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 -1.110528424646 -0.82 0.80 7.0 23.7ms
2 -1.117200498451 -2.18 -1.86 0.80 1.0 17.3ms
3 -1.117251371578 -4.29 -2.70 0.80 1.0 45.1ms
4 -1.117252402310 -5.99 -3.57 0.80 1.0 17.9ms
5 -1.117252497053 -7.02 -3.93 0.80 2.0 20.1ms
6 -1.117252500585 -8.45 -4.66 0.80 1.0 18.5ms
7 -1.117252500796 -9.68 -4.94 0.80 1.0 34.3ms
8 -1.117252500838 -10.37 -5.31 0.80 1.0 22.9ms
9 -1.117252500846 -11.13 -6.30 0.80 1.0 27.3ms
10 -1.117252500846 -12.62 -6.49 0.80 2.0 30.1ms
11 -1.117252500846 -13.86 -6.93 0.80 1.0 20.3ms
12 -1.117252500846 -14.48 -7.52 0.80 1.0 24.0ms
13 -1.117252500846 -14.70 -8.03 0.80 1.0 20.9ms
14 -1.117252500846 + -15.05 -9.22 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117252500846 -9.67 0.80 1.0 10.9ms
2 -1.117252500846 + -15.05 -10.07 0.80 1.0 17.0ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117252500846 │ │ 0.0269316 │ 1.02s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117519304385 -2.46 0.80 1.0 10.8ms
2 -1.117521893092 -5.59 -3.49 0.80 1.0 33.6ms
3 -1.117521920032 -7.57 -4.23 0.80 1.0 17.5ms
4 -1.117521920492 -9.34 -5.15 0.80 1.0 17.6ms
5 -1.117521920507 -10.81 -5.93 0.80 1.0 22.6ms
6 -1.117521920508 -12.16 -6.44 0.80 1.0 18.7ms
7 -1.117521920508 -13.39 -7.43 0.80 1.0 18.9ms
8 -1.117521920508 -14.35 -8.32 0.80 1.0 19.3ms
9 -1.117521920508 + -14.81 -8.92 0.80 1.0 23.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118248561950 -1.69 0.80 1.0 10.9ms
2 -1.118340340176 -4.04 -2.70 0.80 1.0 18.1ms
3 -1.118341307343 -6.01 -3.44 0.80 1.0 26.0ms
4 -1.118341324783 -7.76 -4.36 0.80 1.0 18.3ms
5 -1.118341325368 -9.23 -5.12 0.80 1.0 26.5ms
6 -1.118341325394 -10.57 -5.62 0.80 1.0 26.1ms
7 -1.118341325396 -11.73 -6.68 0.80 1.0 23.4ms
8 -1.118341325396 -13.56 -7.58 0.80 1.0 19.7ms
9 -1.118341325396 -14.75 -8.12 0.80 1.0 19.3ms
10 -1.118341325396 + -14.88 -8.55 0.80 1.0 23.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118341325396 -9.05 0.80 1.0 10.8ms
2 -1.118341325396 + -14.70 -10.36 0.80 1.0 17.0ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118341325396 │ -2.96 │ 0.00297051 │ 863ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118348017582 -3.09 0.80 1.0 10.9ms
2 -1.118348160830 -6.84 -4.11 0.80 1.0 17.1ms
3 -1.118348162307 -8.83 -4.86 0.80 1.0 17.2ms
4 -1.118348162333 -10.59 -5.78 0.80 1.0 17.5ms
5 -1.118348162334 -12.08 -6.55 0.80 1.0 26.5ms
6 -1.118348162334 -13.41 -7.04 0.80 1.0 18.4ms
7 -1.118348162334 -14.81 -8.13 0.80 1.0 18.8ms
8 -1.118348162334 + -15.35 -8.95 0.80 1.0 93.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356019876 -2.61 0.80 1.0 10.8ms
2 -1.118357382893 -5.87 -3.62 0.80 1.0 17.1ms
3 -1.118357396968 -7.85 -4.37 0.80 1.0 616ms
4 -1.118357397214 -9.61 -5.29 0.80 1.0 17.9ms
5 -1.118357397222 -11.10 -6.06 0.80 1.0 18.0ms
6 -1.118357397223 -12.45 -6.55 0.80 1.0 18.3ms
7 -1.118357397223 -13.56 -7.65 0.80 1.0 18.6ms
8 -1.118357397223 -15.05 -8.45 0.80 1.0 18.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357397223 -9.24 0.80 1.0 10.6ms
2 -1.118357397223 + -14.65 -9.82 0.80 1.0 16.9ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118357397223 │ -4.79 │ 4.47256e-5 │ 1.42s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357398684 -4.94 0.80 1.0 10.6ms
2 -1.118357398714 -10.54 -5.96 0.80 1.0 16.9ms
3 -1.118357398714 -12.52 -6.70 0.80 1.0 17.1ms
4 -1.118357398714 -14.21 -7.62 0.80 1.0 17.4ms
5 -1.118357398714 -14.81 -8.39 0.80 1.0 17.7ms
6 -1.118357398714 + -14.57 -8.89 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357400607 -4.40 0.80 1.0 10.7ms
2 -1.118357400953 -9.46 -5.42 0.80 1.0 17.0ms
3 -1.118357400956 -11.45 -6.17 0.80 1.0 17.1ms
4 -1.118357400956 -13.22 -7.09 0.80 1.0 17.3ms
5 -1.118357400956 -14.70 -7.86 0.80 1.0 17.6ms
6 -1.118357400956 + -14.95 -8.35 0.80 1.0 18.1ms
7 -1.118357400956 -15.05 -9.45 0.80 1.0 18.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357400956 -10.08 0.80 1.0 10.9ms
2 -1.118357400956 + -14.45 -11.02 0.80 1.0 17.2ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118357400956 │ -8.43 │ 1.14274e-8 │ 723ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, 1.68714e-10, -3.55326e-11]u"a₀")
Atom(H, [ 1.44318, 1.69466e-10, -3.43672e-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: 1.4863652617502574 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.