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.