Highly accurate vibration–rotation Franck–Condon factors for high levels

Highly accurate vibration–rotation Franck–Condon factors q^ab, for a transition between two diatomic electronic states (a) and (b), are sought. When the potentials of states (a) and (b) are of the RKR type, the computation of q^ab is reduced to that of Franck–Condon integral ?^ab(i) = ∫ ψ^a(r)ψ^b(r) dr in an interval r_i, r_i+1. By using convenient interpolations for the potentials U^a and U^b in the considered interval, this integral becomes ?^ab(i) = ∑ δ (r_i+1 – r_i)ⁿ⁺¹/(n + 1), where the “coupling constants” δ depend uniquely on the eigenvalues E^a and E^b of the considered transition and on the potentials U^a and U^b (the number N of terms depends on the desired accuracy). The method used computes the Franck–Condon factors q^ab without the explicit use of the wave function and by replacing the integrals by simple summations. To test the values of q^ab obtained by this method, the orthogonality rule ∫ ψ_v′ψ_v″ dr = 0 (for v′ ≠ v″) is used for one state or the other. This test, along with other tests, show that the Franck–Condon factors computed by the present method are accurate to nine significant figures for high and low levels.