Convergence and Validation¶
flowchart TD
START["Pick observable O<br/>and target tolerance ε"]
P{"For each<br/>numerical knob p<br/>(E_cut, k-grid, supercell,<br/>Δt, cutoff, …)"}
SWEEP["Sweep p, holding<br/>others fixed"]
PLOT["Plot |O(p) − O(p_max)|<br/>vs p"]
Q{"Below ε ?"}
OK["Lock in p"]
UP["Increase p<br/>(or change method)"]
DONE["All knobs locked<br/>→ production run"]
START --> P --> SWEEP --> PLOT --> Q
Q -->|"yes"| OK
Q -->|"no"| UP --> SWEEP
OK --> P
P -.->|"all done"| DONEThere is exactly one inviolable rule of computational materials science: every result must be converged with respect to every parameter that could affect it. No exceptions. No "the literature uses X so it must be fine". No "we are time-pressed". An unconverged result is not a result; it is a wrong number that happens to be in roughly the right ballpark, and the difference between "right number" and "wrong number near the right number" is the difference between research and noise.
This section is about how to demonstrate convergence on your system, and how to validate against external benchmarks before claiming any new result.
What convergence means¶
A computed property \(P\) depends on a list of numerical parameters \(\theta_1, \theta_2, \ldots\) Convergence means: as each parameter is taken toward its limiting value (typically: large cutoff energy, dense k-grid, large supercell, small timestep, long MD run), the property \(P\) approaches a limit, and the difference from that limit is smaller than the precision you need.
In equations: for each parameter \(\theta_i\), there exists a value \(\theta_i^*\) such that
for all \(\theta_i \geq \theta_i^*\) (assuming "larger is more converged"), and \(\epsilon\) is the precision you have committed to.
The numerical practice is to compute \(P\) at several values of each parameter, plot \(P\) versus the parameter, and visually identify the plateau region where \(P\) no longer changes.
A convergence checklist¶
For each major method, the parameters that must be tested:
DFT (plane-wave codes)¶
- Cutoff energy \(E_\mathrm{cut}\): the plane-wave basis size. Test a range like 200, 300, 400, 500 eV. The property of interest should plateau before you commit to a working value.
- K-point grid: density of \(\mathbf{k}\)-point sampling. Test 2×2×2, 4×4×4, 6×6×6, 8×8×8, etc., scaled appropriately for cell shape.
- Smearing (for metals): smearing scheme (Gaussian, Methfessel-Paxton, Fermi-Dirac) and smearing width. Different schemes have different best-practice widths; check the manual.
- Pseudopotential / PAW dataset: less commonly varied, but if you have access to two compatible datasets, compare. Some elements have notoriously inconsistent published pseudopotentials.
- Exchange-correlation functional: PBE vs. PBEsol vs. SCAN vs. hybrid. Test on a known property to choose, then report which you chose and why.
- Spin polarisation: if your system has unpaired electrons, you must allow spin-polarised solutions and converge them properly. Do not assume the closed-shell solution is the ground state.
DFT (localised basis codes)¶
Same as above, except:
- Basis set: instead of cutoff energy, specify the basis (e.g. 6-31G*, cc-pVDZ, def2-TZVPP). Convergence is tested by going to a larger basis.
Molecular dynamics¶
- Timestep: small enough that the fastest motion (typically a C-H vibration if present, or any fast bond) is well-resolved. Standard rule: at least 20 timesteps per period of the fastest mode. Test by running with different timesteps and checking energy conservation in NVE.
- Equilibration time: how long does the system need to "forget" its initial condition? Test by plotting an observable (temperature, density, RDF) as a function of MD time; the equilibration period is before the observable plateaus.
- Production time: long enough that statistical uncertainty in the property of interest is below your target precision. The number of independent samples scales as \(t_\mathrm{prod} / \tau_\mathrm{corr}\) where \(\tau_\mathrm{corr}\) is the autocorrelation time of your observable. Block averaging helps you estimate the uncertainty honestly.
- Thermostat / barostat coupling: too tight gives spurious dynamics; too loose gives bad temperature control. Test by checking that your derived observables do not depend on the coupling parameter within reasonable ranges.
- System size: defects and finite-size effects can dominate small cells. Test the property at 2-3 different system sizes.
Defect calculations¶
- Supercell size: a defect in a 64-atom supercell is rarely converged; 128 or 216 atoms is a more common target, and for charged defects you may need 432 atoms or more plus image-charge corrections.
- Image-charge correction: for charged defects under periodic boundary conditions, the Makov-Payne or Freysoldt corrections are essential.
- Chemical potentials: for formation energies, the chemical potential of each species must be defined in a way consistent with your thermodynamic boundary conditions.
MLIP training¶
- Training data coverage: does your training set span the configurations your production runs will visit? An MLIP that has never seen a low-energy structure is unreliable on that structure.
- Validation curve: plot training and validation loss as a function of training data size and epochs. Both should plateau.
- Out-of-sample validation: at the very minimum, hold out a set of configurations generated differently from the training set (e.g. trained on AIMD, tested on minimum-energy structures), and report the error there.
- Active learning convergence: if you are using AL, the loop should terminate (in principle) when the model's uncertainty on candidate configurations is below threshold. In practice, monitor the rate at which new configurations are being added and stop when it slows.
Monte Carlo¶
- Equilibration: same considerations as MD; the MC chain needs to forget its starting point.
- Statistical convergence: standard error of the mean of your observables, with proper accounting for autocorrelation between MC steps.
Pause and recall
Before reading on, try to answer these from memory:
- What does it mean for a calculation to be "converged" with respect to a numerical parameter?
- Why must you converge the observable you report rather than just the total energy?
- Why is each numerical knob swept independently while the others are held fixed?
If any of these is shaky, re-read the preceding section before continuing.
How to actually run a convergence study¶
The temptation is to do convergence "on the side" — a few quick checks, record the numbers in a lab notebook, move on. This is fine if you are disciplined. In practice, most students do not record the convergence checks carefully and end up redoing them under viva pressure.
A better workflow:
-
Set up a dedicated convergence directory with subdirectories labelled by the parameter being varied. E.g.
convergence/ecut/300/,convergence/ecut/400/, etc. -
Run the same calculation at each parameter value, varying only the parameter of interest.
-
Extract the target property from each. A short Python script (10-30 lines) suffices.
-
Plot the property versus the parameter as a figure.
-
Include the figure in your thesis (probably in an appendix) and reference it from the methods section.
This makes the convergence study first-class output, not an afterthought.
A minimal Python sketch for the extraction and plotting:
"""Sketch for a convergence study: plot a target property vs. a
varied parameter, saving the figure for the thesis appendix.
"""
from __future__ import annotations
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
def extract_energy(directory: Path) -> float:
"""Pull the final energy from a single converged DFT run."""
# Pseudocode; adapt to your DFT code's output format.
output = (directory / "OUTPUT").read_text()
for line in reversed(output.splitlines()):
if line.strip().startswith("Final total energy"):
return float(line.split()[-2])
raise RuntimeError(f"No energy in {directory}")
def main() -> None:
base = Path("convergence/ecut")
cutoffs: list[int] = [200, 250, 300, 350, 400, 450, 500]
energies: list[float] = [
extract_energy(base / str(ec)) for ec in cutoffs
]
# Use the largest cutoff as reference.
e_ref = energies[-1]
deltas = [e - e_ref for e in energies]
fig, ax = plt.subplots(figsize=(5, 3.5))
ax.plot(cutoffs, np.array(deltas) * 1000.0, "o-") # convert to meV
ax.axhline(0.0, color="k", lw=0.5)
ax.set_xlabel("Plane-wave cutoff (eV)")
ax.set_ylabel("Energy relative to converged (meV/cell)")
ax.set_title("Convergence with respect to cutoff")
fig.tight_layout()
fig.savefig("convergence_ecut.pdf")
if __name__ == "__main__":
main()
Three habits embedded in this sketch that you should adopt:
- One script per convergence study. Reproducible, version-controlled, re-runnable.
- Plot the difference from a reference, not absolute values. The plateau is easier to see in millielectronvolts than in hartrees.
- Save the figure as PDF for the thesis. Do not screenshot. Do not embed PNGs in the printed copy.
How tight does convergence need to be?¶
This depends on the property and the comparison you intend to make. Rule-of-thumb targets:
| Property | Target convergence | Reason |
|---|---|---|
| Total energy per atom | 1-5 meV/atom | For comparing structures |
| Formation energy of defects | 10-20 meV | Comparison with literature |
| Adsorption energy | 10-30 meV | Catalysis benchmarks |
| Band gap | 0.05-0.1 eV | Comparison with experiment |
| Lattice parameter | 0.005-0.01 Å | Comparison with X-ray |
| Elastic constants | 5% | Comparison with experiment |
| Phonon frequencies | 5-10 cm\(^{-1}\) | Comparison with Raman/IR |
| MD diffusion coefficient | 20% | Order-of-magnitude work |
If you are computing differences (e.g. defect formation energies = bulk energy minus defective-cell energy), some of the absolute energy convergence cancels. You may be able to get away with coarser absolute-energy convergence as long as the difference is converged. But test this explicitly — do not assume.
Validation against external benchmarks¶
Convergence shows that your numerical scheme has reached its limit. Validation shows that the limit it has reached is correct. The two are different.
The minimum standard, for any computational thesis: reproduce a known result for your system before claiming a new one.
Examples of acceptable validation benchmarks:
- For DFT: reproduce a published lattice parameter, formation energy, or band gap for the system at hand. If the published number used a different functional, compute with the same functional and check.
- For MD: reproduce a published density at room temperature, or a published diffusion coefficient.
- For an MLIP: validate against held-out DFT data, and additionally against an experimental observable (lattice parameter at finite T, say) where feasible.
- For high-throughput screening: pick 5-10 known systems and check that your pipeline reproduces their database values.
The validation does not need to be exact. PBE band gaps disagree with experiment by 30-50%; that is a well-known systematic error of the method, not a flaw of your calculation. The point of validation is to catch gross errors — wrong stoichiometry, wrong number of valence electrons, mis-specified pseudopotential, wrong functional input — that would render the work meaningless.
A failed validation is a vital warning signal. Do not push past a failed validation. Find the cause. Most of the time it is a setup error (a typo, a misunderstood parameter); occasionally it is a real sign that the method does not apply to your system, and you need to re-think.
What to put in the thesis¶
Convergence and validation belong in the methods section (briefly) and in the appendix (in full). A reasonable structure:
In the methods section:
Convergence tests were performed for plane-wave cutoff (200-500 eV in steps of 50 eV), Monkhorst-Pack k-grid (from 2×2×2 to 12×12×12), and supercell size (54-432 atoms). The values used in production — \(E_\mathrm{cut} = 450\) eV, \(8 \times 8 \times 8\) k-grid, 128-atom supercell — converge the defect formation energy to within 15 meV. Convergence plots are shown in Appendix A.
In the appendix:
- A figure for each parameter, showing the target property vs. the parameter.
- A short paragraph explaining the choice of working value.
- A reference value (literature or your own benchmark) for comparison.
This is enough to convince any reasonable examiner that the calculations are converged. It is also enough that a future researcher could reproduce your choices.
Negative results matter¶
A common mistake is to treat unsuccessful convergence as a failure to be hidden. It is not. A clear demonstration that a particular calculation does not converge with available resources is a valuable result, and one that an honest thesis includes.
Example: you wanted to compute the formation energy of a charged defect in a 64-atom cell, found that the result depends sensitively on the cell size, and could only converge satisfactorily in a 432-atom cell at the limit of your computational budget. The honest thesis includes both observations: "we tested up to 432 atoms and found that smaller cells give artefacts up to 0.3 eV. The 432-atom result is reported with that caveat."
The dishonest thesis omits the smaller cells and reports only the 432-atom number, leaving the reader to wonder whether convergence has been demonstrated.
The instinct will be toward the dishonest version. Resist. Examiners and reviewers can usually smell omitted convergence checks, and the damage to your credibility is greater than the embarrassment of a slightly messy convergence story.
The 'good enough' trap
A common student rationalisation: "my supervisor said this k-grid is fine, so I do not need to test." Two responses. First, the supervisor's advice was for a different system or a different property; check that it applies to yours. Second, even if it does, you still need to run the test and produce the plot. The thesis examiner does not take your supervisor's word for it; they take your evidence.
A final word: convergence is a habit¶
The skill of running convergence studies is, in the end, a habit. Once you have run two or three on different systems, you will start automatically running them on every new project. The first time you don't run one and then have to face the question "how do you know this is converged?" is the time you internalise the rule.
The five capstone projects (Section 7) each require a convergence section. Use them as practice.
Section 5 collects specific pitfalls beyond convergence that bite undergraduate projects.