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 -1.110338151432 -0.82 0.80 7.0 22.6ms
2 -1.117193311459 -2.16 -1.83 0.80 1.0 40.1ms
3 -1.117250382816 -4.24 -2.67 0.80 1.0 17.1ms
4 -1.117251253739 -6.06 -3.45 0.80 1.0 17.2ms
5 -1.117251304799 -7.29 -3.88 0.80 1.0 17.6ms
6 -1.117251309999 -8.28 -4.55 0.80 2.0 20.0ms
7 -1.117251310356 -9.45 -5.02 0.80 1.0 18.4ms
8 -1.117251310380 -10.61 -5.78 0.80 1.0 37.0ms
9 -1.117251310381 -12.19 -6.37 0.80 1.0 19.2ms
10 -1.117251310381 -13.53 -6.55 0.80 1.0 19.3ms
11 -1.117251310381 -13.63 -7.18 0.80 1.0 19.5ms
12 -1.117251310381 + -14.75 -8.15 0.80 1.0 19.4ms
13 -1.117251310381 -15.05 -8.19 0.80 2.0 26.4ms
14 -1.117251310381 -14.95 -9.59 0.80 1.0 20.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -9.66 0.80 1.0 10.6ms
2 -1.117251310381 -14.57 -10.54 0.80 1.0 16.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 725ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518116483 -2.46 0.80 1.0 10.6ms
2 -1.117520704466 -5.59 -3.49 0.80 1.0 16.6ms
3 -1.117520731400 -7.57 -4.23 0.80 1.0 17.0ms
4 -1.117520731863 -9.33 -5.14 0.80 1.0 23.2ms
5 -1.117520731879 -10.81 -5.92 0.80 1.0 17.7ms
6 -1.117520731880 -12.13 -6.42 0.80 1.0 17.9ms
7 -1.117520731880 -13.32 -7.39 0.80 1.0 18.4ms
8 -1.117520731880 + -15.35 -8.18 0.80 1.0 95.1ms
9 -1.117520731880 + -15.05 -8.96 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378323 -1.69 0.80 1.0 10.6ms
2 -1.118339157162 -4.04 -2.70 0.80 1.0 16.9ms
3 -1.118340124335 -6.01 -3.44 0.80 1.0 665ms
4 -1.118340141776 -7.76 -4.36 0.80 1.0 17.4ms
5 -1.118340142360 -9.23 -5.12 0.80 1.0 17.8ms
6 -1.118340142387 -10.57 -5.62 0.80 1.0 18.2ms
7 -1.118340142389 -11.73 -6.68 0.80 1.0 18.6ms
8 -1.118340142389 -13.57 -7.58 0.80 1.0 19.2ms
9 -1.118340142389 -14.54 -8.12 0.80 1.0 19.4ms
10 -1.118340142389 + -15.05 -8.55 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142389 -9.05 0.80 1.0 10.7ms
2 -1.118340142389 -15.35 -10.36 0.80 1.0 17.0ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118340142389 │ -2.96 │ 0.00297053 │ 1.27s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834659 -3.09 0.80 1.0 10.7ms
2 -1.118346977909 -6.84 -4.11 0.80 1.0 17.2ms
3 -1.118346979386 -8.83 -4.86 0.80 1.0 17.4ms
4 -1.118346979412 -10.59 -5.78 0.80 1.0 17.7ms
5 -1.118346979412 -12.09 -6.55 0.80 1.0 18.0ms
6 -1.118346979412 -13.42 -7.04 0.80 1.0 18.4ms
7 -1.118346979412 -14.40 -8.13 0.80 1.0 19.3ms
8 -1.118346979412 + -14.48 -8.95 0.80 1.0 19.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837059 -2.61 0.80 1.0 10.7ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 16.7ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 17.0ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 17.4ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 18.0ms
6 -1.118356214425 -12.44 -6.55 0.80 1.0 27.8ms
7 -1.118356214425 -13.63 -7.65 0.80 1.0 18.8ms
8 -1.118356214425 + -15.35 -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 10.6ms
2 -1.118356214425 -14.65 -9.82 0.80 1.0 16.7ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.47262e-5 │ 493ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 10.5ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 16.8ms
3 -1.118356215917 -12.53 -6.70 0.80 1.0 16.9ms
4 -1.118356215917 -14.11 -7.62 0.80 1.0 17.1ms
5 -1.118356215917 + -15.18 -8.39 0.80 1.0 17.5ms
6 -1.118356215917 + -15.65 -8.89 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356217810 -4.40 0.80 1.0 10.5ms
2 -1.118356218156 -9.46 -5.42 0.80 1.0 16.7ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 25.0ms
4 -1.118356218159 -13.21 -7.09 0.80 1.0 17.3ms
5 -1.118356218159 -14.54 -7.86 0.80 1.0 17.5ms
6 -1.118356218159 -15.65 -8.35 0.80 1.0 18.0ms
7 -1.118356218159 + -14.88 -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 10.5ms
2 -1.118356218159 + -14.57 -11.02 0.80 1.0 16.6ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.07624e-8 │ 430ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, 3.66997e-11, 4.73588e-11]u"a₀")
Atom(H, [ 1.44318, 3.16475e-11, 4.75808e-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: 1.4863655835048666 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.