Pseudopotentials
In this example, we'll look at how to use various pseudopotential (PSP) formats in DFTK and discuss briefly the utility and importance of pseudopotentials.
Currently, DFTK supports norm-conserving (NC) PSPs in separable (Kleinman-Bylander) form. Two file formats can currently be read and used: analytical Hartwigsen-Goedecker-Hutter (HGH) PSPs and numeric Unified Pseudopotential Format (UPF) PSPs.
In brief, the pseudopotential approach replaces the all-electron atomic potential with an effective atomic potential. In this pseudopotential, tightly-bound core electrons are completely eliminated ("frozen") and chemically-active valence electron wavefunctions are replaced with smooth pseudo-wavefunctions whose Fourier representations decay quickly. Both these transformations aim at reducing the number of Fourier modes required to accurately represent the wavefunction of the system, greatly increasing computational efficiency.
Different PSP generation codes produce various file formats which contain the same general quantities required for pesudopotential evaluation. HGH PSPs are constructed from a fixed functional form based on Gaussians, and the files simply tablulate various coefficients fitted for a given element. UPF PSPs take a more flexible approach where the functional form used to generate the PSP is arbitrary, and the resulting functions are tabulated on a radial grid in the file. The UPF file format is documented on the Quantum Espresso Website.
In this example, we will compare the convergence of an analytical HGH PSP with a modern numeric norm-conserving PSP in UPF format from PseudoDojo. Then, we will compare the bandstructure at the converged parameters calculated using the two PSPs.
using AtomsBuilder
using DFTK
using Unitful
using UnitfulAtomic
using PseudoPotentialData
using Plots
Here, we will use a Perdew-Wang LDA PSP from PseudoDojo, which is available via the JuliaMolSim PseudoPotentialData package. See the documentation of PseudoPotentialData for the list of available pseudopotential families.
pseudopotentials_upf = PseudoFamily("dojo.nc.sr.lda.v0_4_1.oncvpsp3.standard.upf");
Such a PseudoFamily
object acts like a dictionary from an element symbol to a pseudopotential file. They can be directly employed to select the appropriate pseudopotential when constructing an ElementPsp
or a model based on an AtomsBase
-compatible system. For the latter see the run_bands
function below for an example.
Note that an alternative to a PseudoFamily
object is in all cases a plain Dict
to map from atomic symbols to the employed pseudopotential file. This we employ in combination with the HGH-type pseudopotentials:
pseudopotentials_hgh = Dict(:Si => "hgh/lda/si-q4.hgh");
First, we'll take a look at the energy cutoff convergence of these two pseudopotentials. For both pseudos, a reference energy is calculated with a cutoff of 140 Hartree, and SCF calculations are run at increasing cutoffs until 1 meV / atom convergence is reached.
The converged cutoffs are 26 Ha and 18 Ha for the HGH and UPF pseudos respectively. We see that the HGH pseudopotential is much harder, i.e. it requires a higher energy cutoff, than the UPF PSP. In general, numeric pseudopotentials tend to be softer than analytical pseudos because of the flexibility of sampling arbitrary functions on a grid.
Next, to see that the different pseudopotentials give reasonably similar results, we'll look at the bandstructures calculated using the HGH and UPF PSPs. Even though the converged cutoffs are higher, we perform these calculations with a cutoff of 12 Ha for both PSPs.
function run_bands(pseudopotentials)
system = bulk(:Si; a=10.26u"bohr")
# These are (as you saw above) completely unconverged parameters
model = model_DFT(system; functionals=LDA(), temperature=1e-2, pseudopotentials)
basis = PlaneWaveBasis(model; Ecut=12, kgrid=(4, 4, 4))
scfres = self_consistent_field(basis; tol=1e-4)
bandplot = plot_bandstructure(compute_bands(scfres))
(; scfres, bandplot)
end;
The SCF and bandstructure calculations can then be performed using the two PSPs, where we notice in particular the difference in total energies.
result_hgh = run_bands(pseudopotentials_hgh)
result_hgh.scfres.energies
Energy breakdown (in Ha):
Kinetic 3.1589969
AtomicLocal -2.1424066
AtomicNonlocal 1.6042750
Ewald -8.4004648
PspCorrection -0.2948928
Hartree 0.5515527
Xc -2.4000879
Entropy -0.0031625
total -7.926189848171
result_upf = run_bands(pseudopotentials_upf)
result_upf.scfres.energies
Energy breakdown (in Ha):
Kinetic 3.0954162
AtomicLocal -2.3650689
AtomicNonlocal 1.3082407
Ewald -8.4004648
PspCorrection 0.3952219
Hartree 0.5521760
Xc -3.1011616
Entropy -0.0032195
total -8.518859913641
But while total energies are not physical and thus allowed to differ, the bands (as an example for a physical quantity) are very similar for both pseudos:
plot(result_hgh.bandplot, result_upf.bandplot, titles=["HGH" "UPF"], size=(800, 400))