Householder transformation in the inverse problem for a complex vibronic analogue of the Fermi resonance

Abstract

In the inverse problem for a complex vibronic analogue of the Fermi resonance, the matrix elements of the electron-vibration interaction should be obtained from experimental data, energies Ek and intensities I_k (k = 1, 2, …, n; n ³ 3), a “conglomerate” of lines in the spectrum. This problem in the direct-coupling model, where the Hamiltonian H^DIR is specified by the energies of the “dark” states Ai and the matrix elements of their coupling with the “bright” state Bi (i = 1, 2, …, n –1), was solved by the author on the basis of algebraic methods. It is shown that the Hamiltonian H^DW of the doorway-coupling model, in which the “bright” state has “interaction” with only 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 which is constructed from the Bi values. The expressions for main elements of the doorway model, namely, the energy of the |DW> state and the matrix element of its coupling with the "bright" state, are obtained. For pyrazine and acetylene molecules, the matrix elements of the Hamiltonian H^DW are calculated using the data of the electronic-vibrational-rotational spectra.