On the effects of spin–orbit coupling on conical intersection seams in molecules with an odd number of electrons. II. Characterizing the local topography of the seam

Characteristic of conical intersections of Born–Oppenheimer potential energy surfaces is η, the dimension of the branching space, the space in which the degeneracy is lifted linearly. In molecules with an odd number of electrons, η=2 for the nonrelativistic Coulomb Hamiltonian, while η=3(5) when the spin–orbit interaction is included and the molecule has (does not have) Cs symmetry. In the η=2 case, the branching space is defined by two vectors: the energy difference gradient vector, g, and the interstate coupling vector, h. g and h can, without loss of generality, be chosen orthogonal. g×h is invariant under the unitary wave function transformation that orthogonalizes g and h. The orthogonal g and h can be used to define an optimal set of coordinates for describing the vicinity of the conical intersection. Here these ideas are generalized to η=3 intersections. In particular, it is shown that g, the energy difference gradient vector, and hr and hi, the real and imaginary parts of the interstate coupling vector, which define the η=3 space, can without loss of generality be chosen orthogonal. It is also shown that g×hr⋅hi is invariant under the unitary wave function transformation that orthogonalizes these vectors. These ideas are illustrated using a portion of the OH(A 2Σ1/2+,X 2Π3/2,1/2)+H2 seam of conical intersection.