Pseudopotentials and basis sets available in QuantumATK

PDF version

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 sets being the 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 QuantumATK 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 PseudoDojo pseudopotential. For LDA we recommend the FHI pseudopotential. For SOLDA we recommend the OMX, it can also give higher accuracy for LDA, however, the required mesh cutoff needs to be tested carefully.

Table 13 Pseudo- and PAW potential types available for the DFT calculators.
Name Notes
FHI Trouiller–Martins type.
HGH Hartwigsen–Goedecker–Hutter type.
OMX Fully relativistic. OMX webpage.
SG15 ONCV type, scalar-relativistic (SG15) and fully relativistic (SG15-SO). SG15 webpage.
PseudoDojo ONCV type, scalar-relativistic (PseudoDojo) and fully relativistic (PseudoDojo-SO). PseudoDojo webpage.
GPAW PAW potentials. GPAW webpage.
JTH PAW potentials. JTH webpage.
Suggested A mixture of GPAW and JTH, chosen to give optimal results for each element.

Through the use of the keyword NormConservingPseudoPotential, it is also possible to specify a pseudopotential from a user-specified file. Not all pseudopotentials are available for all QuantumATK DFT calculators and exchange-correlation functional types. Please see the following table for details on each of the pseudopotential types:

Table 14 Details about the pseudo- and PAW potential types available for the QuantumATK DFT calculators.
Name Introduced Basis types XC-types Cutoff [Ha] Accuracy Efficiency
FHI ATK-2008 LCAO, PW LDA, GGA, MGGA, HGGA 30-100 Low High
HGH ATK-2011 LCAO, PW LDA, GGA, MGGA, HGGA 75-500 Medium Low
OMX ATK-2015 LCAO LDA, GGA, MGGA 75-500 High Low
OMX-SO ATK-2015 LCAO SOLDA, SOGGA, SOMGGA 75-500 High Low
SG15 ATK-2016 LCAO, PW GGA, HGGA 30-220 High Medium
SG15-SO ATK-2016 LCAO, PW GGA, SOGGA, HSOGGA 30-220 High Medium
PseudoDojo O-2018.06 LCAO, PW GGA, HGGA 30-125 High Medium
PseudoDojo-SO P-2019.03 LCAO, PW GGA, SOGGA, HSOGGA 30-125 High Medium
PAW-GPAW Q-2019.12 PW LDA, GGA N/A High Medium
PAW-JTH Q-2019.12 PW LDA, GGA N/A High Medium
PAW-Suggested Q-2019.12 PW GGA N/A High Medium

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.

LCAO basis sets

QuantumATK mainly 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.

SG15/PseudoDojo

Three different basis sets are available for each element when using the SG15/PseudoDojo pseudopotentials; Medium, High, and Ultra. All three derive from the numerical atom-centered basis sets of the FHI-aims package, but have been significantly modified and optimized with respect to computational speed with the DFT: LCAO calculator.

The Medium basis set is default for the SG15/PseudoDojo pseudopotentials, and should be sufficient for most applications. However, if extreme accuracy is needed, the High and Ultra basis sets add more basis functions at the expense of increased computational load.

OMX

The OMX basis sets were developed in Refs. [Oza03] and [OK04], and are optimized towards maximum computational speed and accuracy with the OMX pseudopotentials. The DFT: LCAO calculator offers three different OMX basis set sizes for most elements; Low, Medium, and High, which include increasingly more pseudo-atomic orbitals in the basis set. More details are available on the website for the OpenMX database (2013).

FHI

Four different types of basis functions can be used for the FHI pseudopotentials:

The basis sets available for FHI pseudopotentials are listed below. They are ordered with increasing number of basis orbitals – those with more orbitals are often more accurate, at the expense of increased computational load. The DZP basis set is default.

Visualizing the basis functions

To see the type of the basis sets it is most easy to plot the basis functions, like this:

  • Open the scripter
  • Add an LCAOcalculator block from the Calculators section
  • 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 tests 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 15 Accuracy and performance of PBE pseudopotentials and basis types available for the DFT calculators. Performance is an approximate measure of time spent, relative to FHI-DZP. Lower numbers are faster.
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 3.45 2
SG15 High 1.88 6
SG15 Ultra 2.03 25
SG15 PW 1.32 N/A
PseudoDojo Medium 4.53 2
PseudoDojo High 1.52 6
PseudoDojo Ultra 1.40 25
PseudoDojo PW 1.04 N/A
PAW-GPAW PW 1.44 N/A
PAW-JTH PW 0.86 N/A
PAW-Suggested PW 0.64 N/A

A \(\Delta\) value below 2 meV indicates a state of the art DFT calculation. PAW-Suggested is a combination of the two PAW sets where we have selected the best PAW potential for each element. The performance number given in the Table is an average indication of how much time a calculation of a 64 atom supercell takes, using the different models relative to the FHI-DZP model. Lower numbers are faster. 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 16 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 \(a\) (%) \(B\) (%) \(E_f\) (%) Source
Ultra-Soft PW 0.13 5.0 N/A [GBRV14]
PAW PW 0.15 4.5 N/A [GBRV14]
PseudoDojo Ultra 0.15 1.90 1.83 QuantumATK
PseudoDojo High 0.18 2.12 2.79 QuantumATK
PseudoDojo Medium 0.5 5.32 15.59 QuantumATK
PseudoDojo PW (40 Ha) 0.09 1.86 0.85 QuantumATK
SG15 High 0.3 8.6 4.6 QuantumATK
SG15 Medium 0.6 13.0 12.5 QuantumATK
FHI DZP 3.0 23.2 15.5 QuantumATK
Table 17 Accuracy of selected methods in QuantumATK on Perovskites (P) geometries from Ref. [GBRV14]
Pseudo Basis \(a\) (%) \(B\) (%) \(E_f\) (%) Source
Ultra-Soft PW 0.08 5.5 N/A [GBRV14]
PAW PW 0.13 6.1 N/A [GBRV14]
PseudoDojo Ultra 0.13 1.87 0.91 QuantumATK
PseudoDojo High 0.21 3.04 1.18 QuantumATK
PseudoDojo Medium 0.35 3.68 2.73 QuantumATK
PseudoDojo PW (40 Ha) 0.06 0.66 0.23 QuantumATK
SG15 High 0.3 5.7 1.4 QuantumATK
SG15 Medium 0.4 6.4 2.6 QuantumATK
FHI DZP 3.5 18.3 12.8 QuantumATK

Notes for each pseudopotential type

PseudoDojo Pseudopotentials and basis sets

The PseudoDojo are state of the art norm-conserving pseudopotentials including multiple projectors, semi-core states and non-linear core correction [vSGB+18]. They are only provided for the GGA-PBE functional, and comes in both 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 125 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_pd_medium.png
../../../_images/delta_pd_high.png
../../../_images/delta_pd_ultra.png
../../../_images/delta_pd_pw.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 220 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

PAW data sets

For the Projector Augmented Wave (PAW) method, we need data describing the system inside the augmentation sphere (partial waves, projectors, core densities, ...). This is gathered in a PAW data set file. We support data set files in the .xml format, as specified on the ESL website. We ship two different data sets, the GPAW and JTH [JTH14] data sets, for which we found default wave function cutoffs that are as low as possible without losing accuracy. We made a combined PAW-Suggested data set for GGA, that gives the best results.

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

../../../_images/gpaw_30Hartree.png
../../../_images/jth_30Hartree.png
../../../_images/suggested_default.png

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

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 pre-calculated 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

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

References

[GBRV14](1, 2, 3, 4, 5, 6, 7, 8) 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.
[JTH14]François Jollet, Marc Torrent, and Natalie Holzwarth. Generation of projector augmented-wave atomic data: A 71 element validated table in the xml format. Computer Physics Communications, 185(4):1246 – 1254, 2014. URL: http://www.sciencedirect.com/science/article/pii/S0010465513004359, doi:https://doi.org/10.1016/j.cpc.2013.12.023.
[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.
[Oza03]T. Ozaki. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B, 67:155108, 2003. doi:10.1103/PhysRevB.67.155108.
[OK04]T. Ozaki and H. Kino. Numerical atomic basis orbitals from h to kr. Phys. Rev. B, 69:195113, 2004. doi:10.1103/PhysRevB.69.195113.
[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.
[vSGB+18]M.J. van Setten, M. Giantomassi, E. Bousquet, M.J. Verstraete, D.R. Hamann, X. Gonze, and G.-M. Rignanese. The pseudodojo: Training and grading a 85 element optimized norm-conserving pseudopotential table. Computer Physics Communications, 226:39 – 54, 2018. URL: http://www.sciencedirect.com/science/article/pii/S0010465518300250, doi:https://doi.org/10.1016/j.cpc.2018.01.012.