Wannier centers

III.B. In order to analyze the wavefunction in a chemically more intuitive way, it is useful to localize it. In the framework of AIMD this is, for example, done by calculating the maximally localized Wannier functions (MLWF) and the corresponding expectation values of the position operator for a MLWF basis the so-called maximally localized Wannier centers (MLWCs), see Fig. 1 (67-72). With the help of the MLWC it is possible to compute molecular dipole moments (72-82). Furthermore, it is possible with the MLWC to obtain molecular properties, e.g., IR spectra (75,76,82-85). 

Having now to deal with localized orbitals, they should have a well-defined center, referred to as the Wannier center in PBC and defined as [239,240,242]... 

Figure 8 displays the maximally localized Wannier orbitals computed for one isolated water molecule. We can straight away identify two OH bond orbitals and two lone pair orbitals. The Wannier centers for these orbitals are shown on Fig. 9 they form approximately a tetrahedron around the oxygen of the water molecules. 

By analyzing the Wannier orbitals of a sample of 32 water molecules, Silvestrelli et ah [242] have determined the average molecular dipole moment of water molecules in the liquid phase. They have assigned all Wannier centers... 

Fig. 10. Wannier centers of a snapshot of a simulation of 32 water molecules...

Fig. 11. Radial distribution function between the Wannier center of the 5s electron of a silver atom and the silver nucleus.

Besides water, methanol was investigated with respect to its changing dipole moment [96]. Handgraaf et al. found - despite little alterations in the Wannier center positions - a dipole moment increase of methanol from 1.73 D in the gas phase for a single molecule to 2.54 D in the liquid phase. 

The protein field might be very important to the chemistry of the active site of this and other enzymes [11]. The effect of the environment was estimated by comparing the electronic structure of the complexes in vacuum with those in the presence of the protein. The Wannier functions [37], the centers (WFC) of which represent... 

A maximally localized Wannier function analysis84-86 was performed to better analyze the bonding in our simulations. The maximally localized Wannier functions express the quantum wave function in terms of functions localized at centers, rather than as delocalized plane waves. The positions of these centers give us insight into the localization of charge during the... 

In a fairly early discussion of solids Wannier[38] showed how linear combinations of the AOs could be made that rendered the functions orthogonal while retaining a relatively large concentration on one center. In more modern language we would now say that he used a symmetric orthonormalization of the AO basis. If we symbolize the overlap matrix for the AO basis by S, then any matrix N that satisfies... 

In chemical usage [23, Section 14.11] an electron is said to be delocalized if its molecular orbital cannot be ascribed to a two-center bond otherwise it is localized. It is, however, always possible, but perhaps rarely convenient, to describe the electron distribution in a molecule with delocalized orbitals only. The situation in a covalent insulator such as diamond is similar to the molecular case. There are four valence electrons per atom, and four neighbors. Therefore, it is possible to describe the structure with four two-center, two-electron bonds, and localized Wannier orbitals. But keep in mind that the only physical reality is the resulting charge distribution. This reality can also be described by freely moving Bloch electrons. 

The expectation value r, of the position operator for a MLWF i is thus is often called a Wannier function s center (WFC). With this definition the electronic part of the supercell dipole moment reads... 

If f i a,B (weak confinement), the splitting between the levels in a spherical well is much smaller than the binding energy of a three-dimensional Wannier-Mott exciton. In this case the latter can be assumed to be a rigid particle moving in a spherical well. The variables corresponding to the relative motion of an electron and a hole and the motion of the center of mass effectively separate, and the wavefunction factorizes. Then... 

In the last decade the Wannier exciton emission from direct band gap soniconductors was reconsidered for high count-rate and coincidence-detection scintillation applications and Cul, Hglj, Pbl2, ZnO Ga, and CdS In compounds in powder form were studied (Derenzo et al. 2002). In direct gap semiconductors a favorable combination of a smaller gap and an UV-VIS emission center based on Wannier exciton can provide high scintillation efficiency and subnanosecond radiative lifetimes due to microscopic superradiance effect (Niki 2006, Wilkinson et al. 2004). On the other hand, the Stokes shift of such emission centers is necessarily low (typically below 0.1 eV) and it prevents their usage in the bulk form due to enhanced reabsorption effect, see Figure 4.4. The ZnO Ga has shown the best combination of subnanosecond decay time and emission intensity... 

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. 

In accordance with the theory of induced (band) representations the corresponding Wannier functions in the silicon crystal (four per unit cell) are centered at the middle of the bonds between the nearest Si atoms (Wyckoff position c with site group Gc = Dsa) and transform according to the irrep aig of the site group Dsa, see Sect. 3.3.1. 

The local MP2 electron-correlation method for nonconducting crystals [109] is an extension to crystalline solids of the local correlation MP2 method for molecules (see Sect. 5.1.5), starting from a local representation of the occupied and virtual HF subspaces. The localized HF crystalline orbitals of the occupied states are provided in the LCAO approximation by the CRYSTAL program [23] and based on a Boys localization criterion. The localization technique was considered in Sect. 3.3.3. The label im of the occupied localized Wannier functions (LWF) Wim = Wj(r — Rm) includes the type of LWF and translation vector Rm, indicating the primitive unit cell, in which the LWF is centered (m = 0 for the reference cell). The index i runs from 1 to A i, the number of filled electron bands used for the localization procedure the correlation calculation is restricted usually to valence bands LWFs. The latter are expressed as a linear combination of the Gaussian-type atomic orbitals (AOs) Xfiif Rn) = Xfin numbered by index = 1,..., M M is the number of AOs in the reference cell) and the cell n translation vector... 

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