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 Ik (k = 1, 2, …, n; n ≥ 3), a “conglomerate” of lines in the spectrum. This problem in the direct-coupling model, where the Hamiltonian HDIR 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 basisof algebraic methods. It is shown that the Hamiltonian HDW of the doorway-coupling model, in which the “bright” state has “interaction” with only single distinguished |DW> state, can be obtained from the Hamiltonian HDIR using the Householder triangularization method, namely, by the similarity transformation
 HDW = PHDIRP, 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 HDW are calculated using the data of the electronic-vibrational-rotational spectra.