Hybrid functionals within the all-electron FLAPW method: Implementation and applications of PBE0

Abstract

We present an efficient implementation of the Perdew-Burke-Ernzerhof hybrid functional PBE0 within the full-potential linearized augmented-plane-wave ͑FLAPW͒ method. The Hartree-Fock exchange term, which is a central ingredient of hybrid functionals, gives rise to a computationally expensive nonlocal potential in the one-particle Schrödinger equation. The matrix elements of this exchange potential are calculated with the help of an auxiliary basis that is constructed from products of FLAPW basis functions. By representing the Coulomb interaction in this basis the nonlocal exchange term becomes a Brillouin-zone sum over vector-matrixvector products. The Coulomb matrix is calculated only once at the beginning of a self-consistent-field cycle. We show that it can be made sparse by a suitable unitary transformation of the auxiliary basis, which accelerates the computation of the vector-matrix-vector products considerably. Additionally, we exploit spatial and time-reversal symmetry to identify the nonvanishing exchange matrix elements in advance and to restrict the k summations for the nonlocal potential to an irreducible set of k points. Favorable convergence of the selfconsistent-field cycle is achieved by a nested density-only and density-matrix iteration scheme. We discuss the convergence with respect to the parameters of our numerical scheme and show results for a variety of semiconductors and insulators, including the oxides ZnO, EuO, Al 2 O 3 , and SrTiO 3 , where the PBE0 hybrid functional improves the band gaps and the description of localized states in comparison with the PBE functional. Furthermore, we find that in contrast to conventional local exchange-correlation functionals ferromagnetic EuO is correctly predicted to be a semiconductor.

Precise response functions in all-electron methods: Application to the optimized-effective-potential approach

“…(4)] can be cast into an algebraic matrix equation if the quantities are formulated in terms of an auxiliary basis that is designed to represent wave-function products. For this purpose we introduced the MPB, [34][35][36] which is built from products of LAPW basis functions, giving rise to plane waves exp(iG · r)/ √ in the interstitial region and MT functions M a LP (r)Y LM (r) in the MT sphere of atom a with cutoff values |G| G max and L L max , respectively. (For the present purpose, the MPB functions are independent of k because of the periodicity of the local potential.)…”

“…[24][25][26] We employed the mixed product basis (MPB) [34][35][36] to reformulate the OEP integral equation in terms of a matrix equation that can be solved by standard numerical linear algebra tools. This approach also enables the construction of the local EXX potential without shape approximations.…”

Atomic-Level Structural and Electronic Properties of Hybrid Inorganic–Organic ZnO:Hydroquinone Superlattices Fabricated by ALD/MLD

“…46 Concerning previous computational studies on ZnO, PBE0 band gaps of 3.32 and 3.2 eV have been predicted using full-potential linearized augmented-planewave (FLAPW) and projector-augmented-wave (PAW) basis sets, respectively. [47][48][49][50] The band structure of bulk ZnO calculated at the PBE0/TZVP level of theory is shown in Figure 5a and a comparison to the band structures calculated with the SVP and TZVPP level basis sets is shown in Supporting information. Figure 5 also shows the band-projected electron densities for the highest valence band (VB) and the lowest conduction band (CB).…”

Approximations to the exact exchange potential: KLI versus semilocal

“…(2) is exact and is based on the pseudocharge method 67,68 14 ) We also mention that the SCF convergence could be achieved much more efficiently by using an inner/outer loops procedure similar to the one described in Ref. 73 for the HF method.…”