DOORWAY models in the inverse problem for a complex vibronic analogue of the fermi resonance

Abstract

We solve the inverse problem for the complex Fermi resonance or its vibronic analogue, and to this end we use the matrix XEXt, where E=diag(\Ek\) is a diagonal matrix, Ek are the energies of the observed "conglomerate" of lines, and the intensities of these lines Ik determine the first row of the matrix X, (X1k)2=Ik k=1,2,...,n,n≥3. Hamiltonian matrix of the direct model, HDIR, whose parameters are the energies of pre-diagonalized "dark" states, Ai, and the matrix elements of their coupling to the "bright" state, Bi, (i=1,2,...,n-1), is obtained after the diagonalization of the XEXt block, which belongs to the "dark" states. We show that Hamiltonian matrix with the single doorway state (DW), HDW1, can be obtained from the matrices HDIR or XEXt by first step of the Householder triangularization, i.e. by similarity transformation with a reflection matrix constructed by quantities Bi or Di=(XEXt)1,i+1. For the energy of the first DW1 state, g1, and the matrix element of its coupling to the "bright" state, w1, the use of the Householder transformation gives: g1=Sigman-1i=1B2iAi/(Sigman-1j=1B2j)= Sigmank=1E3kIk/Sigmanl=1E2lIl, |w1|=(Sigman-1i=1B2i)1/2= Sigmank=1E2kIk)1/2. In similar way, using the Householder transformation, the Hamiltonians for the models with several doorway states, HDW2,HDW3,...,HDW(n-1), are successively obtained. The Hamiltonian of the DW(n-2) model is represented by a symmetric tridiagonal matrix HDW(n-1), its diagonal elements gi determine the energies of the DW1-,DW2-,...,DW(n-1) states, and the off-diagonal elements wi determine the corresponding coupling between them. Keywords: vibronic coupling, complex vibronic analogue of the Fermi resonance, inverse problem, Householder transformation, doorway models.