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 -1.110673959928 -0.82 0.80 8.0 25.7ms
2 -1.117204258881 -2.19 -1.87 0.80 1.0 17.2ms
3 -1.117250678339 -4.33 -2.71 0.80 1.0 41.9ms
4 -1.117251290879 -6.21 -3.55 0.80 1.0 17.3ms
5 -1.117251308873 -7.74 -4.05 0.80 1.0 17.6ms
6 -1.117251310207 -8.87 -4.57 0.80 1.0 17.8ms
7 -1.117251310375 -9.78 -5.46 0.80 1.0 18.5ms
8 -1.117251310381 -11.22 -6.38 0.80 1.0 37.4ms
9 -1.117251310381 -12.78 -6.78 0.80 2.0 21.5ms
10 -1.117251310381 -14.88 -7.41 0.80 1.0 19.1ms
11 -1.117251310381 -14.88 -7.88 0.80 1.0 19.1ms
12 -1.117251310381 + -14.88 -8.86 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -9.87 0.80 1.0 93.3ms
2 -1.117251310381 -14.81 -10.57 0.80 1.0 16.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 1.26s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518114554 -2.46 0.80 1.0 10.4ms
2 -1.117520704471 -5.59 -3.49 0.80 1.0 23.7ms
3 -1.117520731416 -7.57 -4.23 0.80 1.0 16.8ms
4 -1.117520731864 -9.35 -5.16 0.80 1.0 17.1ms
5 -1.117520731879 -10.82 -5.94 0.80 1.0 17.5ms
6 -1.117520731879 -12.20 -6.46 0.80 1.0 22.6ms
7 -1.117520731879 -13.39 -7.55 0.80 1.0 18.3ms
8 -1.117520731879 + -14.57 -8.28 0.80 1.0 18.4ms
9 -1.117520731879 -14.54 -8.63 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378354 -1.69 0.80 1.0 10.5ms
2 -1.118339157166 -4.04 -2.70 0.80 1.0 16.5ms
3 -1.118340124339 -6.01 -3.44 0.80 1.0 16.7ms
4 -1.118340141779 -7.76 -4.36 0.80 1.0 22.9ms
5 -1.118340142364 -9.23 -5.12 0.80 1.0 17.6ms
6 -1.118340142390 -10.57 -5.62 0.80 1.0 17.9ms
7 -1.118340142392 -11.73 -6.68 0.80 1.0 18.3ms
8 -1.118340142392 -13.51 -7.58 0.80 1.0 23.5ms
9 -1.118340142392 + -15.35 -8.12 0.80 1.0 18.9ms
10 -1.118340142392 + -Inf -8.55 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142392 -9.04 0.80 1.0 10.6ms
2 -1.118340142392 + -Inf -10.35 0.80 1.0 16.7ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118340142392 │ -2.96 │ 0.00297053 │ 571ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834660 -3.09 0.80 1.0 10.5ms
2 -1.118346977911 -6.84 -4.11 0.80 1.0 23.0ms
3 -1.118346979388 -8.83 -4.86 0.80 1.0 16.8ms
4 -1.118346979414 -10.59 -5.78 0.80 1.0 17.1ms
5 -1.118346979414 -12.09 -6.55 0.80 1.0 17.8ms
6 -1.118346979414 -13.40 -7.04 0.80 1.0 22.5ms
7 -1.118346979414 -14.65 -8.13 0.80 1.0 18.5ms
8 -1.118346979414 + -15.05 -8.94 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837060 -2.61 0.80 1.0 10.5ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 16.5ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 16.7ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 16.9ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 23.4ms
6 -1.118356214425 -12.45 -6.55 0.80 1.0 18.0ms
7 -1.118356214425 -13.61 -7.65 0.80 1.0 18.5ms
8 -1.118356214425 + -14.88 -8.45 0.80 1.0 49.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -9.24 0.80 1.0 10.5ms
2 -1.118356214425 + -Inf -9.82 0.80 1.0 16.9ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.47262e-5 │ 533ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 15.0ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 23.5ms
3 -1.118356215917 -12.53 -6.70 0.80 1.0 19.4ms
4 -1.118356215917 -14.16 -7.62 0.80 1.0 17.2ms
5 -1.118356215917 + -15.35 -8.39 0.80 1.0 22.4ms
6 -1.118356215917 + -14.88 -8.89 0.80 1.0 17.9ms
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 22.7ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 16.8ms
4 -1.118356218159 -13.19 -7.09 0.80 1.0 17.2ms
5 -1.118356218159 + -15.05 -7.86 0.80 1.0 17.6ms
6 -1.118356218159 -14.54 -8.35 0.80 1.0 22.6ms
7 -1.118356218159 -15.18 -9.45 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -10.08 0.80 1.0 10.4ms
2 -1.118356218159 + -14.95 -11.02 0.80 1.0 22.8ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.043e-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, [-0.0431828, 2.11429e-10, -1.24105e-10]u"a₀")
Atom(H, [ 1.44318, 2.13829e-10, -1.25094e-10]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863655847464003 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.