Uniqueness of the collective submanifold in the adiabatic theory of large amplitude collective motion

It is emphasized that the conditions for the existence of a collective submanifold which follow from adiabatic time-dependent Hartree-Fock theory are precisely the conditions for the existence of a manifold of solutions of Hamilton's equations confined to a surface of reduced dimensionality. A constructive procedure, valid in any number of dimensions and involving the concept of the multidimensional valley, is developed to determine whether a given system admits such a manifold. It is extended to include the idea of the approximate manifold, and an application to a generalized landscape model is described.