Adiabatic-to-diabatic transformation matrix quantization

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

The fact that the BO treatments forms two frameworks, the adiabatic and the diabatic ones, and the fact that the two are related through a unitary transformation introduces one of the most exciting results of the present approach, namely, the quantization of the NACM. As we shall see the quantization is an outcome of the requirement that the diabatic potential matrix W as given in equation (6) has to be single-valued just like the diabatic potential matrix V given in equations (14) and (15). There is also a practical reason why W has to be single-valued because otherwise it is impossible to obtain any sensible solution for the nuclear SE. 

---