Wannier symmetry

The systems of valent states and oxidation states introduced by chemists are not merely electron accounting systems. They are the systems which allow us to understand and predict which ratios of elements will form compounds and also suggests what are the likely structures and properties for these compounds (3). In the case of highly covalent compounds, the actual occupancy of the parent orbitals may seem to be very different than that implied from oxidation states if ionicity were high. Nonetheless, even some physicists have recognized the fundamental validity and usefulness of the chemist s oxidation state approach where the orbitals may now be described as symmetry or Wannier orbitals (6). 

P. L. Silvestrelli (1999) Maximally locahzed Waimier functions for simulations with supercells of general symmetry. Phys. Rev. B 59, p. 9703 G. Berghold, C. J. Mundy, A. H. Romero, J. Butter, and M. ParrineUo (2000) General and Efficient Algorithms for Obtaining Maximally-Locahzed Wannier Functions. Phys. Rev. B 61, p. 10040... 

One has a separate operator for each spin orbital so the equation has to be solved several times and (controllable) problems of orthogonality have to be dealt with. Unlike the LSD energy, the LSD-SIC energy is not invariant to a unitary transformation among the occupied orbitals. For example, in a solid the SIC is zero for Bloch functions but not for Wannier functions. This clearly leads to arbitrariness in the application of LSD-SIC in situations involving wavefunctions which are delocalized by symmetry—a topic discussed further below. 

The first two terms do not break the translational symmetry and can fairly simply be incorporated into the calculations. The matrix elements can be calculated using the expression of the Wannier functions in terms of the Bloch functions, and subsequently performing the required integrals analytically in the interstitial region and numerically inside the spheres with expressions that are very similar to those we need for the other lattice-periodic parts of the potential (see, e.g.. Refs. [2,31]). 

We considered two approximate treatments of the DC field, i.e., one where we only included Z of Fig. 5 and equations (48)-(50), and another where the full sawtooth curve z was included. Some representative results are shown in Figs 7 and 8. Since the Wannier functions can be ascribed to individual unit cells, we show in Fig. 7 the number of electrons (relative to the number, 8, for the undistorted system) of each unit cell in the case that the field operator has the symmetry of z of Fig. 5. Not surprisingly, the electrons do show an asymmetric distribution, although the flow from one end of the Born von Karman zone to the other is small. The number of electrons inside the muffin-tin spheres also gives information on the electron redistributions. Thus, for e-E = 0.0002 hartree these numbers are 3.2403 and 3.2413 for the two carbon atoms per unit cell for the operator zi of Fig. 5, and 3.2217 and 3.2575 for the operator z- Here we also see a larger effect for z than for z However, for the z all atomic spheres show the same numbers, so that the charge redistribution of Fig. 6 is restricted to the interstitial region. 

There is an important connection between particle-hole symmetry and the relative parity of the particle-hole pair. Consider a basis state created by the removal of an electron from a valence band Wannier orbital on the repeat unit at i — r/2 and the creation of an electron on a conduction band Wannier orbital at R + r/2. This is illustrated in Fig. 6.1. This particle-hole pair has a centre-of-mass coordinate, R, and a relative coordinate, r ... 

Table 6.1 The classification of the many body singlet exciton states with particle-hole symmetry in terms of their Mott- Wannier exciton quantum numbers (The corresponding triplet states with the same spatial symmetry but opposite particle-hole symmetry have the same quantum numbers)...

In the weak-coupling (Mott-Wannier) limit m = 2 if particle-hole symmetry applies. Otherwise m > 2. In the strong-coupling (Mott-Hubbard) limit m > 2 always. The essentially states are shown schematically in Figs 6.7 and 6.9 for the weak-coupling and strong-coupling limits, respectively. 

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. 

In our examples, aU the induced irreps are simple, excluding the BR corresponding to the 6-sheeted lower valence subband (see Fig. 3.3). This band representation is a composite one as it is formed by two simple band representations d,aig) and (6, t u) induced by 0 2s- and Sr 4pstates, respectively. Analysis of the space symmetry of crystalline orbitals is used to consider the possible centers of localization of chemical bonding in crystals. This task requires the Wannier-function definition and is considered in the next section. 

Symmetry of Localized Crystalline Orbitals. Wannier Functions... 

The background of LOs symmetry analysis is the theory of BRs of space groups G. This analysis is equally applicable to localized Wannier orbitals (LWOs) i.e. to an orthonormal set of LO s. We describe the main principles of this theory related to the examined problem. 

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. 

As the symmetrical orthogonalization procedure (3.122) leaves unchanged the reality and symmetry properties of the functions, the set of orthonormalized functions Wf r) satisfy all the requirements to the localized Wannier functions (reality, symmetry requirements and orthonormality) and (in the case when this set of functions is unique) has to coincide with the latter ... 

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. 

WTAOs are defined as the Wannier functions that are constructed from a set of specially chosen occupied and vacant bands and have a definite symmetry (they are centered on atoms and transform via irreducible representations (irreps) of the corresponding site groups). Thus, the index t in (3.114) may be substituted by several indices - - that reflect the symmetry properties of the WTAOs —... 

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