  We present a family of alchemical perturbation potentials that enable the
calculation of hydration free energies of small to medium-sized molecules in a
concerted single alchemical coupling step instead of the commonly used sequence
of two distinct coupling steps for Lennard-Jones and electrostatic
interactions. The perturbation potentials are based on the softplus function of
the solute-solvent interaction energy designed to focus sampling near entropic
bottlenecks along the alchemical pathway. We present a general framework to
optimize the parameters of alchemical perturbation potentials of this kind. The
optimization procedure is based on the $\lambda$-function formalism and the
maximum-likelihood parameter estimation procedure we developed earlier to avoid
the occurrence of multi-modal distributions of the coupling energy along the
alchemical path. A novel soft-core function applied to the overall
solute-solvent interaction energy rather than individual interatomic pair
potentials critical for this result is also presented. Because it does not
require modifications of core force and energy routines, the soft-core
formulation can be easily deployed in molecular dynamics simulation codes. We
illustrate the method by applying it to the estimation of the hydration free
energy in water droplets of compounds of varying size and complexity. In each
case, we show that convergence of the hydration free energy is achieved
rapidly. This work paves the way for the ongoing development of more
streamlined algorithms to estimate free energies of molecular binding with
explicit solvation.
