Interpolation of multi-sheeted multi-dimensional potential-energy surfaces via a linear optimization procedure

Significant progress has been achieved in recent years with the development of high-dimensional permutationally invariant analytic Born-Oppenheimer potential-energy surfaces, making use of polynomial invariant theory. In this work, we have developed a generalization of this approach which is suitable for the construction of multi-sheeted multi-dimensional potential-energy surfaces exhibiting seams of conical intersections. The method avoids the nonlinear optimization problem which is encountered in the construction of multi-sheeted diabatic potential-energy surfaces from ab initio electronic-structure data. The key of the method is the expansion of the coefficients of the characteristic polynomial in polynomials which are invariant with respect to the point group of the molecule or the permutation group of like atoms. The multi-sheeted adiabatic potential-energy surface is obtained from the Frobenius companion matrix which contains the fitted coefficients. A three-sheeted nine-dimensional adiabatic potential-energy surface of the ²T₂ electronic ground state of the methane cation has been constructed as an example of the application of this method.