Cost of the minimal sets¶
The number of strain patterns in the minimal set equals the number of lattice parameters that the symmetry
leaves free, and no stress-strain scheme can use fewer
(proof sketch). The usual approach, used by VASP's
IBRION = 6 and by most workflows, applies the six single-component strains. Both apply every pattern at
the same step sizes: four by default in minelastic (\(\pm\delta\), \(\pm 2\delta\), as VASP's NFREE = 4) and
two with --stencil central (as NFREE = 2). The saving is therefore the ratio of the pattern counts.
| elastic type | strain patterns | single points (four-point / central) | six single strains (four-point / central) | fewer by |
|---|---|---|---|---|
| triclinic | 6 | 24 / 12 | 24 / 12 | 1 |
| monoclinic | 4 | 16 / 8 | 24 / 12 | 1.5 |
| orthorhombic | 3 | 12 / 6 | 24 / 12 | 2 |
| tetragonal I and II | 2 | 8 / 4 | 24 / 12 | 3 |
| trigonal I and II | 2 | 8 / 4 | 24 / 12 | 3 |
| hexagonal | 2 | 8 / 4 | 24 / 12 | 3 |
| cubic | 1 | 4 / 2 | 24 / 12 | 6 |
Most materials of interest are in the last four rows. The paper also compares the minimal sets with earlier tabulated schemes, which need more configurations for several symmetry types.
What each quantity costs¶
Everything that minelastic computes is built on the same strained cells, so every quantity inherits the factor above. \(n\) is the number of patterns, \(p\) the number of steps per pattern (4 or 2) and \(D\) the number of symmetry-irreducible atomic displacements.
| quantity | minimal set | six single strains |
|---|---|---|
| clamped-ion constants | \(n p\) single points | \(6 p\) single points |
| internal-strain tensor | no extra cost: the forces of the same cells | \(D p\) displaced-atom single points (VASP) |
| clamped-ion piezoelectric tensor \(e^0\) | no extra cost: Berry phase in the same cells | a DFPT run, or \(6 p\) Berry-phase cells |
| relaxed-ion constants by relaxing the ions | \(n p\) relaxations | \(6 p\) relaxations |
relaxed-ion constants, Born charges, total \(e\) and \(d\) (--ionic) |
\(n p + D p\) single points | \(6 p + D p\) single points (VASP IBRION = 6) |
| finite-temperature constants | \(n p\) MD runs (\(p = 2\)) | \(6 p\) MD runs |
For wurtzite AlN (\(n = 2\), \(D = 4\)) and two points per strain, --ionic needs 4 + 8 = 12 single points
where VASP's IBRION = 6 needs 12 + 8 = 20. For fcc Al at finite temperature it is 2 MD runs instead of
12, the largest saving in absolute terms, since every cell is a full molecular-dynamics run.
Measured¶
Wall times of the clamped-ion runs of wurtzite AlN in the tutorials (VASP, Quantum ESPRESSO, ABINIT), with the same settings for both sets (two points per strain, one 32-core node):
| code | minimal set (4 cells) | six single strains (12 cells) | ratio |
|---|---|---|---|
| VASP 6.6.1 | 42 s | 163 s | 3.9 |
| Quantum ESPRESSO 7.6 | 9 s | 27 s | 3.0 |
| Quantum ESPRESSO 7.6, relaxed ions | 22 s | 65 s | 3.0 |
| ABINIT 10.6.7 | 373 s | 1220 s | 3.3 |
The constants of the two sets agree to 0.05 to 0.1 GPa (Validation). In VASP the ratio exceeds the ratio of the cell counts because the cells strained along zz keep the hexagonal symmetry (30 irreducible k points against 93 to 102 for the sheared cells, 6 s against 15 to 18 s per cell). They are half of the minimal set but only a sixth of the six-strain set.