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 -1.110759000128 -0.82 0.80 7.0 3.22s
2 -1.117205425622 -2.19 -1.89 0.80 1.0 1.57s
3 -1.117250416043 -4.35 -2.71 0.80 1.0 19.7ms
4 -1.117251160852 -6.13 -3.48 0.80 1.0 19.6ms
5 -1.117251188092 -7.56 -4.02 0.80 1.0 19.7ms
6 -1.117251190053 -8.71 -4.79 0.80 1.0 19.6ms
7 -1.117251190141 -10.05 -5.99 0.80 1.0 20.0ms
8 -1.117251190143 -11.75 -6.68 0.80 2.0 45.7ms
9 -1.117251190143 + -15.35 -7.44 0.80 1.0 20.8ms
10 -1.117251190143 -14.48 -8.00 0.80 1.0 20.6ms
11 -1.117251190143 + -15.35 -9.17 0.80 1.0 20.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.57 0.80 1.0 2.72s
2 -1.117251190143 -15.05 -10.21 0.80 1.0 72.2ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 16.9s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517996701 -2.46 0.80 1.0 13.1ms
2 -1.117520586063 -5.59 -3.49 0.80 1.0 53.2ms
3 -1.117520612997 -7.57 -4.23 0.80 1.0 19.0ms
4 -1.117520613449 -9.34 -5.15 0.80 1.0 19.1ms
5 -1.117520613464 -10.81 -5.93 0.80 1.0 19.3ms
6 -1.117520613465 -12.16 -6.45 0.80 1.0 33.6ms
7 -1.117520613465 -13.40 -7.47 0.80 1.0 20.3ms
8 -1.117520613465 -14.75 -8.27 0.80 1.0 20.0ms
9 -1.117520613465 + -14.88 -8.66 0.80 1.0 22.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264615 -1.69 0.80 1.0 18.3ms
2 -1.118339043935 -4.04 -2.70 0.80 1.0 18.5ms
3 -1.118340011114 -6.01 -3.44 0.80 1.0 18.5ms
4 -1.118340028554 -7.76 -4.36 0.80 1.0 24.2ms
5 -1.118340029139 -9.23 -5.12 0.80 1.0 19.5ms
6 -1.118340029166 -10.57 -5.62 0.80 1.0 21.5ms
7 -1.118340029168 -11.73 -6.68 0.80 1.0 20.1ms
8 -1.118340029168 -13.54 -7.58 0.80 1.0 25.8ms
9 -1.118340029168 -15.18 -8.12 0.80 1.0 22.9ms
10 -1.118340029168 + -15.65 -8.55 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -9.05 0.80 1.0 13.1ms
2 -1.118340029168 -14.95 -10.36 0.80 1.0 24.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -11.05 0.80 1.0 19.4ms
2 -1.118340029168 -14.48 -11.42 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029168 │ │ 0.00297055 │ 2.45s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 13.2ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 18.3ms
3 -1.118346866236 -8.83 -4.86 0.80 1.0 283ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 19.2ms
5 -1.118346866263 -12.08 -6.55 0.80 1.0 19.2ms
6 -1.118346866263 -13.41 -7.04 0.80 1.0 19.5ms
7 -1.118346866263 -14.75 -8.13 0.80 1.0 20.3ms
8 -1.118346866263 + -Inf -8.95 0.80 1.0 20.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 12.9ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.6ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 20.0ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 18.9ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.1ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.6ms
7 -1.118356101386 -13.67 -7.65 0.80 1.0 23.1ms
8 -1.118356101386 -14.51 -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 -15.65 -9.82 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.1ms
2 -1.118356101386 -15.05 -11.46 0.80 1.0 18.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 805ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.0ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.6ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 25.3ms
4 -1.118356102877 -14.12 -7.62 0.80 1.0 19.0ms
5 -1.118356102877 + -15.18 -8.40 0.80 1.0 19.4ms
6 -1.118356102877 -14.70 -8.89 0.80 1.0 25.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.1ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.4ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 20.9ms
4 -1.118356105120 -13.21 -7.09 0.80 1.0 18.9ms
5 -1.118356105120 -14.57 -7.86 0.80 1.0 19.1ms
6 -1.118356105120 + -Inf -8.35 0.80 1.0 21.8ms
7 -1.118356105120 -15.05 -9.45 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 13.1ms
2 -1.118356105120 -14.54 -11.02 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.0ms
2 -1.118356105120 -15.65 -12.44 0.80 1.0 18.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.03999e-8 │ 463ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -2.90294e-34, -5.3008e-35]u"a₀")
Atom(H, [ 1.44318, -2.77536e-35, 3.9756e-35]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658669451074 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 -8.335310140345 -0.81 0.80 7.3 702ms
2 -8.340901594754 -2.25 -1.69 0.80 1.3 72.9ms
3 -8.341202560063 -3.52 -2.78 0.80 2.7 36.6ms
4 -8.341214223425 -4.93 -3.24 0.80 3.0 42.7ms
5 -8.341214267330 -7.36 -3.90 0.80 1.3 30.8ms
6 -8.341214272300 -8.30 -5.21 0.80 1.7 32.7ms
7 -8.341214272515 -9.67 -5.60 0.80 4.0 47.2ms
8 -8.341214272518 -11.60 -6.22 0.80 2.0 51.4ms
9 -8.341214272518 -12.75 -7.49 0.80 2.7 37.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.33 0.80 1.0 66.2ms
2 -8.341214272518 + -Inf -8.76 0.80 1.0 28.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.21 0.80 1.0 25.0ms
2 -8.341214272518 + -14.27 -9.36 0.80 1.0 28.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 2.72537e-33 │ 0.163458 │ -0.16 │ 25.7s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350403004865 -1.20 0.80 8.7 76.3ms
2 -8.351871108485 -2.83 -1.88 0.80 2.0 36.0ms
3 -8.351925797483 -4.26 -2.79 0.80 3.0 40.6ms
4 -8.351928511425 -5.57 -3.82 0.80 3.0 43.2ms
5 -8.351928514533 -8.51 -5.21 0.80 3.0 42.0ms
6 -8.351928514648 -9.94 -6.06 0.80 4.0 49.3ms
7 -8.351928514648 -12.66 -6.91 0.80 2.7 66.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353391925434 -1.52 0.80 7.7 68.6ms
2 -8.353731994886 -3.47 -2.21 0.80 1.0 29.1ms
3 -8.353757359949 -4.60 -2.98 0.80 3.0 42.0ms
4 -8.353757522614 -6.79 -4.23 0.80 2.0 34.1ms
5 -8.353757531711 -8.04 -5.34 0.80 3.3 53.1ms
6 -8.353757531724 -10.88 -6.43 0.80 3.3 44.0ms
7 -8.353757531724 -13.00 -7.25 0.80 2.7 41.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531724 -8.27 0.80 1.0 25.5ms
2 -8.353757531724 -14.75 -9.17 0.80 1.0 33.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531724 -9.64 0.80 1.0 25.5ms
2 -8.353757531724 -14.75 -9.98 0.80 1.0 29.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531724 -10.21 0.80 1.0 25.5ms
2 -8.353757531724 -14.45 -10.58 0.80 1.0 29.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531724 │ │ │ 0.0156413 │ -0.016 │ 5.81s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353709204710 -3.19 0.80 8.0 68.6ms
2 -8.353756919826 -4.32 -3.69 0.80 1.0 42.7ms
3 -8.353770535953 -4.87 -3.79 0.80 5.3 56.4ms
4 -8.353770537241 -8.89 -4.12 0.80 1.0 28.6ms
5 -8.353770537275 -10.47 -4.48 0.80 1.0 28.9ms
6 -8.353770537385 -9.96 -5.00 0.80 1.0 42.1ms
7 -8.353770537401 -10.80 -5.67 0.80 2.0 34.5ms
8 -8.353770537401 -12.11 -6.50 0.80 2.3 38.1ms
9 -8.353770537401 -13.30 -7.19 0.80 2.7 40.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353791923909 -2.00 0.80 7.3 71.4ms
2 -8.353878701671 -4.06 -2.72 0.80 1.0 29.2ms
3 -8.353895034346 -4.79 -3.35 0.80 4.0 47.0ms
4 -8.353895102402 -7.17 -4.05 0.80 2.0 38.3ms
5 -8.353895110550 -8.09 -4.80 0.80 2.7 39.5ms
6 -8.353895110777 -9.64 -6.12 0.80 2.7 40.3ms
7 -8.353895110779 -11.71 -7.24 0.80 3.7 52.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -7.68 0.80 1.0 26.0ms
2 -8.353895110779 + -Inf -8.71 0.80 1.0 32.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -9.13 0.80 1.0 28.4ms
2 -8.353895110779 + -14.75 -9.37 0.80 1.0 28.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -9.61 0.80 1.0 25.1ms
2 -8.353895110779 -14.75 -9.91 0.80 1.0 31.9ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110779 │ -3.86 │ │ 0.000646169 │ -0.00065 │ 3.03s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353820295504 -3.69 0.80 7.7 68.9ms
2 -8.353883740631 -4.20 -3.88 0.80 1.0 27.5ms
3 -8.353895212291 -4.94 -4.18 0.80 5.7 59.2ms
4 -8.353895212417 -9.90 -4.42 0.80 1.0 28.2ms
5 -8.353895212498 -10.09 -4.91 0.80 1.3 71.6ms
6 -8.353895212516 -10.76 -5.19 0.80 1.3 30.8ms
7 -8.353895212522 -11.22 -6.42 0.80 1.7 32.3ms
8 -8.353895212522 -12.40 -6.48 0.80 3.7 348ms
9 -8.353895212522 + -13.60 -6.24 0.80 1.0 29.0ms
10 -8.353895212522 -13.55 -7.43 0.80 1.3 30.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353825422789 -3.29 0.80 7.7 67.9ms
2 -8.353884335416 -4.23 -3.55 0.80 1.0 27.8ms
3 -8.353895362053 -4.96 -3.93 0.80 5.0 53.6ms
4 -8.353895362266 -9.67 -4.29 0.80 1.3 29.5ms
5 -8.353895362604 -9.47 -4.90 0.80 1.0 27.9ms
6 -8.353895362679 -10.12 -5.45 0.80 2.0 32.9ms
7 -8.353895362686 -11.22 -6.18 0.80 2.0 35.5ms
8 -8.353895362686 -12.66 -6.91 0.80 3.3 40.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.68 0.80 1.0 24.0ms
2 -8.353895362686 + -Inf -8.58 0.80 1.0 26.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.84 0.80 1.0 23.8ms
2 -8.353895362686 + -14.45 -9.14 0.80 1.0 27.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.35 0.80 1.0 23.1ms
2 -8.353895362686 + -14.75 -9.78 0.80 1.0 34.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ 1.07868e-32 │ 1.71831e-6 │ -1.7e-6 │ 3.34s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [9.58602e-36 5.27783 5.27783;
5.27783 9.58602e-36 5.27783;
5.27783 5.27783 9.58602e-36]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: 10.555662916119893 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.