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 -1.110721921403 -0.82 0.80 8.0 3.09s
2 -1.117206965667 -2.19 -1.88 0.80 1.0 1.51s
3 -1.117250658383 -4.36 -2.72 0.80 1.0 19.7ms
4 -1.117251178588 -6.28 -3.59 0.80 1.0 19.6ms
5 -1.117251188940 -7.98 -4.04 0.80 1.0 20.4ms
6 -1.117251190020 -8.97 -4.66 0.80 1.0 20.3ms
7 -1.117251190139 -9.93 -5.76 0.80 1.0 42.4ms
8 -1.117251190143 -11.36 -6.33 0.80 2.0 23.8ms
9 -1.117251190143 -13.35 -7.30 0.80 1.0 21.5ms
10 -1.117251190143 + -Inf -7.65 0.80 1.0 28.3ms
11 -1.117251190143 -15.05 -8.35 0.80 1.0 21.8ms
12 -1.117251190143 + -15.65 -9.44 0.80 1.0 21.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.81 0.80 1.0 3.33s
2 -1.117251190143 -14.88 -10.45 0.80 1.0 99.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 20.1s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517996977 -2.46 0.80 1.0 13.7ms
2 -1.117520586059 -5.59 -3.49 0.80 1.0 67.4ms
3 -1.117520612995 -7.57 -4.23 0.80 1.0 19.6ms
4 -1.117520613449 -9.34 -5.15 0.80 1.0 19.9ms
5 -1.117520613464 -10.81 -5.93 0.80 1.0 20.3ms
6 -1.117520613465 -12.15 -6.44 0.80 1.0 20.5ms
7 -1.117520613465 -13.37 -7.45 0.80 1.0 37.3ms
8 -1.117520613465 -14.81 -8.26 0.80 1.0 21.5ms
9 -1.117520613465 + -Inf -8.93 0.80 1.0 21.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264607 -1.69 0.80 1.0 13.6ms
2 -1.118339043932 -4.04 -2.70 0.80 1.0 19.2ms
3 -1.118340011111 -6.01 -3.44 0.80 1.0 25.2ms
4 -1.118340028552 -7.76 -4.36 0.80 1.0 19.8ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 20.0ms
6 -1.118340029163 -10.57 -5.62 0.80 1.0 25.8ms
7 -1.118340029165 -11.73 -6.68 0.80 1.0 20.8ms
8 -1.118340029165 -13.57 -7.58 0.80 1.0 23.8ms
9 -1.118340029165 -14.75 -8.12 0.80 1.0 21.5ms
10 -1.118340029165 + -14.54 -8.55 0.80 1.0 24.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -9.05 0.80 1.0 20.1ms
2 -1.118340029165 + -14.65 -10.36 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -11.05 0.80 1.0 19.4ms
2 -1.118340029165 -14.75 -11.42 0.80 1.0 28.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029165 │ │ 0.00297055 │ 3.97s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 14.5ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 20.8ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 21.4ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 23.7ms
5 -1.118346866261 -12.09 -6.55 0.80 1.0 24.9ms
6 -1.118346866261 -13.40 -7.04 0.80 1.0 24.4ms
7 -1.118346866261 -14.88 -8.13 0.80 1.0 25.1ms
8 -1.118346866261 + -14.75 -8.95 0.80 1.0 23.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 15.6ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 20.6ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 22.5ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 21.4ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 21.9ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 29.1ms
7 -1.118356101386 -13.67 -7.65 0.80 1.0 30.4ms
8 -1.118356101386 -14.81 -8.45 0.80 1.0 30.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 14.6ms
2 -1.118356101386 -14.65 -9.82 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 15.3ms
2 -1.118356101386 -14.88 -11.46 0.80 1.0 19.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 601ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.9ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 19.6ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 29.3ms
4 -1.118356102877 -14.15 -7.62 0.80 1.0 20.3ms
5 -1.118356102877 + -14.75 -8.39 0.80 1.0 20.4ms
6 -1.118356102877 -14.75 -8.89 0.80 1.0 20.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.9ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 19.5ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 27.3ms
4 -1.118356105120 -13.21 -7.09 0.80 1.0 20.3ms
5 -1.118356105120 -14.51 -7.86 0.80 1.0 20.7ms
6 -1.118356105120 + -15.05 -8.35 0.80 1.0 21.2ms
7 -1.118356105120 -14.95 -9.45 0.80 1.0 21.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 14.1ms
2 -1.118356105120 + -14.75 -11.02 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 14.0ms
2 -1.118356105120 -14.01 -12.44 0.80 1.0 23.9ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.02426e-8 │ 498ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -1.75432e-17, 8.79454e-18]u"a₀")
Atom(H, [ 1.44318, 2.19824e-17, -1.30115e-17]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658676568838 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 -8.335101517142 -0.80 0.80 8.0 850ms
2 -8.340934961357 -2.23 -1.69 0.80 1.3 694ms
3 -8.341201972005 -3.57 -2.78 0.80 2.7 39.8ms
4 -8.341214226835 -4.91 -3.21 0.80 3.3 44.6ms
5 -8.341214266539 -7.40 -3.82 0.80 1.3 30.1ms
6 -8.341214272301 -8.24 -4.91 0.80 2.0 33.7ms
7 -8.341214272513 -9.68 -5.40 0.80 3.3 43.7ms
8 -8.341214272517 -11.33 -5.88 0.80 1.3 48.3ms
9 -8.341214272518 -12.17 -6.99 0.80 2.0 35.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.23 0.80 1.0 663ms
2 -8.341214272518 + -Inf -8.62 0.80 1.0 27.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.14 0.80 1.0 24.7ms
2 -8.341214272518 + -14.75 -9.14 0.80 1.0 27.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.61313e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350571382238 -1.21 0.80 9.3 77.9ms
2 -8.351866330848 -2.89 -1.88 0.80 2.0 33.6ms
3 -8.351926382707 -4.22 -2.82 0.80 3.0 37.9ms
4 -8.351928505900 -5.67 -3.64 0.80 4.0 66.1ms
5 -8.351928514243 -8.08 -4.56 0.80 3.3 41.0ms
6 -8.351928514658 -9.38 -5.39 0.80 3.7 44.3ms
7 -8.351928514669 -10.99 -6.38 0.80 4.0 44.7ms
8 -8.351928514669 -12.83 -7.72 0.80 4.0 54.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353375539432 -1.52 0.80 8.3 71.2ms
2 -8.353727261587 -3.45 -2.21 0.80 1.0 97.0ms
3 -8.353757298734 -4.52 -2.97 0.80 3.7 44.6ms
4 -8.353757522997 -6.65 -4.16 0.80 3.0 38.3ms
5 -8.353757531660 -8.06 -4.99 0.80 3.3 44.3ms
6 -8.353757531706 -10.33 -5.67 0.80 3.0 653ms
7 -8.353757531709 -11.56 -7.06 0.80 3.0 41.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531709 -8.13 0.80 1.0 24.5ms
2 -8.353757531709 + -Inf -8.79 0.80 1.0 27.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531709 -9.03 0.80 1.0 24.0ms
2 -8.353757531709 + -Inf -9.34 0.80 1.0 27.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531709 -9.49 0.80 1.0 24.0ms
2 -8.353757531709 + -14.45 -9.86 0.80 1.0 26.6ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531709 │ │ 1.02109e-14 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353686036548 -3.16 0.80 8.7 66.8ms
2 -8.353753186509 -4.17 -3.78 0.80 1.0 26.1ms
3 -8.353770534162 -4.76 -3.81 0.80 7.3 70.1ms
4 -8.353770537162 -8.52 -4.40 0.80 1.0 27.1ms
5 -8.353770537195 -10.49 -4.53 0.80 2.0 33.4ms
6 -8.353770537367 -9.76 -5.15 0.80 1.3 30.6ms
7 -8.353770537385 -10.73 -5.59 0.80 3.3 42.8ms
8 -8.353770537387 -11.75 -6.48 0.80 1.7 42.0ms
9 -8.353770537387 -13.19 -6.54 0.80 4.0 49.5ms
10 -8.353770537387 -13.85 -8.01 0.80 1.3 34.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353803487341 -2.00 0.80 9.0 81.4ms
2 -8.353884019209 -4.09 -2.69 0.80 1.0 29.7ms
3 -8.353895030346 -4.96 -3.28 0.80 4.7 60.7ms
4 -8.353895104667 -7.13 -4.11 0.80 2.7 40.2ms
5 -8.353895110680 -8.22 -4.93 0.80 3.3 48.1ms
6 -8.353895110782 -9.99 -5.65 0.80 2.7 48.9ms
7 -8.353895110785 -11.46 -7.35 0.80 3.0 44.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110785 -7.75 0.80 1.0 27.5ms
2 -8.353895110785 -14.75 -8.51 0.80 1.0 36.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110785 -8.97 0.80 1.0 26.0ms
2 -8.353895110785 -14.75 -9.04 0.80 1.0 29.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110785 -9.54 0.80 1.0 26.6ms
2 -8.353895110785 + -14.75 -9.58 0.80 1.0 31.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110785 │ -3.86 │ 7.36519e-15 │ 0.000646161 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353825156177 -3.60 0.80 9.0 101ms
2 -8.353888613994 -4.20 -3.83 0.80 1.3 33.4ms
3 -8.353895212274 -5.18 -4.02 0.80 6.3 69.6ms
4 -8.353895212349 -10.12 -4.23 0.80 1.3 39.0ms
5 -8.353895212439 -10.05 -4.69 0.80 1.0 33.8ms
6 -8.353895212498 -10.23 -4.94 0.80 2.0 40.3ms
7 -8.353895212523 -10.61 -5.62 0.80 1.7 34.1ms
8 -8.353895212525 -11.60 -5.99 0.80 3.0 51.9ms
9 -8.353895212525 -12.87 -6.15 0.80 2.3 41.3ms
10 -8.353895212525 + -13.52 -6.06 0.80 2.7 43.1ms
11 -8.353895212525 -12.89 -6.97 0.80 1.0 41.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353843973614 -3.34 0.80 9.3 85.1ms
2 -8.353888882123 -4.35 -3.87 0.80 1.0 37.8ms
3 -8.353895362358 -5.19 -4.09 0.80 6.7 73.6ms
4 -8.353895362656 -9.52 -4.83 0.80 1.0 31.8ms
5 -8.353895362680 -10.62 -5.12 0.80 2.7 43.9ms
6 -8.353895362685 -11.32 -6.36 0.80 1.7 36.1ms
7 -8.353895362686 -12.27 -7.44 0.80 4.0 52.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.32 0.80 1.0 26.4ms
2 -8.353895362686 + -Inf -9.31 0.80 1.0 28.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.75 0.80 1.0 33.0ms
2 -8.353895362686 + -14.75 -10.20 0.80 1.0 31.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -10.44 0.80 1.0 25.4ms
2 -8.353895362686 + -Inf -10.81 0.80 1.0 33.7ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.22782e-15 │ 1.71554e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-2.56047e-18 5.27783 5.27783;
5.27783 2.96736e-18 5.27783;
5.27783 5.27783 3.13562e-18]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: 10.555662925007576 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.