is a three-body correction, etc. Higher-order terms are expressible in a natural way. Truncation of Eq. ( ) at some finite order, n, affords an approximation that we will call MBE(n). This machinery has been exploited as a means to fit classical force fields, especially for water and for ion-water interactions, as well as for high-accuracy calculations of crystal lattice geometries and energies. MBE(n) approximations have also been used to accelerate molecular dynamics simulations, and as a means to limit the complexity of chemical space in applications of machine learning. Finally, MBE(n) forms the basis of the fragment molecular orbital method. For a system with N fragments, MBE(n) generates a combinatorial number of n-body subsystems,