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 -1.110722418724 -0.83 0.80 7.0 25.8ms
2 -1.117199297294 -2.19 -1.90 0.80 1.0 128ms
3 -1.117250265502 -4.29 -2.70 0.80 1.0 20.2ms
4 -1.117251123332 -6.07 -3.37 0.80 1.0 20.5ms
5 -1.117251186749 -7.20 -4.09 0.80 1.0 20.8ms
6 -1.117251190071 -8.48 -4.90 0.80 2.0 23.1ms
7 -1.117251190137 -10.18 -5.34 0.80 1.0 21.4ms
8 -1.117251190143 -11.22 -6.48 0.80 1.0 21.7ms
9 -1.117251190143 -13.05 -6.98 0.80 2.0 24.2ms
10 -1.117251190143 -14.57 -7.82 0.80 1.0 60.0ms
11 -1.117251190143 + -15.35 -7.96 0.80 1.0 22.2ms
12 -1.117251190143 -15.05 -8.64 0.80 1.0 22.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -8.97 0.80 1.0 1.19s
2 -1.117251190143 + -14.70 -10.03 0.80 1.0 19.6ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 3.66s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997302 -2.46 0.80 1.0 13.3ms
2 -1.117520586058 -5.59 -3.49 0.80 1.0 36.8ms
3 -1.117520612993 -7.57 -4.23 0.80 1.0 20.2ms
4 -1.117520613449 -9.34 -5.15 0.80 1.0 20.3ms
5 -1.117520613465 -10.81 -5.93 0.80 1.0 20.8ms
6 -1.117520613465 -12.17 -6.44 0.80 1.0 20.9ms
7 -1.117520613465 -13.36 -7.48 0.80 1.0 21.5ms
8 -1.117520613465 + -14.65 -8.31 0.80 1.0 21.5ms
9 -1.117520613465 -15.35 -8.63 0.80 1.0 33.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264616 -1.69 0.80 1.0 13.2ms
2 -1.118339043930 -4.04 -2.70 0.80 1.0 19.8ms
3 -1.118340011109 -6.01 -3.44 0.80 1.0 19.8ms
4 -1.118340028550 -7.76 -4.36 0.80 1.0 20.6ms
5 -1.118340029134 -9.23 -5.12 0.80 1.0 26.4ms
6 -1.118340029161 -10.57 -5.62 0.80 1.0 21.2ms
7 -1.118340029163 -11.73 -6.68 0.80 1.0 21.4ms
8 -1.118340029163 -13.52 -7.58 0.80 1.0 22.1ms
9 -1.118340029163 + -15.05 -8.12 0.80 1.0 21.9ms
10 -1.118340029163 -15.35 -8.55 0.80 1.0 27.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029163 -9.05 0.80 1.0 13.3ms
2 -1.118340029163 + -14.88 -10.36 0.80 1.0 19.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029163 -11.05 0.80 1.0 13.0ms
2 -1.118340029163 -14.95 -11.43 0.80 1.0 28.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029163 │ │ 0.00297055 │ 850ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721504 -3.09 0.80 1.0 13.4ms
2 -1.118346864756 -6.84 -4.11 0.80 1.0 20.0ms
3 -1.118346866234 -8.83 -4.86 0.80 1.0 19.8ms
4 -1.118346866259 -10.59 -5.78 0.80 1.0 20.3ms
5 -1.118346866260 -12.08 -6.55 0.80 1.0 29.3ms
6 -1.118346866260 -13.42 -7.04 0.80 1.0 21.2ms
7 -1.118346866260 + -Inf -8.13 0.80 1.0 21.5ms
8 -1.118346866260 + -14.65 -8.95 0.80 1.0 21.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.4ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 28.2ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 20.0ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 20.3ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 21.1ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 21.0ms
7 -1.118356101386 -13.64 -7.65 0.80 1.0 26.8ms
8 -1.118356101386 -14.37 -8.45 0.80 1.0 22.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.3ms
2 -1.118356101386 + -14.45 -9.82 0.80 1.0 19.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 20.6ms
2 -1.118356101386 + -14.27 -11.46 0.80 1.0 20.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 600ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.6ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 20.1ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 20.1ms
4 -1.118356102877 -14.65 -7.62 0.80 1.0 29.2ms
5 -1.118356102877 -14.40 -8.40 0.80 1.0 21.7ms
6 -1.118356102877 + -14.54 -8.89 0.80 1.0 21.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.3ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 19.9ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 26.8ms
4 -1.118356105120 -13.23 -7.09 0.80 1.0 20.9ms
5 -1.118356105120 -14.42 -7.86 0.80 1.0 20.7ms
6 -1.118356105120 -15.18 -8.35 0.80 1.0 21.2ms
7 -1.118356105120 -15.35 -9.45 0.80 1.0 21.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 19.1ms
2 -1.118356105120 -15.35 -11.02 0.80 1.0 19.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.3ms
2 -1.118356105120 -14.35 -12.44 0.80 1.0 19.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.1119e-8 │ 529ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘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, 1.369e-17, 8.53454e-18]u"a₀")
Atom(H, [ 1.44318, 8.43752e-18, -3.57672e-17]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.486365864383724 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Recent versions of GeometryOptimization support cell shape optimisations as well by passing variablecell=true to minimize_energy!. See the GeometryOptimization documentation for an example.