  We present an analytical description of the Alchemical Transfer Method (ATM)
for molecular binding using the Potential Distribution Theory (PDT) formalism.
ATM models the binding free energy by mapping the bound and unbound states of
the complex by translating the ligand coordinates. PDT relates the free energy
and the probability densities of the perturbation energy along the alchemical
path to the probability density at the initial state, which is the unbound
state of the complex in the case of a binding process. Hence, the ATM
probability density of the transfer energy at the unbound state is first
related by a convolution operation to the probability densities for coupling
the ligand to the solvent and coupling it to the solvated receptor, for which
analytical descriptions are available with parameters obtained from maximum
likelihood analysis of data from double-decoupling alchemical calculations. PDT
is then used to extend this analytical description along the alchemical
transfer pathway. We tested the theory on the alchemical binding of five guests
to the TEMOA host from the SAMPL8 benchmark set. In each case, the probability
densities of the perturbation energy for transfer along the alchemical transfer
pathway obtained from numerical calculations match those predicted from the
theory and double-decoupling simulations. The work provides a solid theoretical
foundation for alchemical transfer, offers physical insights on the form of the
probability densities observed in alchemical transfer calculations, and
confirms the conceptual and numerical equivalence between the alchemical
transfer and double-decoupling processes.
