Feshbach–Fano formalism in Hilbert space: Application to shape resonances in molecular photoionization

Abstract

An explicit Hilbert-space formulation of Feshbach–Fano theory is described which is particularly well suited for treating the problem of shape resonances in molecular photoionization. The prepared states of Fano and Cooper are employed to resolve the irreducibly infinite degeneracy of molecular electronic continua that arises from the noncentral nature of molecular (body-frame) potentials. Previously defined L2 Stieltjes states generated employing appropriate test functions are shown to converge to these prepared states, avoiding prior construction of the associated degenerate channel functions. A novel choice of zeroth-order state particularly suitable for applications of the L2 Feshbach–Fano method to molecular shape resonances is introduced and shown to avoid many of the difficulties associated with more conventional treatments employing resonance scattering theory. Specifically, the energy of this zeroth-order state is seen to be a weighted average over the correct photoionization cross section, the associated background cross section is shown to vanish for all energies, and calculations of the associated ρ(E) and q(E) parameters are avoided entirely. Computational applications in the static-exchange approximation to selected valence-shell channels in N2, O2, and N2O illustrate the attributes of the method and provide accurate cross sections in each case. The L2 Feshbach–Fano formalism is seen to provide improved results from precisely the same information—pseudostate energies and oscillator strengths—as is required in the conventional L2 Stieltjes development, with no significant increase in computational effort required.