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Many-Body eXpansion (MBX) is a C++ library that implements many-body potential energy functions (PEFs) within the “many-body energy” (MB-nrg) formalism. MB-nrg PEFs integrate an underlying polarizable model with explicit machine-learned representations of many-body interactions to achieve chemical accuracy from the gas to the condensed phases. MBX can be employed either as a stand-alone package or as an energy/force engine that can be integrated with generic software for molecular dynamics and Monte Carlo simulations. MBX is parallelized internally using Open Multi-Processing and can utilize Message Passing Interface when available in interfaced molecular simulation software. MBX enables classical and quantum molecular simulations with MB-nrg PEFs, as well as hybrid simulations that combine conventional force fields and MB-nrg PEFs, for diverse systems ranging from small gas-phase clusters to aqueous solutions and molecular fluids to biomolecular systems and metal-organic frameworks. 

Density functional theory (DFT) has been applied to modeling molecular interactions in water for over three decades. The ubiquity of water in chemical and biological processes demands a unified understanding of its physics, from the single molecule to the thermodynamic limit and everything in between. Recent advances in the development of data-driven and machine-learning potentials have accelerated simulation of water and aqueous systems with DFT accuracy. However, anomalous properties of water in the condensed phase, where a rigorous treatment of both local and non-local many-body (MB) interactions is in order, are often unsatisfactory or partially missing in DFT models of water. In this review, we discuss the modeling of water and aqueous systems based on DFT and provide a comprehensive description of a general theoretical/computational framework for the development of data-driven many-body potentials from DFT reference data. This framework, coined MB-DFT, readily enables efficient many-body molecular dynamics (MD) simulations of small molecules, in both gas and condensed phases, while preserving the accuracy of the underlying DFT model. Theoretical considerations are emphasized, including the role that the delocalization error plays in MB-DFT potentials of water and the possibility to elevate DFT and MB-DFT to near-chemical-accuracy through a density-corrected formalism. The development of the MB-DFT framework is described in detail, along with its application in MB-MD simulations and recent extension to the modeling of reactive processes in solution within a quantum mechanics/MB molecular mechanics (QM/MB-MM) scheme, using water as a prototypical solvent. Finally, we identify open challenges and discuss future directions for MB-DFT and QM/MB-MM simulations in condensed phases.