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 -1.110399550013 -0.82 0.80 6.0 2.55s
2 -1.117191292758 -2.17 -1.83 0.80 1.0 1.31s
3 -1.117249956701 -4.23 -2.64 0.80 1.0 16.0ms
4 -1.117251072359 -5.95 -3.27 0.80 1.0 14.8ms
5 -1.117251178193 -6.98 -3.76 0.80 1.0 14.8ms
6 -1.117251189652 -7.94 -4.90 0.80 1.0 14.9ms
7 -1.117251190137 -9.31 -5.16 0.80 2.0 32.8ms
8 -1.117251190131 + -11.22 -5.14 0.80 1.0 15.2ms
9 -1.117251190142 -10.96 -5.98 0.80 1.0 15.4ms
10 -1.117251190143 -12.15 -6.85 0.80 2.0 21.3ms
11 -1.117251190143 -14.16 -7.34 0.80 1.0 16.4ms
12 -1.117251190143 + -14.57 -7.22 0.80 2.0 16.7ms
13 -1.117251190143 -15.05 -7.87 0.80 1.0 207ms
14 -1.117251190143 -14.70 -8.12 0.80 1.0 16.7ms
15 -1.117251190143 + -14.61 -8.95 0.80 1.0 312ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.61 0.80 1.0 2.31s
2 -1.117251190143 -14.88 -10.13 0.80 1.0 14.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 14.4s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998028 -2.46 0.80 1.0 10.2ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 14.4ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 14.3ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 14.3ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 54.9ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 15.2ms
7 -1.117520613465 -13.37 -7.39 0.80 1.0 15.2ms
8 -1.117520613465 -14.57 -8.19 0.80 1.0 15.2ms
9 -1.117520613465 -14.45 -8.96 0.80 1.0 15.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118242085274 -1.69 0.80 2.0 11.2ms
2 -1.118338829018 -4.01 -2.68 0.80 1.0 14.4ms
3 -1.118340016219 -5.93 -3.42 0.80 1.0 14.4ms
4 -1.118340028780 -7.90 -4.28 0.80 1.0 14.4ms
5 -1.118340029125 -9.46 -4.83 0.80 1.0 20.5ms
6 -1.118340029164 -10.41 -5.40 0.80 1.0 14.8ms
7 -1.118340029167 -11.46 -6.18 0.80 1.0 14.8ms
8 -1.118340029167 -12.67 -7.35 0.80 1.0 14.9ms
9 -1.118340029167 -14.31 -7.70 0.80 2.0 18.8ms
10 -1.118340029167 + -14.57 -8.61 0.80 1.0 15.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -9.49 0.80 1.0 9.93ms
2 -1.118340029167 + -15.65 -10.47 0.80 1.0 14.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -10.99 0.80 1.0 9.85ms
2 -1.118340029167 -15.18 -12.32 0.80 1.0 14.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029167 │ │ 0.00297055 │ 1.90s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721508 -3.09 0.80 1.0 10.0ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 14.5ms
3 -1.118346866236 -8.83 -4.86 0.80 1.0 14.6ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 21.1ms
5 -1.118346866263 -12.08 -6.55 0.80 1.0 15.2ms
6 -1.118346866263 -13.44 -7.04 0.80 1.0 17.5ms
7 -1.118346866263 -14.81 -8.13 0.80 1.0 15.2ms
8 -1.118346866263 -14.51 -8.95 0.80 1.0 17.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 9.80ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 17.2ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 14.4ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 17.1ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 14.7ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 17.4ms
7 -1.118356101386 -13.58 -7.65 0.80 1.0 15.1ms
8 -1.118356101386 + -14.88 -8.45 0.80 1.0 17.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 12.4ms
2 -1.118356101386 + -Inf -9.82 0.80 1.0 14.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 9.88ms
2 -1.118356101386 -13.99 -11.46 0.80 1.0 14.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 428ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 9.86ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 21.0ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 14.4ms
4 -1.118356102877 -13.98 -7.62 0.80 1.0 14.7ms
5 -1.118356102877 + -14.45 -8.39 0.80 1.0 21.5ms
6 -1.118356102877 -14.95 -8.89 0.80 1.0 14.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 9.95ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 14.2ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 17.0ms
4 -1.118356105120 -13.23 -7.09 0.80 1.0 14.5ms
5 -1.118356105120 -14.11 -7.86 0.80 1.0 17.2ms
6 -1.118356105120 -14.29 -8.35 0.80 1.0 14.8ms
7 -1.118356105120 + -13.92 -9.45 0.80 1.0 17.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 9.81ms
2 -1.118356105120 -14.61 -11.02 0.80 1.0 16.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 61.0ms
2 -1.118356105120 + -14.81 -12.44 0.80 1.0 14.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.00381e-8 │ 724ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -4.14067e-34, -3.86873e-34]u"a₀")
Atom(H, [ 1.44318, -1.87155e-34, 2.62541e-34]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658682960796 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 -8.335321424714 -0.81 0.80 7.3 528ms
2 -8.340901192359 -2.25 -1.69 0.80 1.3 52.4ms
3 -8.341202431537 -3.52 -2.78 0.80 2.7 23.8ms
4 -8.341214227734 -4.93 -3.24 0.80 3.3 27.9ms
5 -8.341214267340 -7.40 -3.91 0.80 1.0 18.4ms
6 -8.341214272310 -8.30 -5.16 0.80 1.7 20.8ms
7 -8.341214272515 -9.69 -5.58 0.80 3.7 29.1ms
8 -8.341214272518 -11.57 -6.47 0.80 2.0 21.0ms
9 -8.341214272518 -12.73 -7.31 0.80 3.3 27.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.25 0.80 1.0 48.8ms
2 -8.341214272518 + -14.15 -9.37 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.68 0.80 1.0 16.2ms
2 -8.341214272518 -14.75 -10.09 0.80 1.0 18.3ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ │ 0.163458 │ -0.16 │ 22.4s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350659140825 -1.21 0.80 12.0 62.0ms
2 -8.351917146595 -2.90 -1.87 0.80 4.0 30.9ms
3 -8.351927937150 -4.97 -2.86 0.80 1.7 63.7ms
4 -8.351928467898 -6.28 -4.07 0.80 2.7 23.7ms
5 -8.351928514195 -7.33 -4.39 0.80 3.7 28.7ms
6 -8.351928514584 -9.41 -4.94 0.80 1.7 21.0ms
7 -8.351928514639 -10.26 -6.20 0.80 2.7 24.7ms
8 -8.351928514640 -11.92 -7.21 0.80 4.3 31.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353438013413 -1.53 0.80 11.3 56.7ms
2 -8.353753982472 -3.50 -2.18 0.80 2.3 24.6ms
3 -8.353757341210 -5.47 -3.17 0.80 1.0 18.6ms
4 -8.353757512277 -6.77 -4.25 0.80 2.3 23.0ms
5 -8.353757531681 -7.71 -4.63 0.80 4.0 29.7ms
6 -8.353757531715 -10.47 -5.27 0.80 1.0 23.4ms
7 -8.353757531729 -10.84 -6.03 0.80 2.3 23.1ms
8 -8.353757531730 -12.21 -6.67 0.80 2.7 26.3ms
9 -8.353757531730 -13.71 -7.03 0.80 2.7 24.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531730 -7.73 0.80 1.0 20.7ms
2 -8.353757531730 + -Inf -9.00 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531730 -9.09 0.80 1.0 16.2ms
2 -8.353757531730 -14.45 -9.47 0.80 1.0 22.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531730 -9.72 0.80 1.0 16.3ms
2 -8.353757531730 + -14.75 -10.10 0.80 1.0 22.9ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531730 │ │ │ 0.0156413 │ -0.016 │ 5.11s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353770302153 -3.28 0.80 11.3 58.1ms
2 -8.353770507129 -6.69 -3.95 0.80 1.0 18.3ms
3 -8.353770537293 -7.52 -4.59 0.80 3.7 28.8ms
4 -8.353770537400 -9.97 -5.66 0.80 2.0 22.3ms
5 -8.353770537407 -11.16 -6.55 0.80 3.3 26.9ms
6 -8.353770537407 -13.30 -7.36 0.80 2.7 24.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353858002742 -2.00 0.80 11.3 57.0ms
2 -8.353893529726 -4.45 -2.67 0.80 1.0 18.2ms
3 -8.353895062952 -5.81 -3.53 0.80 2.3 22.3ms
4 -8.353895109260 -7.33 -4.21 0.80 2.3 23.7ms
5 -8.353895110676 -8.85 -4.98 0.80 2.3 22.3ms
6 -8.353895110772 -10.02 -6.08 0.80 3.0 25.8ms
7 -8.353895110773 -12.03 -6.47 0.80 3.0 26.7ms
8 -8.353895110773 -13.30 -7.58 0.80 2.3 26.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110773 -8.39 0.80 1.0 19.9ms
2 -8.353895110773 + -14.75 -8.87 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110773 -9.29 0.80 1.0 16.1ms
2 -8.353895110773 -14.75 -9.46 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110773 -9.84 0.80 1.0 16.1ms
2 -8.353895110773 + -Inf -10.03 0.80 1.0 18.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110773 │ -3.86 │ │ 0.000646176 │ -0.00065 │ 2.35s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353894997514 -3.91 0.80 11.3 57.5ms
2 -8.353895174366 -6.75 -4.43 0.80 1.0 18.5ms
3 -8.353895212434 -7.42 -4.75 0.80 4.7 37.3ms
4 -8.353895212511 -10.11 -5.55 0.80 1.7 20.8ms
5 -8.353895212518 -11.15 -6.06 0.80 2.3 24.0ms
6 -8.353895212519 -12.63 -6.71 0.80 2.0 22.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895101367 -3.42 0.80 10.3 55.2ms
2 -8.353895313207 -6.67 -4.05 0.80 1.0 18.5ms
3 -8.353895362582 -7.31 -4.60 0.80 4.3 30.5ms
4 -8.353895362678 -10.01 -5.71 0.80 1.7 24.7ms
5 -8.353895362686 -11.13 -6.70 0.80 3.3 28.6ms
6 -8.353895362686 -13.71 -7.74 0.80 3.0 27.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.93 0.80 1.0 16.3ms
2 -8.353895362686 + -14.75 -9.72 0.80 1.0 18.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.80 0.80 1.0 20.7ms
2 -8.353895362686 -14.75 -10.19 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -10.32 0.80 1.0 20.9ms
2 -8.353895362686 -14.75 -10.71 0.80 1.0 18.5ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ 2.83689e-32 │ 1.71956e-6 │ -1.7e-6 │ 2.32s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-4.79301e-36 5.27783 5.27783;
5.27783 -4.79301e-36 5.27783;
5.27783 5.27783 -4.79301e-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: 10.555662907518208 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.