Backpropagated frame transformation theory: A reformulation

The energy-dependent frame transformation theory of Gao and Greene [H. Gao and C. H. Greene, Phys. Rev. A 42, 6946 (1990), 10.1103/PhysRevA.42.6946] is extended to yield a quantitatively accurate description of the dissociative recombination process. Evidence is presented to show that direct application of the original theory leads to inaccurate cross sections. A major revision, based on an interaction-free backpropagation of the Born-Oppenheimer solutions, markedly improves the frame transformation theory, reducing its average error by orders of magnitude. The original theory and its extension are tested on the previously explored two-dimensional (2D) model that is tailored to describe the singlet ungerade states of molecular hydrogen. The 2D model can be solved exactly (within the numerical accuracy) without implementing the Born-Oppenheimer approximation. These exact results then serve as a benchmark for the frame transformation theory developed in this paper.