Wannier treatment

The foundations of the Wannier treatment for positron impact are not as solid as for electron impact. The reason is that in the latter case, there are only two open channels near threshold elastic and inelastic scattering and ionization, whereas for positron impact, there is an additional, very strong channel, positronium formation. Furthermore, in the e impact case, the two light particles are much closer to one another in the final state, this casting doubt on the non-overlapping, classical orbits of the model. 

In the classical treatment of near-threshold electron impact ionization developed by Wannier (1953), the repulsion between the two electrons causes them to emerge with very similar energies but in opposite directions along the so-called Wannier ridge. This effect is depicted in Figure 5.6, where it is contrasted with the case for positron impact described below. According to this theory the energy dependence of the ionization cross section for electron impact is predicted to be... 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

The collinear models are also useful close to the two-electron break-up threshold. In 1994 Rost was able to obtain the correct Wannier exponent by a semiclassical treatment of electron impact ionization of hydrogen, another important quantum problem which involves nonintegrable three-body dynamics (see also Rost (1995)). 

The problem in which we are more interested is the quantum mechanics of periodic lattices which have been perturbed by the presence of defects. There is considert able experimental evidence for the association of discrete localized states with lattice defects of one sort or another the introduction of a perturbation into the quantum-mechanical problem should lead naturally to the prediction of these states. Quite recently Slater (18) has generalized a theorem used by Wannier for the discussion of excited states of crystals and through its use has clarified the whole problem of electronic motions in perturbed periodic lattices. It is possible to give an essentially non-mathematical discussion of Slater s treatment, as it is one which lends itself to simple graphical illustration. 

We considered two approximate treatments of the DC field, i.e., one where we only included Z of Fig. 5 and equations (48)-(50), and another where the full sawtooth curve z was included. Some representative results are shown in Figs 7 and 8. Since the Wannier functions can be ascribed to individual unit cells, we show in Fig. 7 the number of electrons (relative to the number, 8, for the undistorted system) of each unit cell in the case that the field operator has the symmetry of z of Fig. 5. Not surprisingly, the electrons do show an asymmetric distribution, although the flow from one end of the Born von Karman zone to the other is small. The number of electrons inside the muffin-tin spheres also gives information on the electron redistributions. Thus, for e-E = 0.0002 hartree these numbers are 3.2403 and 3.2413 for the two carbon atoms per unit cell for the operator zi of Fig. 5, and 3.2217 and 3.2575 for the operator z- Here we also see a larger effect for z than for z However, for the z all atomic spheres show the same numbers, so that the charge redistribution of Fig. 6 is restricted to the interstitial region. 

In this contribution we have concentrated on presenting some fundamental considerations concerning die flieoretical treatment of an infinite, periodic, polymeric chain being exposed to an external electrostatic field. The analysis of a simple Huckel-like model revealed that only under certain circumstances one can base the discussion on Bloch functions and substituting f with a derivative wifli respect to k. Thus, this was not the case when including the field directly in the calculations, but could, e.g., be used when using Wannier functions as basis functions which are continuous functions not only of r but also of k. 

The study of excitons in conjugated polymers has often been inspired by the treatment of excitons in bulk three-dimensional semiconductors (as described in Knox (1963)). A particle-hole excitation from the valence band to the conduction band in a semiconductor leaves a positively charged hole in the valence band and a negatively charged electron in the conduction band. The Coulomb attraction between these particles results in bound states, or excitons. In three-dimensional semiconductors the excitons are usually weakly bound, with large particle-hole separations, and are well described by a hydrogenic model. Excitons in this limit are known as Mott- Wannier excitons. 

Eqn (6.19) is the Schrodinger equation for describing the Mott-Wannier ex-citon wavefunctions and energies. In more sophisticated treatments it is usually known as the Bethe-Salpeter equation. In the following two sections the solutions of this equation will be described. 

In the cases of completely filled or empty bands (semiconductors or insulators), the numerical work can be reduced if one does not substitute the Bloch functions in integral (5.37) in their LCAO form [see equations (5.20) and (5.21)], but replaces the AOs by Wannier fiinctions (see Section 5.1). In this way matrix elements (5.37) retain their k-depen-dence (which is necessary for the correct treatment of the translational symmetry), but this procedure does not involve any special difficulties. By writing the Bloch functions in the form< )... 

For Mott-Wannier excitons, the treatment is similar, only instead of the localized Wannier functions " (r — R ), (/> (i... 

The treatment of correlation, even in periodic polymers, still presents a formidable and an only partially solved problem if the unit cell is large. To solve this, different localization techniques like the Fourier tran ormation of the HF COs to a localized Wannier function (WF) and, if the unit cell is large, localization within the unit cell (using the Boys or Edminston-Ruedenberg technique) helps to decrease the necessary computer time. Further, the optimal organization of the four index transformations and the good vectorization of the programs provides an additional increase in speed. 

The second exact treatment of the order-disorder problem is based on a powerful method due to Kramers and Wannier [1941] and has permitted to Onsager [1944], Houtappel [1950], Wannier [1950], Kac and Ward [1952] and Ter Haar [1954] to obtain closed solutions for the order-disorder problem in two dimensional lattices. It is outside... 

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