Reformulation of $\mathrm{DFT}+U$ as a Pseudohybrid Hubbard Density Functional for Accelerated Materials Discovery

The accurate prediction of the electronic properties of materials at a low computational expense is a necessary condition for the development of effective high-throughput quantum-mechanics (HTQM) frameworks for accelerated materials discovery. HTQM infrastructures rely on the predictive capability of density functional theory (DFT), the method of choice for the first-principles study of materials properties. However, DFT suffers from approximations that result in a somewhat inaccurate description of the electronic band structure of semiconductors and insulators. In this article, we introduce ACBN0, a pseudohybrid Hubbard density functional that yields an improved prediction of the band structure of insulators such as transition-metal oxides, as shown for ${\mathrm{TiO}}_2$, MnO, NiO, and ZnO, with only a negligible increase in computational cost.

Popular Summary

Accurate, inexpensive predictions of the electronic properties of materials have been the holy grail of computational materials science from the first applications of density-functional theory (DFT) in the early 1980s. Despite the enormous success of DFT at describing many physical properties of real systems, its limitations in correctly describing the electronic band structure of insulators are well known. The method is limited by the presence of an unknown correlation term that represents the difference between the true energy of the many-body system of the electrons and the approximate energy that we can compute. In recent years, two competing approaches have unfolded: $\mathrm{DFT}+U$ and hybrid functionals. The first method suffers from an ambiguity in the computation of critical parameters; the second allows for some empiricism and is computationally very expensive.

We introduce the Agapito-Curtarolo-Buongiorno Nardelli pseudohybrid Hubbard density functional as a fast, accurate, and parameter-free alternative to traditional $\mathrm{DFT}+U$ and hybrid exact exchange methods. In this density function, the Hubbard energy of $\mathrm{DFT}+U$ is calculated via the direct evaluation of the local Coulomb and exchange integrals. These values are thus functionals of the electron density and depend directly on the chemical environment and the crystalline field, properly describing insulators and semiconductors. The Agapito-Curtarolo-Buongiorno Nardelli functional satisfies the rather ambitious criteria outlined by Pickett et al. in 1998, and its flexibility allows for the calculation of the local Coulomb and exchange integrals for any atom in the system of interest, yielding, for instance, non-negligible values for the $2p$ lone pair of oxygen in transition-metal oxides or for the $p$ states of the anion in transition-metal chalcogenides. Via the inclusion of these terms, Agapi-Curtarolo-Buongiorno Nardelli corrects for both the band gap and the relative position of the different bands, in particular, the bands deriving from the $d$ orbitals of transition-metal atoms.

In our discussion of the electronic properties of four technologically relevant transition-metal oxides with different $3d$ shell fillings (${\mathrm{TiO}}_2$, MnO, NiO, and ZnO), which show excellent agreement with hybrid functionals, we emphasize that our new pseudohybrid Hubbard density functional is characterized by much lower computational costs.

Figure 1

Comparison between the band structures and projected density of states of ${\mathrm{TiO}}_2$ without on-site interactions $U=0$ (a) and with the converged effective values of $U=0.15\mathrm{eV}$ for Ti and 7.34 for O (b).

Figure 2

Band structure (spin up) of manganese oxide. All energies are relative to the valence-band maximum $E_{\mathrm{V}}$. Effective values of $U=4.67\mathrm{eV}$ for the $\mathrm{Mn}\text{−}3d$ states and 2.68 eV for the $\mathrm{O}\text{−}2p$ states are used in the ACBN0 calculation.

Figure 3

Band structure (spin up) of nickel oxide. All energies are relative to the valence-band maximum $E_{\mathrm{V}}$. The effective values of $U_{3d}=7.63\mathrm{eV}$ for nickel and $U_{2p}=3.0\mathrm{eV}$ for oxygen are used in the ACBN0 calculation.

Figure 4

Comparison between the PBE (gray) and ACBN0 (black) band structures for ZnO. The converged effective Coulomb interactions are Zn $U_{3d}=12.8\mathrm{eV}$ and O $U_{2p}=5.29\mathrm{eV}$. The horizontal grid lines show the DFT (0.85 eV) and ACBN0 (2.91 eV) band gaps. The panel on the right shows the projected DOS for the ACBN0 calculation.

Figure 5

Effect of the zinc on-site Coulomb repulsion $U_{3d}$ on the occupied $3d$ bands. An ad hoc value of $U_{3d}=9.0\mathrm{eV}$ (a) is compared against the converged value of 12.8 eV (b). Both cases use the converged values of $U_{2d}=5.29\mathrm{eV}$ for oxygen.

Figure 6

Fitting the radial component of the PAOs [$r^{-1}{R}_l(r)$, yellow line] as a linear combination of primitive Gaussians (black line). Each contracted Gaussian is the sum of the three primitive Gaussians shown in blue. The Zn $d$ (a) and O $p$ PAOs (b) are taken from the PSlibrary 1.0.0.