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 -1.110529983146 -0.83 0.80 7.0 5.41s
2 -1.117182573581 -2.18 -1.87 0.80 1.0 2.33s
3 -1.117250113827 -4.17 -2.64 0.80 1.0 19.6ms
4 -1.117251120246 -6.00 -3.43 0.80 1.0 69.7ms
5 -1.117251188659 -7.16 -4.44 0.80 1.0 19.9ms
6 -1.117251190128 -8.83 -5.10 0.80 2.0 21.8ms
7 -1.117251190142 -10.84 -5.77 0.80 1.0 31.7ms
8 -1.117251190143 -11.96 -6.67 0.80 1.0 20.9ms
9 -1.117251190143 -13.64 -7.17 0.80 1.0 20.9ms
10 -1.117251190143 -15.35 -7.75 0.80 1.0 23.3ms
11 -1.117251190143 + -14.95 -8.46 0.80 2.0 22.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -8.94 0.80 1.0 4.28s
2 -1.117251190143 -14.75 -10.06 0.80 1.0 18.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 23.5s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997729 -2.46 0.80 1.0 72.0ms
2 -1.117520586046 -5.59 -3.49 0.80 1.0 19.0ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 19.2ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.3ms
5 -1.117520613465 -10.81 -5.92 0.80 1.0 37.9ms
6 -1.117520613465 -12.15 -6.43 0.80 1.0 20.3ms
7 -1.117520613465 -13.37 -7.41 0.80 1.0 20.6ms
8 -1.117520613465 -15.05 -8.24 0.80 1.0 21.1ms
9 -1.117520613465 -15.35 -8.93 0.80 1.0 23.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264597 -1.69 0.80 1.0 13.3ms
2 -1.118339043931 -4.04 -2.70 0.80 1.0 24.6ms
3 -1.118340011110 -6.01 -3.44 0.80 1.0 18.8ms
4 -1.118340028551 -7.76 -4.36 0.80 1.0 19.1ms
5 -1.118340029135 -9.23 -5.12 0.80 1.0 24.7ms
6 -1.118340029162 -10.57 -5.62 0.80 1.0 19.8ms
7 -1.118340029164 -11.73 -6.68 0.80 1.0 84.9ms
8 -1.118340029164 -13.53 -7.58 0.80 1.0 20.7ms
9 -1.118340029164 + -14.95 -8.12 0.80 1.0 534ms
10 -1.118340029164 + -15.65 -8.55 0.80 1.0 21.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -9.05 0.80 1.0 13.1ms
2 -1.118340029164 + -14.75 -10.36 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -11.05 0.80 1.0 13.2ms
2 -1.118340029164 + -14.75 -11.42 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029164 │ │ 0.00297055 │ 3.63s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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 18.5ms
3 -1.118346866234 -8.83 -4.86 0.80 1.0 18.8ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 19.0ms
5 -1.118346866261 -12.09 -6.55 0.80 1.0 19.4ms
6 -1.118346866261 -13.40 -7.04 0.80 1.0 19.6ms
7 -1.118346866261 -15.18 -8.13 0.80 1.0 24.5ms
8 -1.118346866261 -14.95 -8.95 0.80 1.0 33.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.1ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.4ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 18.7ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 18.9ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.4ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.8ms
7 -1.118356101386 -13.60 -7.65 0.80 1.0 26.6ms
8 -1.118356101386 -15.05 -8.45 0.80 1.0 20.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.0ms
2 -1.118356101386 -14.21 -9.82 0.80 1.0 18.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.1ms
2 -1.118356101386 + -15.65 -11.46 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 528ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.3ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.7ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 25.9ms
4 -1.118356102877 -14.26 -7.62 0.80 1.0 19.1ms
5 -1.118356102877 -14.70 -8.39 0.80 1.0 19.5ms
6 -1.118356102877 -14.95 -8.89 0.80 1.0 25.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.2ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.4ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 21.4ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 18.9ms
5 -1.118356105120 -14.95 -7.86 0.80 1.0 19.5ms
6 -1.118356105120 + -14.24 -8.35 0.80 1.0 22.5ms
7 -1.118356105120 -14.18 -9.45 0.80 1.0 20.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 13.2ms
2 -1.118356105120 + -14.54 -11.02 0.80 1.0 21.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.1ms
2 -1.118356105120 -14.81 -12.44 0.80 1.0 21.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 9.96223e-9 │ 476ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -3.0507e-18, 1.37924e-17]u"a₀")
Atom(H, [ 1.44318, -1.39711e-17, 1.08881e-17]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658687038097 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 -8.335338117619 -0.80 0.80 7.3 777ms
2 -8.340978673747 -2.25 -1.70 0.80 1.3 628ms
3 -8.341208638636 -3.64 -2.76 0.80 3.0 33.5ms
4 -8.341214247518 -5.25 -3.33 0.80 3.3 37.8ms
5 -8.341214270311 -7.64 -4.19 0.80 1.3 26.9ms
6 -8.341214272490 -8.66 -5.32 0.80 2.7 52.1ms
7 -8.341214272517 -10.57 -5.68 0.80 3.7 40.2ms
8 -8.341214272518 -11.99 -7.18 0.80 2.0 29.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.60 0.80 1.0 628ms
2 -8.341214272518 -14.45 -7.85 0.80 1.0 24.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.53 0.80 1.0 22.0ms
2 -8.341214272518 -14.75 -8.67 0.80 1.0 24.6ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.6575e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350579107714 -1.21 0.80 8.7 68.6ms
2 -8.351864372241 -2.89 -1.88 0.80 1.3 26.3ms
3 -8.351926541015 -4.21 -2.80 0.80 3.0 72.3ms
4 -8.351928504697 -5.71 -3.58 0.80 3.7 40.8ms
5 -8.351928513903 -8.04 -4.47 0.80 2.7 33.8ms
6 -8.351928514713 -9.09 -5.49 0.80 3.3 38.5ms
7 -8.351928514736 -10.65 -6.30 0.80 4.3 43.4ms
8 -8.351928514736 -13.37 -7.02 0.80 3.0 35.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353386340946 -1.52 0.80 7.7 59.6ms
2 -8.353728595818 -3.47 -2.21 0.80 1.0 24.8ms
3 -8.353757383906 -4.54 -2.98 0.80 3.3 36.9ms
4 -8.353757521592 -6.86 -4.22 0.80 2.3 49.0ms
5 -8.353757531640 -8.00 -5.27 0.80 4.7 44.5ms
6 -8.353757531660 -10.70 -5.88 0.80 3.0 37.1ms
7 -8.353757531660 -12.10 -7.08 0.80 3.0 35.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531660 -7.89 0.80 1.0 22.2ms
2 -8.353757531660 -14.75 -8.46 0.80 1.0 24.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531660 -9.04 0.80 1.0 27.8ms
2 -8.353757531660 + -14.75 -9.26 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531660 -9.65 0.80 1.0 22.2ms
2 -8.353757531660 -14.75 -9.98 0.80 1.0 24.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531660 │ │ 1.00172e-14 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353708351252 -3.19 0.80 8.3 62.0ms
2 -8.353758803863 -4.30 -3.87 0.80 1.0 29.4ms
3 -8.353770532083 -4.93 -3.80 0.80 6.0 54.3ms
4 -8.353770536504 -8.35 -4.11 0.80 1.0 25.3ms
5 -8.353770536906 -9.40 -4.43 0.80 2.0 30.2ms
6 -8.353770537313 -9.39 -5.05 0.80 1.7 31.4ms
7 -8.353770537341 -10.56 -5.51 0.80 2.7 33.5ms
8 -8.353770537343 -11.62 -6.91 0.80 2.3 32.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353730803712 -2.00 0.80 8.7 62.6ms
2 -8.353870463865 -3.85 -2.72 0.80 1.0 24.8ms
3 -8.353895051506 -4.61 -3.33 0.80 4.7 44.0ms
4 -8.353895106071 -7.26 -4.16 0.80 2.3 36.5ms
5 -8.353895110677 -8.34 -4.90 0.80 3.0 36.8ms
6 -8.353895110768 -10.04 -5.92 0.80 3.3 38.8ms
7 -8.353895110771 -11.61 -6.99 0.80 4.0 42.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110771 -7.82 0.80 1.0 22.1ms
2 -8.353895110771 -14.75 -8.61 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110771 -9.04 0.80 1.0 22.4ms
2 -8.353895110771 + -Inf -9.46 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110771 -9.59 0.80 1.0 27.3ms
2 -8.353895110771 -14.75 -10.04 0.80 1.0 25.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110771 │ -3.86 │ 7.87599e-15 │ 0.00064618 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353805527313 -3.53 0.80 7.7 59.6ms
2 -8.353875230603 -4.16 -3.77 0.80 1.0 30.1ms
3 -8.353895212065 -4.70 -3.85 0.80 6.7 56.2ms
4 -8.353895212408 -9.46 -4.05 0.80 1.0 24.9ms
5 -8.353895212436 -10.56 -4.36 0.80 1.3 26.5ms
6 -8.353895212496 -10.22 -4.66 0.80 2.0 32.0ms
7 -8.353895212513 -10.77 -5.07 0.80 1.0 25.8ms
8 -8.353895212516 -11.44 -5.70 0.80 1.7 28.5ms
9 -8.353895212517 -12.31 -6.16 0.80 2.3 35.1ms
10 -8.353895212517 -13.52 -6.44 0.80 1.7 29.3ms
11 -8.353895212517 + -14.45 -6.42 0.80 1.3 29.8ms
12 -8.353895212517 -14.27 -7.52 0.80 1.0 26.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353753728599 -3.21 0.80 8.0 60.8ms
2 -8.353862532770 -3.96 -3.37 0.80 1.0 25.1ms
3 -8.353895355307 -4.48 -3.82 0.80 6.0 57.0ms
4 -8.353895361041 -8.24 -4.23 0.80 1.3 27.6ms
5 -8.353895362620 -8.80 -5.30 0.80 2.0 30.1ms
6 -8.353895362685 -10.19 -5.98 0.80 4.0 41.5ms
7 -8.353895362685 + -13.47 -6.12 0.80 2.7 38.1ms
8 -8.353895362686 -12.49 -7.11 0.80 2.0 31.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.70 0.80 1.0 22.0ms
2 -8.353895362686 -14.75 -8.39 0.80 1.0 29.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.98 0.80 1.0 22.3ms
2 -8.353895362686 -14.45 -9.14 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.57 0.80 1.0 22.1ms
2 -8.353895362686 + -14.45 -9.71 0.80 1.0 25.1ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.65068e-15 │ 1.71948e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-3.05112e-18 5.27783 5.27783;
5.27783 -3.05112e-18 5.27783;
5.27783 5.27783 -3.05112e-18]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: 10.555662909056702 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.