Completely general canonical and microcanonical (energy-resolved) flexible transition state theory (FTST) rate constant expressions for an arbitrary choice of reaction coordinate have recently been derived [Robertson et al. J. Chem. Phys. 2000, 113, 2648.] by the present authors. The rate expressions apply to any definition of the separation distance between fragments in a barrierless recombination (or dissociation) that is held fixed during hindered rotations at the transition state, and to any combination of fragment structure (atom, linear top, nonlinear top). The minimization of the rate constant with respect to this definition can be regarded as optimizing the reaction coordinate within a canonical or microcanonical framework. The expressions are analytic, with the exception of a configuration integral whose evaluation generally requires numerical integration over an integrand which depends on internal angles (from one to five depending on the fragment structures). The primary component of the integrand is the determinant of the inverse G-matrix associated with the external rotations and the relative internal rotation of the fragments. In this paper, we derive closed-forth, analytic expressions for the inverse G-matrix determinant for all combinations of fragment top types for an arbitrary reaction coordinate definition entirely in terms of kinetic energy matrix elements for a centers-of-mass reaction coordinate. For a model potential for CFH2 + H, the effect of optimizing the reaction coordinate definition is displayed, and the optimized coordinate is compared to the traditional center-of-mass definition at the canonical level. The associated rate constant is about a factor of 20% to 45% lower than that obtained using a centers-of-mass reaction coordinate.