Fano model

There are but few models of irreversibility which can be solved exactly. Such a model was originally introduced by Ref. [Friedrichs 1948] but it also goes under the name the Fano model, [Fano 1961 Cohen-Tannoudji 1992], It offers the theory of a single quantum state imbedded in a continuum or a quasi-continuum. It has been used in molecular physics as the Bixon-Jortner model [Bixon 1968] and in solid state physics as an Anderson model [Anderson 1961]. 

The infinite matrix representation of fhe Fano model [7,19,28] can be written in the form... 

From this, however, it should not be concluded that the statistical model of the atom is a very good one. As Fano (1963) has pointed out, I appears only as a logarithm and an error Si in the computation of I shows up as a relative error in the stopping power as (l/5)<5l II. Besides, it is an average quantity and can be approximated reasonably well without knowing the details of the distribution. 

Fano, G., Ortolani, F., Ziosi, L. The density matrix renormalization group method Application to the PPP model of a cyclic polyene chain. J. Chem. Phys. 1998, 108(22), 9246. 

For the purposes of this review it is convenient to focus attention on that class of molecules in which the valence electrons are easily distinguished from the core electrons (e.g., -n electron systems) and which have a large number of vibrational degrees of freedom. There have been several studies of the photoionization of aromatic molecules.206-209 In the earliest calculations either a free electron model, or a molecule-centered expansion in plane waves, or coulomb functions, has been used. Only the recent calculation by Johnson and Rice210 explicitly considered the interference effects which must accompany any process in a system with interatomic spacings and electron wavelength of comparable magnitude. The importance of atomic interference effects in the representation of molecular continuum states has been emphasized by Cohen and Fano,211 but, as far as we know, only the Johnson-Rice calculation incorporates this phenomenon in a detailed analysis. 

In Fig. 22.6(a) we show the spectrum of the transition from the Ba 5d6p F3 state to the 5d5/216d3/2 5/2 states which lie just above the Ba+5d 3/2 limit, and in Fig. 22.6(b) we show the spectrum from the same initial state to the 5d5/214d3/2 5/2 states which lie just below the Ba+ 5d3/2 limit. It is apparent that the envelopes of the excitations are identical. We have been discussing these states as if they were bound states, and in fact for our present purpose they might as well be. First, below the 5d3/2 limit the observed linewidths equal the laser linewidth, and second, there is no visible excitation of the 6seH continua below the 5d3/2 limit. It is also useful to note that a three channel quantum defect treatment of the 5d nd / = 4 series reproduces both the Lu-Fano plot of Fig. 22.2, and the spectra shown in Fig. 22.6. The experimental spectra were reproduced by QDT models using both the a and i channel dipole moment parametrizations described in Chapter 21. 

In the Sharf and Fischer treatment the real manifold is subdivided into n idealized submanifolds where each one conforms with the Bixon-Jortner model. By taking suitable linear combinations of the n idealized submanifolds the problem may be reduced to that of a discrete state and two manifolds, both of which carry intensity but only one of which is associated with a non-zero interaction. The Bixon-Jortner procedure applied to this situation leads to a Fano lineshape superposed on a constant background absorption. 

The diabatic two-state representation for homogeneous CT can be extended to heterogeneous CT processes between a reactant in a condensed-phase solvent and a metal electrode. The system Hamiltonian is then given by the Fano-Anderson model" " " ... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

The contribution of the short-range Coulomb interaction was later taken into account by Fano in (10) for the same classical model. A generalization for the case of liquids has been given in (11). 

Historically the first theoretical approach to autoionisation makes use of the fact that, in the independent electron model, it would simply not occur. The perturbation theory of autoionisation was established by Fano [256], along lines which will be described below. 

The independent electron model serves as the reference basis. Fano s theory of autoionisation consists in describing the consequence of turning on an interaction between a sharp state and the underlying continuum, which are presumed initially to be devoid of correlations. Of course, the perturbation is a hypothetical one, since it cannot really be turned off. The independent electron atom, as such, does not exist. Hypothetical interactions are familiar in perturbation theory. They carry with them the implication that, if they could be removed, the zero-order Hamiltonian which would result can be solved exactly, providing the basis for a perturbative expansion. For a many-electron atom, this is clearly not so, but the idea is nevertheless convenient. It is a case of pretending that,... 

A fundamental issue in the description of even the simplest, isolated autoionising resonance in the parametric approach followed by Fano [391] - and further pursued in K-matrix theory - is that the atom cannot be deperturbed, that is one cannot access the so-called prediagonalised states which are imagined to exist prior to autoionisation being included as a perturbative interaction, since the effect is anyway internal to the atom and cannot truly be turned off. This has the disadvantage that the parameters, once they have been obtained, must still be calculated from an ab initio model of the atom for a full comparison with theory. It might seem that the parametric theory cannot really be checked independently of ab initio calculations whose accuracy is hard to ascertain. 

Such model calculations display a rich variety of effects even on a very simple model. They show that the essential structure of two-dimensional quantum defect plots is preserved, but that conclusions as to the strength of inter-series coupling cannot be reached merely by inspecting Lu-Fano graphs a simultaneous study of the spectra is also required. 

This guided mode is sometimes referred as to the Fano mode [7]. As the permittivity of dielectric materials is usually positive, for the Fano mode to exist, the real part of the permittivity of the metal needs to be negative. For metals following the free-electron model [13] ... 

Measurements by photographic photometry require careful calibration due to the nonlinear response of photographic plates saturation effects can lead to erroneous values. Line profiles can be recorded photoelectrically, if the stability of the source intensity and the wavelength scanning mechanism are adequate. Often individual rotational lines are composed of incompletely resolved spin or hyperfine multiplet components. The contribution to the linewidth from such unresolved components can vary with J (or TV). In order to obtain the FWHM of an individual component, it is necessary to construct a model for the observed lineshape that takes into account calculated level splitttings and transition intensities. An average of the widths for two lines corresponding to predissociated levels of the same parity and J -value (for example the P and R lines of a 1II — 1E+ transition) can minimize experimental uncertainties. A theoretical Lorentzian shape is assumed here for simplicity, but in some cases, as explained in Section 7.9, interference effects with the continuum can result in asymmetric Fano-type lineshapes. 

S. Longhi, Bound states in the continuum in a single-level Fano-Anderson model, Eur. Phys. J. B 57 (2007) 45. 

---