Semiclassical bound-continuum Franck-Condon factors uniformly valid at 4 coinciding critical points: 2 Crossings and 2 turning points

A uniform semiclassical approximation to bound-continuum Franck-Condon factors is derived by applying differential topological mapping techniques on a suitable three dimensional integralrepresentation. This approach even holds uniformly if two potentialcurve intersections (real or complex conjugate) and two turning points come close or coincide. The resulting Franck-Condon matrix element is expressed in terms of two generic swallowtail (A₄) integrals whose unfolding parameters are obtained from a single algebraic equation amenable to fast standard routines. Transitional approximations to this result are shown to cover all previously known approaches and to lead to a generalization of a formula of K. Sando and F. H. Mies 1]. A simple and fast trapezoidal method to evaluate the generic swallowtail integrals and other generic integrals of odd determinacy is presented, which even permits the derivation of numerical error bounds.