Discrete state embedded

Quite often one will find there are multiple resonant states lying in the continua under consideration. Some of them may interfere with each other through their mutual interaction with the same continua. The interference may dramatically alter the profile of the resonance structure from what they d look like if no interference occurred. These resonant states are said to overlap. On the other hand, there is usually more than one available channel within the range of photon energy under study. The formalism for photodetachment calculations should then be able to treat the case of multiple discrete states embedded in multiple continua. The configuration interaction in the continuum (CIC) theory is our theory of choice. 

Extension of Fano s formalism to the general case of multiple discrete states embedded in multiple continua was first conducted by Mies [54]. Like Fano, Mies assumed a prediagonalized basis. By imposing the asymptotic condition for the continuum state from the scattering theory, Mies derived the complete solution to the total continuum problem and gave formulas for energies and widths of resonances. 

Physically this may be visualized as a set of discrete states x with probability pn embedded in a continuous range. If P(x) consists of delta functions alone, i.e., if P(x) = 0, it can also be considered as a probability distribution pn on the discrete set of states x . A mathematical theorem asserts that any distribution on — oo < x < oo can be written in the form (1.3), apart from a third term, which, however, is of rather pathological form and does not appear to occur in physical problems. ... 

In the procedure (Appendix 9B) to evaluate the lineshape (9.40) we use the representation defined by the states 1/) that diagonalize the Hamiltonian in the ( 5), /> ) subspace. Of course any basis can be used for a mathematical analysis. It was important and useful to state the physical problem in terms of the zero-order states 5) and ]/) because an important attribute of the model was that in the latter representation the ground state g) is coupled by the radiation field only to the state 5), which therefore has the status of a doorway state. This state is also referred to as a resonance state, a name used for the spectral feature associated with an underlying picture of a discrete zero-order state embedded in and coupled to a continuous manifold of such states. 

Figure 3.17 demonsfrafes a LICS-STIRAP scheme for maximizing fhe ionization from a discrefe sfafe coupled fo a confinuum by a fwo-phofon transition via a lossy intermediate state whose lifetime is much shorter than the interaction duration [97]. As in three-level STIRAP, the counter-intuitive pulse ordering is most efficient. It achieves almost complete ionization by embedding a third discrete state into the continuum using a third control laser. 

The radiationless decay of a quasidiscrete excited state of an atom or molecule into an ion and electron of the same total energy is called autoionization. The quasidiscrete state must, of course, lie above the first ionization potential of the atom or molecule. The occurrence of autoionization may be inferred from the appearance of absorption spectra or ionization cross-section curves which exhibit line or band structure similar to that expected for transitions between discrete states. However, in the case of autoionization the lines or bands are broadened in inverse proportion to the lifetime of the autoionizing state, as required by the uncertainty principle. In the simple case of one quasidiscrete state embedded in one continuum, the line profile has a characteristic asymmetry which has been shown to be due to wave-mechanical interference between the two channels, i.e., between autoionization and direct ionization. In an extreme case the line profile may appear as a window resonance, i.e., as a minimum in the absorption cross section. 

As a special application of the diabatization techniques of ab initio MO-CI calculations, one should mention the research on resonant states in electron-molecule collisions. The problem concerns the existence of a discrete (usually valence) state of the molecular anion, embedded in a continuum of diffuse states. For a molecule like H2, the electronic Hamiltonian of H J has a bound state at large interatomic distance (since H is stable), while it has no bound state for interatomic distances shorter than 1.4 A. Collisional properties, however, suggest the existence of a broad resonance of character, which may be seen to be due to the coupling of a discrete state of essentially a, ffy valence character with a continuum of diffuse states associated with the (ffg) ground state of the molecule, the outer electron being unbound. This problem may be treated in various modes, but it appears as a challenge to quantum chemists, whose finite basis sets seem to forbid the examination of such a problem. Nevertheless some approximate methods have been proposed (known as stabilization techniques ) to find the... 

Suppose for an atomic system, some discrete states are embedded in a range of eontinuum... 

Pietrzyk, P. and Sojka, Z. (2005) Relativistic density functional calculations of EPR g tensor for i7l CuNO ll species in discrete and zeolite-embedded states, J. Phys. Chem. A, 109, 10571. 

The discrete variational (DV) Xa method is applied to the study of the electronic structure of silicate glasses in embedded model clusters. The effects of the cluster size, the size of embedded imits, and the Si-0-Si bond angles on the electronic states are discussed. Embeddii units drastically improve the description of the electronic state, when compared to the isolated Si044- cluster, which is the structural unit of silicate glasses e.g., the Fermi energy for the embedded cluster becomes smaller when compared to that of the... 

Adsorption energies on metals calculated in a cluster approach often show considerable oscillations with size and shape of the cluster models because such (finite) clusters describe the surface electronic structure insufficiently [257-260]. These models may yield rather different results for the Pauli repulsion between adsorbate and substrate, depending on whether pertinent cluster orbitals localized at the adsorption site are occupied or empty. The discrete density of states is an inherent feature of clusters that may prevent a correct description of the polarizability of a metal surface and thus hinders cluster size convergence of adsorption energies [257]. Even embedding of metal clusters does not offer an easy way out of this dilemma [260,261]. Anyway, the form of conventional moderately large cluster models may be particularly crucial. Such models are inherently two-dimensional with substrate atoms from two or three crystal layers usually taken into accormt thus, a large fraction of atoms at the cluster boundaries lacks proper coordination. 

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