Wannier generation

Probably the most powerful technique for sampling the conformation space of real polymers is the transfer matrix approach in combination with the RIS approximation discussed in the previous chapter. The approach was pioneered by Flory (e.g., [13]), but its potential for generating configurational averages was recognized very early (cf. the reference to E. Montrol in the seminal paper by Kramers and Wannier [14]). In the transfer matrix approach the partition function of a polymer is expressed in terms of a product of matrices, whose elements in the simplest case, e.g., polyethylene, have the form... 

The theory of induced representations of space groups gives the answer to the question of whether it is possible to generate in the space of states of a given energy band the basis of localized functions The answer to this question allows the symmetry connection between delocalized Bloch-type and localized Wannier-type crystalline orbitals to be obtained. This point is discussed in Sect. 3.3. 

Localized crystalUne orbitals (LCO) are generated from a canonical delocahzed Bloch functions (CO). As in the case of the molecules one or other localization criteria is used. The orthonormaUzed LCO in crystals are known as Wannier functions. Wannier functions (WFs) have attracted much attention in solid-state physics since their first introduction in 1937 [43] and up to now. The analytical behavior of Bloch... 

The existence of Wannier functions decreasing exponentially at infinity (for the model of an infinite crystal) is established in many cases. The different locahzation criteria used for the generation of locahzed orbitals in crystals are considered in the next subsection. 

The general method of the most locahzed Wannier-function (WF) generation exists only for the one-dimensional case and was offered by Kohn [57]. 

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. 

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... 

To minimize the essential dependence of results on the basis-set choice and POPAN, it is reasonable to generate Wannier-type atomic functions that are orthogonal and localized on atomic sites. These functions can be generated from Bloch-type functions (after relevant symmetry analysis), see Chap. 3. In the next section it is shown that the results of POPAN with use of Wannier-type atomic functions weakly depend on the basis choice in Bloch function LCAO calculations. 

Localized Wannier functions (LWFs) have been calculated for three upper valence bands in SrTiOs and SrZrOs, represented mainly by O 2p, Sr 4p, and O 2s atomic states (in the case of SrZrOs the last two bands overlap considerably). A total of 15 crystalline orbitals have been used to generate, correspondingly, 15 LWFs per primitive unit cell in both crystals under consideration, three oxygen atoms occupy the same Wyckoff positions, and four LWFs can be attributed to each oxygen atom. It was found by calculations with CRYSTAL03 code [23] that the centroids of four functions are positioned near the center of one oxygen (at distances of about 0.3 A). 

The LCAO approximation allows comparison of the electron-charge distribution in the bulk crystal and on the surface. The application of Wannier functions for this comparison can be found in [781], where the Boys localization criteria was appUed and only the valence-band states were included for WFs generation (for bulk crystals such an approach is discussed in Chap. 9). As was seen in Chap. 9 for bulk crystals, the population analysis using Wannier-type atomic orbitals (WTAO) gives more adequate... 

Let us work in a local representation here This is appropriate since many molecular solids are filled shell systems For notational simplicitiy, designate the Wannier function WiN(r) as the 1 Wannier function about site j Form a complete set of Wannier orbitals describing the ground state of the neutral, N-electron solid in the Hartree-Fock limit We will use them to generate the ion states as well. For a system of N-electrons the Hamiltonian is... 

For the calculation of the correlation energy per unit cell in the ground state of a polymer (either conductor or an insulator) one can use any size-consistent method (perturbation theory /18/, coupled cluster expansion /19/, electron pair theories /20/, etc.). In the case of insulators one can Fourier transform the delocalized Bloch orbitals into site semilocalized Wannier functions (WF-s) and perform the excitations between Wannier functions belonging to near lying sites /4/. (For the generation of optimally localized Wannier functions see /21/.) This procedure is, however,... 

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