Precise density-functional method for periodic structures

A density-functional method for calculations on periodic systems (periodicity in one, two, or three dimensions) is presented in which all aspects of numerical precision are efficiently controlled. Highly accurate and rapidly converging strategies have been implemented for (a) the computation of Hamiltonian matrix elements (by a numerical integration method based on a partitioning of space and application of product Gauss rules), (b) the approximation of integrals over the Brillouin zone (by the quadratic tetrahedron method), (c) the evaluation and processing of the Coulomb potential (via a density-fitting procedure), and (d) the expansion of one-particle states in suitable basis functions (numerical atomic orbitals, Slater-type exponential functions, and plane waves). Absolute precision and convergence are demonstrated for all these aspects and show that the method is a well-suited tool for unambiguous investigations of the density-functional approximation itself. Attention is given, in particular, to basis-set questions. Although the method is of the mixed-basis type, it is demonstrated that plane waves are not necessary; this holds for metals as well as for insulators and semiconductors. By a general prescription, sequences of accurate linear-combination-of-atomic-orbital (LCAO) basis sets can be defined that systematically approach the basis-set limit. This enables the routine application of the inherently efficient LCAO method to all kinds of systems. Exemplary calculations are performed on bulk Si-, g-C (graphite), Na, Ni, Cu, and NaCl, and on a hexagonal monolayer of weakly interacting ${\mathrm{O}}_{2}$ molecules.