On the insufficient sampling problem in the free energy perturbation

We re-examine the underlying theory of the widely-employed free energy perturbation (FEP) method, which is also known as the Zwanzig equation. FEP frequently fails to converge, and this poor performance reflects the problem of insufficient sampling inherent in the theory. Using a harmonic oscillator model, we are able to prove analytically that the deviation between the exact solution and the falsely-converged FEP result obtained from a finite sampling is related to a relaxation energy that the system would release, if it were allowed to relax, after the perturbation is introduced. Upon including this relaxation energy as a correction term to the standard FEP, we are able to obtain exact free energy difference using only trapped finite sampling, i.e. where standard FEP fails even when using Benettu0027s acceptance ratio method. Our new approach of augmenting the free energy theory with the relaxation process hence sheds a new light on the insufficient sampling problem in free energy calculations.