Geometry optimization
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)n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110663137008 -0.82 0.80 9.0 25.7ms
2 -1.117204544119 -2.18 -1.87 0.80 1.0 41.4ms
3 -1.117250717920 -4.34 -2.71 0.80 1.0 18.0ms
4 -1.117251293371 -6.24 -3.57 0.80 1.0 17.5ms
5 -1.117251309009 -7.81 -4.04 0.80 1.0 18.0ms
6 -1.117251310208 -8.92 -4.56 0.80 1.0 18.4ms
7 -1.117251310374 -9.78 -5.34 0.80 1.0 18.9ms
8 -1.117251310381 -11.15 -6.40 0.80 1.0 37.9ms
9 -1.117251310381 -13.14 -6.93 0.80 1.0 19.9ms
10 -1.117251310381 -14.22 -7.49 0.80 2.0 21.7ms
11 -1.117251310381 -14.95 -8.28 0.80 1.0 20.0ms
12 -1.117251310381 -15.65 -9.13 0.80 1.0 19.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -10.15 0.80 1.0 98.7ms
2 -1.117251310381 -14.88 -10.63 0.80 1.0 17.6ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 1.32s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518116291 -2.46 0.80 1.0 10.9ms
2 -1.117520704466 -5.59 -3.49 0.80 1.0 23.3ms
3 -1.117520731400 -7.57 -4.23 0.80 1.0 18.2ms
4 -1.117520731862 -9.34 -5.14 0.80 1.0 18.2ms
5 -1.117520731878 -10.81 -5.92 0.80 1.0 18.4ms
6 -1.117520731878 -12.15 -6.43 0.80 1.0 22.4ms
7 -1.117520731878 -13.36 -7.41 0.80 1.0 19.6ms
8 -1.117520731878 + -14.51 -8.23 0.80 1.0 19.8ms
9 -1.117520731878 -14.57 -8.92 0.80 1.0 19.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378327 -1.69 0.80 1.0 10.9ms
2 -1.118339157176 -4.04 -2.70 0.80 1.0 17.4ms
3 -1.118340124349 -6.01 -3.44 0.80 1.0 19.1ms
4 -1.118340141790 -7.76 -4.36 0.80 1.0 23.5ms
5 -1.118340142374 -9.23 -5.12 0.80 1.0 18.3ms
6 -1.118340142401 -10.57 -5.62 0.80 1.0 18.6ms
7 -1.118340142403 -11.73 -6.68 0.80 1.0 23.1ms
8 -1.118340142403 -13.55 -7.58 0.80 1.0 19.9ms
9 -1.118340142403 -15.35 -8.12 0.80 1.0 21.1ms
10 -1.118340142403 -14.57 -8.55 0.80 1.0 19.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142403 -9.05 0.80 1.0 11.1ms
2 -1.118340142403 + -Inf -10.36 0.80 1.0 17.7ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118340142403 │ -2.96 │ 0.00297053 │ 591ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834667 -3.09 0.80 1.0 10.8ms
2 -1.118346977917 -6.84 -4.11 0.80 1.0 17.5ms
3 -1.118346979394 -8.83 -4.86 0.80 1.0 17.5ms
4 -1.118346979420 -10.59 -5.78 0.80 1.0 17.8ms
5 -1.118346979421 -12.08 -6.55 0.80 1.0 22.6ms
6 -1.118346979421 -13.47 -7.04 0.80 1.0 19.1ms
7 -1.118346979421 -14.22 -8.13 0.80 1.0 19.4ms
8 -1.118346979421 + -14.57 -8.95 0.80 1.0 23.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837061 -2.61 0.80 1.0 10.9ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 17.4ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 23.0ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 18.4ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 18.3ms
6 -1.118356214425 -12.44 -6.55 0.80 1.0 18.5ms
7 -1.118356214425 -13.66 -7.65 0.80 1.0 22.9ms
8 -1.118356214425 -15.05 -8.45 0.80 1.0 19.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -9.24 0.80 1.0 10.6ms
2 -1.118356214425 + -Inf -9.82 0.80 1.0 22.7ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.47262e-5 │ 517ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 11.0ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 17.5ms
3 -1.118356215917 -12.52 -6.70 0.80 1.0 23.1ms
4 -1.118356215917 -14.31 -7.62 0.80 1.0 18.0ms
5 -1.118356215917 -15.65 -8.39 0.80 1.0 18.5ms
6 -1.118356215917 -14.65 -8.89 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356217810 -4.40 0.80 1.0 11.1ms
2 -1.118356218156 -9.46 -5.42 0.80 1.0 17.6ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 17.6ms
4 -1.118356218159 -13.21 -7.09 0.80 1.0 24.3ms
5 -1.118356218159 + -15.35 -7.86 0.80 1.0 18.5ms
6 -1.118356218159 + -14.88 -8.35 0.80 1.0 18.8ms
7 -1.118356218159 -14.45 -9.45 0.80 1.0 30.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -10.08 0.80 1.0 11.5ms
2 -1.118356218159 -14.61 -11.02 0.80 1.0 17.7ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.03434e-8 │ 471ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, [-0.0431828, -9.52297e-11, -8.35513e-11]u"a₀")
Atom(H, [ 1.44318, -9.48043e-11, -8.2537e-11]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863655850699333 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.