  The energy-dependent frame transformation theory of Gao and Greene 1990
[Phys. Rev. A {\bf 42}, 6946 (1990)] is extended to yield 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
back-propagation 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 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.
