Comparison of approximations in density functional theory calculations: Energetics and structure of binary oxides

Abstract

High-throughput first-principles calculations based on density functional theory (DFT) are a powerful tool in data-oriented materials research. The choice of approximation to the exchange-correlation functional is crucial as it strongly affects the accuracy of DFT calculations. This study compares performance of seven approximations, six of which are based on Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) with and without Hubbard U and van der Waals corrections (PBE, PBE+U, PBED3, PBED3+U, PBEsol, and PBEsol+U), and the strongly constrained and appropriately normed (SCAN) meta-GGA on the energetics and crystal structure of elementary substances and binary oxides. For the latter, only those with closed-shell electronic structures are considered, examples of which include $\mathrm{C}{\mathrm{u}}_{2}\mathrm{O}$, $\mathrm{A}{\mathrm{g}}_{2}\mathrm{O}$, MgO, ZnO, CdO, SnO, PbO, $\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$, $\mathrm{G}{\mathrm{a}}_{2}{\mathrm{O}}_{3}$, $\mathrm{I}{\mathrm{n}}_{2}{\mathrm{O}}_{3}$, $\mathrm{L}{\mathrm{a}}_{2}{\mathrm{O}}_{3}$, $\mathrm{B}{\mathrm{i}}_{2}{\mathrm{O}}_{3}$, $\mathrm{Si}{\mathrm{O}}_{2}$, $\mathrm{Sn}{\mathrm{O}}_{2}$, $\mathrm{Pb}{\mathrm{O}}_{2}$, $\mathrm{Ti}{\mathrm{O}}_{2}$, $\mathrm{Zr}{\mathrm{O}}_{2}$, $\mathrm{Hf}{\mathrm{O}}_{2}$, ${\mathrm{V}}_{2}{\mathrm{O}}_{5}$, $\mathrm{N}{\mathrm{b}}_{2}{\mathrm{O}}_{5}$, $\mathrm{T}{\mathrm{a}}_{2}{\mathrm{O}}_{5}$, $\mathrm{Mo}{\mathrm{O}}_{3}$, and $\mathrm{W}{\mathrm{O}}_{3}$. Prototype crystal structures are selected from the Inorganic Crystal Structure Database (ICSD) and cation substitution is used to make a set of existing and hypothetical oxides. Two indices are proposed to quantify the extent of lattice and internal coordinate relaxation during a calculation. The former is based on the second invariant and determinant of the transformation matrix of basis vectors from before relaxation to after relaxation, and the latter is derived from shifts of internal coordinates of atoms in the unit cell. PBED3, PBEsol, and SCAN reproduce experimental lattice parameters of elementary substances and oxides well with few outliers. Notably, PBEsol and SCAN predict the lattice parameters of low dimensional structures comparably well with PBED3, even though these two functionals do not explicitly treat van der Waals interactions. SCAN gives formation enthalpies and Gibbs free energies closest to experimental data, with mean errors (MEs) of 0.01 and \ensuremath{-}0.04 eV, respectively, and root-mean-square errors (RMSEs) are both 0.07 eV. In contrast, all GGAs including those with Hubbard U and van der Waals corrections give 0.1 to 0.2 eV MEs and at least 0.11 eV RMSEs. Phonon contributions of solid phases to the formation enthalpies and Gibbs free energies are estimated to be small at less than \ensuremath{\sim}0.1 eV/atom within the quasiharmonic approximation. The same crystal structure appears as the lowest energy polymorph with different approximations in most of the investigated binary oxides. However, there are some systems where the choice of approximation significantly affects energy differences between polymorphs, or even the order of stability between phases. SCAN is the most reasonable regarding relative energies between polymorphs. The calculated transition pressure between polymorphs of ZnO and $\mathrm{Sn}{\mathrm{O}}_{2}$ is closest to experimental values when PBED3, PBEsol (also PBED3+U and PBEsol+U for ZnO), and SCAN are employed. In summary, SCAN appears to be the best choice among the seven approximations based on the analysis of the energetics and crystal structure of binary oxides, while PBEsol is the best among the GGAs considered and shows a comparably good performance with SCAN for many cases. The use of PBEsol+U alongside PBEsol is also a reasonable choice, given that U corrections are required for several materials to qualitatively reproduce their electronic structures.