A new approach to the construction of first-principles pseudopotentials is described. The method allows transferability to be improved systematically while holding the cutoff radius fixed, even for large cutoff radii. Novel features are that the pseudopotential itself becomes charge-state dependent, the usual norm-conservation constraint does not apply, and a generalized eigenproblem is introduced. The potentials have a separable form well suited for plane-wave solid-state calculations, and show promise for application to first-row and transition-metal systems.

We discuss a method for determining the optimally localized set of generalized Wannier functions associated with a set of Bloch bands in a crystalline solid. By “generalized Wannier functions” we mean a set of localized orthonormal orbitals spanning the same space as the specified set of Bloch bands. Although we minimize a functional that represents the total spread ${\sum }_n〈r^2{〉}_n-〈\mathbf{r}{〉}_n^2$ of the Wannier functions in real space, our method proceeds directly from the Bloch functions as represented on a mesh of $k$ points, and carries out the minimization in a space of unitary matrices $U_{\mathrm{mn}}^{\left(\mathbf{k}\right)}$ describing the rotation among the Bloch bands at each $k$ point. The method is thus suitable for use in connection with conventional electronic-structure codes. The procedure also returns the total electric polarization as well as the location of each Wannier center. Sample results for Si, GaAs, molecular C${}_2$H${}_4$, and LiCl will be presented.

We consider the change in polarization ΔP which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for ΔP is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically ΔP 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.

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized ``Wannier functions'' was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or ``Boys orbitals'' as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

The spontaneous polarization, dynamical Born charges, and piezoelectric constants of the III-V nitrides AlN, GaN, and InN are studied ab initio using the Berry-phase approach to polarization in solids. The piezoelectric constants are found to be up to ten times larger than in conventional III-V and II-VI semiconductor compounds, and comparable to those of ZnO. Further properties at variance with those of conventional III-V compounds are the sign of the piezoelectric constants (positive as in II-VI compounds) and the very large spontaneous polarization.