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

Russian version

DOI: 10.21883/EOS.2022.11.55091.3327-22

Kuzmitsky V. A.¹

¹University of Civil Protection, Ministry for Emergency Situations of Belarus, Minsk, Belarus

Email: llum07@rambler.ru

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Abstract
References
Статистика просмотров

We solve the inverse problem for the complex Fermi resonance or its vibronic analogue, and to this end we use the matrix XEX^t, where E=diag(E_k) is a diagonal matrix, E_k are the energies of the observed "conglomerate" of lines, and the intensities of these lines I_k determine the first row of the matrix X, (X_1k)²=I_k k=1,2,...,n,n≥3. Hamiltonian matrix of the direct model, H^DIR, whose parameters are the energies of pre-diagonalized "dark" states, A_i, and the matrix elements of their coupling to the "bright" state, B_i, (i=1,2,...,n-1), is obtained after the diagonalization of the XEX^t block, which belongs to the "dark" states. We show that Hamiltonian matrix with the single doorway state (DW), H^DW1, can be obtained from the matrices H^DIR or XEX^t by first step of the Householder triangularization, i.e. by similarity transformation with a reflection matrix constructed by quantities B_i or D_i=(XEX^t)_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=Sigma^n-1_i=1B²_iA_i/(Sigma^n-1_j=1B²_j)= Sigmaⁿ_k=1E³_kI_k/Sigmaⁿ_l=1E²_lI_l, |w₁|=(Sigma^n-1_i=1B²_i)^1/2= (Sigmaⁿ_k=1E²_kI_k)^1/2. In similar way, using the Householder transformation, the Hamiltonians for the models with several doorway states, H^DW2,H^DW3,...,H^DW(n-1), are successively obtained. The Hamiltonian of the DW(n-2) model is represented by a symmetric tridiagonal matrix H^DW(n-1), its diagonal elements g_i determine the energies of the DW1-,DW2-,...,DW(n-1) states, and the off-diagonal elements w_i determine the corresponding coupling between them. Keywords: vibronic coupling, complex vibronic analogue of the Fermi resonance, inverse problem, Householder transformation, doorway models.

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