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

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 $E=\operatorname{diag}(\{E_k\})$ is a diagonal matrix, $E_k$ are the energies of the observed “conglomerate” of lines, and the intensities of these lines Ik determine the first row of the matrix $X$, $(X_{1k})^2=I_k$, $k=1,2,\dots,n$, $n\geq3$. Hamiltonian matrix of the direct model, $H^{\mathrm{DIR}}$, whose parameters are the energies of pre-diagonalized “dark” states, $A_i$, and the matrix elements of their coupling to the “bright” state, $B_i$, ($i=1,2,\dots,n-1$), 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), $H^{\mathrm{DWI}}$, can be obtained from the matrices $H^{\mathrm{DIR}}$ 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, $g_1$, and the matrix element of its coupling to the “bright” state, $w_1$, the use of the Householder transformation gives: $g_1=\sum_{i=1}^{n-1}B_i^2A_i/(\sum_{j=1}^{n-1}B_j^2)=\sum_{k=1}^n E_k^3I_k/\sum_{l=1}^n E_l^2 I_l$, $|w_1|=(\sum_{i=1}^{n-1}B_i^2)^{1/2}=(\sum_{k=1}^n E_k^2I_k)^{1/2}$. In similar way, using the Householder transformation, the Hamiltonians for the models with several doorway states, $H^{\mathrm{DW2}},H^{\mathrm{DW3}},\dots,H^{\mathrm{DW}(n-1)}$, are successively obtained. The Hamiltonian of the $\mathrm{DW}(n-2)$ model is represented by a symmetric tridiagonal matrix $H^{\mathrm{DW}(n-1)}$, its diagonal elements $g_i$ determine the energies of the $\mathrm{DW}1$-, $\mathrm{DW}2$-, $\mathrm{DW}(n-1)$ states, and the off-diagonal elements $w_i$ determine the corresponding coupling between them.