Quasi-degenerate levels

However, if we had allowed more general variation in the two-electron, two-orbital problem, in particular permitting the spin-up and spin-down densities to differ by using two 3 values in wavefunction (68), the FON solution would not lie lowest. Furthermore, for three or more quasi-degenerate levels (the true situation in C2) non-magnetic FON solutions never occur. 

For many electron systems, QED corrections must also include many-body contributions. For the time being only a limited number of results, besides semi-empirical extrapolations, are available for heavy elements where a perturbative Za approach (in terms of the electron nucleus interaction) is irrelevant. The reason is not only that the most precise numerical methods developed for the one-electron contributions [34] encounter serious numerical accuracy problems for high angular momentum values but also that, even for two-electron atoms or ions, the standard QED prescription [35] is unable to deal with quasi-degenerate levels. Recent developments [36-37] open new perspectives for getting accurate estimates in two-electron systems without any restriction on the nuclear charge. 

In this section we focus on narrow, close-lying levels of varying nature in diatomic molecules. Such levels may come about due to a quasi-degeneracy of either hyper-fine and rotational levels [45], or between the fine and vibrational levels within the molecular electronic ground state [49] (see Figure 16.1). The transitions between the quasi-degenerate levels correspond to microwave frequencies, which are experimentally accessible, and have narrow linewidths, typically of the relative variation can exceed 10 in such cases. 

Fig. 20. A schematic representation of the emission of an isolated large molecule following internal conversion from the second to the first singlet, a" and 6J1 denote the amplitudes of the second singlet and quasi-degenerate vibrational levels of the first singlet, respectively, in the excited molecular state >/in. /v, and m are the corresponding electronic dipole transition matrix elements coupling < >n and as indicated.

While a single-matrix element, T//, suffices for the simple TSA, an extended treatment may be required in cases of degenerate or quasi-degenerate initial and final states associated with open-shell reactants and/or products, since the ET process may then involve a distribution of thermally populated initial states and several possible final states. Even in such a multistate framework, it may be possible to cast the overall rate constants as a superposition of individual processes, each treated at the level of the TSA. 

Fig. 18.1 A dressed-state model that is used in the text to describe absorption, emission, and elastic (Rayleigh) and inelastic (Raman) light scattering. g) and. v> represent particular vibronic levels associated with the lower (1) and upper (2) electronic states, respectively. These are levels associated with the nuclear potential surfaces of electronic states 1 and 2 (schematically represented hy the parabolas). Rj are radiative continua— 1 -photon-dressed vibronic levels of the lower electronic states. The quasi-continuum L represents a nonradiative channel—the high-energy regime of the vibronic manifold of electronic state 1. Note that the molecular dipole operator /t couples ground (g) and excited (s) molecular states, but the ensuing process occurs between quasi-degenerate dressed states g,k and 5,0).

These nonresonant couplings may be dealt with by a Van Vleck (see Section 4.2) or contact transformation (Nielsen, 1951), which folds their effects into systematically quasi-degenerate groups of states called polyads (see Section 9.4.5). The polyad Heff fit model accounts accurately for the observed energy levels (and many other properties) of an entire family of scaling-related polyads. In effect, the dimensionality of the exact H is drastically reduced in the polyad Heff. This reduction is due to the existence of several approximate constants of motion which permit H to be block diagonalized into families of dynamically decoupled polyad Heff matrices. 

Fine structure splitting and transition moments for radiative and nonradiative spin-orbit coupling induced processes were calculated by Tatchen et al. for several low-lying singlet and triplet states of 4H-pyran-4-thione (7). The spin-independent properties were calculated by the DFT/MRCI method of Grimme and Waletzke, and the spin-dependent part was included at the level of quasi-degenerate perturbation theory... 

Essential changes in the concentrations of charge carriers and accordingly in the potential profile and emission spectra are observed when the quasi-Fermi level difference AF exceeds, e.g., for the superlattice No. 4 the value of 0.9 eV. Then, the chemical potential for electrons in the n-type layers becomes positive and the degeneration begins. For superlattice No. 4i it occurs at a smaller value of AF. 

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