Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110809914508 -0.83 0.80 6.0 3.32s
2 -1.117194838927 -2.19 -1.91 0.80 1.0 1.57s
3 -1.117249776652 -4.26 -2.66 0.80 1.0 19.2ms
4 -1.117251059575 -5.89 -3.33 0.80 1.0 19.0ms
5 -1.117251188349 -6.89 -4.08 0.80 2.0 42.1ms
6 -1.117251190081 -8.76 -5.03 0.80 1.0 19.4ms
7 -1.117251190140 -10.23 -5.47 0.80 2.0 22.0ms
8 -1.117251190142 -11.60 -5.83 0.80 1.0 26.4ms
9 -1.117251190143 -12.07 -6.65 0.80 1.0 20.4ms
10 -1.117251190143 -13.33 -6.98 0.80 2.0 25.0ms
11 -1.117251190143 -14.27 -7.65 0.80 1.0 20.8ms
12 -1.117251190143 + -14.65 -8.18 0.80 2.0 78.7ms
13 -1.117251190143 -14.65 -8.56 0.80 1.0 510ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -8.96 0.80 1.0 3.55s
2 -1.117251190143 + -Inf -9.85 0.80 1.0 25.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 20.2s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997833 -2.46 0.80 1.0 13.5ms
2 -1.117520586052 -5.59 -3.49 0.80 1.0 18.3ms
3 -1.117520612987 -7.57 -4.23 0.80 1.0 74.2ms
4 -1.117520613449 -9.34 -5.14 0.80 1.0 19.3ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.3ms
6 -1.117520613465 -12.15 -6.43 0.80 1.0 19.6ms
7 -1.117520613465 -13.33 -7.41 0.80 1.0 20.0ms
8 -1.117520613465 + -14.88 -8.24 0.80 1.0 37.8ms
9 -1.117520613465 -15.35 -8.92 0.80 1.0 35.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264593 -1.69 0.80 1.0 13.0ms
2 -1.118339043931 -4.04 -2.70 0.80 1.0 19.1ms
3 -1.118340011110 -6.01 -3.44 0.80 1.0 24.9ms
4 -1.118340028551 -7.76 -4.36 0.80 1.0 23.5ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 31.1ms
6 -1.118340029162 -10.57 -5.62 0.80 1.0 19.7ms
7 -1.118340029164 -11.73 -6.68 0.80 1.0 26.1ms
8 -1.118340029164 -13.52 -7.58 0.80 1.0 20.7ms
9 -1.118340029164 + -14.65 -8.12 0.80 1.0 22.8ms
10 -1.118340029164 -14.65 -8.55 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -9.05 0.80 1.0 13.2ms
2 -1.118340029164 -14.75 -10.36 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -11.05 0.80 1.0 13.2ms
2 -1.118340029164 + -15.35 -11.42 0.80 1.0 18.3ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029164 │ │ 0.00297055 │ 2.70s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 13.0ms
2 -1.118346864757 -6.84 -4.11 0.80 1.0 24.0ms
3 -1.118346866234 -8.83 -4.86 0.80 1.0 18.5ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 23.7ms
5 -1.118346866261 -12.08 -6.55 0.80 1.0 19.4ms
6 -1.118346866261 -13.44 -7.04 0.80 1.0 19.6ms
7 -1.118346866261 -14.45 -8.13 0.80 1.0 22.6ms
8 -1.118346866261 -14.75 -8.95 0.80 1.0 22.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 15.2ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.4ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 20.6ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.1ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 21.1ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.7ms
7 -1.118356101386 -13.59 -7.65 0.80 1.0 22.1ms
8 -1.118356101386 -14.26 -8.45 0.80 1.0 20.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 12.8ms
2 -1.118356101386 -14.51 -9.82 0.80 1.0 20.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 12.9ms
2 -1.118356101386 + -15.35 -11.46 0.80 1.0 21.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 521ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 15.9ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.2ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 510ms
4 -1.118356102877 -14.29 -7.62 0.80 1.0 18.7ms
5 -1.118356102877 -14.65 -8.40 0.80 1.0 19.1ms
6 -1.118356102877 + -14.88 -8.89 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 12.8ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.1ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.4ms
4 -1.118356105120 -13.22 -7.09 0.80 1.0 18.7ms
5 -1.118356105120 -14.24 -7.86 0.80 1.0 18.9ms
6 -1.118356105120 + -14.35 -8.35 0.80 1.0 19.2ms
7 -1.118356105120 -14.61 -9.45 0.80 1.0 19.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 14.6ms
2 -1.118356105120 -15.05 -11.02 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 17.0ms
2 -1.118356105120 -15.05 -12.44 0.80 1.0 24.8ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.1795e-8 │ 1.02s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -2.46025e-34, -1.56525e-34]u"a₀")
Atom(H, [ 1.44318, 1.65152e-35, 0]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658617347997 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335338673782 -0.81 0.80 7.3 752ms
2 -8.340904650766 -2.25 -1.69 0.80 1.3 607ms
3 -8.341205447431 -3.52 -2.76 0.80 2.7 35.5ms
4 -8.341214247127 -5.06 -3.32 0.80 3.7 42.4ms
5 -8.341214270391 -7.63 -4.12 0.80 1.3 28.5ms
6 -8.341214272423 -8.69 -5.51 0.80 2.0 31.7ms
7 -8.341214272517 -10.03 -5.72 0.80 4.0 44.6ms
8 -8.341214272518 -11.89 -6.75 0.80 1.0 26.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.48 0.80 1.0 606ms
2 -8.341214272518 -14.05 -7.61 0.80 1.0 26.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.35 0.80 1.0 21.8ms
2 -8.341214272518 -14.75 -8.43 0.80 1.0 24.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350585430828 -1.21 0.80 8.7 64.8ms
2 -8.351864098012 -2.89 -1.88 0.80 1.3 26.2ms
3 -8.351927035488 -4.20 -2.78 0.80 3.3 35.8ms
4 -8.351928496288 -5.84 -3.65 0.80 3.3 37.1ms
5 -8.351928514468 -7.74 -4.66 0.80 3.7 38.4ms
6 -8.351928514757 -9.54 -5.27 0.80 4.3 57.9ms
7 -8.351928514774 -10.77 -6.01 0.80 3.0 35.4ms
8 -8.351928514774 -12.30 -6.51 0.80 3.0 36.2ms
9 -8.351928514774 -13.30 -8.14 0.80 2.7 34.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353386698382 -1.52 0.80 8.0 59.9ms
2 -8.353731325957 -3.46 -2.21 0.80 1.0 24.5ms
3 -8.353756948769 -4.59 -2.99 0.80 4.0 38.6ms
4 -8.353757525071 -6.24 -4.05 0.80 2.7 36.7ms
5 -8.353757531604 -8.18 -5.23 0.80 3.3 37.1ms
6 -8.353757531632 -10.55 -5.94 0.80 3.3 38.1ms
7 -8.353757531632 -12.41 -7.09 0.80 2.3 33.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531632 -7.84 0.80 1.0 21.6ms
2 -8.353757531632 -14.75 -8.22 0.80 1.0 27.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531632 -8.76 0.80 1.0 21.8ms
2 -8.353757531632 -14.75 -9.05 0.80 1.0 24.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531632 -9.43 0.80 1.0 21.8ms
2 -8.353757531632 + -Inf -9.72 0.80 1.0 24.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531632 │ │ 4.34057e-32 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353680217581 -3.21 0.80 8.0 59.2ms
2 -8.353739292811 -4.23 -3.94 0.80 1.0 24.5ms
3 -8.353770531709 -4.51 -3.84 0.80 6.3 53.3ms
4 -8.353770536025 -8.36 -4.10 0.80 1.0 24.3ms
5 -8.353770536790 -9.12 -4.42 0.80 2.0 28.6ms
6 -8.353770537270 -9.32 -5.03 0.80 1.7 27.5ms
7 -8.353770537315 -10.34 -5.78 0.80 2.7 32.9ms
8 -8.353770537317 -11.70 -6.60 0.80 3.3 36.2ms
9 -8.353770537317 -13.50 -6.82 0.80 2.7 41.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353766070898 -2.00 0.80 7.0 56.7ms
2 -8.353873974717 -3.97 -2.70 0.80 1.0 25.0ms
3 -8.353895044663 -4.68 -3.34 0.80 4.3 41.4ms
4 -8.353895104289 -7.22 -4.08 0.80 2.7 32.7ms
5 -8.353895110669 -8.20 -4.91 0.80 3.0 36.3ms
6 -8.353895110762 -10.03 -5.84 0.80 2.0 30.6ms
7 -8.353895110765 -11.51 -6.90 0.80 3.3 36.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110765 -7.97 0.80 1.0 22.1ms
2 -8.353895110765 -14.27 -8.40 0.80 1.0 24.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110765 -8.67 0.80 1.0 21.6ms
2 -8.353895110765 -14.45 -9.12 0.80 1.0 24.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110765 -9.20 0.80 1.0 21.6ms
2 -8.353895110765 + -Inf -9.66 0.80 1.0 29.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110765 │ -3.86 │ │ 0.000646188 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353797151154 -3.49 0.80 8.7 61.6ms
2 -8.353866776454 -4.16 -3.83 0.80 1.0 28.0ms
3 -8.353895211510 -4.55 -3.85 0.80 6.3 53.0ms
4 -8.353895212203 -9.16 -4.03 0.80 1.0 24.8ms
5 -8.353895212376 -9.76 -4.44 0.80 1.3 28.6ms
6 -8.353895212482 -9.98 -4.76 0.80 1.0 24.8ms
7 -8.353895212502 -10.69 -4.97 0.80 1.7 27.9ms
8 -8.353895212512 -11.03 -5.59 0.80 1.7 30.5ms
9 -8.353895212513 -11.77 -6.25 0.80 2.3 31.6ms
10 -8.353895212513 -13.00 -6.37 0.80 3.0 36.0ms
11 -8.353895212513 + -Inf -6.29 0.80 1.0 28.3ms
12 -8.353895212513 -13.75 -7.77 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353784082043 -3.24 0.80 8.3 64.9ms
2 -8.353876698674 -4.03 -3.50 0.80 1.0 24.5ms
3 -8.353895358487 -4.73 -3.88 0.80 6.0 50.4ms
4 -8.353895361880 -8.47 -4.30 0.80 1.3 30.3ms
5 -8.353895362664 -9.11 -5.17 0.80 2.0 29.8ms
6 -8.353895362685 -10.69 -5.76 0.80 2.3 33.3ms
7 -8.353895362686 -11.95 -6.90 0.80 3.3 34.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.79 0.80 1.0 22.0ms
2 -8.353895362686 + -14.75 -8.29 0.80 1.0 24.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.77 0.80 1.0 21.5ms
2 -8.353895362686 + -14.75 -9.08 0.80 1.0 24.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.39 0.80 1.0 21.7ms
2 -8.353895362686 + -14.75 -9.70 0.80 1.0 24.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.72153e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [6.47126e-39 5.27783 5.27783;
5.27783 -4.79301e-36 5.27783;
5.27783 5.27783 -2.39715e-35]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.555662901043988 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.