Householder Transformation in the Inverse Problem for a Complex Vibronic Analog of the Fermi Resonance

The matrix elements of the electron-vibration interaction should be obtained from experimental data, energies E_k, and intensities I_k (k = 1, 2, …, n, n ≥ 3) of a conglomerate of spectral lines in the inverse problem for a complex vibronic analog of the Fermi resonance. This problem in the direct-coupling model, where the Hamiltonian H^DIR is specified by the energies of "dark" states A_i and the matrix elements of their coupling with the "bright" state B_i (i = 1, 2, …, n –1), was solved by the author using algebraic methods. It was shown that the Hamiltonian H^DW of the doorway-coupling model, in which the bright state interacts with only a single distinguished |DW〉-state, can be obtained from the Hamiltonian H^DIR using the Householder triangularization method, namely, by the similarity transformation H^DW = PH^DIRP, where P is the reflection matrix constructed from the Bi values. Expressions for the main elements of the doorway model, i.e., energies of the |DW〉-state and the matrix element of its coupling with the bright state were obtained. Matrix elements of the Hamiltonian H^DW were calculated using molecular electronicvibrational-rotational spectral data for pyrazine and acetylene.