophilic, and they aredirectly related
to equilibrium constants for chemical processes. Thus, from freeenergy differences, we
can compute acid or base ionization constants. We can also quantifythe therapeutic
viability of a candidate drug compound by calculating its inhibition constant or IC50
value from the binding free energy. Another type of free energy often sought is the
free energy as a function of one or more generalized coordinates ina system, such
as the free energy surface as a function of a pair of Ramachandran anglesφandψ
in an oligopeptide. This surface provides a map of the stable conformations of the
molecule, the relative stability of these conformations, and the barrier heights that
must be crossed for a change in conformation.
In this chapter, we describe a variety of widely used techniques that have been
developed for calculating free energies and discuss the relative merits and disadvan-
tages of the methods. The fact that the free energy is a state function, ensuring that
the system can be transformed from one state to another along physical or unphysical
paths without affecting the free energy difference, allows for considerable flexibility in
the design of novel techniques and will be frequently exploited in thedevelopments we
will present.
The techniques described in this chapter are constructed within the framework of
the canonical ensemble with the aim of obtaining Helmholtz free energy differences
∆A. Generalization to the isothermal-isobaric ensemble and the Gibbs free energy
difference ∆Gis, in all cases, straightforward. (For a useful compendium of free energy
calculation methods, readers are referred toFree Energy Calculations, C. Chipot and
A. Pohorille, eds. (2007)).
8.1 Free energy perturbation theory
We begin our treatment of free energy differences by considering the problem of
transforming a system from one thermodynamic state to another. Let these states
be denoted generically asAandB. At the microscopic level, these two states are char-