CHAPTER 05 · BASICS · CORE CONCEPTS

Cutoff and k-point convergence

The standard procedure for converging ecutwfc, ecutrho and the k-grid, with the usual misconceptions.

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This chapter is half of QE. A number that has not passed a convergence test is not a number, and QE will print physically wrong results in a perfectly clean format.

The standard procedure

  1. Converge ecutwfc. Fix a reasonable k-grid and scan the cutoff from 20 to 80 Ry. Judge by the energy change per atom (typically within 1–5 meV/atom), not by the absolute total energy.
  2. Converge ecutrho. For US/PAW, fix ecutwfc and scan from 4x to 12x.
  3. Converge the k-grid. Fix the cutoffs and densify the grid. Metals need far denser grids than insulators.
  4. Converge degauss (metals only). Check the behavior toward the σ → 0 limit (Chapter 06).

Convert Ry to meV/atom when judging:

$$\Delta E\,[\mathrm{meV/atom}] = \frac{|E(n) - E(n_{\max})| \times 13605.7}{N_\mathrm{at}}$$

Typical thresholds:

Goal Criterion
Total-energy differences (phase stability etc.) 1–5 meV/atom
Forces (ML potential training data) ~1 meV/Å per force component (≈ 2×10⁻⁵ Ry/bohr)
Stress (before any vc-relax) ~0.1 kbar

Measured: convergence behavior of PAW silicon

Below are the measured curves from Example E3, using the bundled scripts on silicon (PAW, with ecutrho = 8 × ecutwfc scanned together).

Si convergence: total energy vs ecutwfc and k-grid, force vs ecutwfc
Measured silicon convergence (QE 7.5, PAW). Left: the cutoff scan drops below 1 meV/atom at 40 Ry. Center: the k-grid scan; as an insulator, 6×6×6 is already decent. Right: force convergence on a distorted structure, which lags the energy.

Common misconceptions

  • Absolute total energies have a different zero for every pseudopotential, so comparing them is meaningless. Only differences computed under identical conditions matter.
  • A converged energy does not imply converged forces, stress, or DOS. Verify convergence for the property you actually care about (an energy difference? forces? a band gap? a magnetic moment?). For ML training data, converge on forces.
  • Monotonic convergence in the cutoff is guaranteed by the variational principle; monotonic convergence in the k-grid is not. Non-monotonic k-point behavior is normal.
Common mistakes

Running one convergence test and declaring "this element needs 40 Ry" forever. Cutoff requirements attach to the pseudopotential file, not to the element. New potential, new test. And if a project mixes very different structures (bulk, surface, molecule), standardize on the settings demanded by the most demanding one.

Practical sense for k-grids

  • In K_POINTS (automatic), the six numbers are the Monkhorst-Pack grid nk1 nk2 nk3 and shifts s1 s2 s3. A shift of 1 (half-step offset) can speed convergence for insulators at equal density. However, any nscf that will use the tetrahedron method must have an unshifted, Γ-centered grid.
  • Scale the grid inversely with cell size. Large cells (slabs, supercells) need fewer points; one point suffices along a vacuum direction.
  • Metals need dense grids to resolve the Fermi surface. See the 12×12×12 in Example E5 and 16×16×16 in Example E9.