  The determination of ultra-long-range molecular potential curves has been
reformulated using the Coulomb Greens function to give a solution in terms of
the roots of an analytical determinantal equation. For a system consisting of
one Rydberg atom with fine structure and a neutral perturbing ground state atom
with hyperfine structure, the solution yields potential energy curves and
wavefunctions in terms of the quantum defects of the Rydberg atom and the
electron-perturber scattering phaseshifts and hyperfine splittings. This method
provides a promising alternative to the standard currently utilized method of
diagonalization, which suffers from problematic convergence issues and
nonuniqueness, and can potentially yield a more quantitative relationship
between Rydberg molecule spectroscopy and electron-atom scattering phaseshifts.
