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 -1.110570348676 -0.82 0.80 6.0 3.84s
2 -1.117194566642 -2.18 -1.88 0.80 1.0 1.64s
3 -1.117249713325 -4.26 -2.72 0.80 1.0 20.0ms
4 -1.117251154967 -5.84 -3.43 0.80 2.0 23.7ms
5 -1.117251189291 -7.46 -4.20 0.80 1.0 27.2ms
6 -1.117251190074 -9.11 -4.79 0.80 1.0 25.2ms
7 -1.117251190141 -10.18 -5.88 0.80 1.0 23.9ms
8 -1.117251190143 -11.61 -6.24 0.80 2.0 54.5ms
9 -1.117251190143 -13.11 -7.30 0.80 1.0 20.9ms
10 -1.117251190143 -14.33 -7.74 0.80 2.0 22.6ms
11 -1.117251190143 -14.61 -8.05 0.80 1.0 20.8ms
12 -1.117251190143 + -14.45 -8.56 0.80 1.0 20.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.12 0.80 1.0 3.44s
2 -1.117251190143 + -14.75 -10.25 0.80 1.0 78.4ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 20.3s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997733 -2.46 0.80 1.0 53.8ms
2 -1.117520586050 -5.59 -3.49 0.80 1.0 18.4ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 18.4ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 18.8ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 32.8ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 19.3ms
7 -1.117520613465 -13.34 -7.41 0.80 1.0 19.8ms
8 -1.117520613465 + -14.61 -8.23 0.80 1.0 22.9ms
9 -1.117520613465 -14.42 -8.94 0.80 1.0 20.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264593 -1.69 0.80 1.0 12.8ms
2 -1.118339043932 -4.04 -2.70 0.80 1.0 23.5ms
3 -1.118340011111 -6.01 -3.44 0.80 1.0 18.6ms
4 -1.118340028552 -7.76 -4.36 0.80 1.0 23.5ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 19.2ms
6 -1.118340029163 -10.57 -5.62 0.80 1.0 21.4ms
7 -1.118340029165 -11.73 -6.68 0.80 1.0 19.9ms
8 -1.118340029165 -13.51 -7.58 0.80 1.0 22.6ms
9 -1.118340029165 + -14.45 -8.12 0.80 1.0 20.4ms
10 -1.118340029165 -14.42 -8.55 0.80 1.0 22.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -9.05 0.80 1.0 19.1ms
2 -1.118340029165 -14.54 -10.36 0.80 1.0 18.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -11.05 0.80 1.0 12.9ms
2 -1.118340029165 -14.45 -11.42 0.80 1.0 18.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029165 │ │ 0.00297055 │ 2.96s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 12.9ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 24.2ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 18.4ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 79.9ms
5 -1.118346866261 -12.08 -6.55 0.80 1.0 19.2ms
6 -1.118346866261 -13.46 -7.04 0.80 1.0 518ms
7 -1.118346866261 -15.35 -8.13 0.80 1.0 20.0ms
8 -1.118346866261 -14.65 -8.95 0.80 1.0 19.9ms
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 18.1ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 19.2ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 21.6ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 20.6ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 21.4ms
7 -1.118356101386 -13.59 -7.65 0.80 1.0 21.7ms
8 -1.118356101386 -14.88 -8.45 0.80 1.0 21.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 16.2ms
2 -1.118356101386 + -14.51 -9.82 0.80 1.0 25.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 17.0ms
2 -1.118356101386 + -14.24 -11.46 0.80 1.0 29.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 1.10s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 12.9ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.2ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 19.3ms
4 -1.118356102877 -14.14 -7.62 0.80 1.0 18.7ms
5 -1.118356102877 + -14.42 -8.40 0.80 1.0 18.9ms
6 -1.118356102877 -14.15 -8.89 0.80 1.0 19.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.2ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.2ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.3ms
4 -1.118356105120 -13.19 -7.09 0.80 1.0 18.6ms
5 -1.118356105120 + -14.57 -7.86 0.80 1.0 18.9ms
6 -1.118356105120 -14.57 -8.35 0.80 1.0 28.3ms
7 -1.118356105120 + -14.54 -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 12.7ms
2 -1.118356105120 + -14.75 -11.02 0.80 1.0 18.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 12.8ms
2 -1.118356105120 + -13.93 -12.44 0.80 1.0 18.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 9.51356e-9 │ 454ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, 5.21053e-35, -5.65582e-36]u"a₀")
Atom(H, [ 1.44318, 2.91957e-34, 1.06016e-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: 1.4863658702550802 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 -8.335364288237 -0.81 0.80 7.3 781ms
2 -8.340918541493 -2.26 -1.69 0.80 1.3 612ms
3 -8.341202931456 -3.55 -2.77 0.80 2.7 32.5ms
4 -8.341214229234 -4.95 -3.24 0.80 4.0 39.7ms
5 -8.341214266491 -7.43 -3.89 0.80 1.0 24.1ms
6 -8.341214272247 -8.24 -5.08 0.80 2.0 28.8ms
7 -8.341214272512 -9.58 -5.45 0.80 4.0 55.8ms
8 -8.341214272518 -11.25 -6.28 0.80 2.3 30.7ms
9 -8.341214272518 -12.63 -7.11 0.80 3.0 36.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.82 0.80 1.0 612ms
2 -8.341214272518 + -Inf -9.33 0.80 1.0 23.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.63 0.80 1.0 21.3ms
2 -8.341214272518 + -14.75 -10.07 0.80 1.0 23.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350376665566 -1.20 0.80 9.7 66.9ms
2 -8.351867600764 -2.83 -1.88 0.80 2.0 29.6ms
3 -8.351927855788 -4.22 -2.79 0.80 3.3 36.6ms
4 -8.351928505164 -6.19 -3.77 0.80 3.7 39.7ms
5 -8.351928514442 -8.03 -4.92 0.80 2.7 33.6ms
6 -8.351928514648 -9.69 -5.48 0.80 3.7 40.3ms
7 -8.351928514653 -11.26 -5.98 0.80 3.0 35.8ms
8 -8.351928514654 -12.25 -7.37 0.80 3.0 35.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353378158264 -1.52 0.80 8.0 62.5ms
2 -8.353723906927 -3.46 -2.21 0.80 1.0 24.9ms
3 -8.353757009225 -4.48 -2.99 0.80 4.3 81.1ms
4 -8.353757523591 -6.29 -4.05 0.80 2.3 30.8ms
5 -8.353757531701 -8.09 -5.37 0.80 3.7 38.7ms
6 -8.353757531720 -10.74 -6.04 0.80 3.0 35.1ms
7 -8.353757531720 -12.60 -7.10 0.80 1.7 29.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -7.93 0.80 1.0 21.4ms
2 -8.353757531720 -14.75 -8.54 0.80 1.0 24.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -9.01 0.80 1.0 21.3ms
2 -8.353757531720 -14.75 -9.23 0.80 1.0 41.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -9.63 0.80 1.0 21.5ms
2 -8.353757531720 + -14.75 -9.89 0.80 1.0 24.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531720 │ │ │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353725951306 -3.20 0.80 8.0 58.5ms
2 -8.353762408691 -4.44 -3.90 0.80 1.0 24.2ms
3 -8.353770533289 -5.09 -3.91 0.80 6.0 51.0ms
4 -8.353770536869 -8.45 -4.27 0.80 1.0 28.1ms
5 -8.353770537158 -9.54 -4.63 0.80 2.3 30.6ms
6 -8.353770537388 -9.64 -5.38 0.80 2.0 29.4ms
7 -8.353770537397 -11.04 -6.10 0.80 2.7 35.0ms
8 -8.353770537398 -12.13 -7.31 0.80 3.3 36.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353771718651 -2.00 0.80 8.0 61.7ms
2 -8.353873547666 -3.99 -2.70 0.80 1.0 24.3ms
3 -8.353895041094 -4.67 -3.28 0.80 4.3 41.2ms
4 -8.353895103771 -7.20 -4.14 0.80 2.0 29.3ms
5 -8.353895110696 -8.16 -5.04 0.80 2.7 37.6ms
6 -8.353895110768 -10.14 -5.66 0.80 3.7 38.0ms
7 -8.353895110770 -11.63 -6.70 0.80 2.3 31.5ms
8 -8.353895110770 -13.30 -7.84 0.80 3.3 41.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110770 -8.49 0.80 1.0 21.4ms
2 -8.353895110770 + -Inf -9.58 0.80 1.0 28.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110770 -10.21 0.80 1.0 21.4ms
2 -8.353895110770 + -Inf -10.45 0.80 1.0 24.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110770 -10.73 0.80 1.0 21.6ms
2 -8.353895110770 + -Inf -11.02 0.80 1.0 24.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110770 │ -3.86 │ │ 0.00064618 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353727415014 -3.44 0.80 8.3 63.2ms
2 -8.353850174371 -3.91 -3.64 0.80 1.0 27.4ms
3 -8.353895211553 -4.35 -3.78 0.80 6.3 53.4ms
4 -8.353895211710 -9.80 -3.95 0.80 1.0 26.8ms
5 -8.353895212246 -9.27 -4.33 0.80 1.0 24.4ms
6 -8.353895212445 -9.70 -4.66 0.80 1.0 25.1ms
7 -8.353895212512 -10.17 -5.51 0.80 1.7 27.7ms
8 -8.353895212514 -11.62 -5.40 0.80 3.0 37.7ms
9 -8.353895212516 -11.68 -6.00 0.80 1.0 25.0ms
10 -8.353895212517 -12.57 -6.50 0.80 2.0 29.8ms
11 -8.353895212517 -13.71 -7.09 0.80 2.0 33.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353754536809 -3.30 0.80 8.7 60.6ms
2 -8.353838338476 -4.08 -3.88 0.80 1.0 28.7ms
3 -8.353895361276 -4.24 -3.96 0.80 6.3 53.1ms
4 -8.353895362516 -8.91 -4.33 0.80 1.3 25.9ms
5 -8.353895362579 -10.20 -4.67 0.80 2.3 29.9ms
6 -8.353895362668 -10.05 -5.14 0.80 1.0 27.8ms
7 -8.353895362685 -10.77 -5.78 0.80 3.0 34.1ms
8 -8.353895362686 -12.02 -6.33 0.80 2.0 31.3ms
9 -8.353895362686 -13.14 -6.99 0.80 3.0 36.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.56 0.80 1.0 21.2ms
2 -8.353895362686 -14.75 -8.68 0.80 1.0 27.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.11 0.80 1.0 21.5ms
2 -8.353895362686 + -Inf -9.51 0.80 1.0 24.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.77 0.80 1.0 21.6ms
2 -8.353895362686 -14.45 -10.10 0.80 1.0 24.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.71694e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
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: 10.555662919276667 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.