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 -1.111214155265 -0.83 0.80 9.0 26.5ms
2 -1.117209536450 -2.22 -2.01 0.80 1.0 46.7ms
3 -1.117250517502 -4.39 -2.71 0.80 1.0 17.9ms
4 -1.117251163626 -6.19 -3.56 0.80 1.0 18.2ms
5 -1.117251189451 -7.59 -4.54 0.80 1.0 18.4ms
6 -1.117251190088 -9.20 -4.87 0.80 2.0 21.0ms
7 -1.117251190142 -10.27 -5.74 0.80 1.0 19.4ms
8 -1.117251190143 -12.23 -6.13 0.80 1.0 35.6ms
9 -1.117251190143 -12.73 -7.05 0.80 1.0 20.4ms
10 -1.117251190143 -14.54 -8.01 0.80 1.0 20.0ms
11 -1.117251190143 -14.88 -8.81 0.80 2.0 22.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.32 0.80 1.0 116ms
2 -1.117251190143 -15.65 -10.41 0.80 1.0 19.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 1.33s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998060 -2.46 0.80 1.0 11.2ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 24.3ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 18.6ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.1ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.2ms
6 -1.117520613465 -12.13 -6.42 0.80 1.0 23.4ms
7 -1.117520613465 -13.32 -7.39 0.80 1.0 19.9ms
8 -1.117520613465 -14.75 -8.18 0.80 1.0 20.3ms
9 -1.117520613465 + -14.65 -8.97 0.80 1.0 25.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264592 -1.69 0.80 1.0 11.5ms
2 -1.118339043934 -4.04 -2.70 0.80 1.0 18.1ms
3 -1.118340011113 -6.01 -3.44 0.80 1.0 24.5ms
4 -1.118340028554 -7.76 -4.36 0.80 1.0 18.8ms
5 -1.118340029138 -9.23 -5.12 0.80 1.0 19.4ms
6 -1.118340029165 -10.57 -5.62 0.80 1.0 113ms
7 -1.118340029167 -11.73 -6.68 0.80 1.0 20.8ms
8 -1.118340029167 -13.57 -7.58 0.80 1.0 20.8ms
9 -1.118340029167 -15.35 -8.12 0.80 1.0 20.5ms
10 -1.118340029167 + -14.88 -8.55 0.80 1.0 927ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -9.05 0.80 1.0 11.3ms
2 -1.118340029167 + -Inf -10.36 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -11.04 0.80 1.0 11.2ms
2 -1.118340029167 -14.70 -11.41 0.80 1.0 18.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029167 │ │ 0.00297055 │ 1.72s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 11.3ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 17.9ms
3 -1.118346866236 -8.83 -4.86 0.80 1.0 18.5ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 18.9ms
5 -1.118346866262 -12.08 -6.55 0.80 1.0 18.9ms
6 -1.118346866262 -13.44 -7.04 0.80 1.0 19.5ms
7 -1.118346866262 -15.65 -8.13 0.80 1.0 20.0ms
8 -1.118346866262 -14.75 -8.95 0.80 1.0 19.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 11.0ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 17.9ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 18.1ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 18.5ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 18.9ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.0ms
7 -1.118356101386 -13.57 -7.65 0.80 1.0 19.8ms
8 -1.118356101386 -15.05 -8.45 0.80 1.0 19.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 11.1ms
2 -1.118356101386 -15.35 -9.82 0.80 1.0 24.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 24.0ms
2 -1.118356101386 -14.54 -11.46 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 609ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 11.4ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.0ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 18.1ms
4 -1.118356102877 -14.33 -7.62 0.80 1.0 18.6ms
5 -1.118356102877 -15.35 -8.40 0.80 1.0 19.0ms
6 -1.118356102877 + -15.05 -8.89 0.80 1.0 19.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 11.3ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.3ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.7ms
4 -1.118356105120 -13.18 -7.09 0.80 1.0 18.8ms
5 -1.118356105120 -14.48 -7.86 0.80 1.0 20.0ms
6 -1.118356105120 + -15.05 -8.35 0.80 1.0 19.3ms
7 -1.118356105120 + -15.35 -9.45 0.80 1.0 27.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 11.3ms
2 -1.118356105120 + -14.24 -11.02 0.80 1.0 18.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 11.1ms
2 -1.118356105120 + -14.33 -12.44 0.80 1.0 18.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.11492e-8 │ 533ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘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, [-0.0431829, 4.45861e-12, 2.32107e-12]u"a₀")
Atom(H, [ 1.44318, 4.30856e-12, 2.25565e-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: 1.486365864270879 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.