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 -1.110337550385 -0.82 0.80 7.0 22.6ms
2 -1.117195443698 -2.16 -1.84 0.80 1.0 41.2ms
3 -1.117250303752 -4.26 -2.69 0.80 1.0 17.2ms
4 -1.117251231487 -6.03 -3.36 0.80 1.0 17.5ms
5 -1.117251301448 -7.16 -3.80 0.80 1.0 17.8ms
6 -1.117251309977 -8.07 -4.79 0.80 1.0 18.1ms
7 -1.117251310367 -9.41 -5.03 0.80 2.0 20.7ms
8 -1.117251310379 -10.92 -5.53 0.80 1.0 36.4ms
9 -1.117251310381 -11.69 -7.07 0.80 1.0 19.6ms
10 -1.117251310381 -13.10 -6.84 0.80 2.0 21.1ms
11 -1.117251310381 -14.24 -7.84 0.80 2.0 21.4ms
12 -1.117251310381 + -15.35 -8.19 0.80 1.0 19.5ms
13 -1.117251310381 -15.35 -8.90 0.80 1.0 23.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -9.73 0.80 1.0 94.7ms
2 -1.117251310381 + -15.18 -10.52 0.80 1.0 17.0ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 1.30s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518116448 -2.46 0.80 1.0 10.8ms
2 -1.117520704463 -5.59 -3.49 0.80 1.0 17.1ms
3 -1.117520731397 -7.57 -4.23 0.80 1.0 17.1ms
4 -1.117520731860 -9.33 -5.14 0.80 1.0 22.2ms
5 -1.117520731876 -10.81 -5.92 0.80 1.0 18.0ms
6 -1.117520731877 -12.14 -6.42 0.80 1.0 18.2ms
7 -1.117520731877 -13.38 -7.39 0.80 1.0 18.7ms
8 -1.117520731877 -14.88 -8.19 0.80 1.0 23.8ms
9 -1.117520731877 + -Inf -8.95 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378325 -1.69 0.80 1.0 15.5ms
2 -1.118339157195 -4.04 -2.70 0.80 1.0 17.1ms
3 -1.118340124369 -6.01 -3.44 0.80 1.0 17.1ms
4 -1.118340141809 -7.76 -4.36 0.80 1.0 17.4ms
5 -1.118340142394 -9.23 -5.12 0.80 1.0 21.2ms
6 -1.118340142421 -10.57 -5.62 0.80 1.0 18.3ms
7 -1.118340142423 -11.73 -6.68 0.80 1.0 18.6ms
8 -1.118340142423 -13.57 -7.58 0.80 1.0 19.4ms
9 -1.118340142423 -14.51 -8.12 0.80 1.0 22.4ms
10 -1.118340142423 -14.48 -8.55 0.80 1.0 19.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142423 -9.05 0.80 1.0 15.2ms
2 -1.118340142423 -15.35 -10.36 0.80 1.0 17.1ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118340142423 │ -2.96 │ 0.00297053 │ 579ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834679 -3.09 0.80 1.0 10.5ms
2 -1.118346977929 -6.84 -4.11 0.80 1.0 21.6ms
3 -1.118346979406 -8.83 -4.86 0.80 1.0 17.1ms
4 -1.118346979432 -10.59 -5.78 0.80 1.0 17.3ms
5 -1.118346979433 -12.08 -6.55 0.80 1.0 17.6ms
6 -1.118346979433 -13.45 -7.04 0.80 1.0 21.4ms
7 -1.118346979433 -14.57 -8.13 0.80 1.0 18.7ms
8 -1.118346979433 + -Inf -8.95 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837062 -2.61 0.80 1.0 10.7ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 17.1ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 17.1ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 21.9ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 18.3ms
6 -1.118356214425 -12.45 -6.55 0.80 1.0 18.5ms
7 -1.118356214425 -13.57 -7.65 0.80 1.0 18.6ms
8 -1.118356214425 + -15.05 -8.45 0.80 1.0 22.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -9.24 0.80 1.0 10.5ms
2 -1.118356214425 + -15.05 -9.82 0.80 1.0 16.8ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.47261e-5 │ 501ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 10.6ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 16.8ms
3 -1.118356215917 -12.52 -6.70 0.80 1.0 17.0ms
4 -1.118356215917 -14.35 -7.62 0.80 1.0 22.2ms
5 -1.118356215917 -14.65 -8.39 0.80 1.0 17.7ms
6 -1.118356215917 -15.35 -8.89 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356217810 -4.40 0.80 1.0 10.6ms
2 -1.118356218156 -9.46 -5.42 0.80 1.0 16.8ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 17.0ms
4 -1.118356218159 -13.20 -7.09 0.80 1.0 17.2ms
5 -1.118356218159 + -15.65 -7.86 0.80 1.0 21.3ms
6 -1.118356218159 -14.88 -8.35 0.80 1.0 18.1ms
7 -1.118356218159 + -15.05 -9.45 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -10.08 0.80 1.0 10.7ms
2 -1.118356218159 -14.75 -11.02 0.80 1.0 17.1ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.0419e-8 │ 436ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, -2.46788e-11, 7.74655e-12]u"a₀")
Atom(H, [ 1.44318, -2.30995e-11, 4.11983e-12]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863655847874528 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.