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 -1.110428504884 -0.82 0.80 7.0 5.83s
2 -1.117198408182 -2.17 -1.85 0.80 1.0 2.32s
3 -1.117250272375 -4.29 -2.71 0.80 1.0 23.2ms
4 -1.117251140878 -6.06 -3.42 0.80 1.0 19.1ms
5 -1.117251185056 -7.35 -3.84 0.80 1.0 19.4ms
6 -1.117251189999 -8.31 -5.00 0.80 1.0 19.5ms
7 -1.117251190136 -9.86 -5.22 0.80 2.0 22.3ms
8 -1.117251190142 -11.24 -5.73 0.80 1.0 20.7ms
9 -1.117251190143 -12.05 -6.96 0.80 1.0 20.8ms
10 -1.117251190143 -13.77 -7.35 0.80 2.0 40.8ms
11 -1.117251190143 -14.42 -7.49 0.80 1.0 21.0ms
12 -1.117251190143 + -14.40 -8.01 0.80 1.0 23.0ms
13 -1.117251190143 -14.88 -9.33 0.80 1.0 22.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.82 0.80 1.0 4.12s
2 -1.117251190143 + -14.88 -10.59 0.80 1.0 18.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 23.5s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997999 -2.46 0.80 1.0 13.2ms
2 -1.117520586053 -5.59 -3.49 0.80 1.0 18.6ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 18.6ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 46.0ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.7ms
6 -1.117520613465 -12.15 -6.42 0.80 1.0 19.8ms
7 -1.117520613465 -13.34 -7.40 0.80 1.0 20.1ms
8 -1.117520613465 -15.05 -8.21 0.80 1.0 30.3ms
9 -1.117520613465 + -Inf -8.93 0.80 1.0 21.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264592 -1.69 0.80 1.0 13.1ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 24.1ms
3 -1.118340011112 -6.01 -3.44 0.80 1.0 18.7ms
4 -1.118340028553 -7.76 -4.36 0.80 1.0 18.9ms
5 -1.118340029137 -9.23 -5.12 0.80 1.0 25.2ms
6 -1.118340029164 -10.57 -5.62 0.80 1.0 20.2ms
7 -1.118340029166 -11.73 -6.68 0.80 1.0 22.9ms
8 -1.118340029166 -13.60 -7.58 0.80 1.0 20.9ms
9 -1.118340029166 -15.35 -8.12 0.80 1.0 23.5ms
10 -1.118340029166 + -15.35 -8.55 0.80 1.0 21.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -9.05 0.80 1.0 13.1ms
2 -1.118340029166 + -14.54 -10.36 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -11.05 0.80 1.0 13.1ms
2 -1.118340029166 + -14.31 -11.42 0.80 1.0 24.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029166 │ │ 0.00297055 │ 2.98s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 19.0ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 18.9ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 18.7ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 24.8ms
5 -1.118346866262 -12.08 -6.55 0.80 1.0 19.5ms
6 -1.118346866262 -13.43 -7.04 0.80 1.0 22.7ms
7 -1.118346866262 -15.35 -8.13 0.80 1.0 20.2ms
8 -1.118346866262 -14.65 -8.95 0.80 1.0 23.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.0ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 21.3ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 19.0ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 21.7ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.6ms
6 -1.118356101386 -12.46 -6.55 0.80 1.0 22.4ms
7 -1.118356101386 -13.55 -7.65 0.80 1.0 20.3ms
8 -1.118356101386 -15.35 -8.45 0.80 1.0 22.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.1ms
2 -1.118356101386 + -14.42 -9.82 0.80 1.0 84.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 534ms
2 -1.118356101386 + -15.05 -11.46 0.80 1.0 19.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 1.12s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.1ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.4ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 18.6ms
4 -1.118356102877 -14.35 -7.62 0.80 1.0 18.9ms
5 -1.118356102877 -14.88 -8.40 0.80 1.0 19.3ms
6 -1.118356102877 -15.35 -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 13.2ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.5ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.6ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 19.0ms
5 -1.118356105120 -14.88 -7.86 0.80 1.0 19.2ms
6 -1.118356105120 + -15.18 -8.35 0.80 1.0 19.6ms
7 -1.118356105120 + -15.35 -9.45 0.80 1.0 27.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 17.4ms
2 -1.118356105120 + -13.96 -11.02 0.80 1.0 31.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.0ms
2 -1.118356105120 + -14.65 -12.44 0.80 1.0 18.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.0532e-8 │ 476ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
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, 1.0405e-17, -8.83171e-18]u"a₀")
Atom(H, [ 1.44318, -9.37814e-18, 1.00489e-18]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.486365866575932 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 -8.335089037754 -0.80 0.80 8.0 758ms
2 -8.340929603864 -2.23 -1.69 0.80 1.3 617ms
3 -8.341201947316 -3.56 -2.77 0.80 2.7 33.5ms
4 -8.341214216506 -4.91 -3.19 0.80 3.7 41.5ms
5 -8.341214263673 -7.33 -3.76 0.80 1.3 27.5ms
6 -8.341214272297 -8.06 -5.05 0.80 2.3 60.7ms
7 -8.341214272507 -9.68 -5.24 0.80 3.7 44.7ms
8 -8.341214272517 -11.00 -6.04 0.80 1.0 27.1ms
9 -8.341214272518 -12.14 -7.09 0.80 3.3 39.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.79 0.80 1.0 606ms
2 -8.341214272518 -14.75 -8.50 0.80 1.0 25.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.94 0.80 1.0 23.8ms
2 -8.341214272518 -14.75 -9.23 0.80 1.0 26.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.38722e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350589293661 -1.21 0.80 10.0 76.6ms
2 -8.351864268022 -2.89 -1.88 0.80 1.3 62.4ms
3 -8.351926767172 -4.20 -2.79 0.80 3.0 38.2ms
4 -8.351928506370 -5.76 -3.71 0.80 3.3 41.1ms
5 -8.351928514462 -8.09 -4.86 0.80 3.3 39.8ms
6 -8.351928514629 -9.78 -5.73 0.80 4.7 46.7ms
7 -8.351928514632 -11.56 -6.72 0.80 4.0 60.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353393886508 -1.52 0.80 8.0 67.1ms
2 -8.353732741381 -3.47 -2.21 0.80 1.0 26.2ms
3 -8.353757281606 -4.61 -2.98 0.80 3.3 44.7ms
4 -8.353757521973 -6.62 -4.18 0.80 3.0 35.9ms
5 -8.353757531659 -8.01 -4.85 0.80 3.0 40.3ms
6 -8.353757531731 -10.15 -5.63 0.80 2.0 34.9ms
7 -8.353757531736 -11.27 -6.55 0.80 4.0 44.3ms
8 -8.353757531736 -13.50 -7.84 0.80 2.3 35.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531736 -8.94 0.80 1.0 23.8ms
2 -8.353757531736 -14.75 -9.60 0.80 1.0 26.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531736 -9.98 0.80 1.0 23.4ms
2 -8.353757531736 + -14.75 -10.22 0.80 1.0 30.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531736 -10.48 0.80 1.0 23.8ms
2 -8.353757531736 + -14.45 -10.80 0.80 1.0 26.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531736 │ │ 9.85249e-15 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353663199927 -3.21 0.80 8.0 63.3ms
2 -8.353744693864 -4.09 -3.81 0.80 1.0 24.9ms
3 -8.353770526385 -4.59 -3.76 0.80 6.3 54.8ms
4 -8.353770535237 -8.05 -4.06 0.80 1.3 27.0ms
5 -8.353770536767 -8.82 -4.47 0.80 2.0 29.5ms
6 -8.353770537391 -9.21 -5.30 0.80 2.3 31.3ms
7 -8.353770537411 -10.69 -5.74 0.80 2.3 33.6ms
8 -8.353770537412 -12.13 -5.94 0.80 1.3 36.6ms
9 -8.353770537412 -12.36 -7.10 0.80 2.0 31.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353812551128 -2.00 0.80 9.0 72.2ms
2 -8.353885763179 -4.14 -2.71 0.80 1.0 26.2ms
3 -8.353895068389 -5.03 -3.35 0.80 4.7 47.4ms
4 -8.353895108054 -7.40 -4.27 0.80 2.7 36.0ms
5 -8.353895110700 -8.58 -5.03 0.80 2.3 34.8ms
6 -8.353895110777 -10.11 -6.04 0.80 3.7 40.8ms
7 -8.353895110779 -11.84 -6.70 0.80 3.0 38.8ms
8 -8.353895110779 -13.71 -7.45 0.80 2.7 43.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -7.97 0.80 1.0 23.3ms
2 -8.353895110779 + -14.27 -9.21 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -9.70 0.80 1.0 23.0ms
2 -8.353895110779 + -Inf -10.02 0.80 1.0 25.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -10.42 0.80 1.0 22.9ms
2 -8.353895110779 -14.75 -10.51 0.80 1.0 32.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110779 │ -3.86 │ 8.10604e-15 │ 0.00064617 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353826363365 -3.57 0.80 9.0 72.1ms
2 -8.353886349957 -4.22 -3.75 0.80 1.0 30.7ms
3 -8.353895211169 -5.05 -3.89 0.80 6.0 58.1ms
4 -8.353895212350 -8.93 -4.10 0.80 1.3 27.7ms
5 -8.353895212283 + -10.17 -4.51 0.80 1.3 31.2ms
6 -8.353895212434 -9.82 -4.70 0.80 1.3 28.8ms
7 -8.353895212508 -10.13 -5.16 0.80 1.0 26.7ms
8 -8.353895212521 -10.89 -6.62 0.80 2.7 35.5ms
9 -8.353895212520 + -11.77 -5.51 0.80 4.7 53.2ms
10 -8.353895212521 -11.84 -5.78 0.80 3.3 40.8ms
11 -8.353895212522 -12.24 -6.53 0.80 2.7 37.5ms
12 -8.353895212522 -14.45 -6.59 0.80 1.0 31.2ms
13 -8.353895212522 -13.80 -8.06 0.80 1.0 28.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353840256221 -3.27 0.80 9.3 78.6ms
2 -8.353887931386 -4.32 -3.75 0.80 1.0 26.0ms
3 -8.353895360560 -5.13 -3.94 0.80 6.0 56.7ms
4 -8.353895362507 -8.71 -4.32 0.80 2.0 39.6ms
5 -8.353895362529 -10.65 -4.70 0.80 2.3 32.4ms
6 -8.353895362652 -9.91 -5.09 0.80 1.7 30.3ms
7 -8.353895362685 -10.49 -6.46 0.80 3.0 39.9ms
8 -8.353895362686 -11.92 -6.57 0.80 4.0 48.7ms
9 -8.353895362686 -13.71 -7.01 0.80 1.7 32.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.00 0.80 1.0 23.4ms
2 -8.353895362686 + -Inf -8.99 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.55 0.80 1.0 23.5ms
2 -8.353895362686 + -Inf -9.64 0.80 1.0 30.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -10.18 0.80 1.0 23.8ms
2 -8.353895362686 + -Inf -10.31 0.80 1.0 26.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.09658e-15 │ 1.71707e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [2.76298e-18 5.27783 5.27783;
5.27783 2.76298e-18 5.27783;
5.27783 5.27783 2.76298e-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: 10.555662918173487 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.