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 -1.110651512588 -0.82 0.80 6.0 3.58s
2 -1.117204055434 -2.18 -1.88 0.80 1.0 1.60s
3 -1.117250393650 -4.33 -2.72 0.80 1.0 19.7ms
4 -1.117251154476 -6.12 -3.47 0.80 1.0 19.5ms
5 -1.117251187812 -7.48 -4.02 0.80 1.0 56.6ms
6 -1.117251190085 -8.64 -5.02 0.80 1.0 20.0ms
7 -1.117251190142 -10.24 -6.24 0.80 1.0 20.6ms
8 -1.117251190143 -12.02 -6.62 0.80 2.0 31.6ms
9 -1.117251190143 -15.35 -6.66 0.80 2.0 23.1ms
10 -1.117251190143 -13.95 -7.59 0.80 1.0 21.2ms
11 -1.117251190143 -14.95 -8.41 0.80 2.0 26.1ms
12 -1.117251190143 + -14.54 -8.59 0.80 1.0 23.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.16 0.80 1.0 3.60s
2 -1.117251190143 + -14.35 -10.23 0.80 1.0 72.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 21.2s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997602 -2.46 0.80 1.0 52.3ms
2 -1.117520586047 -5.59 -3.49 0.80 1.0 19.3ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 18.9ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.2ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 34.3ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 19.9ms
7 -1.117520613465 -13.35 -7.41 0.80 1.0 20.8ms
8 -1.117520613465 + -Inf -8.23 0.80 1.0 23.3ms
9 -1.117520613465 -15.05 -8.94 0.80 1.0 21.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264596 -1.69 0.80 1.0 13.2ms
2 -1.118339043932 -4.04 -2.70 0.80 1.0 18.8ms
3 -1.118340011111 -6.01 -3.44 0.80 1.0 24.1ms
4 -1.118340028552 -7.76 -4.36 0.80 1.0 19.4ms
5 -1.118340029137 -9.23 -5.12 0.80 1.0 22.2ms
6 -1.118340029163 -10.57 -5.62 0.80 1.0 20.0ms
7 -1.118340029165 -11.73 -6.68 0.80 1.0 22.9ms
8 -1.118340029165 -13.55 -7.58 0.80 1.0 20.8ms
9 -1.118340029165 -14.57 -8.12 0.80 1.0 81.8ms
10 -1.118340029165 + -15.05 -8.55 0.80 1.0 550ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -9.05 0.80 1.0 13.3ms
2 -1.118340029165 -14.54 -10.36 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -11.05 0.80 1.0 13.4ms
2 -1.118340029165 + -14.21 -11.42 0.80 1.0 18.8ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029165 │ │ 0.00297055 │ 3.60s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 13.3ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 19.0ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 18.9ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 19.1ms
5 -1.118346866261 -12.08 -6.55 0.80 1.0 19.8ms
6 -1.118346866261 -13.43 -7.04 0.80 1.0 19.9ms
7 -1.118346866261 -14.57 -8.13 0.80 1.0 20.4ms
8 -1.118346866261 -14.81 -8.95 0.80 1.0 20.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.9ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 25.9ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 29.0ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.4ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.7ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 20.0ms
7 -1.118356101386 -13.59 -7.65 0.80 1.0 20.4ms
8 -1.118356101386 + -15.05 -8.45 0.80 1.0 20.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.7ms
2 -1.118356101386 + -14.88 -9.82 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.2ms
2 -1.118356101386 -14.65 -11.46 0.80 1.0 19.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 526ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 23.2ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 19.2ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 19.2ms
4 -1.118356102877 -14.15 -7.62 0.80 1.0 19.5ms
5 -1.118356102877 -15.05 -8.39 0.80 1.0 20.0ms
6 -1.118356102877 + -14.21 -8.89 0.80 1.0 26.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.3ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.9ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 22.1ms
4 -1.118356105120 -13.22 -7.09 0.80 1.0 19.5ms
5 -1.118356105120 -14.31 -7.86 0.80 1.0 19.8ms
6 -1.118356105120 + -14.57 -8.35 0.80 1.0 22.7ms
7 -1.118356105120 -15.35 -9.45 0.80 1.0 20.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 16.2ms
2 -1.118356105120 + -14.54 -11.02 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.3ms
2 -1.118356105120 + -15.65 -12.44 0.80 1.0 21.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.12416e-8 │ 479ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, -1.5828e-35, 1.51249e-34]u"a₀")
Atom(H, [ 1.44318, 6.12988e-37, -2.95559e-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: 1.4863658638014283 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 -8.335382768255 -0.81 0.80 7.3 822ms
2 -8.340914647800 -2.26 -1.69 0.80 1.3 643ms
3 -8.341205147400 -3.54 -2.76 0.80 3.0 35.5ms
4 -8.341214238307 -5.04 -3.32 0.80 3.3 40.3ms
5 -8.341214269140 -7.51 -4.06 0.80 1.3 28.0ms
6 -8.341214272420 -8.48 -5.40 0.80 2.0 32.0ms
7 -8.341214272516 -10.02 -5.72 0.80 4.0 60.2ms
8 -8.341214272518 -11.73 -6.49 0.80 2.7 35.7ms
9 -8.341214272518 -13.35 -7.23 0.80 2.3 34.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.86 0.80 1.0 631ms
2 -8.341214272518 + -Inf -9.09 0.80 1.0 25.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.71 0.80 1.0 23.1ms
2 -8.341214272518 -14.75 -9.91 0.80 1.0 25.9ms
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 -8.350572309962 -1.21 0.80 9.0 68.3ms
2 -8.351868558782 -2.89 -1.88 0.80 2.0 30.3ms
3 -8.351927812749 -4.23 -2.79 0.80 3.7 39.3ms
4 -8.351928508492 -6.16 -3.78 0.80 2.7 34.7ms
5 -8.351928514572 -8.22 -5.06 0.80 3.0 36.9ms
6 -8.351928514638 -10.18 -5.91 0.80 4.3 43.6ms
7 -8.351928514639 -11.98 -6.68 0.80 3.7 40.2ms
8 -8.351928514639 -13.67 -7.75 0.80 3.0 36.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353380001256 -1.52 0.80 9.0 87.7ms
2 -8.353727850890 -3.46 -2.21 0.80 1.0 25.7ms
3 -8.353757329999 -4.53 -2.95 0.80 3.7 39.3ms
4 -8.353757521925 -6.72 -4.17 0.80 2.7 32.7ms
5 -8.353757531679 -8.01 -4.91 0.80 3.7 39.9ms
6 -8.353757531728 -10.31 -5.61 0.80 3.0 35.5ms
7 -8.353757531731 -11.64 -6.87 0.80 2.7 34.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531731 -7.60 0.80 1.0 31.0ms
2 -8.353757531731 -14.15 -8.21 0.80 1.0 25.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531731 -8.57 0.80 1.0 22.4ms
2 -8.353757531731 -14.75 -8.91 0.80 1.0 25.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531731 -9.14 0.80 1.0 22.7ms
2 -8.353757531731 + -14.75 -9.43 0.80 1.0 30.6ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531731 │ │ │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353712315727 -3.22 0.80 8.3 63.0ms
2 -8.353758611553 -4.33 -3.94 0.80 1.0 25.2ms
3 -8.353770534123 -4.92 -4.06 0.80 6.3 59.2ms
4 -8.353770537303 -8.50 -4.74 0.80 1.7 29.5ms
5 -8.353770537326 -10.64 -4.92 0.80 2.7 34.9ms
6 -8.353770537403 -10.11 -5.68 0.80 1.0 29.3ms
7 -8.353770537408 -11.28 -6.41 0.80 3.3 38.1ms
8 -8.353770537408 -12.99 -6.60 0.80 2.7 34.1ms
9 -8.353770537408 -14.15 -7.19 0.80 1.7 33.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353771057855 -2.00 0.80 8.3 63.0ms
2 -8.353869291636 -4.01 -2.71 0.80 1.0 29.9ms
3 -8.353895040619 -4.59 -3.29 0.80 5.0 45.4ms
4 -8.353895103055 -7.20 -4.07 0.80 2.0 31.4ms
5 -8.353895110620 -8.12 -4.87 0.80 2.7 35.6ms
6 -8.353895110747 -9.90 -5.73 0.80 3.3 41.8ms
7 -8.353895110749 -11.65 -6.64 0.80 3.0 37.2ms
8 -8.353895110749 -13.29 -7.60 0.80 3.7 42.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110749 -8.28 0.80 1.0 22.8ms
2 -8.353895110749 + -Inf -9.44 0.80 1.0 25.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110749 -10.31 0.80 1.0 22.8ms
2 -8.353895110749 -14.45 -10.26 0.80 1.0 25.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110749 -10.61 0.80 1.0 22.6ms
2 -8.353895110749 -14.45 -10.91 0.80 1.0 26.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110749 │ -3.86 │ │ 0.000646208 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353778939501 -3.50 0.80 8.0 60.8ms
2 -8.353869077309 -4.05 -3.69 0.80 1.0 25.2ms
3 -8.353895211856 -4.58 -3.86 0.80 6.0 57.6ms
4 -8.353895212189 -9.48 -4.05 0.80 1.0 25.4ms
5 -8.353895212413 -9.65 -4.51 0.80 1.0 25.5ms
6 -8.353895212470 -10.25 -4.70 0.80 1.0 28.8ms
7 -8.353895212499 -10.54 -5.54 0.80 1.0 26.0ms
8 -8.353895212504 -11.29 -6.30 0.80 3.3 36.9ms
9 -8.353895212504 -13.04 -6.63 0.80 2.0 32.8ms
10 -8.353895212504 -14.15 -6.90 0.80 2.3 37.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353761780109 -3.27 0.80 8.3 64.9ms
2 -8.353844746076 -4.08 -3.76 0.80 1.0 29.4ms
3 -8.353895360276 -4.30 -3.77 0.80 6.3 54.3ms
4 -8.353895362394 -8.67 -4.07 0.80 1.0 25.9ms
5 -8.353895362370 + -10.62 -4.34 0.80 1.3 30.2ms
6 -8.353895362629 -9.59 -4.77 0.80 1.0 25.7ms
7 -8.353895362676 -10.32 -5.19 0.80 1.7 28.5ms
8 -8.353895362685 -11.03 -6.25 0.80 3.0 34.7ms
9 -8.353895362686 -12.36 -6.76 0.80 3.0 40.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.52 0.80 1.0 22.1ms
2 -8.353895362686 + -14.75 -8.93 0.80 1.0 30.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.23 0.80 1.0 23.6ms
2 -8.353895362686 + -Inf -9.47 0.80 1.0 32.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.75 0.80 1.0 22.4ms
2 -8.353895362686 + -Inf -10.20 0.80 1.0 25.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.7187e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [4.793e-36 5.27783 5.27783;
5.27783 4.793e-36 5.27783;
5.27783 5.27783 4.793e-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: 10.555662912990975 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.