Schrodinger equation triatomic molecules

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p2A, being thus mostly pronounced in n electronic state. Renner developed the system of two coupled Schrodinger equations and solved it for n states in the harmonic approximation by means of the perturbation theory. 

Let us again consider the photodissociation of the linear triatomic molecule with coordinates R and r (Figure 2.1). We want to solve the time-dependent Schrodinger equation (4.1) with the Hamiltonian given in (2.39) and the initial condition (4.4). 

Feit, M.D. and Fleck, J.A., Jr. (1983). Solution of the Schrodinger equation by a spectral method. II. Vibrational energy levels of triatomic molecules, J. Chem. Phys. 78, 301-308. 

M. D. Feit, j. a. Fleck Jr., Solution of the Schrodinger Equation by a Spectral Method 11 Vibrational Energy Levels of Triatomic Molecules, /. Chem. Phys., 78, 301 (1983). 

The large-dimension limit has recently resolved at least some of the difficulties of the molecular model. The molecule-like structure falls out quite naturally from the rigid bent triatomic Lewis configuration obtained in the limit D — oo [5], and the Langmuir vibrations at finite D can be analyzed in terms of normal modes, which provide a set of approximate quantum numbers [6,7]. These results are obtained directly from the Schrodinger equation, in contrast to the phenomenological basis of some of the earlier studies. When coupled with an analysis of the rotations of the Lewis structure, this approach provides an excellent alternative classification scheme for the doubly-excited spectrum [8]. Furthermore, an analysis [7] of the normal modes offers a simple explanation of the connection between the explicitly molecular approaches of Herrick and of Briggs on the one hand, and the hyperspherical approach, which is rather different in its formulation and basic philosophy. 

A. R. Hoy and P. R. Bunker, A precise solution of the rotation bending Schrodinger equation for a triatomic molecule with application to the water molecule, J. Mol. Spectry. 74 1 (1979). 

MORBID [9] is a variational approach for solving the Schrodinger equation for the motion of nuclei of triatomic molecule, and RENNER [10] is an extension of MORBID for electronic states subject to the Renner-coupling. These approaches have been applied for simulating spectra of a large number of triatomic molecules (see below). MORBID and RENNER assume that dipole moment surfaces are given in special analytical representations, which in this paper will be referred to as a MORBID-type and described in detail below. 

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