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 -1.110708439425 -0.82 0.80 6.0 3.70s
2 -1.117206171050 -2.19 -1.88 0.80 1.0 1.75s
3 -1.117250518004 -4.35 -2.73 0.80 1.0 19.7ms
4 -1.117251168375 -6.19 -3.56 0.80 1.0 19.8ms
5 -1.117251188380 -7.70 -4.01 0.80 1.0 47.8ms
6 -1.117251190109 -8.76 -5.12 0.80 1.0 20.6ms
7 -1.117251190141 -10.49 -5.94 0.80 1.0 20.9ms
8 -1.117251190143 -11.78 -6.16 0.80 2.0 30.0ms
9 -1.117251190143 -13.16 -6.40 0.80 1.0 21.2ms
10 -1.117251190143 -13.49 -7.23 0.80 1.0 21.9ms
11 -1.117251190143 -15.35 -7.82 0.80 1.0 23.7ms
12 -1.117251190143 -14.75 -8.31 0.80 1.0 21.7ms
13 -1.117251190143 -15.35 -8.82 0.80 1.0 87.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.13 0.80 1.0 3.78s
2 -1.117251190143 + -15.35 -10.24 0.80 1.0 19.0ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 22.1s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998026 -2.46 0.80 1.0 13.5ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 19.0ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 19.3ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.5ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.7ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 20.0ms
7 -1.117520613465 -13.33 -7.39 0.80 1.0 63.0ms
8 -1.117520613465 -14.75 -8.18 0.80 1.0 21.3ms
9 -1.117520613465 + -15.05 -8.98 0.80 1.0 21.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264594 -1.69 0.80 1.0 13.5ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 33.1ms
3 -1.118340011112 -6.01 -3.44 0.80 1.0 19.4ms
4 -1.118340028552 -7.76 -4.36 0.80 1.0 19.4ms
5 -1.118340029137 -9.23 -5.12 0.80 1.0 19.9ms
6 -1.118340029164 -10.57 -5.62 0.80 1.0 20.1ms
7 -1.118340029166 -11.73 -6.68 0.80 1.0 26.1ms
8 -1.118340029166 -13.55 -7.58 0.80 1.0 21.2ms
9 -1.118340029166 + -14.70 -8.12 0.80 1.0 21.3ms
10 -1.118340029166 -14.57 -8.55 0.80 1.0 24.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -9.05 0.80 1.0 13.6ms
2 -1.118340029166 -14.33 -10.36 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -11.04 0.80 1.0 13.5ms
2 -1.118340029166 -14.37 -11.41 0.80 1.0 19.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029166 │ │ 0.00297055 │ 3.17s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 13.3ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 25.7ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 19.4ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 19.4ms
5 -1.118346866262 -12.08 -6.55 0.80 1.0 25.5ms
6 -1.118346866262 -13.42 -7.04 0.80 1.0 20.5ms
7 -1.118346866262 -14.65 -8.13 0.80 1.0 23.2ms
8 -1.118346866262 -14.81 -8.95 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.5ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.9ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 22.1ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.6ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 22.3ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 20.0ms
7 -1.118356101386 -13.61 -7.65 0.80 1.0 23.2ms
8 -1.118356101386 -14.61 -8.45 0.80 1.0 20.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.3ms
2 -1.118356101386 -14.48 -9.82 0.80 1.0 21.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.3ms
2 -1.118356101386 -14.57 -11.46 0.80 1.0 21.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 535ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 20.2ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.9ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 24.8ms
4 -1.118356102877 -14.11 -7.62 0.80 1.0 19.5ms
5 -1.118356102877 -14.88 -8.39 0.80 1.0 19.8ms
6 -1.118356102877 + -14.57 -8.89 0.80 1.0 22.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 15.9ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.7ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 21.6ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 20.1ms
5 -1.118356105120 -14.75 -7.86 0.80 1.0 20.2ms
6 -1.118356105120 -14.70 -8.35 0.80 1.0 83.9ms
7 -1.118356105120 + -14.29 -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 13.6ms
2 -1.118356105120 -14.57 -11.02 0.80 1.0 19.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.4ms
2 -1.118356105120 + -13.92 -12.44 0.80 1.0 19.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.05409e-8 │ 1.10s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -5.71633e-35, 1.36192e-34]u"a₀")
Atom(H, [ 1.44318, -1.10463e-34, -1.45975e-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: 1.4863658664184176 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 -8.335214985695 -0.80 0.80 7.7 785ms
2 -8.340938092061 -2.24 -1.69 0.80 1.3 620ms
3 -8.341205978179 -3.57 -2.80 0.80 3.0 35.2ms
4 -8.341214235019 -5.08 -3.27 0.80 3.0 37.8ms
5 -8.341214268061 -7.48 -3.94 0.80 1.3 26.7ms
6 -8.341214272418 -8.36 -5.08 0.80 2.3 31.8ms
7 -8.341214272515 -10.01 -5.55 0.80 3.7 38.5ms
8 -8.341214272518 -11.59 -6.29 0.80 2.7 32.4ms
9 -8.341214272518 -12.88 -7.25 0.80 3.0 35.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.35 0.80 1.0 629ms
2 -8.341214272518 -14.45 -9.49 0.80 1.0 26.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.73 0.80 1.0 25.8ms
2 -8.341214272518 + -Inf -10.11 0.80 1.0 30.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 5.69311e-33 │ 0.163458 │ -0.16 │ 30.4s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350570611004 -1.21 0.80 8.7 78.2ms
2 -8.351863723203 -2.89 -1.88 0.80 1.3 31.6ms
3 -8.351926648292 -4.20 -2.80 0.80 3.7 44.8ms
4 -8.351928506508 -5.73 -3.60 0.80 3.7 45.3ms
5 -8.351928514098 -8.12 -4.51 0.80 2.7 38.0ms
6 -8.351928514647 -9.26 -5.36 0.80 3.7 66.1ms
7 -8.351928514658 -10.95 -6.29 0.80 3.3 45.6ms
8 -8.351928514658 -12.63 -7.47 0.80 3.7 47.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353392798392 -1.52 0.80 8.3 76.3ms
2 -8.353730451491 -3.47 -2.21 0.80 1.0 28.2ms
3 -8.353757393306 -4.57 -2.97 0.80 4.0 46.0ms
4 -8.353757521326 -6.89 -4.20 0.80 2.3 35.2ms
5 -8.353757531654 -7.99 -4.91 0.80 3.7 45.7ms
6 -8.353757531714 -10.23 -5.70 0.80 2.7 41.2ms
7 -8.353757531717 -11.54 -6.92 0.80 3.0 42.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531717 -7.95 0.80 1.0 29.2ms
2 -8.353757531717 -14.15 -8.60 0.80 1.0 28.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531717 -8.87 0.80 1.0 24.4ms
2 -8.353757531717 + -14.75 -9.24 0.80 1.0 32.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531717 -9.51 0.80 1.0 24.7ms
2 -8.353757531717 -14.75 -9.72 0.80 1.0 27.6ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531717 │ │ │ 0.0156413 │ -0.016 │ 6.11s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353716973544 -3.20 0.80 7.7 68.0ms
2 -8.353759855993 -4.37 -3.87 0.80 1.0 36.8ms
3 -8.353770534781 -4.97 -3.86 0.80 6.0 61.3ms
4 -8.353770537136 -8.63 -4.18 0.80 1.0 28.7ms
5 -8.353770537174 -10.43 -4.52 0.80 2.3 35.7ms
6 -8.353770537388 -9.67 -5.47 0.80 1.7 38.9ms
7 -8.353770537395 -11.20 -6.50 0.80 3.3 43.3ms
8 -8.353770537395 -13.29 -7.43 0.80 3.0 41.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353769027753 -2.00 0.80 8.3 70.2ms
2 -8.353874403263 -3.98 -2.71 0.80 1.0 28.0ms
3 -8.353895042477 -4.69 -3.26 0.80 4.3 52.6ms
4 -8.353895103146 -7.22 -4.03 0.80 2.3 36.3ms
5 -8.353895110639 -8.13 -4.91 0.80 2.3 38.6ms
6 -8.353895110771 -9.88 -5.50 0.80 3.3 44.5ms
7 -8.353895110775 -11.35 -7.21 0.80 1.7 36.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110775 -7.36 0.80 1.0 24.8ms
2 -8.353895110775 -13.50 -7.48 0.80 1.0 28.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110775 -8.11 0.80 1.0 24.6ms
2 -8.353895110775 -14.05 -8.25 0.80 1.0 28.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110775 -8.73 0.80 1.0 25.1ms
2 -8.353895110775 -14.75 -8.85 0.80 1.0 28.1ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110775 │ -3.86 │ │ 0.000646176 │ -0.00065 │ 2.71s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353781957688 -3.64 0.80 8.0 69.9ms
2 -8.353872241507 -4.04 -3.87 0.80 1.0 28.3ms
3 -8.353895211671 -4.64 -4.07 0.80 6.3 63.3ms
4 -8.353895212407 -9.13 -4.30 0.80 1.0 27.9ms
5 -8.353895212404 + -11.46 -4.79 0.80 1.7 32.0ms
6 -8.353895212496 -10.03 -5.11 0.80 1.7 31.8ms
7 -8.353895212518 -10.66 -5.67 0.80 3.0 37.9ms
8 -8.353895212520 -11.73 -6.05 0.80 2.0 33.6ms
9 -8.353895212520 -12.75 -6.12 0.80 1.3 30.2ms
10 -8.353895212520 -12.71 -6.87 0.80 1.3 30.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353807253253 -3.29 0.80 8.3 71.2ms
2 -8.353876859175 -4.16 -3.77 0.80 1.0 38.4ms
3 -8.353895362316 -4.73 -3.88 0.80 6.0 62.7ms
4 -8.353895362604 -9.54 -4.41 0.80 1.0 28.5ms
5 -8.353895362663 -10.23 -4.63 0.80 1.3 30.0ms
6 -8.353895362680 -10.78 -5.18 0.80 1.0 28.8ms
7 -8.353895362685 -11.27 -6.08 0.80 2.0 34.0ms
8 -8.353895362686 -12.35 -6.57 0.80 3.7 43.0ms
9 -8.353895362686 -13.97 -7.82 0.80 2.0 35.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.54 0.80 1.0 24.5ms
2 -8.353895362686 + -Inf -9.60 0.80 1.0 27.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.76 0.80 1.0 23.7ms
2 -8.353895362686 + -14.75 -10.14 0.80 1.0 34.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -10.38 0.80 1.0 24.4ms
2 -8.353895362686 -14.45 -10.63 0.80 1.0 28.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ 2.04192e-32 │ 1.72451e-6 │ -1.7e-6 │ 2.80s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-1.09851e-35 5.27783 5.27783;
5.27783 -1.01748e-35 5.27783;
5.27783 5.27783 -1.01746e-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: 10.555662886539647 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.