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 -1.110525550886 -0.82 0.80 6.0 5.18s
2 -1.117198231291 -2.18 -1.87 0.80 1.0 2.18s
3 -1.117250394823 -4.28 -2.73 0.80 1.0 19.7ms
4 -1.117251160701 -6.12 -3.54 0.80 1.0 19.4ms
5 -1.117251187307 -7.58 -3.98 0.80 1.0 19.8ms
6 -1.117251190035 -8.56 -4.96 0.80 1.0 45.0ms
7 -1.117251190143 -9.97 -5.94 0.80 2.0 23.5ms
8 -1.117251190143 -12.66 -6.31 0.80 1.0 20.5ms
9 -1.117251190143 -13.22 -6.86 0.80 1.0 21.1ms
10 -1.117251190143 -14.35 -7.49 0.80 1.0 29.7ms
11 -1.117251190143 -14.95 -8.31 0.80 1.0 21.2ms
12 -1.117251190143 + -15.18 -8.56 0.80 2.0 23.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.02 0.80 1.0 4.06s
2 -1.117251190143 + -14.75 -9.95 0.80 1.0 21.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 23.2s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998026 -2.46 0.80 1.0 13.6ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 18.8ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 18.8ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.2ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 40.5ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 19.9ms
7 -1.117520613465 -13.34 -7.40 0.80 1.0 20.5ms
8 -1.117520613465 -14.65 -8.20 0.80 1.0 24.1ms
9 -1.117520613465 + -14.75 -8.96 0.80 1.0 21.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264594 -1.69 0.80 1.0 19.4ms
2 -1.118339043932 -4.04 -2.70 0.80 1.0 19.0ms
3 -1.118340011111 -6.01 -3.44 0.80 1.0 18.9ms
4 -1.118340028551 -7.76 -4.36 0.80 1.0 25.1ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 19.9ms
6 -1.118340029163 -10.57 -5.62 0.80 1.0 19.9ms
7 -1.118340029165 -11.73 -6.68 0.80 1.0 23.5ms
8 -1.118340029165 -13.53 -7.58 0.80 1.0 20.9ms
9 -1.118340029165 + -15.18 -8.12 0.80 1.0 24.7ms
10 -1.118340029165 + -14.70 -8.55 0.80 1.0 20.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -9.05 0.80 1.0 13.3ms
2 -1.118340029165 -14.45 -10.36 0.80 1.0 25.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -11.04 0.80 1.0 13.1ms
2 -1.118340029165 -14.61 -11.41 0.80 1.0 96.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029165 │ │ 0.00297055 │ 3.10s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 645ms
2 -1.118346864757 -6.84 -4.11 0.80 1.0 19.0ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 19.1ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 19.5ms
5 -1.118346866261 -12.09 -6.55 0.80 1.0 19.7ms
6 -1.118346866261 -13.42 -7.04 0.80 1.0 19.6ms
7 -1.118346866261 -14.57 -8.13 0.80 1.0 20.0ms
8 -1.118346866261 -14.65 -8.95 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.2ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.8ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 18.7ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.2ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.5ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 20.1ms
7 -1.118356101386 -13.62 -7.65 0.80 1.0 22.9ms
8 -1.118356101386 -15.05 -8.45 0.80 1.0 30.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 17.8ms
2 -1.118356101386 -14.51 -9.82 0.80 1.0 29.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.1ms
2 -1.118356101386 -15.05 -11.46 0.80 1.0 19.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 1.17s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.3ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.6ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 18.7ms
4 -1.118356102877 -14.18 -7.62 0.80 1.0 19.1ms
5 -1.118356102877 + -15.05 -8.39 0.80 1.0 28.7ms
6 -1.118356102877 + -Inf -8.89 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.1ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.4ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.6ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 18.9ms
5 -1.118356105120 -14.70 -7.86 0.80 1.0 26.7ms
6 -1.118356105120 + -Inf -8.35 0.80 1.0 19.7ms
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 13.0ms
2 -1.118356105120 + -14.88 -11.02 0.80 1.0 21.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.1ms
2 -1.118356105120 -14.00 -12.44 0.80 1.0 18.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.03286e-8 │ 469ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, 1.53043e-17, -7.8439e-18]u"a₀")
Atom(H, [ 1.44318, -1.48095e-18, 2.62412e-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: 1.4863658673355795 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 -8.335362766674 -0.81 0.80 7.7 775ms
2 -8.340910357927 -2.26 -1.69 0.80 1.3 628ms
3 -8.341208629629 -3.53 -2.76 0.80 2.7 33.8ms
4 -8.341214251317 -5.25 -3.36 0.80 3.7 40.8ms
5 -8.341214271066 -7.70 -4.24 0.80 1.7 29.5ms
6 -8.341214272493 -8.85 -5.65 0.80 2.0 31.8ms
7 -8.341214272518 -10.60 -6.05 0.80 4.0 62.3ms
8 -8.341214272518 -12.83 -7.55 0.80 2.0 31.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.34 0.80 1.0 617ms
2 -8.341214272518 + -Inf -8.53 0.80 1.0 25.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.13 0.80 1.0 39.4ms
2 -8.341214272518 -14.15 -9.15 0.80 1.0 25.6ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.53114e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350594038525 -1.21 0.80 9.3 72.1ms
2 -8.351868427684 -2.89 -1.88 0.80 1.3 28.0ms
3 -8.351926761953 -4.23 -2.81 0.80 3.0 71.9ms
4 -8.351928503983 -5.76 -3.59 0.80 3.3 41.5ms
5 -8.351928514033 -8.00 -4.50 0.80 3.7 39.6ms
6 -8.351928514669 -9.20 -5.55 0.80 4.3 43.5ms
7 -8.351928514677 -11.12 -6.38 0.80 4.0 43.8ms
8 -8.351928514677 -13.30 -7.70 0.80 3.0 54.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353408872880 -1.52 0.80 8.0 63.1ms
2 -8.353735038226 -3.49 -2.21 0.80 1.0 25.7ms
3 -8.353757323852 -4.65 -2.98 0.80 3.7 39.8ms
4 -8.353757522251 -6.70 -4.21 0.80 2.7 40.1ms
5 -8.353757531670 -8.03 -4.97 0.80 3.7 42.9ms
6 -8.353757531701 -10.51 -5.73 0.80 2.0 31.4ms
7 -8.353757531703 -11.67 -6.74 0.80 3.7 116ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531703 -7.79 0.80 1.0 22.6ms
2 -8.353757531703 -14.75 -8.80 0.80 1.0 25.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531703 -8.95 0.80 1.0 23.2ms
2 -8.353757531703 + -Inf -9.51 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531703 -9.52 0.80 1.0 23.1ms
2 -8.353757531703 + -14.75 -10.05 0.80 1.0 25.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531703 │ │ 1.03559e-14 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353701371179 -3.22 0.80 7.7 63.7ms
2 -8.353754802552 -4.27 -3.89 0.80 1.0 33.6ms
3 -8.353770533464 -4.80 -4.01 0.80 5.7 54.4ms
4 -8.353770536871 -8.47 -4.44 0.80 1.0 26.1ms
5 -8.353770537327 -9.34 -5.00 0.80 2.7 34.9ms
6 -8.353770537379 -10.28 -5.86 0.80 2.0 30.6ms
7 -8.353770537382 -11.58 -7.01 0.80 3.7 40.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353728389897 -2.00 0.80 7.7 61.6ms
2 -8.353838755693 -3.96 -2.70 0.80 1.0 31.8ms
3 -8.353895047156 -4.25 -3.26 0.80 5.0 47.6ms
4 -8.353895104038 -7.25 -4.12 0.80 1.7 30.0ms
5 -8.353895110726 -8.17 -5.16 0.80 3.3 39.6ms
6 -8.353895110786 -10.22 -5.60 0.80 3.7 40.5ms
7 -8.353895110789 -11.64 -6.53 0.80 2.3 32.7ms
8 -8.353895110789 -12.92 -7.29 0.80 3.0 40.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110789 -8.18 0.80 1.0 23.1ms
2 -8.353895110789 + -Inf -9.19 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110789 -9.78 0.80 1.0 23.2ms
2 -8.353895110789 + -Inf -10.04 0.80 1.0 26.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110789 -10.33 0.80 1.0 23.0ms
2 -8.353895110789 -14.75 -10.60 0.80 1.0 25.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110789 │ -3.86 │ 7.89586e-15 │ 0.000646157 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353798629060 -3.54 0.80 8.7 74.5ms
2 -8.353875326729 -4.12 -3.79 0.80 1.0 26.4ms
3 -8.353895212005 -4.70 -3.88 0.80 6.7 58.6ms
4 -8.353895212385 -9.42 -4.08 0.80 1.0 26.3ms
5 -8.353895212406 -10.67 -4.47 0.80 1.0 30.9ms
6 -8.353895212483 -10.11 -4.68 0.80 1.0 27.9ms
7 -8.353895212523 -10.40 -5.27 0.80 1.0 27.3ms
8 -8.353895212527 -11.41 -5.63 0.80 2.3 34.2ms
9 -8.353895212528 -12.19 -5.92 0.80 2.3 37.2ms
10 -8.353895212528 -13.13 -6.34 0.80 1.0 27.5ms
11 -8.353895212528 -13.60 -7.15 0.80 2.7 37.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353804040369 -3.29 0.80 8.7 66.4ms
2 -8.353875888099 -4.14 -3.82 0.80 1.0 26.4ms
3 -8.353895359475 -4.71 -3.95 0.80 6.0 55.5ms
4 -8.353895362390 -8.54 -4.33 0.80 1.3 33.2ms
5 -8.353895362502 -9.95 -4.71 0.80 2.3 33.5ms
6 -8.353895362679 -9.75 -5.54 0.80 2.3 34.2ms
7 -8.353895362686 -11.15 -6.36 0.80 3.3 39.3ms
8 -8.353895362686 -12.86 -6.62 0.80 3.3 45.7ms
9 -8.353895362686 -13.60 -7.67 0.80 1.0 27.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.04 0.80 1.0 22.9ms
2 -8.353895362686 + -14.45 -8.68 0.80 1.0 31.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.30 0.80 1.0 22.9ms
2 -8.353895362686 + -14.75 -9.64 0.80 1.0 25.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.99 0.80 1.0 22.9ms
2 -8.353895362686 + -14.27 -10.18 0.80 1.0 25.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.35523e-15 │ 1.71522e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-1.83937e-18 5.27783 5.27783;
5.27783 -1.83937e-18 5.27783;
5.27783 5.27783 -1.83937e-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: 10.555662926535945 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.