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 -1.110713580644 -0.82 0.80 7.0 5.38s
2 -1.117190794261 -2.19 -1.86 0.80 1.0 2.24s
3 -1.117250010023 -4.23 -2.69 0.80 1.0 19.9ms
4 -1.117251117467 -5.96 -3.41 0.80 1.0 19.9ms
5 -1.117251188994 -7.15 -4.29 0.80 1.0 20.1ms
6 -1.117251190101 -8.96 -4.95 0.80 1.0 49.4ms
7 -1.117251190142 -10.39 -5.69 0.80 1.0 21.0ms
8 -1.117251190143 -11.83 -6.17 0.80 2.0 23.7ms
9 -1.117251190143 -13.06 -6.75 0.80 1.0 30.8ms
10 -1.117251190143 -13.97 -7.49 0.80 1.0 22.0ms
11 -1.117251190143 + -Inf -8.01 0.80 2.0 26.7ms
12 -1.117251190143 + -14.57 -8.58 0.80 1.0 24.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.31 0.80 1.0 4.17s
2 -1.117251190143 -15.35 -10.54 0.80 1.0 19.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 23.6s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997891 -2.46 0.80 1.0 13.7ms
2 -1.117520586049 -5.59 -3.49 0.80 1.0 19.2ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 19.5ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.7ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 42.8ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 20.1ms
7 -1.117520613465 -13.31 -7.39 0.80 1.0 20.6ms
8 -1.117520613465 -14.54 -8.19 0.80 1.0 29.4ms
9 -1.117520613465 + -14.75 -8.97 0.80 1.0 21.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264594 -1.69 0.80 1.0 19.3ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 19.0ms
3 -1.118340011113 -6.01 -3.44 0.80 1.0 19.1ms
4 -1.118340028553 -7.76 -4.36 0.80 1.0 25.5ms
5 -1.118340029138 -9.23 -5.12 0.80 1.0 19.8ms
6 -1.118340029165 -10.57 -5.62 0.80 1.0 20.1ms
7 -1.118340029167 -11.73 -6.68 0.80 1.0 23.6ms
8 -1.118340029167 -13.55 -7.58 0.80 1.0 21.2ms
9 -1.118340029167 + -Inf -8.12 0.80 1.0 93.3ms
10 -1.118340029167 -14.75 -8.55 0.80 1.0 21.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -9.05 0.80 1.0 13.3ms
2 -1.118340029167 -15.18 -10.36 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -11.04 0.80 1.0 13.4ms
2 -1.118340029167 -14.40 -11.41 0.80 1.0 18.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029167 │ │ 0.00297055 │ 3.84s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 13.3ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 18.7ms
3 -1.118346866236 -8.83 -4.86 0.80 1.0 18.8ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 19.2ms
5 -1.118346866262 -12.09 -6.55 0.80 1.0 19.6ms
6 -1.118346866262 -13.41 -7.04 0.80 1.0 20.1ms
7 -1.118346866262 -14.95 -8.13 0.80 1.0 20.8ms
8 -1.118346866262 + -15.65 -8.95 0.80 1.0 22.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 18.5ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 26.5ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 36.0ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.5ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.9ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.9ms
7 -1.118356101386 -13.57 -7.65 0.80 1.0 20.4ms
8 -1.118356101386 + -14.45 -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 13.1ms
2 -1.118356101386 + -15.05 -9.82 0.80 1.0 25.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.2ms
2 -1.118356101386 + -14.65 -11.46 0.80 1.0 18.8ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 541ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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 31.9ms
4 -1.118356102877 -14.07 -7.62 0.80 1.0 19.7ms
5 -1.118356102877 + -14.40 -8.39 0.80 1.0 19.8ms
6 -1.118356102877 + -14.75 -8.89 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 20.3ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 19.1ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 19.3ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 19.7ms
5 -1.118356105120 -14.54 -7.86 0.80 1.0 20.0ms
6 -1.118356105120 + -14.61 -8.35 0.80 1.0 23.6ms
7 -1.118356105120 -14.35 -9.45 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 17.0ms
2 -1.118356105120 -14.81 -11.02 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 17.2ms
2 -1.118356105120 + -14.26 -12.44 0.80 1.0 19.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.07752e-8 │ 482ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, 6.63485e-18, 2.3712e-17]u"a₀")
Atom(H, [ 1.44318, -1.14047e-17, -1.2841e-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: 1.4863658656678422 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 -8.335309406979 -0.81 0.80 7.3 783ms
2 -8.340913047714 -2.25 -1.69 0.80 1.3 633ms
3 -8.341203687211 -3.54 -2.77 0.80 2.7 35.1ms
4 -8.341214239948 -4.98 -3.27 0.80 3.3 41.5ms
5 -8.341214269756 -7.53 -3.99 0.80 1.3 57.0ms
6 -8.341214272377 -8.58 -5.20 0.80 2.0 32.8ms
7 -8.341214272515 -9.86 -5.54 0.80 3.7 43.6ms
8 -8.341214272518 -11.57 -5.99 0.80 2.0 33.3ms
9 -8.341214272518 -12.46 -6.90 0.80 2.0 33.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.64 0.80 1.0 646ms
2 -8.341214272518 -14.75 -8.74 0.80 1.0 26.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.28 0.80 1.0 23.1ms
2 -8.341214272518 + -Inf -9.63 0.80 1.0 26.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.31831e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350378896419 -1.20 0.80 9.3 73.3ms
2 -8.351864441957 -2.83 -1.88 0.80 1.3 74.4ms
3 -8.351926148000 -4.21 -2.78 0.80 3.0 38.5ms
4 -8.351928501384 -5.63 -3.74 0.80 3.0 37.4ms
5 -8.351928514599 -7.88 -5.31 0.80 3.7 42.0ms
6 -8.351928514653 -10.27 -6.04 0.80 4.3 47.6ms
7 -8.351928514653 -12.54 -7.04 0.80 3.7 61.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353380308430 -1.52 0.80 7.3 61.9ms
2 -8.353727383083 -3.46 -2.21 0.80 1.0 26.6ms
3 -8.353757293852 -4.52 -2.98 0.80 3.7 119ms
4 -8.353757523909 -6.64 -4.14 0.80 3.0 37.8ms
5 -8.353757531697 -8.11 -5.22 0.80 3.3 41.8ms
6 -8.353757531720 -10.65 -5.92 0.80 3.7 42.6ms
7 -8.353757531720 -12.33 -7.10 0.80 3.0 732ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -7.78 0.80 1.0 23.6ms
2 -8.353757531720 + -Inf -8.91 0.80 1.0 26.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -9.27 0.80 1.0 23.4ms
2 -8.353757531720 + -14.75 -9.29 0.80 1.0 26.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -9.68 0.80 1.0 23.4ms
2 -8.353757531720 + -14.75 -9.93 0.80 1.0 26.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531720 │ │ 9.73555e-15 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353707863836 -3.23 0.80 7.7 62.7ms
2 -8.353759036465 -4.29 -3.95 0.80 1.0 26.6ms
3 -8.353770535573 -4.94 -4.10 0.80 6.0 56.4ms
4 -8.353770537300 -8.76 -4.69 0.80 1.0 27.0ms
5 -8.353770537375 -10.13 -5.16 0.80 2.7 36.7ms
6 -8.353770537397 -10.65 -6.66 0.80 2.7 34.6ms
7 -8.353770537398 -12.15 -7.32 0.80 3.7 44.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353790292536 -2.00 0.80 8.0 64.1ms
2 -8.353879917094 -4.05 -2.71 0.80 1.0 26.9ms
3 -8.353895059660 -4.82 -3.34 0.80 4.3 46.0ms
4 -8.353895107456 -7.32 -4.25 0.80 2.0 33.1ms
5 -8.353895110727 -8.49 -5.08 0.80 3.0 39.4ms
6 -8.353895110773 -10.34 -5.86 0.80 3.7 41.5ms
7 -8.353895110774 -11.99 -7.12 0.80 2.3 36.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110774 -7.71 0.80 1.0 29.4ms
2 -8.353895110774 -14.15 -8.21 0.80 1.0 27.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110774 -8.72 0.80 1.0 23.0ms
2 -8.353895110774 -14.27 -9.02 0.80 1.0 26.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110774 -9.44 0.80 1.0 23.2ms
2 -8.353895110774 + -Inf -9.65 0.80 1.0 32.6ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110774 │ -3.86 │ 8.17539e-15 │ 0.000646176 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353821240209 -3.58 0.80 8.7 71.9ms
2 -8.353882208416 -4.21 -3.87 0.80 1.0 26.8ms
3 -8.353895211923 -4.89 -4.04 0.80 6.0 56.6ms
4 -8.353895212351 -9.37 -4.25 0.80 1.0 31.9ms
5 -8.353895212446 -10.02 -4.80 0.80 1.3 29.4ms
6 -8.353895212509 -10.20 -5.25 0.80 1.7 31.6ms
7 -8.353895212519 -11.02 -6.17 0.80 3.0 38.1ms
8 -8.353895212519 -12.35 -6.66 0.80 3.7 46.0ms
9 -8.353895212519 -14.45 -7.06 0.80 1.7 32.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353796486075 -3.32 0.80 8.0 69.6ms
2 -8.353866061506 -4.16 -3.77 0.80 1.0 26.7ms
3 -8.353895361705 -4.53 -3.95 0.80 6.0 56.3ms
4 -8.353895362556 -9.07 -4.37 0.80 1.3 28.2ms
5 -8.353895362648 -10.04 -4.76 0.80 1.7 34.5ms
6 -8.353895362678 -10.53 -5.22 0.80 1.0 27.4ms
7 -8.353895362685 -11.13 -5.87 0.80 2.3 34.4ms
8 -8.353895362686 -12.18 -6.84 0.80 3.7 41.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.46 0.80 1.0 23.1ms
2 -8.353895362686 + -Inf -8.49 0.80 1.0 26.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.97 0.80 1.0 23.3ms
2 -8.353895362686 + -14.75 -9.26 0.80 1.0 26.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.51 0.80 1.0 29.3ms
2 -8.353895362686 + -Inf -9.82 0.80 1.0 26.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 4.74417e-15 │ 1.71902e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-9.62965e-35 5.27783 5.27783;
5.27783 9.62965e-35 5.27783;
5.27783 5.27783 -9.62965e-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: 10.555662909023651 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.