Conical Intersection - Categorization By Symmetry of Intersecting Electronic States

Categorization By Symmetry of Intersecting Electronic States

Conical intersections can occur between electronic states with the same or different point group symmetry, with the same or different spin symmetry. When restricted to a non-relativistic Coulomb Hamiltonian, conical intersections can be classified as symmetry-required, accidental symmetry-allowed, or accidental same-symmetry, according to the symmetry of the intersecting states.

A symmetry-required conical intersection is an intersection between two electronic states carrying the same multidimensional irreducible representation. For example, intersections between a pair of E states at a geometry that has a non-abelian group symmetry(e.g. (C3), C(3v) or D(3h)). It is named symmetry-required because these electronic states will always be degenerate as long as the symmetry is present. Symmetry-required intersections are often associated with Jahn-Teller effect.

An accidental symmetry-allowed conical intersection is an intersection between two electronic states that carry different point group symmetry. It is called accidental because the states may or may not be degenerate when the symmetry is present. Movement along one of the dimensions along which the degeneracy is lifted, the direction of the difference of the energy gradients of the two electronic states, will preserve the symmetry while displacements along the other degeneracy lifting dimension, the direction of the non-adiabatic couplings, will break the symmetry of the molecule. Thus, by enforcing the symmetry of the molecule, the degeneracy lifting effect caused by inter-state couplings is prevented. Therefore the search for a symmetry-allowed intersection becomes a one-dimensional problem and does not require knowledge of the non-adiabatic couplings, significantly simplifying the effort. As a result, all the conical intersections found through quantum mechanical calculations during the early years of quantum chemistry were symmetry-allowed intersections.

An accidental same-symmetry conical intersection is an intersection between two electronic states that carry the same point group symmetry. While this type of intersection was traditionally more difficult to locate, a number of efficient searching algorithms and methods to compute non-adiabatic couplings have emerged in the past decade. It is now understood that same-symmetry intersections play as important a role in non-adiabatic processes as symmetry-allowed intersections.