Algebraic-matrix calculation of vibrational levels of triatomic molecules

TL;DR

This paper presents an algebraic method for accurately computing vibrational spectra of triatomic molecules using a sparse Hamiltonian matrix derived from a product Morse-cosine expansion of the potential energy surface, applicable to both linear and bent geometries.

Contribution

The authors introduce a novel algebraic technique that efficiently computes vibrational levels of triatomic molecules from ab initio data, with explicit mode interactions and sparse matrix representations.

Findings

Accurate vibrational spectra for OCS, HCN, H2O, and NO2.

Spectra in good agreement with experimental data.

Efficient diagonalization of sparse Hamiltonian matrices.

Abstract

We introduce an accurate and efficient algebraic technique for the computation of the vibrational spectra of triatomic molecules, of both linear and bent equilibrium geometry. The full three-dimensional potential energy surface (PES), which can be based on entirely {\it ab initio} data, is parameterized as a product Morse-cosine expansion, expressed in bond-angle internal coordinates, and includes explicit interactions among the local modes. We describe the stretching degrees of freedom in the framework of a Morse-type expansion on a suitable algebraic basis, which provides exact analytical expressions for the elements of a sparse Hamiltonian matrix. Likewise, we use a cosine power expansion on a spherical harmonics basis for the bending degree of freedom. The resulting matrix representation in the product space is very sparse and vibrational levels and eigenfunctions can be obtained by efficient diagonalization techniques. We apply this method to carbonyl sulfide OCS, hydrogen cyanide HCN, water H$_2$O, and nitrogen dioxide NO$_2$. When we base our calculations on high-quality PESs tuned to the experimental data, the computed spectra are in very good agreement with the observed band origins.