6. Brillouin-zone integration and occupations
Position in the course: Lesson 6 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A periodic calculation replaces a continuum of crystal momenta by a weighted quadrature. A band plot along selected symmetry lines is useful for visualization but is not the full Brillouin-zone integral. Total energies and densities require a volume mesh. State explicitly whether spin degeneracy is included in occupations or summed as separate channels; otherwise electron counts can silently differ by two.
Insulators have smoothly varying occupied subspaces away from crossings and can often be integrated with moderate meshes. Metals have a Fermi surface: the zero-temperature occupation jumps as an energy crosses the chemical potential. A coarse mesh can then change electron counts and forces abruptly. Smearing replaces this discontinuity by a smooth weight, improving numerical integration and sometimes SCF behavior, but introduces a width dependence that must be checked together with the k mesh.
Fermi–Dirac occupations arise by minimizing a finite-temperature free energy with respect to f under fixed electron count. Differentiate E−TS−μN: the entropy derivative gives kBT ln[f/(1−f)], so ε−μ+kBT ln[f/(1−f)]=0. Rearranging yields the logistic form above. This derivation gives a thermodynamic interpretation for Fermi–Dirac smearing; other numerical smearings need their own energy corrections and should not automatically be assigned a physical temperature.
For a zero-temperature target, converge the mesh and reduce the width while tracking the intended energy and force convention. For a finite-electronic-temperature target, report the free-energy quantity and temperature. Ionic temperature is a separate variable; warm electronic occupations do not imply that nuclei have sampled a hot ensemble. Surfaces need appropriate in-plane sampling and vacuum; a sparse normal-direction mesh is justified by geometry, not a universal magic setting.
2. Derivation step by step
Read each equality with its assumptions. Atomic units are used for DFT equations unless another unit is stated; TB parameters retain explicit energy and length units. The conjugate transpose is denoted by a dagger, and a prime on a coordinate denotes a separate integration variable.
2.1. Weights and chemical potential are coupled
At fixed temperature, determine μ so that the weighted occupations give the chosen N. Moving μ independently after occupation generation changes electron count. Symmetry reduction combines equivalent k points into larger weights; forgetting those weights is equivalent to integrating a different zone. At band crossings, individual eigenvectors may change abruptly even when the full occupied subspace is smooth. Density construction must respect this subspace and occupations. For a molecule in a sufficiently isolated supercell, a gamma-only calculation can be appropriate, but this is a boundary approximation requiring cell-size checks rather than proof that reciprocal-space sampling is irrelevant.
3. Worked example
At ε=μ, f=1/2 at any nonzero temperature. At ε−μ=kBT ln 3, the exponential equals three and f=1/4. For one weighted state with f=1/2, the entropy is kB ln 2 before any spin multiplier. This provides a useful implementation check.
4. Exercises with explained solutions
Exercise. Derive df/dε and explain why the integration difficulty narrows as T decreases.
Explained solution. Differentiating gives df/dε=−f(1−f)/(kBT). Its largest magnitude is 1/(4kBT) at the chemical potential. The transition becomes sharper as temperature falls, so a mesh adequate at large width may fail at small width.
Further check. State the units and the allowed regime for every parameter in the worked example. Change one assumption and identify which derivation step must be revisited. A correct explanation names the affected constraint, operator, or boundary condition rather than merely saying that the answer changes.
5. Misconceptions and limitations
Occupation smearing cannot repair a poor band structure or remove functional gap error. High-symmetry-path agreement does not prove volume-integral convergence.
The illustration is an original teaching schematic. It is not output from a numerical materials simulation.
6. Connections and sources
Related: localized-basis theory · Molecular electronic structure
ABACUS · VASP · Quantum ESPRESSO · CP2K
- Hohenberg–Kohn, ground-state density theorem (1964)
- Kohn–Sham, self-consistent orbital equations (1965)
- PBE, constrained GGA construction (1996)
The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.
Original analytic teaching diagram under the lesson assumptions; no simulation results.
7. Related theory and practice
Quantum mechanics · Molecular methods · Tight binding · Molecular dynamics · VASP · Quantum-Espresso · CP2K · ABACUS