(Eq. 19)
5.2.2 Force matching with the MS-CG model : The method of force-matching provides a rigorous route to developing a coarse-grained force field directly from forces measured in all-atom simulations. In so far as the multi-body coarse-grained PMF is derived from structure factors that depend on temperature, pressure, and composition, they cannot be transferred to new systems. To avoid this problem, the force-matching approach proposes a variational method in which a coarse-grained force field is systematically developed from all-atom simulations under the correct thermodynamic ensemble [103]. In the statistical framework developed by Izvekov and Voth [103-105], it is possible to develop the exact many-body coarse-grained PMF from a trajectory of atomistic forces with a sufficiently detailed set of basis functions.
The method starts with a collection of sampled configurations from an atomistic simulation of the target system and calculate the reference forces between atoms of a particular type. After decomposing their target force into a short-ranged part approximated by a cubic spline and a long-ranged Coulomb part one solves the overdetermined set of linear equations given by (Eq. 20):