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2.1 Converge the PW energy cutoff

Version and execution status. Core syntax is checked against the official ABACUS v3.9.0 source documentation and example files, reviewed on 2026-10-05. The release list also contains stable v3.10.1 and 3.11 beta tags; v3.9.0 is this course’s reproducible baseline, not a claim about the newest release. These are unexecuted teaching templates. Initial numerical values are candidates for testing, not certified results. No energies, timings or convergence traces below are measurements.

Schematic independent energy and force convergence checks against cutoff, without measured points

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2.1.1 The question and a property-level tolerance

What cutoff is adequate for the quantity you intend to report? The question is not whether 60 Ry is a common number, but whether changing the basis alters the desired observable by less than a declared tolerance. Begin with the complete PW silicon parent, preserve its cell, LDA UPF and fixed occupations, and use a mesh that has itself been checked. A scan at one undersampled mesh cannot certify the eventual high-accuracy calculation.

Select a numerical error budget before collecting data. For a teaching energy comparison, 1 meV per atom is a possible target, not a result. For structural work, specify a force tolerance; for elasticity, specify stress or energy-curvature tolerances. High symmetry can make a force vanish despite an incomplete basis, so include a slightly distorted test geometry if forces are your eventual observable. Mark that geometry as a separate controlled test, not a new equilibrium structure.

2.1.2 Worked input deltas and run directories

Copy the parent’s complete INPUT, STRU and KPT to each run directory. Change only suffix and ecutwfc in the first scan. The following is an INPUT delta, not a complete file:

suffix si_pw_e80_kchecked
ecutwfc 80

Candidate cutoffs are 40, 60, 80 and 100 Ry. Extend the series if the last values fail the target; the series is not a recommendation for an arbitrary UPF. Retain the baseline electronic threshold scf_thr 1e-9 initially, and test a tighter threshold at selected points. Keep ecutrho at the version-documented default for this source norm-conserving model during the first scan; independently assess density-grid requirements if the property or a replacement pseudopotential demands it. Ultrasoft datasets have different density-cutoff requirements and are not covered by silently substituting files.

Avoid reusing an output directory with a different cutoff. A stale density or log can make a mixed-data table look smooth. Store the original input snapshot alongside each final log, and record which values the executable actually read.

2.1.3 Why cutoff cost and stress behave differently

At fixed volume, the density of reciprocal vectors implies approximately

\[N_{\mathrm{PW}}\propto V E_{\mathrm{cut}}^{3/2}.\]

Raising the cutoff from 60 to 100 Ry therefore changes this estimated basis count by \((100/60)^{3/2}\), roughly 2.15. This is an analytic scaling estimate, not a timing benchmark. FFT dimensions change discretely, eigensolver work depends on occupied states, and communication may dominate at a chosen process count.

For cell changes, a fixed energy cutoff does not keep the exact number of reciprocal basis functions fixed. Waves enter or leave the retained set as the cell changes, and an incomplete basis can produce Pulay-like stress errors. Thus a total-energy plateau at one volume is not automatically a stress plateau. For an equation of state, compare neighboring volumes using the same sufficiently high cutoff and verify that the fitted curvature is stable when it is raised.

One often expects a more complete variational basis to lower energy under a fixed Hamiltonian, but numerical grids, incomplete SCF, finite tolerances and changes in other parameters can obscure monotonicity. Do not “fix” nonmonotonic data by sorting energies or discarding inconvenient points. Investigate provenance and electronic convergence first.

2.1.4 A blank scan table and analysis formula

Cutoff, Ry SCF passed? Total E, eV ΔE/atom to high-cutoff reference, meV Force/stress error Wall time
40 — — — — —
60 — — — — —
80 — — — — —
100 — — — — —

The dashes mean unmeasured values. For this two-atom parent, convert a measured energy difference using

\[\Delta e_i=\frac{1000[E_i-E_{\mathrm{ref}}]}{2}\quad\mathrm{meV/atom}.\]

Do not divide by ntype, which is the number of species, not the number of atoms. Label the highest-cutoff point as a finite reference, not as the infinite-basis answer. If the final two or three values all satisfy the criterion, rerun the chosen production cutoff with the final refined k-mesh to check coupled effects.

Use an independently named CSV with cutoff, atom count, energy and convergence status. Any parser must verify the status before taking the final marker. Inspect one parsed record manually against its log; extracting the first occurrence of an energy keyword can accidentally select an intermediate iteration.

2.1.5 Acceptance and interpretation

Accept a cutoff only when electronic noise is smaller than the target, the last several cutoff differences are small, and the final mesh/cutoff combination is stable. For forces or stress, use their own errors rather than the total energy as a proxy. Archive a plot made from the learner’s actual CSV; distinguish that measured plot from the explanatory reciprocal-space figure in this lesson.

When preparing a later defect or surface energy, validate a representative low-coordination or strained geometry too. A cutoff that is adequate for bulk silicon at one lattice constant may still produce an unacceptable error in an adsorption difference or cell stress. The pseudopotential generator’s recommended cutoff is a starting point; your target property defines the final criterion. Resource limits should be stated honestly if the high-cutoff endpoint cannot be afforded, rather than reporting an unsupported convergence claim.

2.1.6 Pitfalls and exercises

Never confuse Ry with eV: 1 Ry is approximately 13.6057 eV. Do not apply a cutoff accepted for one UPF to a different file without a new scan. Do not include nonconverged points in a convergence fit. Avoid changing density cutoff, k-mesh and orbital model in the same series unless you explicitly design a multidimensional study.

  1. Unit exercise. Express 60 Ry in eV and explain why entering 60 as an eV-based code’s cutoff is not the same numerical basis. Answer guidance: approximately 816.34 eV; conventions and pseudopotentials still differ across codes, so equal energy cutoffs alone do not establish equivalence.
  2. Property exercise. A real future energy scan meets its proposed tolerance, but the stress changes significantly on raising the cutoff. Is the calculation ready for cell relaxation? Answer guidance: no. Set a stress tolerance appropriate to the relaxation and continue the cutoff/grid tests. A scalar energy can look stable while its strain derivative remains noisy.

The next orbital lesson deliberately separates the LCAO basis dimension from this PW/grid convergence logic.

2.1.7 Official references and provenance

These references establish file grammar and available options. The explanations, comparison designs, algebraic exercises and figures are original teaching material; no published example energy is presented as a result of this course. Additional learning resources are collected in the course hub references section. The university-linked 2024 guide is supplementary teaching, not a substitute for the pinned input reference.