Wannier orbitals

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). 

Fig. 1. We show on the left-hand side the MLWCs (above) in pink and the HOMO-Wannier orbital (below) together with a ball-and-stick model of a I-DMSO-3-H2O water cluster. On the right the same is depicted for the [PdCl2(NH3)]2 complex.

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. 

Besides Bloch orbitals, Wannier orbitals [76] are also widely used in solid state physics. They are defined as Fourier transformations of the Bloch orbitals, e.g. 

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. 8. Maximally localized Wannier orbitals of one water moelcule...

We can now come back to the aqueous silver atom example [57,151] mentioned in Sect. 5.4. The orbital displayed on Fig. 1 is one of the five maximally localized Wannier orbitals assigned to the silver atom and which has been... 

Localized orbitals have also been used as a tool to extract the infrared spectrum of a solute in solution [194,195,202] or to decompose the IR spetrum in intramolecular and intermolecular contributions [202]. Model electrostatics of solute molecules was also based on localized orbitals [242, 243], not only at the dipolar level [244]. As an extension we also defined molecular states from localized orbitals to study the electronic states of liquid water [245], or of solvated ions [47]. It is also possible to perform CP-MD propagating the Wannier orbitals, by constraining the Kohn-Sham orbitals to stay in a Wannier gauge [246]. 

R. Iftimie and M. E. Tuckerman (2005) Decomposing total IR spectra of aqueous systems into solute and solvent contributions A computational approach using maximally localized Wannier orbitals. J. Chem. Phys. 122, p. 214508... 

The idea of distributed dipole moments has also been transferred to the dynamic domain and we shall discuss recent work from our laboratory in this section in more detail. With the help of maximally localized Wannier functions local dipoles and charges on atoms can be derived. The Wannier functions are obtained by Boys localization scheme [217]. Thus, Wannier orbitals [218] are the condensed phase analogs of localized molecular orbitals known from quantum chemistry. Access to the electronic structure during a CPMD simulation allows the calculation of electronic properties. Through an appropriate unitary transformation U of the canonical Kohn-Sham orbitals maximally localized Wannier functions (MLWFs)... 

The novelty of the layered-metal-plane materials with small m (large 3o ) lies in the relative weakness of the short-ranged intraplanar repulsion caused by the Wannier orbitals overlap within the planes, as compared to the unabated vigor of the screening potentials. The paradox is resolved, the mechanism is natural and anchored in common sense. The theory built on it, as outlined in these pages, has only one essential parameter (d/ao ) and several auxilliary parameters, all of which are readily measured or understood. It should appeal, if only because of its relative simplicity. 

This corresponds to determining a set of LMOs that minimizes the spatial extent, i.e. they are as compact as possible. For extended (periodic) systems described by plane wave basis functions, the equivalent of the Boys LMOs is called Wannier orbitals. Feng el al have shown that the Boys LMOs can be made even more compact by 10-25% by allowing the localized orbitals to be non-orthogonal, but this requires a general optimization procedure, rather than a simple unitary transformation. 

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 ... 

The extrinsic dimerization has two effects. First, it causes an increased intrinsic dimerization, as shown in Fig 4.10. Second, it lifts the degeneracy of the A and B phases, as shown in the plot of the ground state energy in Fig. 4.1. This causes a linear confinement of the soliton-antisoliton pair, because the energy to create a B phase relative to the A phase increases linearly with the soliton-antisoliton separation. This new property of soliton-antisoliton confinenment is illustrated by the localized Wannier orbitals associated with the soliton, and antisoliton, These are obtained from the molecular orbitals associated with the mid-gap electronic states, V n > (described in Section 4.5) by inverting eqn (4.33). Thus,... 

Figure 4.11 shows the probability density of the Wannier orbitals associated with the mid-gap states. Although the relative separation of Wannier orbitals is small with an extrinsic dimerization of = 0.1, the fact that there are two... 

For simplicity, however, we prefer to denote all excitons formed from bound states of conduction band electrons and valence band holes as Mott-Wannier excitons, recognizing that this term includes both small and large radius excitons. We call this limit the weak-coupling limit, as the starting point in the construction of the exciton basis is the noninteracting band limit. As we will see, a real space description of a Mott-Wannier exciton is of a hole in a valence band Wannier orbital bound to an electron in a conduction band Wannier orbital. 

Fig. 6.1. The real-space particle-hole excitation, R- -r/2, R—r/2), labelled 1, from the valence band Wannier orbital at R — r/2 to the conduction band valence orbital at R+r/2. Its degenerate counterpart, R—r/2, R+r/2), connected by the particle-hole transformation, is labelled 2. R = (ve + rh)/2 is the centre-of-mass coordinate and r = (re — Vh) is the relative coordinate. A Mott-Wannier exciton is a bound particle-hole pair in this representation.

Figure 4.11 shows the probability density of the Wannier orbitals associated with the mid-gap states. Although the relative separation of Wannier orbitals is small with an extrinsic dimerization of = 0.1, the fact that there are two distinct Wannier orbitals implies that the argument employed in Section 4.6 -concerning the different characters of the 1 B and 1 B+ states after electron-lattice relaxation - is a general one. Thus, the 1 B state is comprised of spinless electron-hole pairs, while the l B state is comprised of two spin-1/2 objects. These become confined in the presence of extrinsic dimerization. We would therefore expect that, as before, the different character of the PB and 1 S+ states will be evident by the different type of geometrical distortions when electron-electron interactions are included. 

---