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 -1.110722414162 -0.82 0.80 6.0 3.08s
2 -1.117203393511 -2.19 -1.88 0.80 1.0 1.52s
3 -1.117250407472 -4.33 -2.71 0.80 1.0 19.0ms
4 -1.117251159782 -6.12 -3.54 0.80 1.0 18.7ms
5 -1.117251188043 -7.55 -4.07 0.80 1.0 19.0ms
6 -1.117251190050 -8.70 -4.93 0.80 1.0 70.7ms
7 -1.117251190142 -10.03 -5.64 0.80 2.0 21.7ms
8 -1.117251190143 -12.02 -6.23 0.80 1.0 20.5ms
9 -1.117251190143 -13.13 -6.84 0.80 1.0 20.4ms
10 -1.117251190143 -14.75 -7.09 0.80 1.0 37.1ms
11 -1.117251190143 -14.35 -7.80 0.80 1.0 20.9ms
12 -1.117251190143 + -15.05 -8.20 0.80 2.0 22.6ms
13 -1.117251190143 -14.88 -8.54 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.13 0.80 1.0 2.77s
2 -1.117251190143 -14.75 -10.59 0.80 1.0 18.4ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 16.9s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998028 -2.46 0.80 1.0 53.6ms
2 -1.117520586052 -5.59 -3.49 0.80 1.0 18.1ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 19.3ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.9ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 37.1ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 19.6ms
7 -1.117520613465 -13.34 -7.39 0.80 1.0 19.9ms
8 -1.117520613465 -14.61 -8.19 0.80 1.0 22.9ms
9 -1.117520613465 + -14.54 -8.95 0.80 1.0 20.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264594 -1.69 0.80 1.0 19.0ms
2 -1.118339043931 -4.04 -2.70 0.80 1.0 18.1ms
3 -1.118340011111 -6.01 -3.44 0.80 1.0 18.2ms
4 -1.118340028551 -7.76 -4.36 0.80 1.0 24.8ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 18.9ms
6 -1.118340029163 -10.57 -5.62 0.80 1.0 22.1ms
7 -1.118340029164 -11.73 -6.68 0.80 1.0 19.9ms
8 -1.118340029164 -13.57 -7.58 0.80 1.0 23.3ms
9 -1.118340029164 + -14.65 -8.12 0.80 1.0 75.7ms
10 -1.118340029164 -14.54 -8.55 0.80 1.0 21.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -9.05 0.80 1.0 12.8ms
2 -1.118340029164 -14.75 -10.36 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -11.04 0.80 1.0 12.7ms
2 -1.118340029164 + -14.75 -11.41 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029164 │ │ 0.00297055 │ 2.77s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 12.7ms
2 -1.118346864757 -6.84 -4.11 0.80 1.0 18.1ms
3 -1.118346866234 -8.83 -4.86 0.80 1.0 18.2ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 18.4ms
5 -1.118346866261 -12.09 -6.55 0.80 1.0 18.7ms
6 -1.118346866261 -13.40 -7.04 0.80 1.0 19.2ms
7 -1.118346866261 + -15.05 -8.13 0.80 1.0 19.7ms
8 -1.118346866261 -15.05 -8.95 0.80 1.0 19.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 12.6ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 17.9ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 18.0ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 21.7ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.0ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.3ms
7 -1.118356101386 -13.58 -7.65 0.80 1.0 26.9ms
8 -1.118356101386 -15.05 -8.45 0.80 1.0 19.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 12.4ms
2 -1.118356101386 -14.26 -9.82 0.80 1.0 17.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 12.6ms
2 -1.118356101386 -15.35 -11.46 0.80 1.0 18.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 522ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 12.6ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 25.3ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 18.4ms
4 -1.118356102877 -14.21 -7.62 0.80 1.0 18.5ms
5 -1.118356102877 + -Inf -8.39 0.80 1.0 25.9ms
6 -1.118356102877 -14.40 -8.89 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 12.6ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 21.1ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.2ms
4 -1.118356105120 -13.19 -7.09 0.80 1.0 21.2ms
5 -1.118356105120 -15.35 -7.86 0.80 1.0 19.3ms
6 -1.118356105120 -14.65 -8.35 0.80 1.0 21.9ms
7 -1.118356105120 -15.35 -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 12.6ms
2 -1.118356105120 + -14.70 -11.02 0.80 1.0 20.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 12.6ms
2 -1.118356105120 + -14.18 -12.44 0.80 1.0 21.3ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.00339e-8 │ 480ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -2.66007e-34, 1.81779e-34]u"a₀")
Atom(H, [ 1.44318, -1.45977e-34, 2.87795e-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: 1.4863658683119065 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 -8.335337914268 -0.81 0.80 7.3 662ms
2 -8.340904622607 -2.25 -1.69 0.80 1.3 68.1ms
3 -8.341203542758 -3.52 -2.78 0.80 2.7 35.7ms
4 -8.341214239655 -4.97 -3.28 0.80 3.3 36.3ms
5 -8.341214270191 -7.52 -4.05 0.80 1.3 27.9ms
6 -8.341214272382 -8.66 -5.24 0.80 1.7 50.3ms
7 -8.341214272517 -9.87 -5.76 0.80 3.3 36.1ms
8 -8.341214272518 -11.86 -6.41 0.80 2.3 32.7ms
9 -8.341214272518 -13.13 -7.45 0.80 2.7 32.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.60 0.80 1.0 63.1ms
2 -8.341214272518 + -14.75 -8.84 0.80 1.0 24.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.22 0.80 1.0 22.2ms
2 -8.341214272518 + -14.15 -9.65 0.80 1.0 24.3ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 6.61225e-33 │ 0.163458 │ -0.16 │ 28.2s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350582569246 -1.21 0.80 9.0 64.3ms
2 -8.351863479281 -2.89 -1.88 0.80 1.3 26.4ms
3 -8.351926952422 -4.20 -2.79 0.80 3.0 34.6ms
4 -8.351928508864 -5.81 -3.69 0.80 3.7 37.1ms
5 -8.351928514324 -8.26 -4.75 0.80 2.0 75.1ms
6 -8.351928514626 -9.52 -5.55 0.80 3.7 40.5ms
7 -8.351928514631 -11.27 -6.49 0.80 4.0 39.4ms
8 -8.351928514631 -13.00 -7.34 0.80 3.3 39.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353385914366 -1.52 0.80 8.0 59.7ms
2 -8.353730092311 -3.46 -2.21 0.80 1.0 25.0ms
3 -8.353757370555 -4.56 -2.98 0.80 3.3 63.8ms
4 -8.353757520955 -6.82 -4.26 0.80 1.3 27.5ms
5 -8.353757531726 -7.97 -5.43 0.80 3.7 40.0ms
6 -8.353757531736 -10.98 -6.33 0.80 2.7 35.4ms
7 -8.353757531736 -13.10 -7.27 0.80 2.3 32.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531736 -8.00 0.80 1.0 22.8ms
2 -8.353757531736 -14.75 -9.27 0.80 1.0 31.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531736 -9.47 0.80 1.0 22.4ms
2 -8.353757531736 -14.45 -9.67 0.80 1.0 25.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531736 -9.95 0.80 1.0 22.6ms
2 -8.353757531736 -14.75 -10.20 0.80 1.0 25.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531736 │ │ │ 0.0156413 │ -0.016 │ 5.85s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353706356655 -3.23 0.80 7.7 75.1ms
2 -8.353758538760 -4.28 -3.89 0.80 1.0 25.5ms
3 -8.353770536111 -4.92 -4.10 0.80 6.3 53.3ms
4 -8.353770537343 -8.91 -4.60 0.80 1.3 27.2ms
5 -8.353770537379 -10.44 -5.01 0.80 2.0 30.6ms
6 -8.353770537412 -10.48 -6.34 0.80 2.3 46.6ms
7 -8.353770537412 -12.06 -7.27 0.80 3.7 40.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353783168063 -2.00 0.80 7.7 58.3ms
2 -8.353875657708 -4.03 -2.71 0.80 1.0 30.5ms
3 -8.353895026630 -4.71 -3.31 0.80 4.3 42.0ms
4 -8.353895104292 -7.11 -4.13 0.80 2.3 32.8ms
5 -8.353895110620 -8.20 -4.79 0.80 3.0 36.8ms
6 -8.353895110771 -9.82 -6.10 0.80 2.0 34.2ms
7 -8.353895110777 -11.19 -6.82 0.80 3.7 41.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110777 -7.99 0.80 1.0 23.0ms
2 -8.353895110777 + -Inf -8.77 0.80 1.0 26.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110777 -9.41 0.80 1.0 27.0ms
2 -8.353895110777 + -Inf -9.60 0.80 1.0 25.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110777 -9.88 0.80 1.0 27.3ms
2 -8.353895110777 -14.75 -10.15 0.80 1.0 25.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110777 │ -3.86 │ │ 0.000646171 │ -0.00065 │ 3.28s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353807888758 -3.51 0.80 8.0 59.3ms
2 -8.353878266987 -4.15 -3.68 0.80 1.0 30.0ms
3 -8.353895210073 -4.77 -3.89 0.80 5.7 50.8ms
4 -8.353895211132 -8.98 -4.08 0.80 1.0 25.5ms
5 -8.353895212432 -8.89 -4.72 0.80 1.0 28.7ms
6 -8.353895212506 -10.13 -4.86 0.80 1.7 28.9ms
7 -8.353895212510 -11.39 -4.81 0.80 1.0 26.0ms
8 -8.353895212515 -11.23 -5.40 0.80 1.0 26.1ms
9 -8.353895212520 -11.33 -5.83 0.80 3.0 36.9ms
10 -8.353895212520 -12.45 -5.85 0.80 1.7 30.5ms
11 -8.353895212519 + -11.83 -5.49 0.80 1.0 26.7ms
12 -8.353895212520 -11.81 -6.01 0.80 2.0 35.4ms
13 -8.353895212514 + -11.21 -5.24 0.80 2.3 33.8ms
14 -8.353895212518 -11.48 -5.35 0.80 3.0 257ms
15 -8.353895212521 -11.53 -6.15 0.80 1.3 48.1ms
16 -8.353895212521 -12.84 -7.23 0.80 1.7 30.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353785123244 -3.32 0.80 7.7 57.9ms
2 -8.353875871618 -4.04 -3.53 0.80 1.0 24.9ms
3 -8.353895361902 -4.71 -3.96 0.80 5.7 69.5ms
4 -8.353895362173 -9.57 -4.33 0.80 1.3 26.6ms
5 -8.353895362616 -9.35 -4.95 0.80 1.7 27.8ms
6 -8.353895362681 -10.19 -5.48 0.80 2.0 30.3ms
7 -8.353895362685 -11.37 -6.00 0.80 2.7 33.7ms
8 -8.353895362686 -12.44 -7.06 0.80 2.3 31.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.99 0.80 1.0 22.4ms
2 -8.353895362686 -14.75 -8.28 0.80 1.0 25.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.79 0.80 1.0 27.8ms
2 -8.353895362686 + -Inf -9.05 0.80 1.0 25.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.40 0.80 1.0 80.4ms
2 -8.353895362686 + -14.45 -9.68 0.80 1.0 25.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ 2.22376e-32 │ 1.717e-6 │ -1.7e-6 │ 4.18s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [4.79301e-36 5.27783 5.27783;
5.27783 4.79301e-36 5.27783;
5.27783 5.27783 4.79301e-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: 10.555662920850136 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.