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 -1.110567899608 -0.82 0.80 9.0 49.9ms
2 -1.117203456948 -2.18 -1.86 0.80 1.0 17.0ms
3 -1.117250771588 -4.33 -2.71 0.80 1.0 17.0ms
4 -1.117251299042 -6.28 -3.61 0.80 1.0 17.3ms
5 -1.117251309225 -7.99 -4.05 0.80 1.0 17.7ms
6 -1.117251310171 -9.02 -4.48 0.80 1.0 18.0ms
7 -1.117251310374 -9.69 -5.66 0.80 1.0 36.4ms
8 -1.117251310381 -11.18 -6.26 0.80 2.0 21.2ms
9 -1.117251310381 -13.62 -6.56 0.80 1.0 19.2ms
10 -1.117251310381 -13.75 -7.31 0.80 1.0 22.6ms
11 -1.117251310381 -14.51 -7.87 0.80 1.0 26.4ms
12 -1.117251310381 + -14.75 -9.42 0.80 1.0 39.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -9.56 0.80 1.0 92.8ms
2 -1.117251310381 -14.81 -10.34 0.80 1.0 16.8ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 1.29s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518116119 -2.46 0.80 1.0 10.9ms
2 -1.117520704460 -5.59 -3.49 0.80 1.0 17.2ms
3 -1.117520731398 -7.57 -4.23 0.80 1.0 17.4ms
4 -1.117520731859 -9.34 -5.14 0.80 1.0 22.4ms
5 -1.117520731874 -10.81 -5.93 0.80 1.0 18.0ms
6 -1.117520731875 -12.16 -6.43 0.80 1.0 18.0ms
7 -1.117520731875 -13.39 -7.41 0.80 1.0 21.4ms
8 -1.117520731875 + -15.35 -8.23 0.80 1.0 19.3ms
9 -1.117520731875 + -15.35 -8.90 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378330 -1.69 0.80 1.0 15.2ms
2 -1.118339157214 -4.04 -2.70 0.80 1.0 16.9ms
3 -1.118340124388 -6.01 -3.44 0.80 1.0 17.2ms
4 -1.118340141829 -7.76 -4.36 0.80 1.0 17.2ms
5 -1.118340142413 -9.23 -5.12 0.80 1.0 21.0ms
6 -1.118340142440 -10.57 -5.62 0.80 1.0 18.0ms
7 -1.118340142442 -11.73 -6.68 0.80 1.0 18.5ms
8 -1.118340142442 -13.52 -7.58 0.80 1.0 22.4ms
9 -1.118340142442 + -15.65 -8.12 0.80 1.0 19.6ms
10 -1.118340142442 + -14.81 -8.55 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142442 -9.05 0.80 1.0 10.7ms
2 -1.118340142442 + -15.35 -10.36 0.80 1.0 16.8ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118340142442 │ -2.96 │ 0.00297053 │ 581ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834691 -3.09 0.80 1.0 16.0ms
2 -1.118346977941 -6.84 -4.11 0.80 1.0 16.8ms
3 -1.118346979418 -8.83 -4.86 0.80 1.0 17.0ms
4 -1.118346979444 -10.59 -5.78 0.80 1.0 17.2ms
5 -1.118346979444 -12.08 -6.55 0.80 1.0 21.1ms
6 -1.118346979444 -13.38 -7.04 0.80 1.0 18.2ms
7 -1.118346979444 + -14.88 -8.13 0.80 1.0 18.6ms
8 -1.118346979444 -15.35 -8.95 0.80 1.0 22.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837064 -2.61 0.80 1.0 10.6ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 16.8ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 21.7ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 17.3ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 17.7ms
6 -1.118356214425 -12.45 -6.55 0.80 1.0 21.0ms
7 -1.118356214425 -13.59 -7.65 0.80 1.0 18.7ms
8 -1.118356214425 -14.95 -8.45 0.80 1.0 18.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -9.24 0.80 1.0 15.1ms
2 -1.118356214425 -15.35 -9.82 0.80 1.0 16.8ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.4726e-5 │ 498ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 10.8ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 21.7ms
3 -1.118356215917 -12.52 -6.70 0.80 1.0 16.9ms
4 -1.118356215917 -14.19 -7.62 0.80 1.0 17.3ms
5 -1.118356215917 + -14.51 -8.39 0.80 1.0 20.8ms
6 -1.118356215917 -14.45 -8.89 0.80 1.0 18.2ms
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 21.3ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 17.1ms
4 -1.118356218159 -13.19 -7.09 0.80 1.0 17.4ms
5 -1.118356218159 + -15.65 -7.86 0.80 1.0 17.7ms
6 -1.118356218159 -15.35 -8.35 0.80 1.0 21.4ms
7 -1.118356218159 -14.65 -9.45 0.80 1.0 18.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -10.08 0.80 1.0 10.5ms
2 -1.118356218159 + -14.65 -11.02 0.80 1.0 21.4ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.06298e-8 │ 451ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, -9.22611e-12, 7.98642e-12]u"a₀")
Atom(H, [ 1.44318, -7.19352e-12, 8.26986e-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: 1.4863655840003336 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.