Quasi-particle states

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

As mentioned in Section 2, the CPs of solids have to be calculated on the quasi-particle scheme. In order to calculate the quasi-particle states, non-local and energy-dependent self-energy in Equation (13) must be evaluated in a real system. In practice, the exact self-energy for real systems are impossible to compute, and we always resort to approximate forms. A more realistic but relatively simple approximation to the selfenergy is the GWA proposed by Hedin [7]. In the GW A, the self-energy operator in Equation (12) is... 

It has been suggested that quasi-particle wave functions do not deviate much from LDA wave functions [26], Furthermore, in the evaluation of momentum densities shown in Figure 9, the characteristics of the quasi-particle states dominantly reflect on the occupation number densities which should be evaluated by using the general quasi-particle Green s function. In GWA, however, the corresponding occupation number densities are... 

The expansion of the quasi-particle states thus assumes the form... 

The discussion of the parity operator can now be repeated verbatim with states, replaced by the molecular states ,, and the understanding that the set, are associated with the quasi-particle solutions of the quantum field theory of the chiral medium. In the next section we discuss some models of these molecular (quasi-particle) states. 

The interpretation of experimental results based on the one electron picture also raises fundamental questions. It has been shown that the low energy elementary excitations in metals can be described as quasi-particles. By making suitable many-body corrections one can convert the one electron states into quasi-particle states. For excitations from inner shells, which become possible when the excitation energy is high, the change of state of one electron is accompanied by a rearrangement of the states of many other electrons in the same core. This is a complicated many-body problem that can not be handled by the simple methods of band calculation. To what extent should one include many electron effects when an electron is excited from a deep band state remains an open question. 

We have developed in previous work a practical scheme for calculating the self-energy corrections and quasi-particle states in periodic crystals. One main iijgredient of this scheme is the short-ran e property of E in r-r [, which has been proven by Sham and Kohn on the basis of general considerations of many-body perturbation theory. As a consequence, S(r,r E) will have matrix elements in a set of local orbitals (Wannier, LCAO s, muffin-tin etc.) appreciably different from zero only about the diagonal, r = r. ... 

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 our previous diamond work we determined the bare HF part of E by making a Slater-Koster fit to an existing HF band calculation, while the correlation part was determined, as in the present Si work, by evaluating the matrix elements in an explicit basis set, which represents a zeroth-order approximation to the actual quasi-particle states. 

The current conservation guarantees consistency in the approximation for both one (g)- and two-particle (W)-propagators in eq. (3.7). Just as in our previous C work, we therefore use the same basis set as an approximation for the actual quasi-particle states everywhere in eq.(3.7). Of course, as discussed above, this procedure is approximate in that it is a pragmatic "short-cut" in the self-consistency circle for the quasi-particle states. 

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