  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.
