Quasi-particle band structure

The calculated gaps of the band structures are 5.5 and 5.8 eV at the most stable conformation. These values are somewhat smaller than the one obtained by Kasowski et a/.80 using a linear combination of muffin tin orbitals in their LDA calculation. It should be mentioned that these values are still essentially smaller than the gap of 7.2 eV which we have obtained for our quasi particle band structure at the MP2 level.67 One should observe also that the lower limit of the conduction band is essentially higher (—1.5-2.0 eV), than in the HF + MP2 calculation ( — 5.7 eV67). Being, however, negative in both approximations (GKS and PZ, respectively) it still indicates a possibility of n-doping. 

Fig. 2.5. Ah initio calculated quasi-particle band structure, (a) truns-polyacetylene, where the valence and conduction bands are denoted as tt and tt, respectively, and the four bands below the valence band are formed from the three sp hybrids and the hydrogen Is orbital, (b) poly(paro-phenylene vinylene), where the valence, 7Ti, conduction, ttJ, and nonbonding bands, 7T2 and > are shown. Reprinted with permission from M. Rohlling and S. G. Louie, Phys. Rev. Lett. 82, 1959, 1999. Copyright 1999 by the American Physical Society.

The third collective effect (which exists only in eompounds with lattiee periodieity of the 4f ions) is a quasi-particle band structure at low temperatures, which exhibits... 

Fig. 10 displays our results for the quasi-particle band-structure in Si for two symmetry directions in an absolute energy scale. The full lines give our HF results. The HF approximation gives a direct gap of 6.8 (eV) compared to 3.3-3.4 (eV) in optical experiments, and a valence-bani gWidth of 13.5 (eV) compared to 12-13 (eV) in photoemission data (see also Table II). The dashed lines (TDSHF) are the dynamically correlated bands, where the two-particle Green s function contains both RPA (including local-field effects) and e-h attraction effects. The direct gap is reduced to... 

Sec. 5.3 Electronic Polaron Model and Quasi-Particle Band Structure... 

ELECTRONIC POLARON MODEL AND THE QUASI-PARTICLE BAND STRUCTURE OF POLYMERS... 

At correlation calculations of the total energies per unit cell or the correlation corrected (quasi particle) band structures one has to start from the HF CO. Therefore we shortly review (without any derivations) this formalism in its LCAO form. "" In the case of quasi one-dimensional (ID) polymers the Hamiltonian of the linear chain can be written as... 

CORRELATION CORRECTED (QUASI PARTICLE) BAND STRUCTURE BASED ON THE ELECTRONIC POLARON MODEL... 

The gap in alternating trans-polyacetylene is described on the basis of its quasi-particle band structure and the question of the Bloch-type conduction in DNA (either through doping or the possibility of intrinsic conduction due to charge transfer from the sugar rings to the nucleotide bases) is discussed. This is followed by a brief discussion of the electronic structure of disordered polypeptide chains. 

The R ions form a periodic lattice, which leads for the 4f electrons together with the conduction electrons to the formation of quasi-particle bands, i.e. the electrons are in a coherent state. Since the magnetic moments either vanish (in the non-magnetic Kondo state) or form themselves a periodic magnetic structure (Kondo systems with magnetic order) there is no elastic scattering of the conduction electrons and therefore Pn,(0) = 0. This is different at high temperatures, where even in a periodic lattice one has disordered moments, which scatter elastically. The coefficient Ai can be calculated analytically. One finds A = j j + with the resistivity in the unitarity limit... 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

A partial justihcation for the interpretation of the KS eigenvalues as starting point for approximations to quasi-particle energies, common in band-structure calculations, can be given by comparing the KS equation with other self-consistent equations of many-body physics. Among the simplest such equations are the Hartree equation... 

There are also QP (quasi particle correlated band structure) calculations for polyethylene (PE)66,67 and polytetrafluorethylene (teflon).67 In the PE case a G-31G and dementi s double basis,68 respectively, was applied. In both calculations66,67 a full geometry optimization was performed. With the G-31G basis a gap of 10.3 eV was obtained, it increased, however, with the poorer double basis of dementi to 11.6 eV, ev max(0 (= —JP) lies at — 8.2 eV while the experimental values of the ionization potential are at 7.6-8.8 eV.69 On the other hand the gap value estimated on the basis of experiment is at 8.8 eV,69... 

The term heavy fermion was coined by Steglich, since the quasi-particle is a fermion (i.e. obeys Fermi statistics) and its mass is extremely large, several hundreds to thousand times larger than that of a free electron. This mass is measured by the electronic specific heat constant, since y/yo = m /m where yo is the unenhanced electronic specific heat constant as calculated from the band structure and m is the free-electron mass and m is the effective mass of the electron. Since yo is of the order... 

Here C ° and E denote a zeroth-order approximation for the quasi-particle states. In our Si calculation this zeroth-order approximation was extracted from an empirically fitted pseudopotential band-structure (see ref.4 and 35). This bandstructure is fitted in terms of a fourth-nearest neighbor (in the fcc-lattice sites) overlap model of bonding and antibond ng orbitals as described n our earlier work on optical properties and impurity screening. Also the calculation of the two-particle Green s function is based on this bandstructure and follows closely the impurity studies (for details see in particular, ref.35). 

In the actual calculation the complicated side chains R were substituted by an H atom. For the first step (calculation of the HF band structures) a 6-3IG basis set (double C + polarization function on both the carbon and hydrogen atoms) was applied. The valence and conduction bands obtained in this way were then corrected using the generalized electronic polaron model [quasi-particle (QP) band structures see Section 5.3]. The lowest sin et-exciton enei es (at K = 0) were then calculated using the QP one-electron levels and performing the three steps described in the previous section after equation (8.22). Table 8.1 shows the results obtained in this way for both PTS and TCDU. ... 

One can define a generalized Koopmans theorem if, by calculating quantity (8.27), one takes into account correlation effects and determines a quasi-particle (QP) band structure based on the generalized electronic polaron model (see Section 5.3). One can then write for a polymer... 

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