Direct Generation of Wannier Functions by Downfolding, Polynomial Approximation, and Symmetrical Orthonormalization

Localized, minimal basis sets are useful for numerous purposes in electronic-structure calculations. Such a basis may pick a single band or a complex of bands, say the occupied bands, and may uncover the bonding, enable real-space order-N calculations, provide understanding, or be used to construct low-energy Hubbard Hamiltonians for correlated systems. We shall explain how to generate such basis sets directly by Lowdin downfolding (partitioning) from a large, complete, basis set of energy-independent, highly localized orbitals such as AOs or LMTOs, and subsequent removal of the energy dependence of the downfolded orbitals by the N-ization technique (1,2). When applied to energy-dependent partial waves, rather than AOs or LMTOs, this gives rise to the NMTO method (3-6). An orbital of the minimal basis set is simply the original orbital, dressed by a cloud of those orbitals which have been removed from the original set by downfolding. If the minimal basis set is chosen to span particular bands, symmetrical orthogonormalization yields a set of localized Wannier functions. Wannier functions which are maximally localized in some other sense may be obtained by a subsequent unitary transformation for a local cluster (1,2). For most correlated d- and f-electron systems, no further localization is achieved by the last step (1,2,7). Examples from NMTO calculations will be presented (8-12).