Explicit expression of the franck-condon factors in terms of the potentials of the two states

The calculus of the overlap integral for two states represented by the vibrational wave functions ψ ν′a and ψ ν″b is reduced to that of the Franck–Condon integral ℒ(0, x) = ∫ 0x ψ ν′aψ ν″b (t) dt. It is proved that for “numerical potentials” (as well as for a Dunham potential), this integral is given on each interval by a simple analytic expression in terms of the two potentials. The Franck–Condon factors are well determined by “coupling constants” related uniquely to the coordinates of the turning points of the potentials. An application to the band system BIIXΣ of Nα2 is compared with the usual numerical methods.