where the parameters c I ia are the CIS coefficients for excited state |Ψ CIS I and the superscript [R] denotes a nuclear derivative. The quantity a|i represents the overlap between an unperturbed virtual orbital |a and a perturbed occupied orbital, |i . It is clear from Eq. ( ) that the nonadiabatic coupling h 0I vanishes at any crossing point between the HF ground state and a CIS excited state (E 0 = E I ), hence the degeneracy is lifted in the direction of g 0I only and the branching space is one-dimensional. So long as the excitation energy gap E I -E 0 is not exactly zero, however, the coupling h 0I does not generally vanish. This makes the topology of the potential surface rather complicated in regions close to crossing points R CX ; see Appendix B.