k θ in total to be optimized, and µ n (R) is one of N R cubic spline basis functions of the inter-particle distance variable. For CG models that have rotational symmetry around one or more axes, the number of angular variables required to specify the pair configuration is reduced. For example, for uniaxial CG particles with one axis of rotational symmetry, only one Euler angle is needed to describe rotation between the molecular coordinate frames, reducing the number of variables in Eq. ( ) by two. Explicit formulas for the cubic spline basis functions and expressions for the force and torque basis vectors are given in section SIV of the SM.