Monkhorst

Monkhorst was one of the researchers who, together with H. J. Pack, introduced a practical scheme for sampling the Brillouin zone in periodic solids. The Monkhorst–Pack grid is a systematic method for choosing a finite set of k-points to approximate Brillouin-zone integrals that arise in electronic structure calculations.

The core idea is to construct a uniform grid in reciprocal space by partitioning each reciprocal lattice vector into a specified number of divisions. The resulting set of k-points covers the Brillouin zone with periodicity preserved and can be generated with or without a shift. A gamma-centered variant places grid points symmetrically about the gamma point, while other shifts can optimize convergence for particular systems or properties. The grid is typically specified by three integers, one for each reciprocal lattice direction, which control the density of sampling.

Monkhorst–Pack grids are widely used in density functional theory, Hartree–Fock, and other electronic-structure methods to convert continuous Brillouin-zone integrations into discrete sums. They balance accuracy and computational effort, often enabling faster convergence than random or nonuniform sampling. The grids may be combined with symmetry reduction to evaluate only irreducible k-points, further reducing computational cost without sacrificing accuracy.

The approach has become standard in many plane-wave and localized-basis codes, including popular packages in solid-state physics and materials science. While exceptionally effective for many insulators and semiconductors, additional care—such as smearing techniques or denser grids—may be required for metals to accurately capture Fermi-surface effects. The Monkhorst–Pack framework remains a foundational tool for Brillouin-zone integration.