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 -1.110877010206 -0.82 0.80 6.0 5.09s
2 -1.117207879415 -2.20 -1.90 0.80 1.0 2.17s
3 -1.117250492859 -4.37 -2.73 0.80 1.0 19.4ms
4 -1.117251158049 -6.18 -3.48 0.80 1.0 19.2ms
5 -1.117251187596 -7.53 -4.00 0.80 1.0 19.4ms
6 -1.117251190096 -8.60 -5.08 0.80 1.0 42.7ms
7 -1.117251190139 -10.36 -5.47 0.80 2.0 22.8ms
8 -1.117251190143 -11.48 -6.04 0.80 1.0 20.7ms
9 -1.117251190143 -12.50 -6.88 0.80 1.0 20.8ms
10 -1.117251190143 -14.57 -8.06 0.80 1.0 29.2ms
11 -1.117251190143 -14.81 -8.27 0.80 2.0 22.7ms
12 -1.117251190143 -14.81 -8.74 0.80 1.0 21.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.43 0.80 1.0 3.92s
2 -1.117251190143 -14.61 -10.65 0.80 1.0 18.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 22.0s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997422 -2.46 0.80 1.0 13.4ms
2 -1.117520586058 -5.59 -3.49 0.80 1.0 18.8ms
3 -1.117520612993 -7.57 -4.23 0.80 1.0 18.7ms
4 -1.117520613449 -9.34 -5.15 0.80 1.0 18.9ms
5 -1.117520613464 -10.81 -5.93 0.80 1.0 47.3ms
6 -1.117520613465 -12.16 -6.44 0.80 1.0 19.6ms
7 -1.117520613465 -13.36 -7.46 0.80 1.0 20.1ms
8 -1.117520613465 + -14.65 -8.29 0.80 1.0 20.6ms
9 -1.117520613465 -14.65 -8.64 0.80 1.0 133ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264611 -1.69 0.80 1.0 18.8ms
2 -1.118339043934 -4.04 -2.70 0.80 1.0 691ms
3 -1.118340011113 -6.01 -3.44 0.80 1.0 19.9ms
4 -1.118340028554 -7.76 -4.36 0.80 1.0 20.7ms
5 -1.118340029138 -9.23 -5.12 0.80 1.0 20.9ms
6 -1.118340029165 -10.57 -5.62 0.80 1.0 20.7ms
7 -1.118340029167 -11.73 -6.68 0.80 1.0 23.8ms
8 -1.118340029167 -13.57 -7.58 0.80 1.0 22.7ms
9 -1.118340029167 -14.88 -8.12 0.80 1.0 22.1ms
10 -1.118340029167 + -15.65 -8.55 0.80 1.0 22.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -9.05 0.80 1.0 13.7ms
2 -1.118340029167 -14.65 -10.36 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -11.05 0.80 1.0 17.0ms
2 -1.118340029167 + -14.48 -11.42 0.80 1.0 26.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029167 │ │ 0.00297055 │ 3.72s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 18.3ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 32.8ms
3 -1.118346866236 -8.83 -4.86 0.80 1.0 18.6ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 19.0ms
5 -1.118346866262 -12.08 -6.55 0.80 1.0 20.3ms
6 -1.118346866262 -13.42 -7.04 0.80 1.0 19.7ms
7 -1.118346866262 + -14.95 -8.13 0.80 1.0 20.1ms
8 -1.118346866262 -14.95 -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 21.7ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.7ms
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 19.2ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.6ms
7 -1.118356101386 -13.59 -7.65 0.80 1.0 20.1ms
8 -1.118356101386 + -14.75 -8.45 0.80 1.0 23.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.1ms
2 -1.118356101386 -15.35 -9.82 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 18.3ms
2 -1.118356101386 -14.65 -11.46 0.80 1.0 18.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 544ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.5ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.4ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 26.9ms
4 -1.118356102877 -14.31 -7.62 0.80 1.0 18.9ms
5 -1.118356102877 -14.65 -8.39 0.80 1.0 19.3ms
6 -1.118356102877 + -14.65 -8.89 0.80 1.0 26.3ms
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 22.4ms
4 -1.118356105120 -13.22 -7.09 0.80 1.0 19.0ms
5 -1.118356105120 -14.51 -7.86 0.80 1.0 22.6ms
6 -1.118356105120 -14.70 -8.35 0.80 1.0 19.8ms
7 -1.118356105120 + -14.57 -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 13.1ms
2 -1.118356105120 + -14.61 -11.02 0.80 1.0 22.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.0ms
2 -1.118356105120 + -14.07 -12.44 0.80 1.0 22.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.05305e-8 │ 480ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, 1.40756e-17, 2.12399e-17]u"a₀")
Atom(H, [ 1.44318, -2.84706e-17, 7.1188e-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: 1.4863658665814297 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 -8.335336452226 -0.81 0.80 7.7 794ms
2 -8.340916760798 -2.25 -1.69 0.80 1.3 614ms
3 -8.341207867673 -3.54 -2.76 0.80 3.0 36.2ms
4 -8.341214241733 -5.20 -3.29 0.80 3.3 39.9ms
5 -8.341214269096 -7.56 -4.03 0.80 1.3 27.2ms
6 -8.341214272449 -8.47 -5.08 0.80 2.3 33.3ms
7 -8.341214272516 -10.17 -5.63 0.80 3.0 57.0ms
8 -8.341214272518 -11.79 -6.62 0.80 2.0 31.3ms
9 -8.341214272518 -13.10 -7.68 0.80 3.0 39.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.76 0.80 1.0 603ms
2 -8.341214272518 + -14.45 -9.84 0.80 1.0 25.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -10.49 0.80 1.0 36.3ms
2 -8.341214272518 + -Inf -10.63 0.80 1.0 25.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.19985e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350577947275 -1.21 0.80 8.7 66.9ms
2 -8.351866384418 -2.89 -1.88 0.80 2.0 30.2ms
3 -8.351926603712 -4.22 -2.81 0.80 4.0 38.9ms
4 -8.351928486606 -5.73 -3.65 0.80 3.0 37.1ms
5 -8.351928513783 -7.57 -4.54 0.80 3.7 39.1ms
6 -8.351928514621 -9.08 -5.11 0.80 4.0 43.7ms
7 -8.351928514650 -10.55 -6.48 0.80 3.3 36.4ms
8 -8.351928514650 -12.13 -6.94 0.80 4.0 42.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353405870630 -1.52 0.80 8.3 64.6ms
2 -8.353735044771 -3.48 -2.21 0.80 1.0 25.7ms
3 -8.353757389927 -4.65 -2.99 0.80 3.3 38.4ms
4 -8.353757524851 -6.87 -4.17 0.80 2.7 33.3ms
5 -8.353757531684 -8.17 -4.98 0.80 3.3 40.0ms
6 -8.353757531721 -10.44 -5.77 0.80 3.0 35.3ms
7 -8.353757531723 -11.75 -6.97 0.80 2.7 36.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531723 -8.02 0.80 1.0 40.9ms
2 -8.353757531723 + -Inf -8.75 0.80 1.0 25.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531723 -9.05 0.80 1.0 22.7ms
2 -8.353757531723 + -Inf -9.37 0.80 1.0 25.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531723 -9.50 0.80 1.0 22.7ms
2 -8.353757531723 -14.75 -9.87 0.80 1.0 25.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531723 │ │ 1.00978e-14 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353678629823 -3.19 0.80 7.3 59.9ms
2 -8.353742275898 -4.20 -3.90 0.80 1.0 25.7ms
3 -8.353770535156 -4.55 -3.77 0.80 6.3 62.2ms
4 -8.353770536972 -8.74 -4.00 0.80 1.0 26.1ms
5 -8.353770537115 -9.84 -4.31 0.80 1.0 25.9ms
6 -8.353770537380 -9.58 -5.02 0.80 1.0 25.9ms
7 -8.353770537398 -10.74 -5.51 0.80 2.3 36.4ms
8 -8.353770537399 -12.08 -6.45 0.80 1.7 29.7ms
9 -8.353770537399 -12.96 -6.82 0.80 3.7 40.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353748782877 -2.00 0.80 8.3 66.5ms
2 -8.353872511011 -3.91 -2.70 0.80 1.0 25.7ms
3 -8.353895020491 -4.65 -3.25 0.80 3.7 42.1ms
4 -8.353895100534 -7.10 -3.98 0.80 2.7 33.0ms
5 -8.353895110625 -8.00 -4.97 0.80 2.7 41.0ms
6 -8.353895110785 -9.80 -5.47 0.80 2.3 34.3ms
7 -8.353895110790 -11.28 -6.57 0.80 3.0 36.7ms
8 -8.353895110790 -12.89 -7.13 0.80 3.0 42.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110790 -7.79 0.80 1.0 22.6ms
2 -8.353895110790 + -Inf -8.90 0.80 1.0 25.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110790 -9.43 0.80 1.0 22.9ms
2 -8.353895110790 + -Inf -9.84 0.80 1.0 25.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110790 -10.19 0.80 1.0 22.7ms
2 -8.353895110790 -14.45 -10.54 0.80 1.0 31.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110790 │ -3.86 │ 8.13869e-15 │ 0.000646155 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353801233787 -3.69 0.80 8.3 63.4ms
2 -8.353872436949 -4.15 -3.81 0.80 1.0 31.2ms
3 -8.353895211652 -4.64 -4.08 0.80 6.0 54.8ms
4 -8.353895212232 -9.24 -4.31 0.80 1.3 27.0ms
5 -8.353895212500 -9.57 -4.78 0.80 1.3 27.5ms
6 -8.353895212522 -10.65 -5.44 0.80 1.0 29.6ms
7 -8.353895212528 -11.22 -5.91 0.80 3.7 39.4ms
8 -8.353895212529 -12.66 -6.33 0.80 1.3 28.3ms
9 -8.353895212529 -13.67 -6.30 0.80 1.3 28.0ms
10 -8.353895212529 -13.75 -7.18 0.80 1.0 30.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353818066179 -3.33 0.80 8.0 63.9ms
2 -8.353880227418 -4.21 -3.89 0.80 1.0 25.3ms
3 -8.353895362194 -4.82 -4.02 0.80 6.3 62.2ms
4 -8.353895362576 -9.42 -4.51 0.80 1.0 25.7ms
5 -8.353895362666 -10.04 -4.83 0.80 1.7 29.0ms
6 -8.353895362682 -10.80 -5.34 0.80 1.0 25.7ms
7 -8.353895362685 -11.47 -5.87 0.80 2.0 36.2ms
8 -8.353895362686 -12.34 -6.98 0.80 2.3 33.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.57 0.80 1.0 24.5ms
2 -8.353895362686 -14.27 -8.39 0.80 1.0 30.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.96 0.80 1.0 22.6ms
2 -8.353895362686 + -14.75 -9.34 0.80 1.0 25.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.57 0.80 1.0 22.8ms
2 -8.353895362686 -14.75 -10.00 0.80 1.0 25.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.23941e-15 │ 1.716e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-4.60235e-18 5.27783 5.27783;
5.27783 -4.60272e-18 5.27783;
5.27783 5.27783 -3.22048e-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: 10.55566292116595 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.