Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems

The Born–Oppenheimer treatment leads to the adiabatic framework where the non-adiabatic terms are the physical entities responsible for the coupling between adiabatic states. The main disadvantage of this treatment is in the fact that these coupling terms frequently become singular thus causing difficulties in solving the relevant Schroedinger equation for the motion of the nuclei that make up the molecular systems. In this review, we present the line integral approach which enables the formation of the adiabatic-to-diabatic transformation matrix that yields the friendlier diabatic framework. The review concentrates on the mathematical conditions that allow the rigorous derivation of the adiabatic-to-diabatic transformation matrix and its interesting physical properties. One of the findings of this study is that the non-adiabatic coupling terms have to be quantized in a certain manner in order to yield single-valued diabatic potentials. Another important feature revealed is the existence of the topological matrix, which contains all the topological features of a given molecular system related to a closed contour in configuration space. Finally, we present an approximation that results from the Born–Oppenheimer treatment which, in contrast to the original Born–Oppenheimer approximation, contains the effect of the non-adiabatic coupling terms. The various derivations are accompanied by examples which in many cases are interesting by themselves.