We present a method for computing free-energy differences using thermodynamic integration with a neural network potential that interpolates between two target Hamiltonians. The interpolation is defined at the sample distribution level, and the neural network potential is optimized to match the corresponding equilibrium potential at every intermediate time-step. Once the interpolating potentials and samples are well-aligned, the free-energy difference can be estimated using (neural) thermodynamic integration. To target molecular systems, we simultaneously couple Lennard-Jones and electrostatic interactions and model the rigid-body rotation of molecules. We report accurate results for several benchmark systems: a Lennard-Jones particle in a Lennard-Jones fluid, as well as the insertion of both water and methane solutes in a water solvent at atomistic resolution using a simple three-body neural-network potential.

Solvation free energy, and notably hydration free energy, is generally recognized as a fundamental thermodynamic quantity of interest in computational chemistry. Defined as the work done when transferring a molecule from the gas phase to the solution (water in the case of hydration free energy), it enables the computation of several key physicochemical properties of molecules, such as solubility, partition coefficients, activity coefficients, and binding free energies in solutions [1, 2]. These properties are of great importance to the pharmaceutical, environmental, and materials sciences [3-9], prompting the organization of blind prediction SAMPL challenges [10-12] with hydration free energy as one of the main targets. In addition, Mobley et al. compiled and curated a FreeSolv database of experimentally measured hydration free energies for small neutral molecules in water [13, 14]. A wide spectrum of methods is available to calculate solvation free energy, ranging from traditional approaches such as continuum solvation models [15, 16] to recent machine learning (ML) algorithms [17-26] and their combinations [27-29]. The alchemical methods with Molecular Dynamics (MD) simulations [14, 30, 31] are typically assumed to be highly accurate but computationally expensive [32, 33]. However, both the fidelity and the efficiency highly depend on the explicitly treated degrees of freedom and the employed potential energy model.

Atomic partial charges are crucial parameters in molecular dynamics (MD) simulation, dictating the electrostatic contributions to intermolecular energies, and thereby the potential energy landscape. Traditionally, the assignment of partial charges has relied on surrogates of \textit{ab initio} semiempirical quantum chemical methods such as AM1-BCC, and is expensive for large systems or large numbers of molecules. We propose a hybrid physical / graph neural network-based approximation to the widely popular AM1-BCC charge model that is orders of magnitude faster while maintaining accuracy comparable to differences in AM1-BCC implementations. Our hybrid approach couples a graph neural network to a streamlined charge equilibration approach in order to predict molecule-specific atomic electronegativity and hardness parameters, followed by analytical determination of optimal charge-equilibrated parameters that preserves total molecular charge. This hybrid approach scales linearly with the number of atoms, enabling, for the first time, the use of fully consistent charge models for small molecules and biopolymers for the construction of next-generation self-consistent biomolecular force fields. Implemented in the free and open source package \texttt{espaloma\_charge}, this approach provides drop-in replacements for both AmberTools \texttt{antechamber} and the Open Force Field Toolkit charging workflows, in addition to stand-alone charge generation interfaces. Source code is available at \url{https://github.com/choderalab/espaloma_charge}.

The importance of solvation or hydration mechanism and accompanying free energy change has made various in silico calculation methods for the solvation energy one of the most important application in computational chemistry[1-25]. The solvation free energy directly influences many chemical properties in solvated phases and plays a dominant role in various chemical reactions: drug delivery[2, 16, 18, 26], organic synthesis[27], electrochemical redox reactions[28-31], etc. The atomistic computer simulation approaches for the solvent and the solute molecules directly offer the microscopic structure of the solvation shell, which surrounds the solutes molecule[7, 8, 13, 17, 18, 32]. The solvation shell structure could provide us detailed physicochemical information like microscopic mechanisms on solvation or the interplay between the solvent and the solute molecules when we use an appropriate force field and molecular dynamics parameters. However, those explicit solvation methods we stated above need an extensive amount of numerical calculations since we have to simulate each individual molecule in the solvated system. The practical problems on the explicit solvation model restrict its applications to classical molecular mechanics simulations[7, 8, 17] or a limited number of QM/MM approaches[13, 32]. For classical mechanics approaches for macromolecules or calculations for small compounds at quantum-mechanical level, the idea of implicit solvation enables us to calculate solvation energy with feasible time and computational costs when one considers a given solvent as a continuous and isotropic medium in the Poisson-Boltzmann equation[1, 3-6, 9, 11, 15, 23, 24]. Many theoretical advances have been introduced to construct the continuum solvation model, which involves parameterized solvent properties: the polarizable continuum model (PCM)[9], the conductor-like screening model (COSMO)[1] and its variations[6, 33], generalized Born approximations like solvation model based on density (SMD)[5] or solvation model 6, 8, 12, etc. (SMx)[4, 11]. The structure-property relationship (SPR) is rather a new approach, which predicts the solvation free energy with a completely different point of view when compared to computer simulation approaches with precisely defined theoretical backgrounds[34, 35].