The problem of construction of the Wannier functions (WFs) in a restricted Hilbert space of eigenstates of the one-electron Hamiltonian H (forming the so-called low-energy part of the spectrum) can be formulated in several different ways. One possibility is to use the projector-operator techniques, which pick up a set of trial atomic orbitals and project them onto the given Hilbert space. Another possibility is to employ the downfolding method, which eliminates the high-energy part of the spectrum and incorporates all related to it properties into the energy-dependence of an effective Hamiltonian. We show that by modifying the high-energy part of the spectrum of the original Hamiltonian H, which is rather irrelevant to the construction of the WFs in the low-energy part of the spectrum, these two methods can be formulated in an absolutely exact and identical form, so that the main difference between them is reduced to the choice of the trial orbitals. Concerning the latter part of the problem, we argue that an optimal choice for the trial orbitals can be based on the maximization of the site-diagonal part of the density matrix. This idea is illustrated on a simple toy model consisting of only two bands, as well as on a more realistic example of t2g bands in V2 O3. Using the model analysis, we explicitly show that a bad choice of the trial orbitals can be linked to the discontinuity of phase of the Bloch waves in the reciprocal space, which leads to the delocalization of the WFs in the real space. Nevertheless, such a discontinuity does not necessary contribute to the matrix elements of H in the Wannier basis. Similar tendencies are seen in more realistic calculations for the t2g bands in V2 O3, though with some variations caused by the multi-orbital effects. An analogy with the search of the ground state of a many-electron system is also discussed.
|Journal||Physical Review B - Condensed Matter and Materials Physics|
|Publication status||Published - 2007|