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 -1.110648120007 -0.82 0.80 6.0 3.00s
2 -1.117203114096 -2.18 -1.87 0.80 1.0 1.59s
3 -1.117250477379 -4.32 -2.71 0.80 1.0 18.5ms
4 -1.117251157619 -6.17 -3.51 0.80 1.0 18.4ms
5 -1.117251187310 -7.53 -3.98 0.80 1.0 20.0ms
6 -1.117251190022 -8.57 -5.10 0.80 1.0 40.5ms
7 -1.117251190142 -9.92 -5.48 0.80 2.0 21.6ms
8 -1.117251190143 -11.97 -6.27 0.80 1.0 20.6ms
9 -1.117251190143 -13.22 -7.00 0.80 1.0 20.1ms
10 -1.117251190143 -14.26 -7.62 0.80 1.0 26.6ms
11 -1.117251190143 -14.75 -8.05 0.80 1.0 20.3ms
12 -1.117251190143 + -14.40 -8.30 0.80 1.0 20.6ms
13 -1.117251190143 -14.95 -8.61 0.80 2.0 25.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.02 0.80 1.0 2.76s
2 -1.117251190143 -15.35 -9.90 0.80 1.0 18.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 16.4s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998075 -2.46 0.80 1.0 12.6ms
2 -1.117520586052 -5.59 -3.49 0.80 1.0 18.0ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 51.1ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 18.4ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 18.6ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 18.8ms
7 -1.117520613465 -13.38 -7.40 0.80 1.0 34.0ms
8 -1.117520613465 -14.65 -8.20 0.80 1.0 19.9ms
9 -1.117520613465 + -15.35 -8.95 0.80 1.0 19.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264595 -1.69 0.80 1.0 12.3ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 17.5ms
3 -1.118340011112 -6.01 -3.44 0.80 1.0 23.8ms
4 -1.118340028553 -7.76 -4.36 0.80 1.0 18.1ms
5 -1.118340029138 -9.23 -5.12 0.80 1.0 25.0ms
6 -1.118340029164 -10.57 -5.62 0.80 1.0 19.0ms
7 -1.118340029166 -11.73 -6.68 0.80 1.0 21.5ms
8 -1.118340029166 -13.59 -7.58 0.80 1.0 19.9ms
9 -1.118340029166 -14.54 -8.12 0.80 1.0 21.8ms
10 -1.118340029166 + -Inf -8.55 0.80 1.0 20.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -9.05 0.80 1.0 12.4ms
2 -1.118340029166 + -14.48 -10.36 0.80 1.0 24.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -11.05 0.80 1.0 60.2ms
2 -1.118340029166 + -15.18 -11.42 0.80 1.0 18.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029166 │ │ 0.00297055 │ 2.40s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 12.7ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 17.9ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 18.0ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 18.4ms
5 -1.118346866262 -12.09 -6.55 0.80 1.0 18.7ms
6 -1.118346866262 -13.41 -7.04 0.80 1.0 19.1ms
7 -1.118346866262 -14.42 -8.13 0.80 1.0 20.1ms
8 -1.118346866262 -15.05 -8.95 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.0ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.0ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 17.9ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.8ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 25.5ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.1ms
7 -1.118356101386 -13.64 -7.65 0.80 1.0 19.6ms
8 -1.118356101386 -14.57 -8.45 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 12.3ms
2 -1.118356101386 -14.61 -9.82 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 12.4ms
2 -1.118356101386 + -14.75 -11.46 0.80 1.0 17.8ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 776ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 19.7ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 17.8ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 18.0ms
4 -1.118356102877 -14.27 -7.62 0.80 1.0 18.3ms
5 -1.118356102877 + -Inf -8.40 0.80 1.0 25.0ms
6 -1.118356102877 -15.35 -8.89 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 12.5ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 20.3ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.0ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 18.1ms
5 -1.118356105120 -15.65 -7.86 0.80 1.0 20.8ms
6 -1.118356105120 -15.18 -8.35 0.80 1.0 18.9ms
7 -1.118356105120 -15.18 -9.45 0.80 1.0 25.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 15.2ms
2 -1.118356105120 -14.29 -11.02 0.80 1.0 17.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 12.6ms
2 -1.118356105120 + -14.88 -12.44 0.80 1.0 20.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.07584e-8 │ 458ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -4.27805e-34, -4.36545e-34]u"a₀")
Atom(H, [ 1.44318, -6.77119e-35, -4.35792e-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: 1.486365865605996 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 -8.335333784646 -0.80 0.80 7.3 654ms
2 -8.340976976498 -2.25 -1.70 0.80 1.3 66.6ms
3 -8.341206328234 -3.64 -2.78 0.80 2.7 30.9ms
4 -8.341214246140 -5.10 -3.24 0.80 3.3 35.8ms
5 -8.341214269586 -7.63 -3.95 0.80 1.0 24.3ms
6 -8.341214272366 -8.56 -5.01 0.80 1.7 27.4ms
7 -8.341214272513 -9.83 -5.39 0.80 3.3 54.1ms
8 -8.341214272518 -11.34 -6.65 0.80 2.0 30.0ms
9 -8.341214272518 -12.71 -7.06 0.80 3.0 36.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.59 0.80 1.0 61.9ms
2 -8.341214272518 -14.75 -8.65 0.80 1.0 24.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.23 0.80 1.0 22.1ms
2 -8.341214272518 -14.75 -9.45 0.80 1.0 24.4ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 3.38808e-33 │ 0.163458 │ -0.16 │ 26.9s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350591370774 -1.21 0.80 8.7 64.7ms
2 -8.351851191050 -2.90 -1.89 0.80 1.0 25.2ms
3 -8.351927032813 -4.12 -2.71 0.80 3.3 36.2ms
4 -8.351928511076 -5.83 -3.72 0.80 3.7 38.2ms
5 -8.351928514359 -8.48 -4.88 0.80 2.0 30.4ms
6 -8.351928514644 -9.55 -5.97 0.80 4.3 41.5ms
7 -8.351928514645 -11.95 -6.72 0.80 3.7 38.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353410023175 -1.53 0.80 7.7 81.7ms
2 -8.353733179306 -3.49 -2.21 0.80 1.0 24.7ms
3 -8.353757369392 -4.62 -2.98 0.80 3.3 35.9ms
4 -8.353757521204 -6.82 -4.21 0.80 2.0 29.6ms
5 -8.353757531702 -7.98 -5.19 0.80 4.0 40.3ms
6 -8.353757531725 -10.64 -5.70 0.80 3.3 35.2ms
7 -8.353757531726 -11.82 -6.58 0.80 2.3 31.5ms
8 -8.353757531726 -13.41 -7.80 0.80 3.3 42.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531727 -8.91 0.80 1.0 22.4ms
2 -8.353757531726 + -14.45 -9.54 0.80 1.0 26.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531727 -9.70 0.80 1.0 21.9ms
2 -8.353757531727 + -Inf -10.06 0.80 1.0 28.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531726 -10.23 0.80 1.0 22.2ms
2 -8.353757531726 + -Inf -10.54 0.80 1.0 25.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531726 │ │ │ 0.0156413 │ -0.016 │ 5.45s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353716591672 -3.21 0.80 7.7 56.9ms
2 -8.353760074031 -4.36 -3.80 0.80 1.0 24.8ms
3 -8.353770534487 -4.98 -3.93 0.80 5.3 49.2ms
4 -8.353770537203 -8.57 -4.34 0.80 1.3 41.6ms
5 -8.353770537300 -10.02 -4.81 0.80 2.0 29.6ms
6 -8.353770537400 -10.00 -5.79 0.80 1.7 28.2ms
7 -8.353770537403 -11.55 -6.83 0.80 2.7 34.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353796589590 -2.00 0.80 7.7 62.7ms
2 -8.353881216350 -4.07 -2.71 0.80 1.0 25.3ms
3 -8.353895035580 -4.86 -3.36 0.80 3.7 38.7ms
4 -8.353895105435 -7.16 -4.13 0.80 2.0 33.8ms
5 -8.353895110652 -8.28 -4.84 0.80 2.7 34.9ms
6 -8.353895110778 -9.90 -6.42 0.80 3.0 35.3ms
7 -8.353895110780 -11.91 -7.05 0.80 4.7 46.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110780 -8.12 0.80 1.0 22.5ms
2 -8.353895110780 + -14.75 -9.02 0.80 1.0 25.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110780 -9.67 0.80 1.0 22.2ms
2 -8.353895110780 + -14.75 -9.90 0.80 1.0 25.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110780 -10.19 0.80 1.0 22.2ms
2 -8.353895110780 + -14.75 -10.47 0.80 1.0 25.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110780 │ -3.86 │ │ 0.000646168 │ -0.00065 │ 2.63s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353821428151 -3.52 0.80 8.0 58.0ms
2 -8.353881314098 -4.22 -3.64 0.80 1.0 29.1ms
3 -8.353895209560 -4.86 -3.87 0.80 5.7 49.9ms
4 -8.353895211103 -8.81 -4.08 0.80 1.0 25.2ms
5 -8.353895212402 -8.89 -4.70 0.80 1.0 25.3ms
6 -8.353895212512 -9.96 -5.07 0.80 2.3 34.3ms
7 -8.353895212521 -11.06 -5.65 0.80 1.7 29.3ms
8 -8.353895212522 -11.94 -5.88 0.80 1.7 30.2ms
9 -8.353895212522 -12.69 -6.84 0.80 1.7 31.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353804957152 -3.33 0.80 7.7 57.4ms
2 -8.353879691275 -4.13 -3.70 0.80 1.0 29.3ms
3 -8.353895362114 -4.80 -3.95 0.80 5.7 49.8ms
4 -8.353895362320 -9.68 -4.34 0.80 1.0 25.2ms
5 -8.353895362647 -9.49 -4.81 0.80 1.0 28.6ms
6 -8.353895362681 -10.48 -5.07 0.80 1.7 29.1ms
7 -8.353895362685 -11.35 -5.67 0.80 1.3 27.1ms
8 -8.353895362686 -12.26 -6.45 0.80 2.3 34.6ms
9 -8.353895362686 -13.35 -6.89 0.80 2.7 34.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.34 0.80 1.0 22.8ms
2 -8.353895362686 -14.45 -9.12 0.80 1.0 28.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.40 0.80 1.0 22.1ms
2 -8.353895362686 + -Inf -9.63 0.80 1.0 29.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.91 0.80 1.0 22.3ms
2 -8.353895362686 -14.75 -10.14 0.80 1.0 28.7ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ 1.03276e-32 │ 1.71676e-6 │ -1.7e-6 │ 3.45s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [5.08747e-36 5.27783 5.27783;
5.27783 5.08747e-36 5.27783;
5.27783 5.27783 5.08747e-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: 10.555662919818642 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.