Aluminium supercell using density-functional theory
In this example we will optimise the structure of a rattled aluminium system using density-functional theory.
First we build a rattled aluminium system:
using AtomsBuilder
using Unitful
system = rattle!(bulk(:Al; cubic=true), 0.2u"Å")FlexibleSystem(Al₄, periodicity = TTT):
cell_vectors : [ 4.05 0 0;
0 4.05 0;
0 0 4.05]u"Å"
Atom(Al, [0.0358777, -0.0220648, -0.14554]u"Å")
Atom(Al, [-0.12069, 2.09902, 2.03422]u"Å")
Atom(Al, [ 1.93682, 0.0292324, 1.97468]u"Å")
Atom(Al, [ 2.08519, 2.03462, 0.0076098]u"Å")
Next we create a calculator employing the density-functional toolkit to compute energies and forces at using the LDA density functional. As pseudopotentials we use the PseudoDojo as available in the PseudoPotentialData package.
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.lda.v0_4_1.oncvpsp3.standard.upf")
model_kwargs = (; functionals=LDA(), temperature=1e-3, pseudopotentials)
basis_kwargs = (; kgrid=(3, 3, 3), Ecut=10.0)
scf_kwargs = (; mixing=KerkerMixing())
calc = DFTKCalculator(; model_kwargs, basis_kwargs, scf_kwargs)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.lda.v0_4_1.oncvpsp3.standard.upf"), temperature=0.001, Ecut=10.0, kgrid=(3, 3, 3))Note, that the numerical parameters above are chosen rather crudely in order to give a fast runtime on CI systems. For production calculations one would require larger computational parameters.
We perform the structure optimisation using the LBFGS solver from Optim with solver parameters adapted for our geometry optimisation setting. This is selected by passing the GeometryOptimization.OptimLBFGS solver as the third argument. The verbosity=2 flag makes sure we get output from both the geometry optimisation as well as the inner SCF solver.
using GeometryOptimization
GO = GeometryOptimization
results = minimize_energy!(system, calc, GO.OptimLBFGS();
tol_forces=1e-4u"eV/Å", verbosity=2)
nothing Downloading artifact: pd_nc_sr_lda_standard_0.4.1_upf
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.439173664572 -1.17 0.80 8.6 5.15s
2 -9.441263972125 -2.68 -1.68 0.80 1.0 8.65s
3 -9.441495024051 -3.64 -2.66 0.80 2.7 452ms
4 -9.441530902947 -4.45 -3.43 0.80 3.6 594ms
5 -9.441531289836 -6.41 -3.92 0.80 2.9 517ms
6 -9.441531327880 -7.42 -4.68 0.80 2.3 425ms
7 -9.441531329799 -8.72 -5.36 0.80 2.9 777ms
8 -9.441531329849 -10.31 -6.00 0.80 2.7 496ms
9 -9.441531329852 -11.54 -6.45 0.80 2.6 491ms
10 -9.441531329852 -12.45 -7.24 0.80 2.0 399ms
11 -9.441531329852 -13.80 -7.82 0.80 2.9 543ms
12 -9.441531329852 + -14.45 -8.87 0.80 2.6 466ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.441531329852 -8.93 0.80 1.0 2.17s
2 -9.441531329852 + -Inf -9.83 0.80 1.0 387ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -9.441531329852 │ │ 0.0103152 │ 44.7s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.441582580124 -3.22 0.80 1.0 255ms
2 -9.441583790299 -5.92 -3.87 0.80 1.0 306ms
3 -9.441583849522 -7.23 -4.33 0.80 3.3 529ms
4 -9.441583852183 -8.57 -4.82 0.80 2.1 413ms
5 -9.441583853128 -9.02 -5.82 0.80 2.3 468ms
6 -9.441583853142 -10.86 -6.13 0.80 3.5 859ms
7 -9.441583853143 -12.11 -7.01 0.80 1.5 320ms
8 -9.441583853143 -13.23 -7.54 0.80 3.4 587ms
9 -9.441583853143 + -Inf -8.14 0.80 2.4 398ms
10 -9.441583853143 -14.45 -8.76 0.80 2.4 526ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.427251295590 -1.15 0.80 2.8 707ms
2 -9.443983441763 -1.78 -1.92 0.80 1.0 290ms
3 -9.444477529903 -3.31 -2.32 0.80 2.9 538ms
4 -9.444520421702 -4.37 -2.75 0.80 2.1 454ms
5 -9.444534589781 -4.85 -3.85 0.80 2.2 369ms
6 -9.444534888748 -6.52 -4.08 0.80 3.8 696ms
7 -9.444534909180 -7.69 -4.94 0.80 1.3 307ms
8 -9.444534910069 -9.05 -5.82 0.80 3.3 580ms
9 -9.444534910069 -12.24 -5.85 0.80 3.5 585ms
10 -9.444534910076 -11.16 -7.18 0.80 1.2 298ms
11 -9.444534910076 -12.91 -7.29 0.80 4.5 1.07s
12 -9.444534910076 -14.15 -8.21 0.80 1.2 311ms
13 -9.444534910076 -14.45 -8.54 0.80 3.2 564ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444534910076 -8.49 0.80 1.0 267ms
2 -9.444534910076 + -Inf -9.75 0.80 1.0 282ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -9.444534910076 │ -2.52 │ 0.00212173 │ 14.1s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.421059530787 -1.10 0.80 3.8 535ms
2 -9.442135566296 -1.68 -1.88 0.80 1.0 552ms
3 -9.442714730553 -3.24 -2.28 0.80 2.9 526ms
4 -9.442771002498 -4.25 -2.79 0.80 2.0 457ms
5 -9.442782446520 -4.94 -3.76 0.80 2.4 420ms
6 -9.442782639774 -6.71 -4.18 0.80 3.7 613ms
7 -9.442782651583 -7.93 -5.25 0.80 2.0 384ms
8 -9.442782651907 -9.49 -5.83 0.80 3.7 653ms
9 -9.442782651908 -11.79 -6.24 0.80 2.4 432ms
10 -9.442782651909 -11.99 -7.05 0.80 2.0 359ms
11 -9.442782651909 -13.39 -7.45 0.80 3.1 871ms
12 -9.442782651909 -14.75 -8.40 0.80 2.2 411ms
13 -9.442782651909 -14.75 -9.03 0.80 3.5 567ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.432465271968 -1.22 0.80 2.8 406ms
2 -9.444189230724 -1.93 -1.93 0.80 1.0 278ms
3 -9.444613594443 -3.37 -2.34 0.80 3.0 539ms
4 -9.444654380812 -4.39 -2.70 0.80 2.0 496ms
5 -9.444672331618 -4.75 -3.84 0.80 1.6 357ms
6 -9.444672632996 -6.52 -3.96 0.80 4.4 938ms
7 -9.444672664664 -7.50 -4.52 0.80 1.1 297ms
8 -9.444672668727 -8.39 -5.79 0.80 2.9 441ms
9 -9.444672668797 -10.15 -5.83 0.80 4.1 772ms
10 -9.444672668803 -11.25 -6.39 0.80 1.0 289ms
11 -9.444672668803 -12.11 -7.01 0.80 2.2 433ms
12 -9.444672668803 -13.43 -7.70 0.80 2.5 516ms
13 -9.444672668803 -14.75 -8.89 0.80 2.8 497ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444672668803 -9.42 0.80 1.0 551ms
2 -9.444672668803 + -Inf -9.99 0.80 1.0 272ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -9.444672668803 │ -3.86 │ 0.0001232… │ 14.7s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444672662646 -3.69 0.80 1.0 276ms
2 -9.444672817833 -6.81 -4.45 0.80 1.0 271ms
3 -9.444672823145 -8.27 -4.84 0.80 3.3 575ms
4 -9.444672823398 -9.60 -5.19 0.80 2.4 412ms
5 -9.444672823582 -9.74 -6.29 0.80 1.6 621ms
6 -9.444672823586 -11.43 -6.41 0.80 3.8 712ms
7 -9.444672823587 -12.32 -7.45 0.80 1.3 294ms
8 -9.444672823587 -13.85 -8.07 0.80 3.5 587ms
9 -9.444672823587 + -14.75 -8.03 0.80 2.6 501ms
10 -9.444672823587 + -Inf -9.13 0.80 1.2 292ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444670204705 -3.06 0.80 1.0 281ms
2 -9.444673014305 -5.55 -3.83 0.80 1.0 314ms
3 -9.444673110123 -7.02 -4.21 0.80 3.2 812ms
4 -9.444673114717 -8.34 -4.56 0.80 2.2 380ms
5 -9.444673118012 -8.48 -5.63 0.80 1.8 338ms
6 -9.444673118079 -10.18 -5.78 0.80 3.8 700ms
7 -9.444673118088 -11.03 -6.92 0.80 1.3 299ms
8 -9.444673118088 -12.63 -7.30 0.80 4.0 695ms
9 -9.444673118088 + -14.27 -7.31 0.80 1.7 322ms
10 -9.444673118088 -14.27 -8.30 0.80 1.0 304ms
11 -9.444673118088 -14.45 -8.87 0.80 3.5 624ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444673118088 -9.53 0.80 1.0 255ms
2 -9.444673118088 -14.75 -10.14 0.80 1.0 283ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -9.444673118088 │ -6.35 │ 1.59566e-5 │ 11.2s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444673116184 -4.48 0.80 1.0 259ms
2 -9.444673120433 -8.37 -5.38 0.80 1.0 274ms
3 -9.444673120540 -9.97 -5.68 0.80 3.1 835ms
4 -9.444673120550 -11.01 -6.14 0.80 1.9 398ms
5 -9.444673120552 -11.64 -7.14 0.80 2.3 430ms
6 -9.444673120552 -13.47 -7.51 0.80 3.5 574ms
7 -9.444673120552 -14.75 -8.15 0.80 1.6 340ms
8 -9.444673120552 + -Inf -9.11 0.80 2.9 484ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444673059205 -3.89 0.80 1.0 286ms
2 -9.444673122987 -7.20 -4.79 0.80 1.0 310ms
3 -9.444673124597 -8.79 -5.10 0.80 3.1 790ms
4 -9.444673124746 -9.83 -5.55 0.80 1.9 384ms
5 -9.444673124780 -10.47 -6.53 0.80 2.2 403ms
6 -9.444673124780 -12.20 -6.90 0.80 3.3 591ms
7 -9.444673124780 -13.43 -7.87 0.80 1.8 336ms
8 -9.444673124780 -14.75 -8.64 0.80 3.6 612ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -9.444673124780 -9.27 0.80 1.0 297ms
2 -9.444673124780 -14.45 -9.79 0.80 1.0 548ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 4 │ -9.444673124780 │ -8.17 │ 1.60413e-6 │ 8.95s │
└─────┴─────────────────┴───────────┴────────────┴────────┘Some calculators (such as DFTK) are able to adapt to the keyword arguments and parameters passed to minimize_energy!. In this case the SCF tolerance is automatically adapted according to the convergence parameters (here tol_forces) passed to minimize_energy!.
The final energy is
results.energy-9.444673124780216 EₕWe can view the final structure
results.systemFlexibleSystem(Al₄, periodicity = TTT):
cell_vectors : [ 4.05 0 0;
0 4.05 0;
0 0 4.05]u"Å"
Atom(Al, [-0.0281957, 0.0227147, -0.0447411]u"Å")
Atom(Al, [-0.028197, 2.04769, 1.98024]u"Å")
Atom(Al, [ 1.9968, 0.0226954, 1.98024]u"Å")
Atom(Al, [ 1.9968, 2.04771, -0.0447663]u"Å")
Some statistics about the optimisation
results.statsSciMLBase.OptimizationStats
Number of iterations: 4
Time in seconds: 91.932264
Number of function evaluations: 9
Number of gradient evaluations: 5
Number of hessian evaluations: 0or the details about the selected algorithm:
results.algOptim.LBFGS{Nothing, LineSearches.InitialHagerZhang{Float64}, LineSearches.BackTracking{Float64, Int64}, Optim.var"#20#22"}(10, LineSearches.InitialHagerZhang{Float64}
ψ0: Float64 0.01
ψ1: Float64 0.2
ψ2: Float64 2.0
ψ3: Float64 0.1
αmax: Float64 Inf
α0: Float64 1.0
quadstep: Bool true
verbose: Bool false
, LineSearches.BackTracking{Float64, Int64}
c_1: Float64 0.0001
ρ_hi: Float64 0.5
ρ_lo: Float64 0.1
iterations: Int64 1000
order: Int64 2
maxstep: Float64 0.8
cache: Nothing nothing
, nothing, Optim.var"#20#22"(), Optim.Flat(), true)The final state of the calculator object is also accessible via results.state and could be employed for postprocessing using the framework of the calculator. E.g. in the case of DFTK, the results.state is what DFTK calls an scfres and could just be used to plot a density of states or plot bands or compute response properties.