On the characterization of three-state conical intersections using a group homomorphism approach: the two-state degeneracy spaces.

Second-order degenerate perturbation theory, in conjunction with the group homomorphism method for describing a similarity transformation, are used to characterize the subspace of two-state conical intersections contained in the branching space of a three-state conical intersection. It is shown by explicit calculation, using the lowest three-state conical intersection of (CH)3N2, that a second-order treatment yields highly accurate absolute energies, even at significant distances from the reference point of three-state intersection. The excellent agreement between the second order and ab initio results depends on the average energy component, which is computed using 5 first-order terms and 15 second-order terms. The second-order absolute energy change over the range rho = 0.0-0.3 au, where rho is the distance from the three-state conical intersection in the branching space coordinates, is approximately 6500 and 9500 cm(-1) for the E(1=2) and E(2=3) seams, respectively, with the maximum ab initio energy deviation from degeneracy of 200 cm(-1) occurring at rho = 0.3 au. The characteristic parameters gIJ and hIJ are also predicted to great accuracy, even at large rho, with the error growing to only 10-15% at rho = 0.3 au.