Wannier function completeness

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

The complete substitution reaction in water was calculated by Pagliai et al. in 2003 [189]. The authors investigated hydrogen bonds from Wannier functions and the electron localization function (ELF) during the reaction. They found the charge at the transition states to be delocalized and, as a result, weakened and shorter lived hydrogens bonds. Similar results were obtained in other investigations [188, 190]. 

The Wannier functions obtained in Eq. 6.25 are not unique, because it is possible to mix the Bloch states of different band numbers by a unitary matrix The resulting Wannier functions are also a complete representation of the electronic structure, although their localisation features are different ... 

The function corresponding to the upper band (dashed curve) is determined up to a linear combination of the three functions transforming via irrep. One can see that both Wannier functions are almost completely localized around one of the oxygen atoms, which confirms the ionic character of MgO compound. 

It is well known that the Wannier functions Wi(R) vanish exponentially as R —> 00 in crystals with completely filled bands. Since the vectors r and r he in the reference (zeroth) primitive unit cell, the products of the Wannier functions on the right-hand side of (4.100) fall off exponentially with increasing Therefore, we may expect the total lattice sum in (4.100) for the offdiagonal elements Pr,r Rn) of the DM to also vanish exponentially with increasing It shonld be noted that in metals, the DM decays according to a power law. 

There are basically two possible approaches to computing. In the case of insulators or semiconductors with a noncrossing completely filled valence band and empty conduction band, one can always use not the delocalized Bloch functions but localized Wannier functions (for their construction see Section 5.1). On the other hand, in the case of metallic polymers like (SN) with a partially filled valence band, one cannot construct Wannier functions and so one must resort to other methods (see Section 5.3). 

Therefore the system of Wannier functions possesses the completeness property in the same space in which the system of Bloch functions is complete. 

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... 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

In the cases of completely filled or empty bands (semiconductors or insulators), the numerical work can be reduced if one does not substitute the Bloch functions in integral (5.37) in their LCAO form [see equations (5.20) and (5.21)], but replaces the AOs by Wannier fiinctions (see Section 5.1). In this way matrix elements (5.37) retain their k-depen-dence (which is necessary for the correct treatment of the translational symmetry), but this procedure does not involve any special difficulties. By writing the Bloch functions in the form< )... 

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