A method for characterizing the degeneracy preserving seam space in the vicinity of a three state conical intersection is introduced. Second order degenerate perturbation theory is used to construct an approximately diabatic Hamiltonian whose eigenenergies and eigenstates accurately describe the vicinity of the three state conical intersection in its full dimensionality. The perturbative analysis enables the large number, , of unique second order parameters needed to construct this accurate Hamiltonian to be determined from ab initio data at a limited number of nuclear configurations, with being minimal. Using the minimum energy three state conical intersection of the pyrazolyl radical , the potential of this approach is illustrated. A Hamiltonian comprised of the ten characteristic (linear) parameters and over order parameters is constructed and used to determine the locus of the conical intersection seam as well as to describe the 18 dimensional space in the vicinity of that point of intersection. Our results demonstrate the ability of this methodology to quantitatively reproduce the ab initio potential energy surfaces near a three state conical intersection.
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The projection scheme was employed because of difficulties that arose in assigning the phase of the ab initio derivative couplings computed at particular displaced geometries employing a projection scheme. This normally routine procedure may be complicated by the presence of near degeneracies or other nonperturbative effects that cause the ab initio derivative couplings to exhibit substantial deviations from the first order results, causing definitive assignments of phase to be problematical. Since these effects are not observed for any of the displaced geometries where at , this projection scheme was chosen to be the basis of comparison.
