The conditions believed to be necessary and sufficient for obtaining (1.1) from the quantum mechanical EOM were indeed shown to lead to kinetic equations for the manifold populations. However the forms of these equations and the rates coefficients depend on the statistical properties of the coupling elements. We also found that only the two lowest moments of the distribution of coupling elements affect the time evolution.
In the molecular problems described in the introduction, the distribution of coupling elements may behave different for differently physical situations. When the manifolds in Fig. 2 are composed of vibrational rotational states of large molecules <δV2> > <V>2 because matrix elements between such highly oscillatory wavefunctions are rapidly and irregularly varying. When the manifolds are composed of states associated with the kinetic energy of relative motion between molecular fragments the coupling is a smooth function of energy and <δV2 << <V>2.When two manifolds are coupled through a single state, a separable coupling situation follows.
Finally we mention that the same procedure yields also expression for absorption and scattering lineshapes associated with these physical situations.