Multichannel quantum theory for propagation of second order transition amplitudes

We study the Green's function representation of second order transition amplitudes for the transition from an initial to a final quantum state that occurs because both are weakly coupled to a manifold of intermediate states. These processes apply to Raman scattering, two-photon absorption, fluorescence from dissociating molecules, electron stimulated desorption, etc., and the transition amplitudes are called generalized Raman amplitudes. The generalized Raman transition amplitudes are expressed in terms of matrix elements of a multichannel Green's function whose determination requires the simultaneous generation of the regular and irregular solutions of the multichannel Schrödinger equation for the intermediate state manifold. However, the numerical propagation of the generalized Raman transition amplitudes through classically forbidden regions requires, in effect, the simultaneous propagation (in the same direction) of both the regular and irregular solutions of the intermediate manifold Schrödinger equation, and use of standard multichannel scattering methods lead to numerical instabilities. We introduce new methods for maintaining both the stability and linear independence of the regular and irregular multichannel intermediate manifold eigenfunctions as they are one-way propagated along a reaction coordinate with standard quantum scattering methods. The methods may be used with systems having asymptotically open or closed channels or both in the intermediate state manifold. First order transition amplitudes, such as state selected photodissociation amplitudes to fragment states of the intermediate state manifold, emerge as a by-product of our algorithm, and the computation of the second order generalized Raman amplitudes scales roughly as double the computation time required for the first order amplitudes.