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 -1.110341668710 -0.82 0.80 7.0 22.5ms
2 -1.117193550726 -2.16 -1.83 0.80 1.0 40.4ms
3 -1.117250159509 -4.25 -2.68 0.80 1.0 17.7ms
4 -1.117251255359 -5.96 -3.40 0.80 1.0 18.2ms
5 -1.117251306858 -7.29 -3.93 0.80 1.0 18.3ms
6 -1.117251310219 -8.47 -4.67 0.80 1.0 18.8ms
7 -1.117251310376 -9.80 -5.38 0.80 1.0 34.5ms
8 -1.117251310381 -11.33 -6.09 0.80 1.0 20.1ms
9 -1.117251310381 -12.69 -6.80 0.80 1.0 19.5ms
10 -1.117251310381 -13.76 -7.42 0.80 2.0 21.8ms
11 -1.117251310381 + -Inf -8.28 0.80 1.0 20.1ms
12 -1.117251310381 -14.45 -8.66 0.80 2.0 25.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -9.24 0.80 1.0 94.7ms
2 -1.117251310381 -15.35 -10.43 0.80 1.0 17.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 1.25s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518116422 -2.46 0.80 1.0 10.8ms
2 -1.117520704464 -5.59 -3.49 0.80 1.0 17.3ms
3 -1.117520731399 -7.57 -4.23 0.80 1.0 17.6ms
4 -1.117520731863 -9.33 -5.14 0.80 1.0 23.1ms
5 -1.117520731878 -10.81 -5.92 0.80 1.0 18.3ms
6 -1.117520731879 -12.14 -6.42 0.80 1.0 18.7ms
7 -1.117520731879 -13.34 -7.39 0.80 1.0 22.1ms
8 -1.117520731879 -14.65 -8.18 0.80 1.0 19.7ms
9 -1.117520731879 -14.88 -8.97 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378324 -1.69 0.80 1.0 15.8ms
2 -1.118339157168 -4.04 -2.70 0.80 1.0 17.5ms
3 -1.118340124341 -6.01 -3.44 0.80 1.0 17.6ms
4 -1.118340141781 -7.76 -4.36 0.80 1.0 18.0ms
5 -1.118340142366 -9.23 -5.12 0.80 1.0 22.2ms
6 -1.118340142393 -10.57 -5.62 0.80 1.0 18.6ms
7 -1.118340142395 -11.73 -6.68 0.80 1.0 19.1ms
8 -1.118340142395 -13.59 -7.58 0.80 1.0 23.1ms
9 -1.118340142395 -14.65 -8.12 0.80 1.0 19.7ms
10 -1.118340142395 + -15.05 -8.55 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142395 -9.05 0.80 1.0 16.0ms
2 -1.118340142395 + -15.05 -10.36 0.80 1.0 17.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142395 -11.04 0.80 1.0 10.8ms
2 -1.118340142395 -15.05 -11.41 0.80 1.0 22.3ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340142395 │ │ 0.00297053 │ 692ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834662 -3.09 0.80 1.0 10.8ms
2 -1.118346977912 -6.84 -4.11 0.80 1.0 17.4ms
3 -1.118346979389 -8.83 -4.86 0.80 1.0 96.6ms
4 -1.118346979415 -10.59 -5.78 0.80 1.0 18.4ms
5 -1.118346979416 -12.08 -6.55 0.80 1.0 18.4ms
6 -1.118346979416 -13.42 -7.04 0.80 1.0 18.4ms
7 -1.118346979416 -14.51 -8.13 0.80 1.0 649ms
8 -1.118346979416 + -15.18 -8.95 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837060 -2.61 0.80 1.0 10.9ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 17.4ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 17.4ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 17.7ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 18.3ms
6 -1.118356214425 -12.45 -6.55 0.80 1.0 18.6ms
7 -1.118356214425 -13.56 -7.65 0.80 1.0 19.1ms
8 -1.118356214425 -15.35 -8.45 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -9.24 0.80 1.0 10.6ms
2 -1.118356214425 -14.57 -9.82 0.80 1.0 17.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -10.62 0.80 1.0 10.7ms
2 -1.118356214425 -14.45 -11.46 0.80 1.0 17.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.47263e-5 │ 1.27s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 10.7ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 17.0ms
3 -1.118356215917 -12.52 -6.70 0.80 1.0 17.1ms
4 -1.118356215917 -14.26 -7.62 0.80 1.0 17.5ms
5 -1.118356215917 + -15.05 -8.40 0.80 1.0 18.1ms
6 -1.118356215917 -14.70 -8.89 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356217810 -4.40 0.80 1.0 10.7ms
2 -1.118356218156 -9.46 -5.42 0.80 1.0 36.2ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 17.4ms
4 -1.118356218159 -13.20 -7.09 0.80 1.0 17.4ms
5 -1.118356218159 -14.26 -7.86 0.80 1.0 18.1ms
6 -1.118356218159 + -14.61 -8.35 0.80 1.0 18.6ms
7 -1.118356218159 -15.18 -9.45 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -10.08 0.80 1.0 10.6ms
2 -1.118356218159 + -Inf -11.02 0.80 1.0 17.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -11.67 0.80 1.0 10.7ms
2 -1.118356218159 + -15.18 -12.44 0.80 1.0 17.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.09976e-8 │ 525ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘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, -2.60839e-11, 3.48063e-11]u"a₀")
Atom(H, [ 1.44318, -2.59595e-11, 3.4504e-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: 1.486365582621881 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.