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A method is given for the numerical calculation of energy surface integrals within the Brillouin zone like density of states, conductivity, susceptibility, dielectric function etc. The Brillouin zone is divided into tetrahedrons in which the integrand is interpolated linearly. I n this way the integration can be done analytically avoiding the histogram method. Several similar methods are discussed with regard to the quotient of accuracy and effort.Es wird eine Methode zur numerischen Berechnung von Integralen iiber Energieflachen in der Brillouin-Zone wie Zustandsdichte, Leitfihigkeit, Suszeptibilitat, dielektrische Funktion u. a. angegeben. Die Brillouin-Zone wird in Tetraeder eingeteilt, in denen der Integrand linear interpoliert wird. Dadurch kann die Integration unter Vermeidung der Histogramm-Methode analytisch durchgefuhrt werden. Mehrere ahnliche Methoden werden hinsichtlich des Verhaltnisses von Genauigkeit und Aufwand diskutiert.

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We consider a model, given by two interacting electrons in an external harmonic potential, that can be solved analytically for a discrete and infinite set of values of the spring constant. The knowledge of the exact electronic density allows us to construct the exact exchange-correlation potential and exchange-correlation energy by inverting the Kohn-Sham equation. The exact exchange-correlation potential and energy are compared with the corresponding quantities, obtained for the same densities, using approximate density functionals, namely the local density approximation and several generalized gradient approximations. We consider two values of the spring constant in order to study the system in the low correlation case (high value of the spring constant) and in the high correlation case (low value of the spring constant). In both cases, the exchange-correlation potentials corresponding to approximate density functionals differ from the exact one over the entire spatial range. The approximate correlation potentials bear no resemblance to the exact ones. The exchange energy for generalized gradient approximation functionals is much improved compared to the result obtained within the local density approximation but the correlation energy is only a little improved.1290

A comprehensive density-functional theory (DFT)-based investigation of rhombohedral (ABC)-type graphene stacks with finite and infinite layer numbers and zero or finite static electric fields applied perpendicular to the surface is presented. Electronic band structures and field-induced charge densities are critically compared with related literature data including tight-binding and DFT approaches as well as with our own results on (AB) stacks. It is found that the undoped AB bilayer has a tiny Fermi line consisting of one electron pocket around the K point and one hole pocket on the line K-. In contrast to (AB) stacks, the breaking of translational symmetry by the surface of finite (ABC) stacks produces a gap in the bulklike states for slabs up to a yet unknown critical thickness N semimet 10, while ideal (ABC) bulk (β graphite) is a semimetal. Unlike in (AB) stacks, the ground state of (ABC) stacks is shown to be topologically nontrivial in the absence of an external electric field. Consequently, surface states crossing the Fermi level must unavoidably exist in the case of (ABC)-type stacking, which is not the case in (AB)-type stacks. These surface states in conjunction with the mentioned gap in the bulklike states have two major implications. First, electronic transport parallel to the slab is confined to a surface region up to the critical layer number N semimet . Related implications are expected for stacking domain walls and grain boundaries. Second, the electronic properties of (ABC) stacks are highly tunable by an external electric field. In particular, the dielectric response is found to be strongly nonlinear and can, e.g., be used to discriminate slabs with different layer numbers. Thus, (ABC) stacks rather than (AB) stacks with more than two layers should be of potential interest for applications relying on the tunability by an electric field.