Validate the quantity a band calculation is meant to predict

Start with the Hamiltonian or energy functional, nuclear geometry, electron count, spin configuration, and symmetry constraints. Changing any of these can change the physical question. Numerical controls then approximate the chosen question: basis size or cutoff, wavevector integration, self-consistency tolerance, and interpolation resolution. Compare convergence sequences while holding the physical choices fixed. If a structure relaxes differently at each setting, its changes can mask the numerical trend in the electronic observable.

Convergence of one observable does not imply convergence of every derivative or small energy difference. A coarse path can reproduce broad bandwidths but miss a tiny pocket between path points. A fitted tight-binding model can match energies while misrepresenting curvature or eigenvector geometry. Independent implementations of the same functional test reproducibility and numerical precision; agreement between them does not by itself establish that the functional predicts the experimental excitation spectrum accurately.

The fundamental independent-band gap is the lowest conduction energy anywhere in the zone minus the highest valence energy anywhere in the zone. A direct gap compares the two at the same wavevector. The smallest direct gap can exceed the fundamental gap when the extrema occur at different momenta. A high-symmetry path is a useful diagnostic but cannot guarantee that it contains either extremum; full-zone searches or a justified symmetry argument are needed.

A calculated Kohn–Sham gap, a quasiparticle gap, and an optical onset are different observables. Geometry, temperature, electron-phonon coupling, excitonic effects, and many-body self-energy can affect comparisons. Report which quantity is actually computed before judging disagreement. For metals, verify total occupations, spin channels, smearing dependence, and Fermi-surface convergence rather than using a smooth density-of-states curve as a substitute for an electron-count audit.

A useful validation record includes the target observable and tolerance, convergence changes, total filling, relevant symmetry or degeneracy checks, and the physical model used for comparison. Distinguish a theorem such as Bloch organization from a controlled weak-potential approximation or an empirical parameter fit. Disagreement may expose a numerical failure, a bad physical approximation, or comparison with a different observable; another digit of printed eigenvalue precision does not identify which explanation applies.

The root-finding lab demonstrates a genuine bound for a much simpler problem. Reset and set target c to 2 and upper bound U to 2, then Step once; predict bracket error bound 0.5. Reset and set c to 2 and U to 4, then Step once; predict bound 1. Its midpoint bounds belong only to bisection of x²−c on a known bracket. The instrument does not certify density-functional errors, band interpolation, or an assumed convergence exponent.