Pseudopotentials and basis sets available in QuantumATK

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The accuracy of the calculations with the LCAO model in QuantumATK depends on the pseudopotential and basis sets used for the calculations. In recent years we have put a large effort into providing a robust set which can provide good accuracy at reasonably computational resources. Thus, there is an evolution in quality of the pseudopotentials we provide, with the best set being the latest SG15 pseudopotentials available from the ATK-2016 release and PseudoDojo pseudopotentials available from the O-2018.06 release. The theoretical background of the basis sets and pseudopotentials are described in the ATK Manual, and the purpose of this note is to guide the user towards the best setting.

Pseudopotentials

The pseudopotentials available for the QuantumATK software are listed in the below table. We see that it shows an evolution with quality of the pseudopotentials for each release. However, not all exchange correlation functionals are available for each type of pseudopotential. For GGA and SOGGA we recommend to use the SG15 pseudopotential. For LDA we recommend the FHI pseudopotential since LDA by itself is rather approximate. For SOLDA we recommend the OMX, it can also give higher accuracy for LDA, however, the required mesh cutoff need to be tested carefully.

Table 13 Pseudopotential types available for the ATK-DFT calculator.
Name Introduced Exchange-Correlation Cutoff [Ha] Accuracy Efficiency
FHI ATK-2008 LDA, GGA, MGGA 30-100 Low High
HGH ATK-2011 LDA, GGA, MGGA 75-500 Medium Low
OMX ATK-2015 LDA, GGA, MGGA 75-500 High Low
OMX ATK-2015 SOLDA, SOGGA, SOMGGA 75-500 High Low
SG15 ATK-2016 GGA 30-220 High Medium
SG15-SO ATK-2016 GGA, SOGGA 30-220 High Medium
PseudoDojo O-2018.6 GGA 30-125 Medium High

Warning

The higher values for OMX pseudopotentials, about 300 Hartree or more, are especially relevant for pseudopotentials with semi-core states. They often need these very high mesh cutoffs to give accurate forces.

Basis sets

ATK uses a numerical LCAO basis sets. The accuracy depends on the number of orbitals and the range of the orbitals. Increasing the number of orbitals and their range decrease the efficiency and increase the memory requirement. For the latest SG15 and PseudoDojo pseudopotentials we provide basis sets which systematically improve the accuracy. For the other pseudopotentials the improvement in accuracy is less systematic. To see the type of the basis sets it is most easy to

  • Open the scripter
  • Add a New Calculator block
  • Select the ATK-DFT calculator
  • Go to the Basis set tab at click the plot widget 2dplot_icon in the line where you define the basis set.
../../../_images/basis_widget.png

You will then get the following plot.

../../../_images/basisset_si.png

Accuracy tests for elemental solids

To give an overall idea of the accuracy of each pseudopotential and basis sets we have performed so-called \(\Delta\)-tests [LBBjorkman+16] (see the web-site Comparing Solid State DFT Codes, Basis Sets and Potentials). These test checks accuracy for the pseudopotential and basis set compared with state of the art all-electron calculations. The tests check the accuracy of the pseudopotential and basis sets in reproducing the equation of state of all the elemental solids in the periodic system using the GGA-PBE exchange-correlation functional.

Table 14 Accuracy and performance of PBE pseudopotential and basis types available for the ATK-DFT calculator.
Pseudo Basis \(\Delta\) (meV) Performance
FHI SZP 39.4 0.5
FHI DZP 18.2 1
FHI DZDP 19.1 2
HGH Tier4 12.8 6
OMX Medium 8.0 3
OMX High 2.2 10
SG15 Medium 3.5 2
SG15 High 1.9 6
SG15 Ultra 2.0 25

A \(\Delta\) value below 2 meV indicates a state of the art DFT calculation. The perfomance number given in the Table is an average indication of how fast a calculation of a 64 atom supercell of the different models relative to the FHI-DZP model. There are large variations among the elements and the performance number should only be used as a rough guide.

Accuracy tests for mixed solids

The \(\Delta\)-tests are done for ideal systems, in order to illustrate that these results are transferable to general systems we have performed a number of additional accuracy tests. The first test is for a number Rock Salt (RS) and Perovskite(P) structures provided in Ref. [GBRV14]. The reference points are FHI-aims all electron calculations. We see the deviations for the \(a\) (lattice constant), \(B\) (Bulk modulus) and \(E_f\) (formation energy) compared to the Ref. [GBRV14] in the below Table.

Table 15 Accuracy of selected methods in QuantumATK on RockSalt (RS) geometries [GBRV14]. The formation energy is not available for the Ultra-soft and PAW plane-wave calculations. The reference is a fully converged all-electron calculation using FHI-aims.
Pseudo Basis \(a\) (%) \(B\) (%) \(E_f\) (%)
Ultra-Soft PW 0.1 5.0  
PAW PW 0.2 4.5  
SG15 High 0.3 8.6 4.6
SG15 Medium 0.6 13.0 12.5
FHI DZP 3.0 23.2 15.5
Table 16 Accuracy of selected methods in QuantumATK on Perovskites (P) geometries from Ref. [GBRV14]
Pseudo Basis \(a\) (%) \(B\) (%) \(E_f\) (%)
Ultra-Soft PW 0.1 5.5  
PAW PW 0.1 6.1  
SG15 High 0.3 5.7 1.4
SG15 Medium 0.4 6.4 2.6
FHI DZP 3.5 18.3 12.8

Defect formation energies

The above tests have been for bulk structures and in the following we investigate the accuracy for more open structures. For this purpose we have investigated the formation energies of a silicon vacancy. The calculation was performed for a 216 atom cell and the coordinates were not relaxed. To improve the LCAO calculation it is possible to include a so-called ghost atom at the defect site, i.e. put LCAO orbitals at the defect site. The calculations show that to reach plane-wave accuracy we need to go to the Ultra basis set. For the High basis set the accuracy is ~0.1 eV, which is typically the general accuracy of DFT for such properties.

Table 17 Accuracy of selected methods in QuantumATK for calculating the defect formation energy of an unrelaxed vacancy in Silicon.
Pseudo Basis \(E_f\) (eV) - with ghost \(E_f\) (eV)
PAW PW 3.96 3.96
SG15 Ultra 4.00 4.12
SG15 High 4.07 4.21
SG15 Medium 4.08 4.22

Defect reaction energies

We next present calculations for the reaction energies of defects in InGaAs. In this case we study energy differences upon defect migration. For instance, the first reaction is a Be atom at an In position (BeIn) migrating to an interstitial position (Bei) leaving a vacancy behind (VIn) as you see in the below table. All atomic positions are in this case relaxed and we do not use any ghost atoms. Again we see that the Ultra basis set provides plane-wave accuracy while the level of accuracy of the High basis set is ~0.1 eV. We also test the accuracy if relaxations are done at the SG15 medium level and formation energies are calculated with the High and Ultra basis sets. We see the accuracy in the parenthesis of the below table. In this case the simulations are much faster. We see that this procedure provides essentially the same accuracy. Thus, we recommend to relax geometries at the Medium level and check total energies and forces with higher accuracy basis sets. If the forces are substantial, relaxation with the higher accuracy basis set is needed.

Table 18 Accuracy of selected methods in QuantumATK for calculating the defect reaction energies in InGaAs
Reaction PAW-PW SG15 Ultra SG15 High SG15 Medium
InGaAs – BeIn \(\rightarrow\) InGaAs – BeiVIn 2.54 2.53(2.56) 2.64(2.66) 2.80
InGaAs – BeGa \(\rightarrow\) InGaAs – BeiVGa 2.50 2.51(2.54) 2.58(2.60) 2.75
InGaAs – Bei \(\rightarrow\) InGaAs – BeGaGai -0.63 -0.68(-0.68) -0.72(-0.73) -0.93
InGaAs - Bei \(\rightarrow\) InGaAs – BeInIni 0.31 0.28(0.30) 0.24(0.24) 0.13
InGaAs - BeAs \(\rightarrow\) InGaAs – BeGaGaAs -0.68 -0.66(-0.66) -0.63(-0.63) -0.63

Notes for each pseudopotential type

FHI pseudopotentials and basis sets

The FHI pseudopotentials are the oldest pseudopotentials in QuantumATK. These pseudopotentials are generated with the Fritz-Haber Institute (FHI) pseudopotential code. They are numerical norm-conserving pseudopotentials with a single projector for each angular momentum and without semicore states. For a number of elements the exclusion of semi-core states gives a low accuracy.

The basis sets provided with the FHI pseudopotentials use generic parameters, which are not optimized for each element and generally not of high accuracy. However, the basis sets and pseudopotentials are rather efficient and for some elements can give satisfactory accuracy. We only recommend the SZP and DZP basis sets. Below are given \(\Delta\) values for these settings and the table can be used to estimate the accuracy of the model for various elements.

../../../_images/delta_fhi_szp.png
../../../_images/delta_fhi_dzp.png

HGH Pseudopotentials and basis sets

The Hartwigsen–Goedecker–Hutter (HGH) type pseudopotentials are also norm-conserving pseudopotentials, however, include multiple projectors and semi-core states. The pseudopotentials use analytical functions which are slightly lower accuracy than numerical pseudopotentials. These pseudopotentials require high values for the mesh cutoff for some elements.

For each pseudopotential we provide a hierarchy of basis sets. The basis sets were optimized to describe the total energy of dimers with different bondlengths. This was an early version of our basis set optimization tool and not all elements are well described.

Generally the pseudopotentials and basis sets have better accuracy than the FHI pseudopotentials but can be computationally heavy and require high mesh cutoffs. The sets are only provided for backwards compatibility and are no longer recommended.

../../../_images/delta_hgh_tier4.png

OMX Pseudopotentials and basis sets

The OMX pseudopotentials are from the OpenMX packages. They are fully relativistic, multiple projectors and with semi-core states. Thus, these are generally high accuracy pseudopotentials which can be used for both LDA, GGA and spin-orbit calculations. The drawback of the pseudopotentials is that they for some elements require very high mesh cutoffs.

The basis sets provided are from the original openMX package. This means that they are not calculated on the fly, and therefore only accurate for the exchange-correlation potential for which they were calculated. Thus, if other exchange-correlation functionals than LDA-PZ or GGA-PBE are used, the accuracy will be lower. For our other pseudopotentials the basis sets are generated on the fly with the selected exchange-correlation potential, thus, even if the pseudopotential is not generated for the exchange-correlation functional the basis set will conform to the exchange-correlation functional.

For the OMX pseudopotentials, we have defined two basis set types, Medium and High. It is also possible for the user to define custom basis sets through the atomic_species keyword, see OpenMXBasisSet.

The basis sets was precalculated using GGA-PBE or LDA-PZ and included together with the pseudopotential. Thus, if other exchange-correlation functionals than LDA-PZ or GGA-PBE are used, the accuracy will be low.

For the other pseudo potentials the basis sets are calculated on the fly, using the selected exchange-correlation potential. i.e. only core electrons are treated with the GGA-PBE functional, while valence electrons are treated with the selected exchange-correlation functional.

Below we provide the \(\Delta\) values for the Medium and High basis sets. For the High basis set we also provide the \(\Delta\) value when a 100 Ha mesh cutoff is used. This, gives an indication for which elements the default mesh cutoff needs to be increased.

../../../_images/delta_omx_medium.png
../../../_images/delta_omx_high.png
../../../_images/delta_omx_high_100Ha.png

SG15 Pseudopotentials and basis sets

The SG15 are state of the art norm-conserving pseudopotentials including multiple projectors, semi-core states and non-linear core correction [Ham13][SG15]. They are only provided for the GGA-PBE functional, and comes in a scalar relativistic and a fully relativistic version. The latter is used for calculations including spin-orbit. The pseudopotentials are smooth and all elements converge with a mesh cutoff below 100 Ha.

A large effort has been put into providing high accuracy basis sets. Three sets are provided: Ultra, High and Medium. The basis sets have been optimized to describe bulk systems, dimers and trimer systems.

Below are given the \(\Delta\)-tests for the different elements.

../../../_images/delta_sg15_medium.png
../../../_images/delta_sg15_high.png
../../../_images/delta_sg15_ultra.png

References

[GBRV14](1, 2, 3, 4) Kevin F. Garrity, Joseph W. Bennett, Karin M. Rabe, and David Vanderbilt. Pseudopotentials for high-throughput DFT calculations. Computational Materials Science, 81:446 – 452, 2014. doi:10.1016/j.commatsci.2013.08.053.
[Ham13]D. R. Hamann. Optimized norm-conserving vanderbilt pseudopotentials. Phys. Rev. B, 88:085117, Aug 2013. doi:10.1103/PhysRevB.88.085117.
[LBBjorkman+16]Kurt Lejaeghere, Gustav Bihlmayer, Torbjörn Björkman, Peter Blaha, Stefan Blügel, Volker Blum, Damien Caliste, Ivano E. Castelli, Stewart J. Clark, Andrea Dal Corso, Stefano de Gironcoli, Thierry Deutsch, John Kay Dewhurst, Igor Di Marco, Claudia Draxl, Marcin Dułak, Olle Eriksson, José A. Flores-Livas, Kevin F. Garrity, Luigi Genovese, Paolo Giannozzi, Matteo Giantomassi, Stefan Goedecker, Xavier Gonze, Oscar Gr\r anäs, E. K. U. Gross, Andris Gulans, François Gygi, D. R. Hamann, Phil J. Hasnip, N. A. W. Holzwarth, Diana Iuşan, Dominik B. Jochym, François Jollet, Daniel Jones, Georg Kresse, Klaus Koepernik, Emine Küçükbenli, Yaroslav O. Kvashnin, Inka L. M. Locht, Sven Lubeck, Martijn Marsman, Nicola Marzari, Ulrike Nitzsche, Lars Nordström, Taisuke Ozaki, Lorenzo Paulatto, Chris J. Pickard, Ward Poelmans, Matt I. J. Probert, Keith Refson, Manuel Richter, Gian-Marco Rignanese, Santanu Saha, Matthias Scheffler, Martin Schlipf, Karlheinz Schwarz, Sangeeta Sharma, Francesca Tavazza, Patrik Thunström, Alexandre Tkatchenko, Marc Torrent, David Vanderbilt, Michiel J. van Setten, Veronique Van Speybroeck, John M. Wills, Jonathan R. Yates, Guo-Xu Zhang, and Stefaan Cottenier. Reproducibility in density functional theory calculations of solids. Science, 2016. doi:10.1126/science.aad3000.
[SG15]M. Schlipf and F. Gygi. Optimization algorithm for the generation of oncv pseudopotentials. Computer Physics Communications, 196:36 – 44, 2015. doi:10.1016/j.cpc.2015.05.011.