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 -1.110407911661 -0.82 0.80 7.0 25.2ms
2 -1.117194555932 -2.17 -1.84 0.80 1.0 17.5ms
3 -1.117250292061 -4.25 -2.67 0.80 1.0 45.4ms
4 -1.117251247577 -6.02 -3.49 0.80 1.0 18.2ms
5 -1.117251306354 -7.23 -3.87 0.80 2.0 20.5ms
6 -1.117251310028 -8.43 -4.56 0.80 1.0 18.4ms
7 -1.117251310375 -9.46 -5.22 0.80 1.0 18.7ms
8 -1.117251310381 -11.23 -5.97 0.80 1.0 21.2ms
9 -1.117251310381 -12.60 -6.91 0.80 1.0 40.1ms
10 -1.117251310381 -13.92 -7.24 0.80 2.0 21.6ms
11 -1.117251310381 + -14.75 -8.01 0.80 1.0 19.8ms
12 -1.117251310381 -14.57 -8.74 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251310381 -9.56 0.80 1.0 106ms
2 -1.117251310381 + -14.45 -10.09 0.80 1.0 17.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117251310381 │ │ 0.0269317 │ 1.32s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117518116434 -2.46 0.80 1.0 10.8ms
2 -1.117520704461 -5.59 -3.49 0.80 1.0 17.1ms
3 -1.117520731396 -7.57 -4.23 0.80 1.0 24.0ms
4 -1.117520731860 -9.33 -5.14 0.80 1.0 18.0ms
5 -1.117520731875 -10.81 -5.92 0.80 1.0 18.3ms
6 -1.117520731876 -12.13 -6.42 0.80 1.0 18.4ms
7 -1.117520731876 -13.36 -7.39 0.80 1.0 24.4ms
8 -1.117520731876 -14.35 -8.18 0.80 1.0 19.6ms
9 -1.117520731876 + -15.65 -8.97 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247378325 -1.69 0.80 1.0 17.5ms
2 -1.118339157202 -4.04 -2.70 0.80 1.0 17.3ms
3 -1.118340124376 -6.01 -3.44 0.80 1.0 17.1ms
4 -1.118340141817 -7.76 -4.36 0.80 1.0 17.4ms
5 -1.118340142401 -9.23 -5.12 0.80 1.0 23.1ms
6 -1.118340142428 -10.57 -5.62 0.80 1.0 19.2ms
7 -1.118340142430 -11.73 -6.68 0.80 1.0 18.8ms
8 -1.118340142430 -13.54 -7.58 0.80 1.0 19.5ms
9 -1.118340142430 + -14.95 -8.12 0.80 1.0 24.2ms
10 -1.118340142430 + -15.18 -8.55 0.80 1.0 19.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340142430 -9.05 0.80 1.0 14.3ms
2 -1.118340142430 + -14.75 -10.36 0.80 1.0 45.4ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118340142430 │ -2.96 │ 0.00297053 │ 622ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346834684 -3.09 0.80 1.0 10.8ms
2 -1.118346977933 -6.84 -4.11 0.80 1.0 16.8ms
3 -1.118346979411 -8.83 -4.86 0.80 1.0 23.8ms
4 -1.118346979436 -10.59 -5.78 0.80 1.0 17.7ms
5 -1.118346979437 -12.09 -6.55 0.80 1.0 18.0ms
6 -1.118346979437 -13.38 -7.04 0.80 1.0 18.4ms
7 -1.118346979437 -14.81 -8.13 0.80 1.0 23.6ms
8 -1.118346979437 + -14.95 -8.95 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354837063 -2.61 0.80 1.0 10.7ms
2 -1.118356200096 -5.87 -3.62 0.80 1.0 16.8ms
3 -1.118356214170 -7.85 -4.37 0.80 1.0 24.0ms
4 -1.118356214417 -9.61 -5.29 0.80 1.0 17.5ms
5 -1.118356214425 -11.10 -6.06 0.80 1.0 17.7ms
6 -1.118356214425 -12.45 -6.55 0.80 1.0 18.5ms
7 -1.118356214425 -13.58 -7.65 0.80 1.0 24.1ms
8 -1.118356214425 + -14.57 -8.45 0.80 1.0 19.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356214425 -9.24 0.80 1.0 10.8ms
2 -1.118356214425 + -14.95 -9.82 0.80 1.0 23.3ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118356214425 │ -4.79 │ 4.47261e-5 │ 513ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356215887 -4.94 0.80 1.0 11.0ms
2 -1.118356215917 -10.54 -5.96 0.80 1.0 17.1ms
3 -1.118356215917 -12.52 -6.70 0.80 1.0 17.3ms
4 -1.118356215917 -14.70 -7.62 0.80 1.0 24.8ms
5 -1.118356215917 -14.75 -8.39 0.80 1.0 18.0ms
6 -1.118356215917 + -14.51 -8.89 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356217810 -4.40 0.80 1.0 10.8ms
2 -1.118356218156 -9.46 -5.42 0.80 1.0 17.1ms
3 -1.118356218159 -11.45 -6.17 0.80 1.0 17.1ms
4 -1.118356218159 -13.22 -7.09 0.80 1.0 17.3ms
5 -1.118356218159 -14.81 -7.86 0.80 1.0 23.0ms
6 -1.118356218159 -14.65 -8.35 0.80 1.0 18.2ms
7 -1.118356218159 -15.35 -9.45 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356218159 -10.08 0.80 1.0 16.7ms
2 -1.118356218159 + -14.65 -11.02 0.80 1.0 16.8ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118356218159 │ -8.43 │ 1.10343e-8 │ 449ms │
└─────┴─────────────────┴───────────┴────────────┴────────┘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, 1.051e-11, -3.29436e-10]u"a₀")
Atom(H, [ 1.44318, 1.92381e-11, -3.18972e-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: 1.4863655824897546 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.