) are the mapped CG coordinates corresponding to the FG configuration r n , and the angle brackets denote an average over the equilibrium ensemble of the FG model. The torque residual is scaled by a factor 1/η, where η has units of length, so that it has same units as the force residual. Optimization of the CG force field requires only that a FG simulation be conducted to give the FG forces and torques as a function of FG positions. For each equilibrium FG simulation configuration used for CG parametrization, the FG positions are mapped onto CG positions and orientations to calculate the CG forces and torques used to evaluate the force and torque residual in Eq. (38).