The turnover theory for activated rate processes, is extended to multidimensional systems. The theory derived in this paper accounts for the competition between intramolecular and intermolecular relaxation. The extent of chaotic motion of the system modes directly affects the rate of energy diffusion in the system. The more chaos, the faster the energy diffusion and the larger the rate. The dependence of the rate on the intramolecular coupling strength is well accounted for. The theory is applied to a model two-dimensional system studied previously by Straub and Berne [J. Chem. Phys. 85, 2999 (1986)]. The theory, which is the multidimensional generalization of the one-dimensional Pollak, Grabert, and Hanggi (PGH) turnover theory [J. Chem. Phys. 91, 4073 (1989)] accounts well for the rate even in the case of extreme anisotropic friction. The theory is cast in terms of the collective normal modes of the system and the bath and is thus applicable also to memory friction.