Relation between the local harmonic formulation and the generalized valley formulation of the exact decoupling conditions in the adiabatic theory of large amplitude collective motion

In recent work, partially recapitulated, we have shown how the conditions for the existence of exactly decoupled classical motion in the adiabatic limit can be transformed into a constructive procedure, designated the generalized valley method, for the determination of the submanifold on which the motion occurs. We prove, for any number of decoupled coordinates, that where the conditions for exact decoupling in the adiabatic limit are satisfied, this formulation is equivalent to the local harmonic one which is widely discussed in the literature. The two formulations are however equivalent, in the general case where the decoupling is not exact, only for the one-dimensional submanifold. An alternative form of the local harmonic formulation, which does not utilize the metric properties associated with the mass tensor, and can therefore be extended to a general hamiltonian, is also discussed briefly.