How energy adjustments and corrections are calculated on the Materials Project (MP) website.
To better model energies across diverse chemical spaces, we apply several adjustments to the total calculated energy of each material. These adjustments fall into two different sets, each of which is described in a different subsection. One set, consisting of anion and GGA/GGA+U mixing scheme corrections, and another consisting of only GGA/GGA+U/r2SCAN mixing scheme corrections. The former is used in the in the current and legacy data, while the latter is only present in releases after the addition of r2SCAN calculations (post v2022.10.28). Both of are used to produce ComputedStructureEntry objects, and mixed phase diagrams.
Description of the methodology used to estimate Gibbs free energies of formation at finite (T>0 K) temperature. This is an option available within the Phase Diagram App.
Background
Methodology
Citations
References
Anion and GGA/GGA+U Mixing
Details on anion and GGA/GGA+U mixing scheme corrections
This correction scheme assumes independent, linear corrections associated with each corrected element. For example, would receive both a '' and an 'oxide' correction (as explained below), while elemental would receive no corrections. For complete details of our correction scheme, refer to Wang et al.
Methodology
1) Anion corrections
For many elements that take on negative oxidation states in solids, differences in electron localization between the elements and the solid can result in substantial errors in formation energies computed from DFT calculations. This is especially true for elements that are gaseous in their standard state - O2, N2, Cl2, F2, and H2.
To address this, we adjust the energies of materials containing certain elements by applying a correction to anionic species, as explained in ref [1]. Specifically, we apply energy corrections to 14 anion species -- 'oxide', 'peroxide', 'superoxide', S, F, Cl, Br, I, N, H, Se, Si, Sb, and Te. In the case of oxygen-containing compounds, separate corrections are applied to oxides, superoxides, and peroxides based on the specific bonding environment of oxygen in the material, as determined from nearest-neighbor bond lengths (e.g., <1.35 Å for superoxide, <1.49 Å for 'peroxide', and 'oxide' otherwise). Thus, Na2O receives an 'oxide' correction while NaO2 receives a `superoxide' correction.
Anion corrections are applied to a material only when it contains a corrected element as an anion. For example, the 'H' correction is applied to LiH but not to H2O. A specie is classified as an anion if its estimated oxidation state (when available) is negative, or if it is the most electronegative element in the formula.
2) GGA / GGA+U Mixing Corrections
Some compounds are better modeled with a U correction term to the density functional theory Hamiltonian while others are better modeled without (i.e., regular GGA). Energies from calculations with the +U correction are not directly comparable to those without. To obtain better accuracy across chemical systems, we use GGA+U when appropriate, GGA otherwise, and mix energies from the two calculation methodologies by adding an energy correction term to the GGA+U calculations to make them comparable to the GGA calculations.
Specifically, we use GGA+U for oxide and fluoride compounds containing any of the transition metals V, Cr, Mn, Fe, Co, Ni, W, and Mo, and GGA for everything else. More details on this method can be found in refs. [1,2]
Accuracy of Total Energies
To estimate the accuracy of our total energy calculations, we compute reaction data and compare against experimental data. Note that this data set was compiled using a lower k-point mesh and pseudopotentials with fewer electrons than the current Materials Project parameter set.
Estimating errors in calculated reaction energies
The accuracy of calculated reaction energies depends on the chemical system investigated. In general, GGA calculations have similar errors among chemically similar systems. Hence, reaction energies between chemically similar systems (e.g., a reaction where the reactants and products are all oxides, such as MgO + Al2O3→MgAl2O4 tend to have smaller errors than reactions between chemically dissimilar systems (e.g., between metals and insulators).
Figure 1: Errors in Calculated Formation Energies for 413 binaries in the Kubaschewski Tables. Energies are normalized to per mol atom.
To provide a quantitative indicator of the error we may expect from the reaction calculator, we have computed the reaction energies of 413 binaries in the Kubaschewski Tables formed with Group V, VI and VII anions. Figure 1 shows the errors in the calculated formation energies (compared to the experimental values) for these compounds. The mean absolute error (MAE) is around 14 kJ mol−1. 75% of the calculated formation energies are within 20 kJ mol−1. We also found that compounds of certain elements tend to have larger errors. For example, Bi, Co, Pb, Eu, U, Tl and W compounds often have errors larger than 20 kJ mol−1.
It should be noted that while an MAE of 14 kJ mol−1 is significantly higher than the desired chemical accuracy of 4 kJ mol−1, it compares fairly well with the performance of most quantum chemistry calculations [3]. Other than the most computationally expensive model chemistries such as G1-G3 and CBS, the reaction energy errors of most computational chemistry model chemistries are well above 10 kJ mol−1.
For oxidation of the elements into binary compounds, an average error of ~4% or 33 kJ/mol-O2 is typical.[^9] For conventional ternary oxide formation from the elements, we have found a mean relative absolute error of about 2%. [4]
Sources of error
The largest contribution to the error comes from the inability of the GGA to fully describe electronic exchange and correlation effects. In addition, there is some error associated with neglecting zero-point effects and with comparing 0K, 0atm computations with room-temperature enthalpy experiments. The latter effect was estimated to contribute less than 0.03 eV/atom by Lany. [5] The stability of antiferromagnetic compounds may be underestimated, as the majority of our calculations are performed ferromagnetically only. The effect of magnetism may be small (under 10 meV/atom) or large (100 meV/atom or greater), depending on the compound. For compounds with heavy elements, relativistic effects may lead to greater-than-expected errors.
GGA errors on reaction energies between chemically similar compounds
We recently conducted a more in-depth study comparing GGA (+U) reaction energies of ternary oxides from binary oxides on 135 compounds. [6]
The main conclusions are:
The error in reaction energies for the binary oxide to ternary oxides reaction energies are an order of magnitude lower than for the more often reported formation energies from the element. An error intrinsic to GGA (+U) is estimated to follow a normal distribution centered in zero (no systematic underestimation or overestimation) and with a standard deviation around 24 meV/at.
When looking at phase stability (and for instance assessing if a phase is stable or not), the relevant reaction energies are most of the time not the formation energies from the elements but reaction energies from chemically similar compounds (e.g., two oxides forming a third oxide). Large cancellation of errors explain this observation.
The +U is necessary for accurate description of the energetics even when reactions do not involve change in formal oxidation states
Citations
To cite the calculation methodology, please reference the following works:
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, pp. 2295-2310. DOI:10.1016/j.commatsci.2011.02.023
A. Jain, G. Hautier, S.P. Ong, C. Moore, C.C. Fischer, K.A. Persson, G. Ceder, Accurate Formation Enthalpies by Mixing GGA and GGA+U calculations, Physical Review B, vol. 84, 2011, p. 045115. DOI:10.1103/PhysRevB.84.045115
References
[1]: Wang, A., Kingsbury, R.S., Horton, M., Jain, A., Ong, S.P., Dwaraknath, S., Persson, K. A framework for quantifying uncertainty in DFT energy corrections. Scientific Reports 11 (2021), 15496. DOI: 10.1038/s41598-021-94550-5
[2]: A. Jain, G. Hautier, S.P. Ong, C. Moore, C.C. Fischer, K.A. Persson, G. Ceder, Formation Enthalpies by Mixing GGA and GGA+U calculations, Physical Review B, vol. 84 (2011), 045115.
[3]: J.B. Foresman, A.E. Frisch, Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian, Gaussian. (1996).
[4]: A. Jain, S.-a Seyed-Reihani, C.C. Fischer, D.J. Couling, G. Ceder, W.H. Green, Ab initio screening of metal sorbents for elemental mercury capture in syngas streams, Chemical Engineering Science. 65 (2010) 3025-3033.
[5]: S. Lany, Semiconductor thermochemistry in density functional calculations, Physical Review B. 78 (2008) 1-8.
[6]: G. Hautier, S.P. Ong, A. Jain, C. J. Moore, G. Ceder, Accuracy of density functional theory in predicting formation energies of ternary oxides from binary oxides and its implication on phase stability, Physical Review B, 85 (2012), 155208
Overview of how chemical potential diagrams (CPDs) are constructed and visualized. These are available as part of the Phase Diagram App.
Introduction
The chemical potential diagram is the mathematical dual to the compositional phase diagram. To create the diagram, convex minimization is performed in energy (E) vs. chemical potential (μ) space by taking the lower convex envelope of hyperplanes. Accordingly, “points” on the compositional phase diagram become N-dimensional convex polytopes (domains) in chemical potential space.
For more information on this specific implementation of the algorithm, please cite/reference the paper below:
Two dimensional (2-D) chemical potential diagram for the V-S chemical system. Energies are DFT-calculated energies directly acquired from MP database.
Three dimensional (3-D) chemical potential diagram for the V-S-O chemical system. Energies are DFT-calculated energies directly acquired from MP database.
Relationship to predominance diagrams
Relationship between 3-D chemical potential diagram and predominance diagrams, which are 2-D views of the full three-dimensional chemical potential diagram surface. Figure by Matthew McDermott.
Citations
Methodology
Todd, P. K., McDermott, M. J., Rom, C. L., Corrao, A. A., Denney, J. J., Dwaraknath, S. S., Khalifah, P. G., Persson, K. A., & Neilson, J. R. (2021). Selectivity in Yttrium Manganese Oxide Synthesis via Local Chemical Potentials in Hyperdimensional Phase Space. Journal of the American Chemical Society, 143(37), 15185-15194. https://doi.org/10.1021/jacs.1c06229
References
[1] Yokokawa, H. “Generalized chemical potential diagram and its applications to chemical reactions at interfaces between dissimilar materials.” JPE 20, 258 (1999). https://doi.org/10.1361/105497199770335794
[1] Todd, P. K., McDermott, M. J., Rom, C. L., Corrao, A. A., Denney, J. J., Dwaraknath, S. S., Khalifah, P. G., Persson, K. A., & Neilson, J. R. (2021). Selectivity in Yttrium Manganese Oxide Synthesis via Local Chemical Potentials in Hyperdimensional Phase Space. Journal of the American Chemical Society, 143(37), 15185-15194. https://doi.org/10.1021/jacs.1c06229
GGA/GGA+U/r2SCAN Mixing
Details on the GGA/GGA+U/r2SCAN mixing scheme corrections
An updated energy correction scheme [1] is used to allow for the mixing of GGA, GGA+U, and r2SCAN calculations. This is constructed by considering all electronic energies to be the sum of a reference energy, and a relative energy. The reference energy (Eref) for each functional is defined as the (empirically corrected) electronic energy of the GGA(+U) ground-state structure at each point in composition space. The energy of a material associated with either functional can then be expressed as a difference relative to a specific reference energy (ΔEref). The formation energy of a material is calculated in the usual way by subtracting the electronic energies of the elemental endpoints in each respective functional. It should be noted that ΔEref is calculated from the differences in polymorph energies, and consequently does not depend on the elemental endpoint energies. While the updated mixing scheme is similar to the previous scheme involving only GGA and GGA+U calculations, it extends the approach to be amenable to any two functionals without relying on pre-fitted energy correction parameters.
Mixing Rules
The two rules used to construct mixed GGA/GGA+U/r2SCAN phase diagrams are as follows:
Start with a GGA(+U) convex energy hull. Replace GGA(+U) energies with r2SCAN energies by adding their to the corresponding GGA(+U) reference energy.
Construct the convex energy hull using formation energy calculated using r2SCAN energies, only when r2SCAN calculations exist for every reference structure (i.e. every stable GGA(+U) structure). In this case, add any missing GGA(+U) materials by adding their to the corresponding r2SCAN reference energy.
For more detailed information on the mixing scheme and its benchmarks, see the original publication in Ref .
References
[1] Kingsbury, R.S., Rosen, A.S., Gupta, A.S. et al. A flexible and scalable scheme for mixing computed formation energies from different levels of theory. npj Comput Mater 8, 195 (2022).
Figure 1. Rules for mixing GGA(+U) (blue) and r2SCAN (red) energies onto a single phase diagram. (left) r2SCAN energies are placed onto the GGA(+U) hull by referencing them to the r2SCAN energy of the GGA(+U) ground state via ΔEref. A, B, C, ad D represent different polymophs at a single composition, and polymorph A is the ground state. (right) r2SCAN formation energies are used to build the convex hull only when there are r2SCAN calculations for every GGA(+U) ground state.
Phase Diagrams (PDs)
A description of the methodology for constructing and interpreting compositional phase diagrams from the Materials Project (MP) website and API.
Introduction
A phase diagram is a calculation of the thermodynamic phase equilibria of multicomponent systems. It is an important tool in materials science for revealing 1) thermodynamic stability of compounds, 2) predicted equilibrium chemical reactions, and 3) processing conditions for synthesizing materials. However, the experimental determination of a phase diagram is an extremely time-consuming process, requiring careful synthesis and characterization of all phases in a chemical system.
Computational modeling tools, such as the density functional theory (DFT) methods used by the Materials Project, can accelerate compositional phase diagram construction significantly. By calculating the energies of all known compounds in a given chemical system (e.g. the lithium/iron/oxygen chemical system, Li-Fe-O), we can determine the phase diagram for that system at a temperature of K and pressure of atm. Furthermore, for systems comprised of predominantly solid phases open with respect to a gaseous element, approximations can be made as to the finite temperature and pressure phase diagrams.
In this section, we will describe the theory/methodology behind the calculation of compositional phase diagrams.
Methodology
This section will discuss how to construct phase diagrams from DFT-calculated energies. This is exact process done by the Materials Project (MP) for computing formation energies, thermodynamic stability, and phase diagrams. This methodology has been implemented in Python within the pymatgen package. Please see for brief examples of how to build phase diagrams on your own.
Calculating formation energy
The formation energy, , is the energy change upon reacting to form a phase of interest from its constituent components. The components typically used are the constituent elements. For a phase composed of components indexed by , the formation energy can be calculated as follows:
where is the total energy of the phase of interest, is the total number of moles of component , and is the total energy of component . Note that is often referred to as the chemical potential of the component, however, this is only rigorously true when working with Gibbs free energies, .
Example:
For barium titanate, BaTiO, the formation energy would be calculated as:
Typically, formation energies are normalized on a per-atom basis by dividing by the number of atoms in 1 mole of formula. For example, for BaTiO, the normalized per-atom formation energy would be calculated by dividing the above by 5 atoms.
Constructing the compositional phase diagram
The convex hull approach
To construct a phase diagram, one needs to compare the relative thermodynamic stability of phases belonging to the system using an appropriate free energy model. For an isothermal, isobaric, closed system, the relevant thermodynamic potential is the Gibbs free energy, , which can be expressed as a Legendre transform of the enthalpy, , and internal energy, , as follows:
where is the temperature of the system, is the entropy of the system, is the pressure of the system, is the volume of the system, and is the number of atoms of species in the system.
For systems comprising primarily of condensed phases, the term can be neglected and at 0K, the expression for simplifies to just . Normalizing with respect to the total number of particles in the system, we obtain . By taking the convex hull of for all phases belonging to the M-component system and projecting the stable nodes into the - dimension composition space, one can obtain the 0 K phase diagram for the closed system at constant pressure. The convex hull of a set of points is the smallest convex set containing the points. For instance, to construct a 0 K, closed system phase diagram, the convex hull is taken on the set of points in space with being related to the other composition variables by .
Evaluating thermodynamic stability
Figure 2 is an example of a calculated binary A-X phase diagram at 0 K and 0 atm. Binary phase diagrams show the complete convex hull for the system, where the y-axis is the formation energy per atom and the x-axis is the composition.
The blue lines show the convex hull construction, which connects stable phases (circles). Unstable phases will always appear above the convex hull line (squares); one measure of the thermodynamic stability of an arbitrary compound is its distance from the convex hull line (), which predicts the decomposition energy of that phase into the most stable phases.
Accuracy of Calculated Phase Diagrams
In general, we can expect that compositional phase diagrams comprising of predominantly solid phases to be reproduced fairly well by our calculations. However, it should be noted that there are inherent limitations in accuracy in the DFT calculated energies. Furthermore, our calculated phase diagrams are at 0 K and 0 atm, and differences with non-zero temperature phase diagrams are to be expected.
For grand potential phase diagrams, further approximations are made as to the entropic contributions . They are therefore expected to be less accurate, but nonetheless provide useful insights on general trends.
Code (pymatgen)
While the Materials Project website has a phase diagram app (), and PhaseDiagram objects can also be obtained directly from the API (), two code snippets are provided below that show how to use the API and pymatgen to construct and plot your own phase diagrams with Python.
GGA/GGA+U
Constructing mixed GGA/GGA+U phase diagrams can be done directly with the corrected ComputedStructureEntry objects from the API.
GGA/GGA+U/R2SCAN
Constructing a mixed GGA/GGA+U/R2SCAN phase diagram requires corrections to be reapplied locally. This is because the corrected ComputedStructureEntry object obtained from the thermodynamic data endpoint of the API for a given material is from its home chemical system phase diagram (i.e. Si-O for SiO2, or Li-Fe-O for Li2FeO3).
Unlike the previous GGA/GGA+U only mixing scheme, the updated scheme does not guarantee the same correction to an entry in phase diagrams of different chemical systems. In other words, the energy correction applied to the entry for silicon (mp-149) in the Si-O phase diagram is not guaranteed to be the same for the one in the Si-O-P phase diagram.
For more details on the correction scheme and its logic, see the section or the original publication .
Citations
Methodology (I)
Methodology (II)
.
References
[1] Bartel, C.J. Review of computational approaches to predict the thermodynamic stability of inorganic solids. J Mater Sci57, 10475–10498 (2022).
[2] V. Raghavan, Fe-Li-O Phase Diagram, ASM Alloy Phase Diagrams Center, P. Villars, editor-in-chief; H. Okamoto and K. Cenzual, section editors; , ASM International, Materials Park, OH, 2006.
[3]:
[4] Kingsbury, R.S., Rosen, A.S., Gupta, A.S. et al. A flexible and scalable scheme for mixing computed formation energies from different levels of theory. npj Comput Mater 8, 195 (2022).
