MonkhorstPack

MonkhorstPack refers to a widely used scheme for generating a uniform grid of k-points in the Brillouin zone for periodic solids. The method is named after its developers, Harold J. Monkhorst and James D. Pack, who introduced it in 1976 to improve the efficiency and accuracy of Brillouin-zone integrations in electronic-structure calculations.

In practice, a Monkhorst-Pack grid is defined by a triad of integers (N1, N2, N3) that specify the number of divisions along the three reciprocal-lattice directions. The grid may also include an optional shift vector, which allows the grid to be centered differently (for example, gamma-centered grids) or to be offset to optimize sampling for metallic or insulating systems. The resulting k-points form a regular, symmetry-adapted set in reciprocal space, and many points are symmetry-equivalent.

A central advantage of the Monkhorst-Pack approach is that only irreducible k-points need to be evaluated directly, with symmetry operations used to generate the rest. This reduces computational cost while maintaining accuracy in calculated properties such as total energy, forces, and electronic structure. The scheme is compatible with plane-wave and pseudopotential methods and is implemented in major first-principles codes.

The method remains a standard tool in solid-state physics and materials science, often complemented by convergence testing with respect to the k-point grid density. While primarily designed for periodic crystals, the basic idea of using a regular, symmetry-aware sampling grid continues to influence k-point generation approaches in electronic-structure calculations.