The electronic non-adiabatic coupling matrix: A numerical study of the curl condition and the quantization condition employing the Mathieu equation

In this article, we discuss the electronic nonadiabatic coupling matrix, τ, which under certain conditions is characterized by two interesting features: (1) its components fulfill an extended Curl equation (Chem. Phys. Lett. 1975, 35, 112, (see Appendix 1)) and (2) it is quantized in the sense that the topological D matrix, presented as an exponentiated line integral over the τ matrix, is a unitary diagonal matrix (Chem. Phys. Lett. 2000, 319, 489). These features can be shown to exist if the relevant group of states forms a Hilbert subspace, namely, a group of states that are strongly coupled with each other but are only weakly coupled with all other states. The numerical study is carried out applying the eigenfunctions of the Mathieu equation.