Validation¶
pytest tests runs the test suite in about 20 seconds. Nearly all of it runs on data shipped in tests/,
so no DFT code is needed.
Algorithm¶
- The paper's strain sets and extraction formulas are reproduced for all nine elastic types. Their minimality is confirmed by exhaustive search.
- Random symmetry-adapted tensors are recovered exactly from synthetic stresses with nonlinear terms. This holds in standard and random orientations, for every extraction method, with the central and four-point formulas.
- With ASE's EMT potential, the minimal and six-strain sets agree for clamped ions, relaxed ions and the internal-strain tensor.
- The symmetry completion of the internal-strain, piezoelectric and force-constant tensors is exact. The completed EMT force constants equal those from displacing every atom (1e-8 eV/Å\(^2\)).
- The internal-strain tensor of hcp Cu (EMT) from the two patterns of the minimal set equals brute force
(all six single strains, all atoms) to 4e-5 eV/Å, and the tensor from the displaced atoms to 4e-6 eV/Å.
The predicted relaxation per unit strain equals that of relaxing the atoms (
examples/emt_internal_strain.py). - For alpha-quartz with the BKS rigid-ion model, the relaxed-ion constants and the piezoelectric tensor from
clamped single points (
ionic=True) equal those from relaxing the ions in the strained cells to 0.1 GPa and 2e-4 C/m\(^2\), and the Born charges equal the model charges (examples/bks_quartz_piezo.py). - The VASP and Quantum ESPRESSO writers and readers are tested. The pw.x lattice construction was also
checked against pw.x's own output for every
ibrav(1, 2, 3, -3, 4, 5, -5, 6 to 11, -9, 91, 12, -12, 13, -13 and 14, withcelldmand withA, B, C).
Real VASP data¶
From the raw \(\pm\delta\) stresses of IBRION = 6 runs (tests/data), minelastic reproduces the tensors that VASP
printed to 5e-5 GPa. This covers unmodified VASP and patched VASP builds that apply the minimal sets.
The minimal and six-strain sets agree for AlN (0.14 GPa), Se in two orientations (up to 0.07 GPa) and Al
(1.05 GPa with a 31x31x31 k-mesh).
The paper's benchmark runs¶
tests/data_paper holds the stresses of the runs behind Table IV of the paper, one material per elastic
type, plus CaWO4, Bi and BaTiO3:
| elastic type | material |
|---|---|
| triclinic | albite |
| monoclinic | β-Ga2O3 |
| orthorhombic | TiSi2 |
| tetragonal I | CaMoO4 |
| tetragonal II | rutile TiO2 |
| trigonal I | dolomite |
| trigonal II | α-quartz |
| hexagonal | Mg |
| cubic | Al |
Each material has a run with VASP's six strains and a run with the paper's minimal set (a patched VASP). From the raw stresses, minelastic reproduces the clamped-ion tensor VASP printed. The six-strain runs agree exactly (below 1e-3 GPa). The minimal runs agree within 0.2 GPa, because the paper's implementation uses a different finite-difference estimator. With VASP's ionic relaxation term added, the results reproduce the published Table IV:
- the eigenvalues agree within 0.2 GPa;
- the individual constants agree within 0.16 GPa where the run frame is the paper's. For α-quartz the agreement is up to the sign of c14, which depends on how the a axes are placed.
The minimal and six-strain sets agree within 0.6 GPa (Al: 1.05 GPa). The β-Ga2O3 runs are sibling runs of the published ones, within about 2%.
Ionic route and piezoelectricity¶
The reference is VASP's own IBRION = 6 with LEPSILON, for wurtzite AlN (tests/data_ionic,
tests/data_berry):
- the ionic contribution to C agrees to 3e-4 GPa (of 77.7);
- with the DFPT Born charges, the ionic piezoelectric term agrees to 1e-5 C/m\(^2\);
- the Berry-phase Born charges agree with DFPT to 0.004 |e|;
- the all-Berry-phase total e and d agree with the DFPT route to 0.006 C/m\(^2\) (e33 = 1.454) and 0.05 pC/N (d33 = 5.30);
- the clamped-ion piezoelectric tensor from the Berry-phase dipoles of the 4 minimal strained cells agrees with DFPT to 0.005 C/m\(^2\) (e33 = -0.424 C/m\(^2\)).
End-to-end runs with real codes¶
Each code below was run on the inputs written by minelastic generate, and minelastic collect (or
minelastic.calculate) read the results. Agreement is with an independent reference computed by the same
code with the same settings.
| code | system | check | agreement |
|---|---|---|---|
| VASP 6.6.1 | wurtzite AlN | clamped ions, 4 vs 12 cells; relaxed ions, 4 relaxations vs --ionic vs VASP's IBRION = 6 |
0.06 GPa; 0.7 GPa |
| VASP 6.6.1 | wurtzite AlN | --ionic with Berry phase: Born charges, e, d vs VASP's IBRION = 6 + LEPSILON |
0.004 |e|, 0.006 C/m\(^2\), 0.05 pC/N |
| VASP 6.6.1 | wurtzite AlN | internal-strain tensor from the forces of the 4 strained cells, and from the stresses of the displaced atoms, vs the tensor printed by VASP's IBRION = 6 (largest entry 11.1 eV/Å) |
0.04 eV/Å; 0.06 eV/Å |
| VASP 6.6.1 | α-quartz (P3_221, 9 atoms) | --ionic with Berry phase vs VASP's IBRION = 6 + LEPSILON: Born charges (not symmetric), C_ion, ionic piezoelectric term (with the DFPT Born charges), clamped-ion C |
0.011 |e|; 0.3 GPa of 230; 0.0002 C/m\(^2\); 2.3 GPa of 310 (steps 0.01 vs 0.015) |
| VASP 6.6.1 | NiO, AFM-II (MAGMOM) |
R-3m in the rhombohedral AFM-II cell: 4 cells vs VASP's IBRION = 6 (12 cells), including c14 |
1.2 GPa (0.4 %) |
| VASP 6.6.1 | NiO, AFM-I, cubic 8-atom cell (MAGMOM) |
only the moments lower Fm-3m to P4/mmm: 4 vs 12 cells | 1.6 GPa |
| VASP 6.6.1 | fcc Al, 32 atoms | KSPACING grid of minelastic vs VASP's own grid |
identical |
| VASP 6.6.1 | fcc Al, 32 atoms, 300 K | generate --md, Langevin MD, collect --md vs an independent average of vasprun.xml (ASE) |
identical |
| Quantum ESPRESSO 7.6 | wurtzite AlN | clamped ions, 2 vs 6 patterns; relaxed ions; internal strain | 0.05 GPa; 0.1 GPa; 0.007 eV/Å |
| Quantum ESPRESSO 7.6 | fcc Al, 32 atoms, 300 K | generate --md, Berendsen MD, collect --md vs ASE's reader of the pw.x output |
identical |
| Quantum ESPRESSO 7.6 | wurtzite AlN | --ionic (12 single points) vs relaxing the ions |
1.1 GPa (0.3 %) |
| Quantum ESPRESSO 7.6 | NiO, AFM-I, cubic 8-atom cell (species labels Ni1/Ni2) | only the moments lower Fm-3m to P4/mmm: 4 vs 12 cells | 0.85 GPa |
| ABINIT 10.6.7 | wurtzite AlN | clamped ions, 4 vs 12 cells | 0.05 GPa |
| ABINIT 10.6.7 | wurtzite AlN | clamped ions and --ionic relaxed ions vs ABINIT's DFPT tensors (anaddb) |
0.6 GPa (0.15 %) |
| ABINIT 10.6.7 | NiO, AFM-I, cubic 8-atom cell (spinat) |
only the moments lower Fm-3m to P4/mmm: 4 vs 12 cells | 1.5 GPa |
| MACE-MPA-0 (ASE) | wurtzite AlN | clamped ions, 2 vs 6 patterns; ionic=True vs relax_ions=True |
0.1 GPa; 0.3 GPa |
| CP2K 2026.2 (ASE) | diamond Si, 64 atoms, \(\Gamma\) point | clamped ions, 1 vs 6 patterns; ionic=True gives c44 = 79.7 GPa against the clamped 102.6 GPa |
0.05 GPa |
The two MD runs check the finite-temperature pipeline only: their k-meshes (3x3x3 and 2x2x2 for 32 atoms)
are far too coarse for a metal, so the constants themselves are not converged; the static lattice at the
same settings gives the same values.
In the NiO AFM-I runs the cell is kept cubic (only its volume is relaxed), so a cubic form would be chosen
if the moments were not read; the tetragonal constants differ by up to 40 GPa (c11 - c33 = -33 GPa in VASP
and pw.x, -40 GPa in ABINIT). The 4- and 12-cell results differ by about the non-hydrostatic residual stress
of the cubic cell (up to 0.9 GPa).
The FHI-aims reader is tested on an FHI-aims output distributed with ASE; FHI-aims itself was not run. The
generic interface is tested with synthetic files only.