Special points for Brillouin-zone integrations¶
Why this mattered¶
Before Monkhorst and Pack, Brillouin-zone integration was a practical bottleneck in electronic-structure calculations: accurate band energies, charge densities, and response properties required sampling periodic functions over reciprocal space, but naive sampling was expensive and inconsistent. Their 1976 paper supplied a systematic way to generate symmetry-compatible “special points” that made these integrals converge efficiently. The key shift was not merely a better quadrature rule; it turned k-point sampling into a reproducible, scalable numerical infrastructure for crystalline solids.
That mattered because modern first-principles materials science depends on doing many such integrations reliably. Monkhorst-Pack meshes became a standard component of density-functional theory workflows, especially in plane-wave and pseudopotential calculations, where total energies, forces, stresses, band structures, and densities of states all depend on Brillouin-zone sampling. The method helped make solid-state calculations less artisanal: instead of hand-selecting points for each crystal, researchers could specify a grid and systematically improve convergence.
Its influence is visible in later breakthroughs that required routine, comparable calculations across many materials: ab initio molecular dynamics, computational phase diagrams, defect and surface calculations, and eventually high-throughput materials databases. The paper did not create density-functional theory or modern materials informatics, but it supplied one of the numerical conventions that allowed those fields to become automated and cumulative. In that sense, Monkhorst-Pack sampling became part of the hidden standard machinery of computational condensed-matter physics: rarely the headline result, but essential to making the headline results trustworthy.
Abstract¶
A method is given for generating sets of special points in the Brillouin zone which provides an efficient means of integrating periodic functions of the wave vector. The integration can be over the entire Brillouin zone or over specified portions thereof. This method also has applications in spectral and density-of-state calculations. The relationships to the Chadi-Cohen and Gilat-Raubenheimer methods are indicated.
Related¶
- enables → Phosphorene: An Unexplored 2D Semiconductor with a High Hole Mobility — Monkhorst-Pack special k-points enabled accurate Brillouin-zone sampling in the density-functional calculations of phosphorene.
- cite ← Phosphorene: An Unexplored 2D Semiconductor with a High Hole Mobility — The phosphorene calculations use Monkhorst-Pack special k-points to sample the Brillouin zone in density-functional simulations.