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 -1.110713985514 -0.82 0.80 7.0 4.96s
2 -1.117205800062 -2.19 -1.88 0.80 1.0 2.24s
3 -1.117250573291 -4.35 -2.73 0.80 1.0 21.1ms
4 -1.117251171500 -6.22 -3.56 0.80 1.0 20.8ms
5 -1.117251188375 -7.77 -4.02 0.80 1.0 21.2ms
6 -1.117251190063 -8.77 -4.92 0.80 1.0 20.4ms
7 -1.117251190140 -10.11 -5.61 0.80 1.0 44.0ms
8 -1.117251190143 -11.54 -6.31 0.80 2.0 24.4ms
9 -1.117251190143 -13.70 -6.81 0.80 1.0 21.9ms
10 -1.117251190143 -14.18 -7.47 0.80 1.0 21.7ms
11 -1.117251190143 + -15.18 -8.33 0.80 1.0 32.4ms
12 -1.117251190143 -14.75 -8.78 0.80 1.0 22.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.48 0.80 1.0 3.28s
2 -1.117251190143 -14.95 -10.56 0.80 1.0 20.4ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 20.3s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998050 -2.46 0.80 1.0 14.7ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 20.3ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 20.3ms
4 -1.117520613448 -9.33 -5.14 0.80 1.0 20.6ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 50.2ms
6 -1.117520613465 -12.13 -6.42 0.80 1.0 21.5ms
7 -1.117520613465 -13.35 -7.39 0.80 1.0 21.4ms
8 -1.117520613465 -14.48 -8.17 0.80 1.0 21.4ms
9 -1.117520613465 + -15.35 -8.97 0.80 1.0 125ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264592 -1.69 0.80 1.0 14.6ms
2 -1.118339043935 -4.04 -2.70 0.80 1.0 20.1ms
3 -1.118340011114 -6.01 -3.44 0.80 1.0 19.9ms
4 -1.118340028555 -7.76 -4.36 0.80 1.0 628ms
5 -1.118340029139 -9.23 -5.12 0.80 1.0 21.0ms
6 -1.118340029166 -10.57 -5.62 0.80 1.0 21.2ms
7 -1.118340029168 -11.73 -6.68 0.80 1.0 21.8ms
8 -1.118340029168 -13.55 -7.58 0.80 1.0 21.9ms
9 -1.118340029168 + -14.70 -8.12 0.80 1.0 21.4ms
10 -1.118340029168 -14.65 -8.55 0.80 1.0 21.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -9.05 0.80 1.0 14.2ms
2 -1.118340029168 -14.95 -10.36 0.80 1.0 19.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -11.04 0.80 1.0 14.6ms
2 -1.118340029168 -14.65 -11.41 0.80 1.0 20.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029168 │ │ 0.00297055 │ 3.61s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721508 -3.09 0.80 1.0 14.8ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 20.4ms
3 -1.118346866237 -8.83 -4.86 0.80 1.0 20.6ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 20.1ms
5 -1.118346866263 -12.09 -6.55 0.80 1.0 20.9ms
6 -1.118346866263 -13.42 -7.04 0.80 1.0 21.3ms
7 -1.118346866263 -14.48 -8.13 0.80 1.0 21.5ms
8 -1.118346866263 + -15.35 -8.95 0.80 1.0 32.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 14.5ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 20.4ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 20.2ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 20.5ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 20.8ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 29.1ms
7 -1.118356101386 -13.61 -7.65 0.80 1.0 21.6ms
8 -1.118356101386 + -14.88 -8.45 0.80 1.0 21.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 14.6ms
2 -1.118356101386 -14.75 -9.82 0.80 1.0 24.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 14.4ms
2 -1.118356101386 + -14.26 -11.46 0.80 1.0 20.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 555ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 14.5ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 19.8ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 28.8ms
4 -1.118356102877 -13.97 -7.62 0.80 1.0 21.3ms
5 -1.118356102877 + -15.05 -8.39 0.80 1.0 21.0ms
6 -1.118356102877 -15.35 -8.89 0.80 1.0 21.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 14.8ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 20.0ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 24.2ms
4 -1.118356105120 -13.24 -7.09 0.80 1.0 22.0ms
5 -1.118356105120 -14.06 -7.86 0.80 1.0 24.5ms
6 -1.118356105120 + -14.27 -8.35 0.80 1.0 21.2ms
7 -1.118356105120 -14.37 -9.45 0.80 1.0 24.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 14.3ms
2 -1.118356105120 -14.27 -11.02 0.80 1.0 24.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 18.1ms
2 -1.118356105120 -15.18 -12.44 0.80 1.0 20.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 9.31809e-9 │ 507ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -2.21776e-17, 1.02282e-17]u"a₀")
Atom(H, [ 1.44318, -3.60991e-18, -6.20722e-18]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658711094292 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 -8.335135051145 -0.80 0.80 7.7 450ms
2 -8.340933076118 -2.24 -1.69 0.80 1.3 311ms
3 -8.341205674568 -3.56 -2.77 0.80 2.3 38.5ms
4 -8.341214217751 -5.07 -3.26 0.80 3.3 45.1ms
5 -8.341214266368 -7.31 -3.93 0.80 2.0 34.2ms
6 -8.341214272310 -8.23 -4.98 0.80 2.3 62.2ms
7 -8.341214272503 -9.71 -5.18 0.80 3.7 47.7ms
8 -8.341214272517 -10.84 -6.46 0.80 1.3 30.8ms
9 -8.341214272518 -12.24 -6.93 0.80 3.3 45.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.09 0.80 1.0 301ms
2 -8.341214272518 -14.75 -9.14 0.80 1.0 28.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.71 0.80 1.0 24.9ms
2 -8.341214272518 -14.45 -10.07 0.80 1.0 29.3ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.50846e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350382173156 -1.20 0.80 9.3 79.6ms
2 -8.351862982460 -2.83 -1.88 0.80 1.3 32.0ms
3 -8.351927736407 -4.19 -2.79 0.80 3.3 97.2ms
4 -8.351928498069 -6.12 -3.73 0.80 3.7 44.4ms
5 -8.351928514487 -7.78 -4.86 0.80 3.3 45.6ms
6 -8.351928514646 -9.80 -5.44 0.80 4.0 48.3ms
7 -8.351928514653 -11.20 -6.49 0.80 3.3 44.3ms
8 -8.351928514653 -12.91 -7.32 0.80 3.3 45.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353362835142 -1.52 0.80 7.7 69.4ms
2 -8.353719112182 -3.45 -2.21 0.80 1.0 52.7ms
3 -8.353757350179 -4.42 -2.98 0.80 4.0 48.0ms
4 -8.353757521997 -6.76 -4.20 0.80 2.3 37.2ms
5 -8.353757531663 -8.01 -4.93 0.80 3.7 47.8ms
6 -8.353757531718 -10.26 -5.64 0.80 2.3 38.7ms
7 -8.353757531721 -11.51 -6.97 0.80 3.0 43.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531721 -7.61 0.80 1.0 32.6ms
2 -8.353757531721 -14.45 -8.20 0.80 1.0 29.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531721 -8.73 0.80 1.0 25.6ms
2 -8.353757531721 + -Inf -8.96 0.80 1.0 29.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531721 -9.39 0.80 1.0 25.3ms
2 -8.353757531721 -14.45 -9.52 0.80 1.0 35.9ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531721 │ │ 1.00313e-14 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353696155843 -3.23 0.80 9.0 83.7ms
2 -8.353754455477 -4.23 -3.94 0.80 1.0 29.0ms
3 -8.353770535505 -4.79 -4.19 0.80 6.3 65.1ms
4 -8.353770537310 -8.74 -4.91 0.80 1.3 30.7ms
5 -8.353770537395 -10.07 -5.62 0.80 3.3 46.6ms
6 -8.353770537398 -11.50 -6.05 0.80 3.0 41.4ms
7 -8.353770537399 -12.25 -7.18 0.80 2.7 38.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353785651484 -2.00 0.80 9.7 80.1ms
2 -8.353881592639 -4.02 -2.72 0.80 1.0 28.8ms
3 -8.353895073861 -4.87 -3.32 0.80 4.7 57.1ms
4 -8.353895106042 -7.49 -4.12 0.80 2.3 37.7ms
5 -8.353895110705 -8.33 -4.97 0.80 3.0 41.8ms
6 -8.353895110760 -10.26 -5.94 0.80 2.7 43.8ms
7 -8.353895110762 -11.65 -6.86 0.80 3.7 48.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110762 -7.30 0.80 1.0 25.5ms
2 -8.353895110762 -14.75 -8.48 0.80 1.0 35.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110762 -9.01 0.80 1.0 25.2ms
2 -8.353895110762 -14.75 -9.41 0.80 1.0 28.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110762 -9.71 0.80 1.0 24.5ms
2 -8.353895110762 + -Inf -10.04 0.80 1.0 29.1ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110762 │ -3.86 │ 7.73997e-15 │ 0.000646191 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353835836886 -3.60 0.80 9.3 86.2ms
2 -8.353887577304 -4.29 -3.79 0.80 1.0 28.9ms
3 -8.353895212340 -5.12 -3.98 0.80 6.7 70.7ms
4 -8.353895212439 -10.00 -4.19 0.80 1.0 29.3ms
5 -8.353895212487 -10.33 -4.63 0.80 1.3 35.2ms
6 -8.353895212503 -10.78 -4.88 0.80 1.3 30.8ms
7 -8.353895212511 -11.09 -6.46 0.80 1.3 32.3ms
8 -8.353895212512 -12.18 -6.43 0.80 4.0 49.4ms
9 -8.353895212512 + -Inf -6.58 0.80 1.0 33.2ms
10 -8.353895212512 -14.45 -6.83 0.80 1.0 29.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353810067640 -3.32 0.80 9.3 86.4ms
2 -8.353883216464 -4.14 -3.80 0.80 1.0 28.2ms
3 -8.353895358234 -4.92 -3.86 0.80 6.3 67.5ms
4 -8.353895362233 -8.40 -4.24 0.80 1.3 37.2ms
5 -8.353895362453 -9.66 -4.57 0.80 2.7 40.5ms
6 -8.353895362652 -9.70 -5.05 0.80 1.7 33.9ms
7 -8.353895362679 -10.57 -5.34 0.80 2.7 40.1ms
8 -8.353895362686 -11.21 -6.75 0.80 2.3 42.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.12 0.80 1.0 25.8ms
2 -8.353895362686 -13.80 -7.84 0.80 1.0 28.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.34 0.80 1.0 25.6ms
2 -8.353895362686 + -Inf -8.66 0.80 1.0 29.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.02 0.80 1.0 25.8ms
2 -8.353895362686 -14.75 -9.28 0.80 1.0 29.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.15165e-15 │ 1.71469e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-1.1726e-19 5.27783 5.27783;
5.27783 -1.16887e-19 5.27783;
5.27783 5.27783 -9.79142e-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: 10.555662925174708 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.