Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-met allic systems. We further discuss the case of density matrices for met allic systems at positive electronic temperature. A few other possible applications are also discussed.
In @cite_124 , Kohn proved the rapid decay of the Wannier functions for one-dimensional, one-particle Schr "odinger operators with periodic and symmetric potentials with non-intersecting energy bands. This type of Hamiltonian describes one-dimensional, centrosymmetric crystals. Kohn's main result takes the following form: where @math denotes a Wannier function (here @math is the distance from the center of symmetry) and @math is a suitable positive constant. In the same paper (page 820) Kohn also points out that for free electrons (not covered by his theory, which deals only with insulators) the decay is very slow, being like @math .
@cite_124
2100781154 1965675553 1991050432 2074950413
The spatial decay properties of Wannier functions and related quantities have been investigated using analytical and numerical methods. We find that the form of the decay is a power law times an exponential, with a particular power-law exponent that is universal for each kind of quantity. In one dimension we find an exponent of @math for Wannier functions, @math for the density matrix and for energy matrix elements, and @math or @math for different constructions of nonorthonormal Wannier-like functions. The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified. The one-dimensional Schr "odinger equation with a periodic and symmetric potential is considered, under the assumption that the energy bands do not intersect. The Bloch waves, @math , and energy bands, @math , are studied as functions of the complex variable, @math . In the complex plane, they are branches of multivalued analytic and periodic functions, @math , and @math , with branch points, @math , off the real axis. A simple procedure is described for locating the branch points. Application is made to the power series and Fourier series developments of these functions. The analyticity and periodicity of @math has some consequences for the form of the Wannier functions. In particular, it is shown that for each band there exists one and only one Wannier function which is real, symmetric or antisymmetric under an appropriate reflection, and falling off exponentially with distance. The rate of falloff is determined by the distance of the branch points @math from the real axis. We consider the change in polarization P which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for P is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically P can be interpreted as a displacement of the center of charge of the Wannier functions. The formulation is successfully applied to compute the piezoelectric tensor of GaAs in a first-principles pseudopotential calculation.