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

## TL;DR

A new method called DFT+α improves predictions of germanium's properties compared to traditional functionals, at lower computational cost.

## Contribution

DFT+α is introduced as a semiempirical correction scheme targeting specific orbital interactions in germanium.

## Key findings

- DFT+α corrects the band gap and orbital character in germanium.
- DFT+α achieves accurate lattice and elastic properties at lower cost than hybrid functionals.

## Abstract

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.