Estimating the Relative Free Energy of Different Molecular States with Respect to a Single Reference State

We have investigated the feasibility of predicting free energy differences between a manifold of molecular states from a single simulation or ensemble representing one reference state. Two formulas that are based on the so-called λ- coupling parameter approach are analyzed and compared:  (i) expansion of the free energy F(λ) into a Taylor series around a reference state (λ = 0), and (ii) the so-called free energy perturbation formula. The results obtained by these extrapolation methods are compared to exact (target) values calculated by thermodynamic integration for mutations in two molecular systems:  a model dipolar diatomic molecule in water, and a series of para-substituted phenols in water. For moderate charge redistribution (≈0.5 e), both extrapolation methods reproduce the exact free energy differences. For free energy changes due to a change of atom type or size, the Taylor expansion method fails completely, while the perturbation formula yields moderately accurate predictions. Both extrapolation ...