Updated Branching Plane for Finding Conical Intersections without Coupling Derivative Vectors.

The conical intersections (CIs) form a (f-2)-dimensional hyperspace on which two diabatic potential energy surfaces (PESs) belonging to the same symmetry cross, where f is the internal degree of freedom. The branching plane (BP) is a (two-dimensional) plane defined by the difference gradient vector (DGV) and the coupling derivative vector (CDV), and on the BP, the degeneracy of the two adiabatic PESs is lifted. The properties of the BP are often used in the exploration of the conical intersection hyperspace, such as determination of the minimum energy CI or the first-order saddle point in CI. Although both DGV and CDV are necessary to construct the BP in general, CDV is not always available depending on ab initio methods and programs. Therefore, we developed an approach for optimizing critical points on the CI hypersurface without CDV by using a BP updating method, which was shown to be accurate and very useful for minimum energy and saddle point optimization and for the minimum energy path following within the CI hypersurface in numerical tests for C6H6 and C5H8N(+).