Conical Intersection - Local Characterization of Conical Intersections

Local Characterization of Conical Intersections

In an ideal system of two dimensionalities, this can occur at one molecular geometry. If the potential energy surfaces are plotted as functions of the two coordinates, they form a cone centered at the degeneracy point. This is shown in the picture on the right, where the upper and lower potential energy surfaces are plotted in different colors. The name conical intersection comes from this observation.

In diatomic molecules, the number of vibrational degrees of freedom is 1. Without the necessary two dimensions required to form the cone shape, conical intersections cannot exist in these molecules. Instead, the potential energy curves experience avoided crossings.

In molecules with three or more atoms, the number of degrees of freedom for molecular vibrations is at least 3. In these systems, when spin-orbit interaction is ignored, the degeneracy of conical intersection is lifted through first order by displacements in a two dimensional subspace of the nuclear coordinate space.

The two-dimensional degeneracy lifting subspace is referred to as the branching space or branching plane. This space is spanned by two vectors, the difference of energy gradient vectors of the two intersecting electronic states(the g vector), and the non-adiabatic coupling vector between the these two states(the h vector). Because the electronic states are degenerate, the wave functions of the two electronic states are subject to an arbitrary rotation. Therefore, the g and h vectors are also subject to a related arbitrary rotation, despite the fact that the space spanned by the two vectors is invariant. To enable a consistent representation of the branching space, the set of wave functions that makes the g and h vectors orthogonal is usually chosen. This choice is unique up to the signs and switchings of the two vectors, and allows these two vectors to have proper symmetry when the molecular geometry is symmetric.

The degeneracy is preserved through first order by differential displacements that are perpendicular to the branching space. The space of non-degeneracy-lifting displacements, which is the orthogonal complement of the branching space, is termed the seam space. Movement within the seam space will take the molecule from one point of conical intersection to an adjacent point of conical intersection.

For an open shell molecule, when spin-orbit interaction is added to the, the dimensionality of seam space is reduced.