Vibrational Assignment by the Matrix Element Method

The matrix element method is based on the fact that if two potential curves intersect exactly once (at E = Ec, R = Rc), then the matrix element connecting near-degenerate perturbing levels can be factored (see Section 3.3.1), 

The correctness of a trial vibrational numbering of electronic state 2 is then indicated by the constancy of the ratio of the observed matrix element to the calculated vibrational overlap. This method is similar to the Franck-Condon method for establishing the absolute vibrational numbering. There, one compares observed transition intensities to sets of Franck-Condon factors calculated for a series of trial numberings. The key to the success of both methods is the existence of at least one relative minimum or maximum among the observed Hi,v,2,v matrix elements or transition intensities. 

New Tv and Bv functions are obtained by adding n (n = 1,2,3. some larger integer) to each trial vibrational quantum number of the perturber 

Each time the trial numbering is increased by 1, the trial potential is shifted to u e lower energy and to slightly smaller internuclear distance, 

When the matrix element method fails, two possibilities for establishing the vibrational numbering remain, ab initio He(R) functions and isotope shifts. When Ec appears to he above the highest observed perturbing level, isotope shifts are the method of choice. However, if He(R) is available, then a modified matrix element method may prove successful. Each trial numbering determines hence He (Rj ial). The calculated vibrational overlap should be equal to the observed perturbation matrix element divided by He (R ial). However, if 

---