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 -1.110765970214 -0.83 0.80 7.0 3.84s
2 -1.117199372043 -2.19 -1.89 0.80 1.0 1.64s
3 -1.117250262024 -4.29 -2.66 0.80 1.0 20.3ms
4 -1.117251109877 -6.07 -3.35 0.80 1.0 23.3ms
5 -1.117251186145 -7.12 -4.14 0.80 1.0 25.5ms
6 -1.117251190084 -8.40 -4.95 0.80 2.0 26.7ms
7 -1.117251190138 -10.27 -5.38 0.80 1.0 23.8ms
8 -1.117251190143 -11.29 -6.69 0.80 1.0 25.4ms
9 -1.117251190143 -12.99 -6.89 0.80 2.0 42.0ms
10 -1.117251190143 -14.45 -7.78 0.80 1.0 21.1ms
11 -1.117251190143 + -15.65 -7.90 0.80 2.0 22.8ms
12 -1.117251190143 -14.88 -8.38 0.80 1.0 21.1ms
13 -1.117251190143 + -Inf -9.15 0.80 1.0 21.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.85 0.80 1.0 3.43s
2 -1.117251190143 + -14.88 -10.63 0.80 1.0 74.4ms
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 -1.117517998058 -2.46 0.80 1.0 13.3ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 59.5ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 18.8ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.1ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 33.1ms
6 -1.117520613465 -12.13 -6.42 0.80 1.0 19.7ms
7 -1.117520613465 -13.35 -7.39 0.80 1.0 20.0ms
8 -1.117520613465 -14.54 -8.18 0.80 1.0 22.8ms
9 -1.117520613465 + -14.75 -8.97 0.80 1.0 20.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264593 -1.69 0.80 1.0 18.5ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 18.4ms
3 -1.118340011112 -6.01 -3.44 0.80 1.0 18.5ms
4 -1.118340028552 -7.76 -4.36 0.80 1.0 24.1ms
5 -1.118340029137 -9.23 -5.12 0.80 1.0 19.4ms
6 -1.118340029164 -10.57 -5.62 0.80 1.0 22.2ms
7 -1.118340029166 -11.73 -6.68 0.80 1.0 20.2ms
8 -1.118340029166 -13.56 -7.58 0.80 1.0 22.9ms
9 -1.118340029166 -15.35 -8.12 0.80 1.0 20.5ms
10 -1.118340029166 + -14.65 -8.55 0.80 1.0 22.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -9.05 0.80 1.0 13.2ms
2 -1.118340029166 + -14.65 -10.36 0.80 1.0 18.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -11.04 0.80 1.0 13.1ms
2 -1.118340029166 + -15.05 -11.41 0.80 1.0 21.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029166 │ │ 0.00297055 │ 2.91s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 13.0ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 25.2ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 18.7ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 24.3ms
5 -1.118346866262 -12.08 -6.55 0.80 1.0 19.5ms
6 -1.118346866262 -13.43 -7.04 0.80 1.0 22.1ms
7 -1.118346866262 -14.57 -8.13 0.80 1.0 20.1ms
8 -1.118346866262 -14.88 -8.95 0.80 1.0 22.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 75.8ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.4ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 535ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.2ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.3ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.8ms
7 -1.118356101386 -13.58 -7.65 0.80 1.0 20.3ms
8 -1.118356101386 + -Inf -8.45 0.80 1.0 20.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 14.5ms
2 -1.118356101386 + -15.05 -9.82 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 14.0ms
2 -1.118356101386 + -14.26 -11.46 0.80 1.0 19.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ 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 13.8ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 19.1ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 25.5ms
4 -1.118356102877 -14.22 -7.62 0.80 1.0 26.2ms
5 -1.118356102877 + -14.95 -8.40 0.80 1.0 27.1ms
6 -1.118356102877 + -14.45 -8.89 0.80 1.0 27.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.0ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.7ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.7ms
4 -1.118356105120 -13.23 -7.09 0.80 1.0 19.1ms
5 -1.118356105120 -14.31 -7.86 0.80 1.0 19.7ms
6 -1.118356105120 + -14.70 -8.35 0.80 1.0 19.9ms
7 -1.118356105120 + -14.61 -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 12.9ms
2 -1.118356105120 -14.16 -11.02 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 12.8ms
2 -1.118356105120 -14.09 -12.44 0.80 1.0 18.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.20243e-8 │ 489ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, 8.68619e-35, 6.68848e-35]u"a₀")
Atom(H, [ 1.44318, 1.8469e-34, -4.20125e-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: 1.4863658608783799 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 -8.335357114006 -0.81 0.80 7.7 765ms
2 -8.340914067453 -2.26 -1.69 0.80 1.3 623ms
3 -8.341207675210 -3.53 -2.76 0.80 2.7 31.6ms
4 -8.341214238091 -5.18 -3.28 0.80 3.3 36.9ms
5 -8.341214268822 -7.51 -4.00 0.80 1.3 26.3ms
6 -8.341214272290 -8.46 -5.07 0.80 2.0 29.2ms
7 -8.341214272515 -9.65 -5.52 0.80 3.3 37.9ms
8 -8.341214272518 -11.58 -6.82 0.80 2.0 28.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.51 0.80 1.0 620ms
2 -8.341214272518 -14.05 -7.61 0.80 1.0 24.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.36 0.80 1.0 21.6ms
2 -8.341214272518 + -Inf -8.45 0.80 1.0 24.2ms
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 -8.350386906504 -1.20 0.80 9.0 117ms
2 -8.351865791044 -2.83 -1.88 0.80 1.3 27.8ms
3 -8.351925780596 -4.22 -2.79 0.80 3.3 37.6ms
4 -8.351928499448 -5.57 -3.65 0.80 3.0 38.3ms
5 -8.351928514601 -7.82 -4.86 0.80 3.7 38.9ms
6 -8.351928514768 -9.78 -5.56 0.80 4.0 59.0ms
7 -8.351928514768 -12.43 -7.40 0.80 2.0 32.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353386564355 -1.52 0.80 7.7 60.7ms
2 -8.353730378086 -3.46 -2.21 0.80 1.0 25.1ms
3 -8.353757327440 -4.57 -2.97 0.80 3.7 45.3ms
4 -8.353757525686 -6.70 -4.21 0.80 2.0 32.1ms
5 -8.353757531622 -8.23 -5.31 0.80 4.3 42.8ms
6 -8.353757531637 -10.84 -6.44 0.80 2.7 36.2ms
7 -8.353757531637 -13.03 -7.61 0.80 3.3 44.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531637 -8.36 0.80 1.0 22.7ms
2 -8.353757531637 + -14.45 -9.63 0.80 1.0 30.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531637 -9.84 0.80 1.0 22.7ms
2 -8.353757531637 + -Inf -10.12 0.80 1.0 25.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531637 -10.24 0.80 1.0 22.8ms
2 -8.353757531637 + -Inf -10.77 0.80 1.0 25.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531637 │ │ 4.34057e-32 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353730130608 -3.21 0.80 8.3 68.0ms
2 -8.353763781480 -4.47 -3.90 0.80 1.0 26.9ms
3 -8.353770533779 -5.17 -3.90 0.80 6.0 57.2ms
4 -8.353770536775 -8.52 -4.21 0.80 1.0 26.9ms
5 -8.353770537134 -9.44 -4.65 0.80 2.3 33.2ms
6 -8.353770537316 -9.74 -5.63 0.80 2.0 33.6ms
7 -8.353770537321 -11.28 -6.76 0.80 3.0 45.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353748886089 -2.00 0.80 7.7 64.8ms
2 -8.353853514962 -3.98 -2.70 0.80 1.0 27.1ms
3 -8.353895044681 -4.38 -3.30 0.80 4.7 47.4ms
4 -8.353895105009 -7.22 -4.18 0.80 2.0 32.5ms
5 -8.353895110705 -8.24 -4.97 0.80 2.7 38.5ms
6 -8.353895110776 -10.15 -5.71 0.80 3.0 38.2ms
7 -8.353895110778 -11.72 -6.74 0.80 2.7 36.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110778 -7.28 0.80 1.0 23.2ms
2 -8.353895110778 -14.27 -8.31 0.80 1.0 32.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110778 -8.74 0.80 1.0 24.0ms
2 -8.353895110778 -14.45 -9.00 0.80 1.0 27.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110778 -9.26 0.80 1.0 23.6ms
2 -8.353895110778 + -14.75 -9.49 0.80 1.0 27.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110778 │ -3.86 │ │ 0.000646171 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353807289735 -3.47 0.80 8.0 67.4ms
2 -8.353875514763 -4.17 -3.78 0.80 1.0 32.0ms
3 -8.353895210681 -4.71 -3.79 0.80 6.0 59.8ms
4 -8.353895212052 -8.86 -3.98 0.80 1.0 27.9ms
5 -8.353895212325 -9.56 -4.48 0.80 1.0 27.9ms
6 -8.353895212476 -9.82 -4.72 0.80 1.0 31.6ms
7 -8.353895212510 -10.47 -5.07 0.80 1.0 28.3ms
8 -8.353895212519 -11.08 -5.32 0.80 2.7 37.7ms
9 -8.353895212521 -11.59 -6.18 0.80 1.0 31.4ms
10 -8.353895212521 -12.99 -6.01 0.80 3.7 45.1ms
11 -8.353895212521 -12.98 -6.66 0.80 1.0 29.2ms
12 -8.353895212521 -14.05 -6.98 0.80 1.0 33.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353758934112 -3.22 0.80 8.7 69.4ms
2 -8.353862987843 -3.98 -3.65 0.80 1.0 32.3ms
3 -8.353895359390 -4.49 -3.70 0.80 6.0 58.5ms
4 -8.353895362181 -8.55 -3.94 0.80 1.0 27.5ms
5 -8.353895361914 + -9.57 -4.24 0.80 1.3 29.6ms
6 -8.353895362598 -9.17 -4.76 0.80 1.3 33.9ms
7 -8.353895362673 -10.12 -5.18 0.80 2.7 38.0ms
8 -8.353895362685 -10.93 -6.02 0.80 2.7 37.1ms
9 -8.353895362686 -12.14 -6.96 0.80 3.7 43.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.85 0.80 1.0 24.0ms
2 -8.353895362686 + -Inf -8.82 0.80 1.0 27.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.08 0.80 1.0 23.6ms
2 -8.353895362686 + -14.45 -9.47 0.80 1.0 26.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.61 0.80 1.0 24.0ms
2 -8.353895362686 + -14.75 -10.04 0.80 1.0 27.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.71992e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-6.3593e-37 5.27783 5.27783;
5.27783 -6.3593e-37 5.27783;
5.27783 5.27783 -6.3593e-37]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: 10.555662907880127 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.