The q-Centroid Approximation

As remarked in Chapter 3, simply formulating the non-Born-Oppenheimer (BO) coupling element 

As one should expect from the terminology, the earliest application of the q-centroid, strictly speaking, the r-centroid method was made in diatomic spectroscopy [131], although in a completely different physical sense. The q-centroid approximation presents a nontrivial generalization of the r-centroid approach of diatomic spectroscopy to the case of the nonradiative decay of polyatomic molecules. The importance of studying the q-dependence of the nonadiabatic coupling element was emphasized by Freed and Gelbart [132] and Freed and lin [27]. 

In this chapter, we shall extend the description given in Ref. [27]. This will include the effect of Duschinsky mixing for some (totally symmetric) modes. We shall also give an extensive discussion of the dependence of the electronic matrix element on the nuclear coordinates. 

In this section, we will justify the use of such favored nuclear configuration by means of which the electronic matrix element should be determined. We shall do this by considering the expansion of the electronic wavefimctions pj and pi derived in Section 1.3  

Substituting the expansion (5.2) of electronic wavefiinction (pj and P( into the matrix element (5.1) and collecting terms of the same order in X after squaring the matrix element, we obtain 

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