Dynamical and statistical effects of the intrinsic curvature of internal space of molecules

The Hamilton dynamics of a molecule in a translationally and/or rotationally symmetric field is kept rigorously constrained in its phase space. The relevant dynamical laws should therefore be extracted from these constrained motions. An internal space that is induced by a projection of such a limited phase space onto configuration space is an intrinsically curved space even for a system of zero total angular momentum. In this paper we discuss the general effects of this curvedness on dynamics and structures of molecules in such a manner that is invariant with respect to the selection of coordinates. It is shown that the regular coordinate originally defined by Riemann is particularly useful to expose the curvature correction to the dynamics and statistical properties of molecules. These effects are significant both qualitatively and quantitatively and are studied in two aspects. One is the direct effect on dynamics: A trajectory receives a Lorentz-like force from the curved space as though it was placed in a magnetic field. The well-known problem of the trapping phenomenon at the transition state is analyzed from this point of view. By showing that the trapping force is explicitly described in terms of the curvature of the internal space, we clarify that the physical origin of the trapped motion is indeed originated from the curvature of the internal space and hence is not dependent of the selection of coordinate system. The other aspect is the effect of phase space volume arising from the curvedness: We formulate a general expression of the curvature correction of the classical density of states and extract its physical significance in the molecular geometry along with reaction rate in terms of the scalar curvature and volume loss (gain) due to the curvature. The transition state theory is reformulated from this point of view and it is applied to the structural transition of linear chain molecules in the so-called dihedral angle model. It is shown that the curvature effect becomes large roughly linearly with the size of molecule.