An accurate alternative to hybrid functionals for germanium: DFT+$\alpha$

The accuracy of bulk property predictions in density functional theory (DFT) calculations depends on the choice of 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 semiconductor properties. Although HSE improves PBE's gap error, it fails to reproduce germanium's correct $\Gamma$-L indirect and $\Gamma$-$\Gamma$ 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+$\alpha$, a semi-empirical correction scheme applied selectively to 4s-like orbitals. For germanium, DFT+$\alpha$ restores the proper ordering and orbital character of the band edges and yields accurate lattice constant, bulk modulus, elastic constants and phonon frequencies at a fraction of hybrid-functional computational cost.