Free Energy Estimators

A free energy estimator, in the context of statistical mechanics and thermodynamics, is a theoretical or computational tool used to calculate or estimate the free energy of a system. Free energy is a fundamental concept in thermodynamics and is related to the ability of a system to do work. It is often denoted by the letter F or A (Helmholtz free energy), depending on the context.

Free Energy Perturbation (FEP)

A method based on statistical mechanics for computing free energy differences from MD or MC simulations.

The FEP method, as introduced by Robert W. Zwanzig in 1954, relates the free energy difference between an initial (reference) and a final (target) state of a system to an average of a function of their energy difference evaluated by sampling for the initial state (Zwanzig equation):

\[ \Delta F(\mathbf{A} \rightarrow \mathbf{B}) = F_\mathbf{B} - F_\mathbf{A} = -k_\mathrm{B} T \ln \left \langle \exp \left ( - \frac{E_\mathbf{B} - E_\mathbf{A}}{k_\mathrm{B} T} \right ) \right \rangle _\mathbf{A} \]

Where \(T\) is the temperature, \(k_B\) is Boltzmann’s constant, and the angular brackets (\(\langle \rangle\)) denote an average over a simulation run for state A.

The difference between states A and B could be the:

1. Atom types involved

2. Differences of geometry

Then the free energy difference (\(\Delta F\)) in 1. would be obtained for “mutating” one molecule onto another, while a free energy map along one or more reaction coordinates would be obtained for 2..

This free energy map along some reaction coordinate is also known as a potential of mean force (PMF).

Important

Free energy perturbation calculations only converge properly when the difference between the two states is small enough. Therefore it is usually divided the perturbation into a series of smaller “windows”, which are computed independently.