High-order expansion of T₂ × t₂ Jahn-Teller potential-energy surfaces in tetrahedral molecules

Methods from Jahn-Teller theory and invariant theory have been combined for the construction of analytic diabatic potential-energy surfaces of triply degenerate states in tetrahedral molecules. The potentials of a threefold degenerate electronic state of T₂ symmetry, subject to the T ₂ × t₂ or T₂ × (t ₂+t₂) Jahn-Teller effect in a three-dimensional or six-dimensional space of nuclear coordinates, respectively, are considered. The permutation symmetry of four identical nuclei is taken into account in the polynomial expansion of the diabatic surfaces. Symmetry adapted polynomials up to high orders are explicitly given and a simple combinatorial scheme was developed to express terms of arbitrary order as products of a small number of polynomials which are invariant under the permutation of identical nuclei. The method is applied to the methane cation in its triply degenerate ground state. The parameters of the analytic surfaces have been fitted to accurate ab initio data calculated at the full-valence CASSCF/MRCI/cc-pVTZ level. A three-sheeted six-dimensional analytic potential-energy surface of the ²T ₂ ground state of CH₄⁺ is reported, which involves terms up to eighth order in the degenerate stretching coordinate, up to 12th order in the degenerate bending coordinate, and up to fourth order in the stretch-bend coupling.