2.3 Metal k-points and smearing

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.

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2.3.1 Why a metal needs a two-dimensional scan

A metal has a Fermi surface: small shifts of k-points or eigenvalues can move states across the chemical potential. Coarse sampling with sharp occupations therefore produces noisy Brillouin-zone integrals and difficult SCF behavior. Smearing smooths the occupation transition, but a large width changes the quantity being approximated. This lesson uses a one-atom FCC aluminum cell rather than imposing silicon’s insulating settings on a metal.

The goal is to converge the target energy or stress jointly in k-mesh and smearing width. Faster SCF at a broad width is not evidence that the narrow-width limit is correct. Conversely, reducing the width at one coarse mesh can worsen sampling noise. The mesh and width are coupled controls, so design a small grid of runs rather than one “best” number.

2.3.2 Complete aluminum teaching input

INPUT

INPUT_PARAMETERS
suffix al_pw_k8_s002
calculation scf
basis_type pw
ntype 1
pseudo_dir ../data/pseudo
ecutwfc 60
scf_thr 1e-9
scf_nmax 200
nspin 1
nbands 8
smearing_method gauss
smearing_sigma 0.002
mixing_type broyden
mixing_beta 0.4
ks_solver cg

STRU

ATOMIC_SPECIES
Al 26.98 Al.pz-vbc.UPF

LATTICE_CONSTANT
7.50

LATTICE_VECTORS
0.0 0.5 0.5
0.5 0.0 0.5
0.5 0.5 0.0

ATOMIC_POSITIONS
Direct
Al
0.0
1
0.0 0.0 0.0 0 0 0

KPT

K_POINTS
0
Gamma
8 8 8 0 0 0

Al.pz-vbc.UPF is the actual filename used in the official charge-mixing aluminum example, not an attached file. Obtain that public scientific asset from its documented source, inspect its element/valence/functional header and save its hash. The explicit vectors here replace the official example’s latname fcc shortcut to make geometry transparent. The 7.50-bohr scale is a fixed teaching geometry, not a predicted equilibrium value. No numerical orbital is needed for PW.

2.3.3 Widths, occupations and band capacity

smearing_sigma is an energy width in Ry. The candidate 0.002 Ry is approximately 0.0272 eV; it is not 0.002 eV. The baseline supports gauss, mp, mp2, mv/cold, and fd among its documented occupation choices. Keep one scheme fixed during a convergence series and read that scheme’s definition before comparing its energy terms. Gaussian broadening is not automatically a physical electronic temperature.

For Fermi–Dirac occupations, the energy scale enters

\[f(\epsilon)=\frac{1}{1+\exp[(\epsilon-\mu)/\sigma]},\qquad\sigma=k_BT.\]

The formula explains why reducing the width sharpens the Fermi surface. Do not convert every non-Fermi–Dirac scheme into a thermodynamic temperature. When comparing smearing schemes, identify whether the logged energy includes a smearing correction or a free-energy-like contribution; use the same well-defined quantity throughout and verify the narrow-width trend rather than asserting exact equivalence of energy labels.

nbands 8 provides extra states for this small teaching cell, but adequacy must be checked. The upper computed states should have negligibly small occupation at the adopted width. Increasing the width can populate higher states, requiring more bands. A top band with substantial occupation means the calculation truncates the occupation tail; a converged SCF residual cannot fix that physical truncation.

2.3.4 Joint workflow and a blank measurement grid

First converge the cutoff for aluminum’s UPF; silicon’s accepted cutoff cannot be reused as proof. Then test candidate meshes 4, 8 and 12 in each direction at widths 0.010, 0.005 and 0.002 Ry. These values define an initial experiment, not a completed benchmark. Extend the grid if it fails the property tolerance. Keep the cell, UPF, occupation scheme, band count and electronic threshold controlled.

Mesh	σ=0.010 Ry	σ=0.005 Ry	σ=0.002 Ry
4×4×4	unmeasured	unmeasured	unmeasured
8×8×8	unmeasured	unmeasured	unmeasured
12×12×12	unmeasured	unmeasured	unmeasured

For every entry record convergence status, the chosen energy definition, Fermi energy, top-band occupation, stress if requested and time. Compare mesh refinement at fixed width, then width reduction at the fine meshes. Symmetry reduces the number of evaluated points, so record both the intended full mesh and the actual irreducible count. Repeat a selected accepted point from fresh inputs to distinguish reproducibility from an accidental restart dependency.

2.3.5 Acceptance and interpretation of the limit

A useful acceptance rule demands that the fine-mesh target quantity change little on both further mesh refinement and reasonable width reduction, with electronic noise smaller still. For stress-driven relaxation, inspect stress independently. Do not choose the width solely for the shortest iteration count. If a broad width is used to stabilize an initial density, perform the final comparison at a separately validated width and converge again.

Do not infer a physical density of states from an SCF integration grid alone; DOS analysis has its own sampling and broadening choices. Similarly, a smearing-induced fractional occupation does not make a system metallic by itself. Interpret the band crossings and density of states along with the occupation model. Preserve enough bands to resolve the occupied tail and any properties of unoccupied states that will later be discussed.

Archive the two-dimensional table even if only one final parameter pair is published. It explains why the pair was chosen and provides a starting point when cell size, strain or magnetic state changes.

2.3.6 Pitfalls and exercises

A common failure is setting fixed occupations for a metal because the silicon template used it. Another is shrinking the width until SCF fails and then blaming only charge mixing. Check whether the mesh resolves the narrow occupation transition and whether enough bands are available. Always distinguish width in Ry from reported energy in eV.

Temperature exercise. For the fd scheme only, convert 0.002 Ry to an approximate temperature using \(k_B=8.6173\times10^{-5}\) eV/K. Answer guidance: the energy is about 0.0272 eV, so the corresponding scale is about 316 K. This is an analytic conversion, not evidence the ionic system is at that temperature; Gaussian smearing does not share the same thermodynamic interpretation.
Convergence exercise. The energy changes strongly between 8 and 12 meshes at the smallest width but weakly at the largest. What should you do? Answer guidance: refine the mesh at the small width and assess its limit; the broad width can mask Fermi-surface sampling errors. Do not declare convergence from the broad-width row alone.

If the underlying metal calculation still oscillates, use SCF mixing diagnostics without abandoning this sampling audit.

2.3.7 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.