Free-energy calculations

If system can be in more than one macro-state, then the relative probability of finding the system in either state depends on their free energy difference. An important example is the situation where a system can be in more than one phase: at constant P and T these phases are equally likely when their (Gibbs) free energies are the same. Knowledge of the free energy is therefore important. However, free energies cannot be obtained by normal MC or MD sampling: special simulation techniques are required. This chapter describes these techniques. First, we discuss the general problem of locating first-order phase transitions. We then explain how free energy differences can be computed by thermodynamic integration, either “natural” or “Hamiltonian”. A separate discussion is devoted to the computation of chemical potentials using the so-called particle insertion method. Often, histogram methods are used to estimate the variation of the free energy as a function of some order parameters/reaction coordinates. In this context, we discuss the overlapping-distribution method and the acceptance-ratio approach. We then move to biased sampling techniques and discuss the “umbrella-sampling” method, Wang-Landau sampling and meta-dynamics. A separate section discusses how free-energy profiles can be recovered by piecing histograms together. We conclude this chapter with a discussion of non-equilibrium methods to estimate free-energy differences.