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 -1.111279530057 -0.83 0.80 8.0 23.1ms
2 -1.117213199636 -2.23 -2.00 0.80 1.0 16.9ms
3 -1.117250686719 -4.43 -2.74 0.80 1.0 39.0ms
4 -1.117251167973 -6.32 -3.61 0.80 1.0 17.9ms
5 -1.117251189090 -7.68 -4.44 0.80 1.0 17.7ms
6 -1.117251190089 -9.00 -4.89 0.80 2.0 20.0ms
7 -1.117251190142 -10.28 -5.71 0.80 1.0 18.5ms
8 -1.117251190143 -11.96 -6.62 0.80 1.0 33.7ms
9 -1.117251190143 -13.38 -7.15 0.80 2.0 21.0ms
10 -1.117251190143 -14.70 -8.05 0.80 1.0 19.1ms
11 -1.117251190143 + -15.05 -8.71 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.10 0.80 1.0 97.1ms
2 -1.117251190143 + -14.54 -10.63 0.80 1.0 16.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 1.20s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998050 -2.46 0.80 1.0 10.6ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 22.3ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 17.0ms
4 -1.117520613448 -9.33 -5.14 0.80 1.0 17.2ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 17.7ms
6 -1.117520613465 -12.13 -6.42 0.80 1.0 21.3ms
7 -1.117520613465 -13.33 -7.39 0.80 1.0 18.4ms
8 -1.117520613465 + -15.18 -8.18 0.80 1.0 18.9ms
9 -1.117520613465 -14.57 -8.97 0.80 1.0 22.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264593 -1.69 0.80 1.0 10.4ms
2 -1.118339043936 -4.04 -2.70 0.80 1.0 16.9ms
3 -1.118340011115 -6.01 -3.44 0.80 1.0 22.2ms
4 -1.118340028555 -7.76 -4.36 0.80 1.0 17.3ms
5 -1.118340029140 -9.23 -5.12 0.80 1.0 17.7ms
6 -1.118340029167 -10.57 -5.62 0.80 1.0 21.1ms
7 -1.118340029169 -11.73 -6.68 0.80 1.0 18.8ms
8 -1.118340029169 -13.54 -7.58 0.80 1.0 19.5ms
9 -1.118340029169 + -15.18 -8.12 0.80 1.0 18.9ms
10 -1.118340029169 -14.57 -8.55 0.80 1.0 22.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029169 -9.05 0.80 1.0 10.5ms
2 -1.118340029169 + -14.57 -10.36 0.80 1.0 21.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029169 -11.04 0.80 1.0 10.4ms
2 -1.118340029169 -14.65 -11.41 0.80 1.0 16.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029169 │ │ 0.00297055 │ 678ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721508 -3.09 0.80 1.0 10.6ms
2 -1.118346864760 -6.84 -4.11 0.80 1.0 17.0ms
3 -1.118346866237 -8.83 -4.86 0.80 1.0 17.0ms
4 -1.118346866263 -10.59 -5.78 0.80 1.0 17.2ms
5 -1.118346866264 -12.08 -6.55 0.80 1.0 17.6ms
6 -1.118346866264 -13.38 -7.04 0.80 1.0 32.2ms
7 -1.118346866264 + -14.70 -8.13 0.80 1.0 18.5ms
8 -1.118346866264 -15.05 -8.95 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 10.4ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 16.7ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 22.3ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 17.4ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 17.6ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 18.0ms
7 -1.118356101386 -13.64 -7.65 0.80 1.0 21.9ms
8 -1.118356101386 -14.81 -8.45 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 10.3ms
2 -1.118356101386 -14.75 -9.82 0.80 1.0 21.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 10.4ms
2 -1.118356101386 -14.65 -11.46 0.80 1.0 16.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 603ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 10.5ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 16.8ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 16.8ms
4 -1.118356102877 -14.18 -7.62 0.80 1.0 22.4ms
5 -1.118356102877 + -14.88 -8.39 0.80 1.0 17.6ms
6 -1.118356102877 + -14.88 -8.89 0.80 1.0 18.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 10.4ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 16.7ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 16.8ms
4 -1.118356105120 -13.22 -7.09 0.80 1.0 20.6ms
5 -1.118356105120 -14.42 -7.86 0.80 1.0 17.8ms
6 -1.118356105120 -14.81 -8.35 0.80 1.0 18.0ms
7 -1.118356105120 + -14.95 -9.45 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 10.4ms
2 -1.118356105120 + -14.88 -11.02 0.80 1.0 16.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 15.6ms
2 -1.118356105120 + -14.81 -12.44 0.80 1.0 17.3ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.06918e-8 │ 521ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘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, -3.26926e-12, -9.16529e-13]u"a₀")
Atom(H, [ 1.44318, -3.10429e-12, -1.18257e-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: 1.4863658659791918 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.