Macroscopic polarization as a discrete Berry phase of the Hartree-Fock wave function: The single-point limit

When describing a crystalline system using periodic boundary conditions, the Bloch vector assumes discrete values on a regular mesh in the first Brillouin zone. This mesh is used here in a fundamental way to express the dielectric polarization as a discrete Berry phase of the single-determinant wave function. The present discrete formulation can be used to recover the results of the well-established continuum theory [R. Resta, Rev. Mod. Phys. 66. 899 (1994)], but also provides some other findings. In particular, when the crystal cell is taken as very large (supercell), the Brillouin zone is very small and a single reciprocal point is enough to provide the Berry phase, which measures the macroscopic polarization. The properties of this highly unusual single-paint discrete Berry phase are thoroughly investigated. As a simple test case, we provide the calculation of the Hartree-Fock dynamical charge of MgO: the result is compared to experiment and to previous calculations within density-functional theory. [S0163-1829(98)06524-2].