Wannier variational method

It is a variational method based on the first-principles approach, t.e. without preliminary knowledge of Bloch-type delocalized functions. The localized functions with the symmetry of Wannier functions and depending on some number of parameters are used. 

An attempt to overcome the disadvantages mentioned was made in [42] where a variational method of Wannier-type function generation was suggested. This method is applicable with the different localization criteria, the Bloch functions can be calculated both in LCAO and in PW basis, the full s3Tnmetry is taken into account. In the next section we consider it in more detail. 

To demonstrate the reliability of the proposed variational method let us consider two examples of its applications given in [42] - the Wannier-function generation in silicon and MgO crystals. 

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wannier-type atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set. 

In Table 9.19 we compare the local characteristics of the electronic structure determined for the Ti02 crystal with the use of the projection technique and those obtained in an earlier study [623], in which the Wannier-type atomic frmctions of the minimal valence basis set of titanium and oxygen atoms were constructed by the variational method. In [623], the Bloch functions were calculated by the DFT method... 

The character of the locahzation of Wannier functions depends on the analytical properties of Bloch states (as a function of the wavevector) that are essentially determined by the nature of the system under consideration. One can arbitrary change only the form of an unitary transformation of Bloch functions. It is just this arbitrariness that is used in the variational approach [42] to assure the best localization of Wannier functions. The accuracy of the Wannier functions obtained by the proposed method is determined solely by the accuracy of the Bloch functions and the size of the supercell used. As the calculations have shown, the proposed method is reUable and useful in the problem of generation of the locaUzed Wannier functions. In the two examples... 

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