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Procedure: design a convergence study

A convergence study establishes how numerical choices affect a reported physical quantity. Define the quantity and tolerance first, then vary a controlled parameter while preserving all other inputs.

Complete guide

Define a comparison that answers the paper

For a phase-energy comparison, converge the energy difference using comparable cells and normalization. For relaxation, inspect force and stress stability. For a band edge, transport coefficient, or phonon mode, converge that observable directly.

A stable total energy does not imply every response property has converged. Keep the target tolerance in your project description or method notes.

Run a controlled series

  1. Duplicate a reference setup and give each variant a name containing the varied parameter and unit.
  2. Vary ecutwfc with a compatible ecutrho. Keep geometry, pseudopotentials, k mesh, spin, occupations, and SCF threshold fixed.
  3. At a sufficiently high cutoff, vary the uniform k mesh. Inspect smearing sensitivity independently for metals.
  4. For low-dimensional systems, vary actual vacuum separation without changing the slab geometry or in-plane cell.
  5. Run the necessary combinations near the chosen settings to check that separately converged choices remain sufficient together.
  6. Retain every run and calculate the difference from the most resolved reference. Export the observations as a numerical table for plotting.

Record a minimal results table

Replace placeholders with observed values; the table is a recording template, not simulated data. For the same composition and cell, delta per atom = (E - E_reference) / atom_count. Convert Ry to the unit reported in the paper consistently.

Do not subtract raw total energies of unrelated compositions as if they were a formation energy. Such comparisons require defined reference chemical potentials and stoichiometry.

Example / commands
cutoff_Ry,k_mesh,energy_Ry,atoms,delta_meV_per_atom,run_id
40,4x4x4,<measured>,2,<derived>,<run-id>
50,4x4x4,<measured>,2,<derived>,<run-id>
60,4x4x4,<measured>,2,<derived>,<run-id>

Choose a stopping rule

ObservableInspectFrequent mistake
Energy differenceChange between successively more resolved calculationsChoosing a universal threshold without regard to the effect being studied.
Forces and stressLargest relevant component and equilibrium geometryRelaxing at low cutoff, then assuming a high-cutoff single point fixes the geometry.
Band extremak sampling and the actual location of extremaAssuming the band gap always lies on the plotted high-symmetry path.
PhononsSCF/DFPT thresholds, k/q meshes, acoustic behaviorRemoving negative modes cosmetically before understanding them.
TransportDense electronic sampling, interpolation, chemical potential and temperatureTreating an interpolated smooth curve as evidence of convergence.

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