We use DFT as implemented in the Vienna Ab Initio Simulation Package (VASP) software [1] to evaluate the total energy of compounds. For the exchange-correlational functional, we employ a mix of Generalized Gradient Approximation (GGA) and GGA+U, or a mix of GGA, GGA+U, and r2SCAN. Both mixing schemes are described here. All calculations are performed at 0 K and 0 atm. All computations are performed with spin polarization on and with magnetic ions in a high-spin ferromagnetic initialization (the system can of course relax to a low spin state during the DFT relaxation). For a select number of materials, alternate spin states are searched for. Details on this can be found in the Magnetic Properties section.

Input structures are sourced from many different places, including the Inorganic Crystal Structure Database (ICSD). [2] We relax all cell and atomic positions in our calculation two times in consecutive runs. When multiple crystal structures are present for a single chemical composition, we attempt to evaluate all unique structures as determined by an affine mapping technique. [3]

More detailed information on the GGA/GGA+U and r2SCAN calculations run by the Materials Project can be found in the following two subsections:

References

[1]: Kresse, G. & Furthmuller, J., 1996. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B, 54, pp.11169-11186.

[2]: G. Bergerhoff, The inorganic crystal-structure data-base, Journal Of Chemical Information and Computer Sciences. 23 (1983) 66-69.

[3]: R. Hundt, J.C. Schön, M. Jansen, CMPZ - an algorithm for the efficient comparison of periodic structures, Journal Of Applied Crystallography. 39 (2006) 6-16.

Calculation Parameters

We use the Projector Augmented Wave (PAW) method for modeling core electrons with an energy cutoff of 520 eV. This cutoff corresponds to 1.3 times the highest cutoff recommended among all the pseudopotentials we use (more details can be found in the pseudopotentials section). A baseline k-point mesh of 1000/(number of atoms in the cell) is used for all computations. Specifically, the Monkhorst-Pack method is used for the k-point choices (with $\Gamma$-centered for hexagonal cells), and the tetrahedron method is used to perform the k-point integration. It is important to note that Pymatgen has the ability change those default parameters if they are not adequate for the computation (e.g., switch to another k-point integration scheme). Some details of our calculation method can be found in ref [1]; however, the Materials Project has updated many parameters as documented throughout the Methodology sections. The most up-to-date input sets can be .

Total energy convergence

As mentioned, we currently employ a k-point mesh of 1000 per reciprocal atom (pra). However, we have performed a convergence test of total energy with respect to k-point density and convergence energy difference for a subset of chemically diverse compounds for a previous parameter set, which employed a smaller k-point mesh of 500 pra. Using a 500 pra k-point mesh, the numerical convergence for most compounds tested was within 5 meV/atom, and 96% of compounds tested were converged to within 15 meV/atom. Results for the new parameter set will be better due to the denser k-point mesh employed. Convergence will depend on chemical system; for example, oxides were generally converged to less than 1 meV/atom.

Structure convergence

The energy difference for ionic convergence is set to 0.0005 * natoms in the cell. Data on expected accuracy on cell volumes can be found in a previous paper. We have found these parameters to yield well-converged structures in most instances; however, if the structures are to be used for further calculations that require strictly converged atomic positions and cell parameters (e.g. elastic constants, phonon modes, etc.), we recommend that users re-optimize the structures with tighter cutoffs or in force convergence mode.

Authors

1. Shyue Ping Ong

References

[1]: A. Jain, G. Hautier, C. Moore, S.P. Ong, C.C. Fischer, T. Mueller, K.A. Persson, G. Ceder., A High-Throughput Infrastructure for Density Functional Theory Calculations, Computational Materials Science. vol. 50 (2011) 2295-2310.

[2]: L. Wang, T. Maxisch, G. Ceder, Oxidation energies of transition metal oxides within the GGA+U framework, Physical Review B. 73 (2006) 1-6.

Hubbard U Values

It is well-known that first principles calculations within the local density approximation (LDA) or generalized gradient approximation (GGA) lead to considerable error in calculated redox reaction energies of many transition metal compounds. This error arises from the self-interaction error in LDA and GGA, which is not canceled out in redox reactions where an electron is transferred between significantly different environments, such as between a metal and a transition metal or between a transition metal and oxygen or fluorine. Extensive discussion of this issue can be found in the following works. [1-4]

In the Materials Project, we have calibrated $U$ values for many transition metals of interest using the approach outlined in Wang et al.'s work . At the present moment, values have only been calibrated for transition metal oxide systems. values were calibrated for the following elements: , , , , , , and . The choice of systems to which we apply was largely determined by our experience and by systematic benchmarking. It is very likely that we will expand calibration of values to more chemical systems in the future.

In the Materials Project, for an oxide or fluoride material with a transition element listed previously, with the VASP input settings constructed according to the logic defined in .

Note that for fluorides, the value gets set to the one calibrated from the oxide system, although in principle our architecture allows different values to be set for oxides and fluorides respectively.

Calibration and Values

The values were obtained by fitting to experimental binary formation enthalpies as described in Wang et al.'s work. This method is simple yet accurately reproduces phase stabilities. A least squares method of obtaining the correct value was used, as follows:

1. For each non-overlapping formation energy reaction considered, we find the region where the formation energy error passes zero. For the system, this includes the following:

The full list of U values used is described in the table below. For oxides and fluorides containing any of the elements, only GGA+U calculations are performed.

Caveats

The U values are calibrated for phase stability analyses, and should be used with care if applied to obtain other properties such as band structures. Also, the U values depend on the pseudopotential used. Further, typically, U values should be site specific, however in our approach, U values were applied to all sites with an element listed above, and only to the d-orbitals. A discussion of the pseudopotentials used in the Materials Project can be found .

References

[1]: F. Zhou, M. Cococcioni, C. A. Marianetti, D. Morgan and G. Ceder. First-principles prediction of redox potentials in transition-metal compounds with LDA+U. Physical Review B, 2004, 70, 235121.

[2]: M. Cococcioni, S. de Gironcoli, Linear response approach to the calculation of the effective interaction parameters in the LDA+U method. Physical Review B, 2005, 71, 035105.

[3]: L. Wang, T. Maxisch, & G. Ceder. Oxidation energies of transition metal oxides within the GGA+U framework. Physical Review B. 2006, 73, 195107,

[4]: A. Jain, G. Hautier, S. P. Ong, C. Moore, C. Fischer, K. A. Persson, & G. Ceder. Formation enthalpies by mixing GGA and GGA + U calculations. Physical Review B, 2011, 84(4), 045115.

[5]: M. Wang, A. Navrotsky Enthalpy of formation of LiNiO2, LiCoO2 and their solid solution, LiNi1-xCoxO2, Solid State Ionics, vol. 166, no. 1-2, pp. 167-173, Jan. 2004.

Since database release v2022.10.28 the Materials Project has incorporated metaGGA functionals into its core dataset in the form of r2SCAN calculations. Part of this includes a that allows for the mixing of GGA, GGA+U and r2SCAN results in its thermodynamic data.

All r2SCAN data is obtained from a two-step workflow which is comprised of an initial GGA structure optimization, followed by final optimization with r2SCAN. The first step allows for the generation of an initial guess of the structure and charge density at a lower computational cost, speeding up the subsequent metaGGA calculation. More specifically, PBESol is used as the GGA functional for the first optimization step. For more details on the workflow see Ref .

Information regarding calculation parameters, convergence, and pseudopotential choices can also be found in the following subsections:

Calculation Parameters

We use the projector-augmented wave (PAW) or modeling core electrons with an energy cutoff of 680 eV. K-point grids were generated automatically by VASP using KSPACING values ranging from 0.22/Å to 0.44/Å. Specifically, the Monkhorst-Pack method is used for grid generation (with $\Gamma$-centered for hexagonal cells), and the tetrahedron method is used to perform the k-point integrations. These were determined from the GGA-estimated bandgap of each material based on the work by Wisesa et al. [1]. More details regarding the calculation method can be found in ref [2]; however, the Materials Project has updated many parameters as documented throughout the Methodology sections. The most up-to-date input sets can be found here.

Convergence

Plane-wave energy cutoff and k-point density settings were selected such that formation energies converged within approximately 1 meV/atom for a benchmark set of 21 materials and were selected to be conservatively high :

References

[1] P. Wisesa, K. A. McGill, and T. Mueller, Efficient generation of generalized Monkhorst-Pack grids through the use of informatics, Phys. Rev. B 93, 1 (2016).

[2] R. Kingsbury, A. S. Gupta, C. J. Bartel, J. M. Munro, S. Dwaraknath, M. Horton, and K. A. Persson Phys. Rev. Materials 6, 013801 (2022)

All calculations used pseudopotentials from the “PBE PAW datasets version 54” set released in September 2015; a list of the specific POTCAR symbols used for each element is provided below. Although these pseudopotentials were developed for use with the PBE functional, their use with SCAN is common practice because no SCAN-specific pseudopotentials are available for use in VASP.

References

[1] R. Kingsbury, A. S. Gupta, C. J. Bartel, J. M. Munro, S. Dwaraknath, M. Horton, and K. A. Persson Phys. Rev. Materials 6, 013801 (2022)

Pseudopotentials are used to reduce computation time by replacing the full electron system in the Coulombic potential by a system only taking explicitly into account the "valence" electrons (i.e., the electrons participating into bonding) but in a pseudopotential. This approach not only reduces the electron number but also the energy cutoff necessary (this is critical in plane-wave-based computations). All computations in the materials project have been performed using a specific type of very efficient pseudopotentials: the projector augmented wave (PAW) pseudopotentials. We used the library of PAW pseudopotentials provided by VASP but for a given element there are often several possibilities in the VASP library. This wiki presents how the choices between the different pseudopotential options were made.

The strategy

As a test set, we ran all elements and binary oxides present in the ICSD with the available PAW pseudopotentials. As it is difficult to test for all properties (structural, electronic, etc...), we chose to be inclusive and to select the pseudopotential with the largest number of electrons (high e) except if convergence issues were seen on our test set, or if previous experience excluded a specific pseudopotential. We also excluded pseudopotentials with too large an energy cutoff.

We also compared to recommendations from the VASP manual present in .

Finally, as we had energies for elements and binary oxides, we compared binary oxide formation energies with the available pseudopotentials. The oxygen molecule energy was obtained from Wang et al. Please note that this data is pure GGA and some chemistries (e.g., transition metals) will give extremely bad formation energy results in GGA. This is not an issue with the pseudopotential but with the functional, so we do not focus on that issue in this wiki.

Pseudopotential comments and choice

1st-row elements

Usually, they have three pseudopotentials: a soft _s, a hard _h, and a standard. The standard is recommended by VASP and will be used for all. The hard ones have extremely high cut-offs (700 eV)

alkali and alkali-earth

The table below indicates our choices. Basically, we chose all high e- pseudopotentials except for Na where we excluded Na_sv due to its very high cutoff (700 eV).

d-elements, transition metals

The table below shows the details on the PSP choices. All high e- PSPs have been chosen except for Pd which had convergences problem with the high e- PSP in PdO.

main group

Si, P, Cl, S will be used in their standard form (not hard) as suggested by VASP manual.

The Al_h psp was found to be definitely wrong in terms of band structure. There were "ghost" states found in the DOS.

Pb is interesting as the high e- psp shows significantly higher error in formation energies. We kept the high e- psp (Pb_d), but it might be interesting to study this a little more. One hypothesis relies on a recent result showing that lead oxide formation energies need the use of spin-orbit coupling to be accurate. Our computations do not include any relativistic corrections for valence electrons. However, spin-orbit coupling is taken into account during the psp construction. This would explain why a psp with more core electrons (treated indirectly with spin-orbit coupling) would give more accurate results than a psp with fewer electrons.

Bi_d shows a convergence problem, so the decision on Bi has been postponed to further analysis.

Finally, Po and At, while referred to in the VASP manual, are not present in the VASP PAW library.

rare-earth, f-electrons

These are probably the most problematic to use as pseudopotentials. Here is what the VASP manual says about them:

Due to self-interaction errors, f-electrons are not handled well by presently available density functionals. In particular, partially filled states are often incorrectly described, leading to large errors for Pr-Eu and Tb-Yb where the error increases in the middle (Gd is handled reasonably well, since 7 electrons occupy the majority shell). These errors are DFT and not VASP related. Particularly problematic is the description of the transition from an itinerant (band-like) behavior observed at the beginning of each period to localized states towards the end of the period. For the elements, this transition occurs already in La and Ce, whereas the transition sets in for Pu and Am for the elements. A routine way to cope with the inabilities of present DFT functionals to describe the localized electrons is to place the electrons in the core. Such potentials are available and described below. Furthermore, PAW potentials in which the states are treated as valence states are available, but these potentials are not expected to work reliable when the electrons are localized.

In summary, the pseudopotentials can either include or not include f electrons; how accurate including them or not is depends on the nature of the bonding for each particular system (localized or not).

What we found is that convergence issues are often seen for high electron psp (e.g., Pr, Nd, Sm). Also, some pseudopotentials (e.g., Er_2, Eu_2) freeze too many electrons and therefore have issues with oxidation states that make one of the frozen electron participate in bonding (e.g., Eu2O3, Er2O3). Finally, there is a major problem with Tb. Only Tb_3 exists but Tb is known to also form Tb4+ compounds (e.g., TbO2). For those Tb4+ compounds, this psp is likely to be extremely wrong. There is currently no fix for this except waiting for someone to develop a PAW Tb_4 psp.

transuranides, f-electrons

U, Ac, Th, Pa, Np, Pu, Am

Following VASP suggestion, we decided to use the standard (and not the soft) version for all those pseudopotentials.

Citation

To cite the Materials Project, please reference the following work:

A. Jain, G. Hautier, C. J. Moore, S. P. Ong, C. C. Fischer, T. Mueller, K. A. Persson, and G. Ceder, A high-throughput infrastructure for density functional theory calculations, Computational Materials Science, vol. 50, 2011, pp. 2295-2310.

Authors

1. Geoffroy Hautier

References

[1]: P.E. Blöchl, Physical Review B 50, 17953-17979 (1994).

[2]: R. Ahuja, A. Blomqvist, P. Larsson, P. Pyykkö, and P. Zaleski-Ejgierd, Physical Review Letters 106, 1-4 (2011).