Construction of a general semilocal exchange-correlation hole model: Application to nonempirical meta-GGA functionals

Using a reverse-engineering method, we construct a meta-generalized gradient approximation (meta-GGA) angle-averaged exchange-correlation (XC) hole model which has a general applicability. It satisfies known exact hole constraints and can exactly recover the exchange-correlation energy density of any reasonable meta-GGA exchange-correlation energy functional satisfying a minimal set of exact properties. The hole model is applied to several nonempirical meta-GGA functionals: the Tao-Perdew-Staroverov-Scuseria (TPSS), the revised TPSS (revTPSS), and the recently Balanced LOCalization (BLOC) meta-GGA [L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem. Theory Comput. 9, 2256 (2013)]. The empirical M06-L meta-GGA functional is also considered. Real-space analyses of atoms and ions as well as wave-vector analyses of jellium surface energies show that the meta-GGA hole models, in particular the BLOC one, are very realistic and can reproduce many features of benchmark XC holes. In addition, the BLOC hole model can be used to estimate with good accuracy the Coulomb hole radius of small atoms and ions. Thus, the proposed meta-GGA hole models provide a valuable tool to validate in detail existing meta-GGA functionals, and can be further used in the development of density functional theory methods beyond the semilocal level of theory.