Conical intersections are known to play a key role in nonrelativistic, spin-conserving electronically non-adiabatic processes. If we include the spin-orbit interaction, we alter the situation qualitatively. By coupling states of different spin-multiplicity, the spin-orbit interaction gives rise to spin-nonconserving processes. In addition, the spin-orbit interaction produces a more subtle but no less significant effect when the molecule in question has an odd number of electrons. In the nonrelativistic case, the branching space, the space in which the conical topography is evinced, has dimension 2, which is also the case when the spin-orbit interaction is included, provided the molecule has an even number of electrons. However for a molecule with an odd number of electrons the dimension of the branching space, ν, is five in general or three when the system is restricted to Cs symmetry. Recently, considerable progress has been made in describing this novel class of conical intersections. An effective algorithm for locating such points of conical intersection based on ab initio wave functions has been developed. Complementing the work, an analytic representation of the energies and derivative couplings in the vicinity of these points of conical intersection has been determined based on degenerate perturbation theory. These efforts will make it possible to describe nonadiabatic processes driven by conical intersections, for which the spin-orbit interaction cannot be neglected, on the same footing that has been so useful in the nonrelativistic case. In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons is reviewed.
