Quasi-particle wave functions

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... 

In the above analysis, and Y are expressed in the principal coordinate system, but in general, it is necessary to transform these vector functions from the principal coordinate system to the particle coordinate system through a rotation. The vector quasi-spherical wave functions can also be defined for biaxial media (e 7 y z) by considering the expansion of the tangential vector function T>c (3,a)Va + T>is j3, a)vjj in terms of vector spherical harmonics. 

The expressions of the elements of the Qaiis matrix are similar but with M and 7V in place of M and iV, respectively. Using the properties of the vector quasi-spherical wave functions (cf. (1.47)) it is simple to show that for Siz = Si, the present approach leads to the T-matrix solution of an isotropic particle. 

As noticed from this expression, the CP calculation has to be basically carried out on the quasi-particle picture. Formally, quasi-particle energies and wave functions have to be evaluated by solving... 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

Narrow bands arise when the overlap of the atomic wave functions is small (as for 5 f s). In this case, the dispersion E(k) is strongly reduced and the bandwidth W becomes very small (zero, in the case of no overlap). The electron charge density, caused by these wave functions, is high in the core region of Fig. 12, and the quasi-particles spend most of their life there, nearly bound to the atom. In case the charge density is all confined within the core region (as for 4f in lanthanides), then the bond description loses its meaning and the atomic description holds. 

These problems can be solved if one starts from the (untruncated ) Foldy-Wouthuysen transformation for a free particle, the only case for which the transformation is known anal3d ically, and incorporates the effects of the external potential on top. Along these lines, the so-called Douglas-Kroll-HeB (DKH) method [61-64] is constructed which is probably the most successful quasi-relativistic method in wave function based quantum chemistry. No details will be given here since this topic has been extensively discussed in volume 1 [34] of this series. Meanwhile several density functional implementations exist based on the Douglas-Kroll-HeJ3 approach [39-45]. In recent years. 

Nevertheless, the calculation described above for the quasi-particle gap in alternating fra j-polyacetylene proves that one can start from a high-quality HF wave function for infinite systems as well (in contrast to... 

One can, however, impose certain conditions on the coefficients resulting in a simpler form of the quasi-particle commutator Qi. First, the requirement that the two-electron wave functions created by form an orthonormal set requires ... 

The effect of inter-site hopping is then introduced into the system. The manifold of basis states are limited to those in which the local correlations have been diagonalized. The wave functions for the composite particles then obey Bloch s theorem, which results in the formation of a dispersion relation consisting of two bands for the quasi-bosons the first band describes spinless quasi-boson excitations, the second band describes the magnetic quasi-bosons. Although these composite particles are bosons in that they commute on different sites, they nevertheless have local occupation numbers which are Fermi-Dirac like. 

It is at once evident from the above expression that the particle s oscillation at a frequency i o between the two energy levels Ek results in a field splitting of each level into two sublevels E k (f2o/2) ft, spaced a distance fti o apart. Such split energy levels, referred to as quasi-energy levels, have a common origin associated with the properties of the periodic wave function of a two-level system in an external periodic field of constant amplitude (Zel dovich 1973). 

As discussed above, when an infinite graphene sheet is cut to form a quasi-one-dimensional graphene nanoribbon with a finite width and infinite length, the 7T-electrons wave function is confined along the direction perpendicular to the axis of the ribbon and is forced to vanish at large distances along this direction. These particle in a box like boundary conditions induce... 

Theoretical aspects related to the above problems have been elaborated and discussed in detail in the Ab initio theory of complex electronic ground state of superconductors , published recently [52a, b]. The main theoretical point is a generalization of the BOA by sequence of canonical - base function transformations. This formalism is equivalent to our original one, based on quasi-particle transformation treatment [53]. The final electronic wave function is explicitly dependent on nuclear coordinates Q and nuclear momenta P. Emerging new quasiparticles, i.e. nonadiabatic fermions, are explicitly dependent on nuclear dynamics. As a result, the effect of nuclear dynamics can be calculated as corrections to the clamped nuclei ground state electronic energy, the one-particle spectrum and the two-particle term, i.e. to the electron correlation energy ... 

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