Extended Born - Oppenheimer equation for tri - state system

We present explicit form of non adiabatic coupling (NAC) elements of nuclear Schroedinger equation (SE) for a coupled tri state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic diabatic transformation (ADT) angles are the mixing angles of electronic basis, ADT matrix transforms NAC terms to exactly zeros and brings diabatic form of SE. ADT and NAC matrices satisfy a curl condition and find a novel relation among the mixing angles for irrotational case. We also find that extended Born Oppenheimer (EBO) equations have meaningful solution and can reproduce numerically exact results only when the equations are gauge invariant.