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.oncvpsp3.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.oncvpsp3.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 -1.110669362739 -0.82 0.80 8.0 25.3ms
2 -1.117205801158 -2.18 -1.89 0.80 1.0 17.3ms
3 -1.117251917369 -4.34 -2.71 0.80 1.0 42.5ms
4 -1.117252481067 -6.25 -3.55 0.80 1.0 18.2ms
5 -1.117252499234 -7.74 -4.07 0.80 1.0 18.3ms
6 -1.117252500653 -8.85 -4.58 0.80 1.0 18.7ms
7 -1.117252500837 -9.73 -5.67 0.80 1.0 19.1ms
8 -1.117252500846 -11.08 -5.98 0.80 2.0 38.1ms
9 -1.117252500846 -12.43 -6.40 0.80 1.0 19.9ms
10 -1.117252500846 -13.38 -7.16 0.80 1.0 19.8ms
11 -1.117252500846 -15.05 -7.81 0.80 1.0 20.2ms
12 -1.117252500846 -14.88 -8.79 0.80 1.0 19.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117252500846 -9.17 0.80 1.0 12.8ms
2 -1.117252500846 + -15.05 -9.84 0.80 1.0 17.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117252500846 │ │ 0.0269316 │ 1.01s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117519304460 -2.46 0.80 1.0 24.2ms
2 -1.117521893084 -5.59 -3.49 0.80 1.0 17.4ms
3 -1.117521920023 -7.57 -4.23 0.80 1.0 17.5ms
4 -1.117521920484 -9.34 -5.14 0.80 1.0 17.8ms
5 -1.117521920500 -10.81 -5.92 0.80 1.0 22.9ms
6 -1.117521920500 -12.14 -6.43 0.80 1.0 18.5ms
7 -1.117521920500 -13.33 -7.41 0.80 1.0 19.7ms
8 -1.117521920500 -15.05 -8.26 0.80 1.0 19.8ms
9 -1.117521920500 + -14.88 -8.94 0.80 1.0 34.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118248561954 -1.69 0.80 1.0 10.9ms
2 -1.118340340266 -4.04 -2.70 0.80 1.0 17.1ms
3 -1.118341307433 -6.01 -3.44 0.80 1.0 28.2ms
4 -1.118341324873 -7.76 -4.36 0.80 1.0 19.8ms
5 -1.118341325458 -9.23 -5.12 0.80 1.0 18.3ms
6 -1.118341325485 -10.57 -5.62 0.80 1.0 18.7ms
7 -1.118341325486 -11.73 -6.68 0.80 1.0 23.9ms
8 -1.118341325487 -13.48 -7.58 0.80 1.0 19.5ms
9 -1.118341325486 + -14.37 -8.12 0.80 1.0 19.5ms
10 -1.118341325486 -14.81 -8.55 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118341325486 -9.05 0.80 1.0 10.9ms
2 -1.118341325486 -15.18 -10.36 0.80 1.0 17.2ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118341325486 │ -2.96 │ 0.0029705 │ 854ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118348017637 -3.09 0.80 1.0 23.4ms
2 -1.118348160885 -6.84 -4.11 0.80 1.0 17.4ms
3 -1.118348162362 -8.83 -4.86 0.80 1.0 17.4ms
4 -1.118348162388 -10.59 -5.78 0.80 1.0 17.7ms
5 -1.118348162388 -12.09 -6.55 0.80 1.0 22.7ms
6 -1.118348162389 -13.45 -7.04 0.80 1.0 18.7ms
7 -1.118348162389 -14.45 -8.13 0.80 1.0 19.0ms
8 -1.118348162389 -14.95 -8.95 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356019884 -2.61 0.80 1.0 11.4ms
2 -1.118357382894 -5.87 -3.62 0.80 1.0 17.6ms
3 -1.118357396968 -7.85 -4.37 0.80 1.0 17.5ms
4 -1.118357397214 -9.61 -5.29 0.80 1.0 17.8ms
5 -1.118357397222 -11.10 -6.06 0.80 1.0 680ms
6 -1.118357397223 -12.45 -6.55 0.80 1.0 18.4ms
7 -1.118357397223 -13.61 -7.65 0.80 1.0 18.9ms
8 -1.118357397223 + -15.35 -8.45 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357397223 -9.24 0.80 1.0 10.9ms
2 -1.118357397223 -14.57 -9.82 0.80 1.0 17.0ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118357397223 │ -4.79 │ 4.47254e-5 │ 1.50s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357398685 -4.94 0.80 1.0 10.9ms
2 -1.118357398714 -10.54 -5.96 0.80 1.0 17.7ms
3 -1.118357398714 -12.52 -6.70 0.80 1.0 17.7ms
4 -1.118357398714 -14.12 -7.62 0.80 1.0 18.0ms
5 -1.118357398714 + -15.18 -8.39 0.80 1.0 18.3ms
6 -1.118357398714 + -14.37 -8.89 0.80 1.0 18.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357400607 -4.40 0.80 1.0 11.0ms
2 -1.118357400953 -9.46 -5.42 0.80 1.0 17.3ms
3 -1.118357400956 -11.45 -6.17 0.80 1.0 17.5ms
4 -1.118357400956 -13.21 -7.09 0.80 1.0 17.8ms
5 -1.118357400956 -14.27 -7.86 0.80 1.0 18.3ms
6 -1.118357400956 + -14.75 -8.35 0.80 1.0 18.8ms
7 -1.118357400956 + -15.35 -9.45 0.80 1.0 20.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357400956 -10.08 0.80 1.0 11.0ms
2 -1.118357400956 -15.65 -11.02 0.80 1.0 18.3ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118357400956 │ -8.43 │ 1.17057e-8 │ 712ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, [-0.0431826, 1.19528e-10, 4.18946e-10]u"a₀")
Atom(H, [ 1.44318, 1.17402e-10, 4.23082e-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: 1.486365260711087 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.