(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):