An Accurate Alternative to Hybrid Functionals for Germanium: DFT+α

The accuracy of bulk-property predictions in density functional theory (DFT) calculations depends on the choice of the exchange-correlation functional. While the Perdew-Burke-Ernzerhof (PBE) functional systematically overestimates lattice parameters and strongly underestimates electronic band gaps, hybrid functionals such as Heyd-Scuseria-Ernzerhof (HSE) offer better overall agreement across a broad range of materials. Using germanium as a critical test case, we challenge the ability of both functionals to capture the semiconductor properties. Although HSE improves PBE's gap error, it fails to reproduce germanium's correct Γ-L indirect and Γ-Γ band gaps simultaneously. Noting that the PBE-underestimated energy separation between the 4p valence-band maximum and 4s conduction-band minimum causes unphysical sp mixing, we propose DFT+α, a semiempirical correction scheme applied selectively to 4s-like orbitals. For germanium, DFT+α restores the proper ordering and orbital character of the band edges and yields accurate lattice constants, bulk modulus, elastic constants, and phonon frequencies at a fraction of hybrid-functional computational cost.