Calculations Manual

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Overview

This manual explains our calculation methods, adjustments made to our calculated energies, and errors in our calculations.

Total Energy Calculation Details

We use density functional theory as implemented in the Vienna Ab Initio Simulation Package (VASP) software to evaluate the total energy of compounds. For the exchange-correlational functional, we employ a mix of Generalized Gradient Approximation (GGA) and GGA+U, as described in <a href="#GGA+U calculations">section 3.2</a>. We use the Projector Augmented Wave (PAW) method for modeling core electrons with an energy cutoff of 1.3 times the default specified by the pseudopotential. All calculations are performed at 0K and 0atm. Most compounds with magnetic ions are computed with a ferromagnetic initialization only, although some compounds have also been initialized with antiferromagnetic orderings. A comprehensive overview of our calculation method can be found in ref [1].

Crystal structures

We use input structures from the Inorganic Crystal Structure Database (ICSD)[8], and relax all cell and atomic positions in our calculation. 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.[2]

Total energy convergence

We performed a convergence test of total energy with respect to k point density and convergence energy difference for a subset of chemically diverse compounds. The numerical convergence for most compounds tested was within 5 meV/atom, and 96% of compounds tested were converged to within 15 meV/atom. Convergence may depend on chemical system; for example, oxides were generally converged to less than 1 meV/atom.[1]

Total Energy Adjustments

To better model energies across diverse chemical spaces, we apply several adjustments to the total energy. These adjustments are described below.</p>

Gases, liquids, and elements

Our total energy calculations are for periodic solids at 0K. For elements that are liquid or gaseous in their standard state, our raw calculations do not represent the same phase as standard experimental data. Rather than calculating the liquid/gas energies directly, we adjust the energies of several elements that are liquid or gaseous at room temperature using Wang's method.[3] The best-tested fit is for oxygen gas reacting to form to oxides. We have adjusted energies of the following compounds:

  • Br liquid
  • O2, H2, N3, Cl2, F2
  • CO2, SO2, SO3, NO2

The adjusted energies outside of O2 are not as well-tested and calculations involving these elements should be taken with greater caution. For example, the H2 energy was adjusted to better reproduce metal hydride formation at the expense of metal hydroxide formation.

We have identified several elements for which an energy adjustment can help correct reaction energies to binary and ternary compounds.[1] Currently, we apply energy adjustments to the following elements:

  • P, S, C

These adjustments generally help better-reproduce reduction energies of these elements (e.g., FeP formation), but hurt the accuracy of calculations in which these elements act as cations (e.g., P2O5 formation).

Many calculation tools provided by the Materials Genome provide an option that disregards most of the energy adjustments listed above.

GGA+U calculations

Some compounds are better modeled with a U correction term to the density functional theory Hamiltonian while others are better modeled without (i.e., straight 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 when appropriate, GGA+U 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. The idea behind this approach is to split reactions into sub-reactions that are well-modeled by GGA, well-modeled by GGA+U, or a binary formation reaction that can be estimated from known experimental data. More details on this method can be found in ref [4].

Accuracy of Total Energies

To estimate the accuracy of our total energy calculations, we compute reaction data and compare against experimental data.

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 <a href="KtablesBinaries.txt">413 binaries</a> 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.[5] 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.[6] 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.[7] 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.

Citation

To cite the calculation methodology, please reference the following works:

  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, accepted for publication, Computational Materials Science. (2011).
  2. 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, (to be submitted).

References

  • 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, accepted for publication, Computational Materials Science. (2011).
  • 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.
  • L. Wang, T. Maxisch, G. Ceder, Oxidation energies of transition metal oxides within the GGA+U framework, Physical Review B. 73 (2006) 1-6.
  • 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, (to be submitted).
  • J.B. Foresman, A.E. Frisch, Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian, Gaussian. (1996)
  • 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.
  • S. Lany, Semiconductor thermochemistry in density functional calculations, Physical Review B. 78 (2008) 1-8.
  • G. Bergerhoff, The inorganic crystal-structure data-base, Journal Of Chemical Information and Computer Sciences. 23 (1983) 66-69.
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    1. 1.0 1.1 1.2 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, accepted for publication, Computational Materials Science. (2011).