Do all pieces make a whole?: Thiele cumulants and the free energy decomposition

The partitioning of the free energy into additive contributions originating from different groups of atoms or force field terms has the potential to provide relationship between structure and biological activity of molecules. In this article, the theoretical foundation for the free energy decomposition in the free energy perturbation (FEP) methodology is formulated using Thiele cumulants, a powerful tool from the arsenal of probability theory and mathematical statistics. We establish that rigorous decomposition of the free energy into its components is precluded by the presence of mixed potential energy terms in Thiele cumulants of second and higher orders. However, we also show that the resultant nonadditivity error can be reduced to an arbitrary value by increasing the number of FEP steps. Consequently, the whole system can be in the limit of small perturbation steps adequately represented by the sum of its constituent parts.