The occupation dependent DFT-1/2 method

There has been a high demand in rectifying the band gap under-estimation problem in density functional theory (DFT), while keeping the computational load at the same level as local density approximation. DFT-1/2 and shell DFT-1/2 are useful attempts, as they correct the spurious electron self-interaction through the application of self-energy potentials, which pull down the valence band. Nevertheless, the self-energy potential inevitably disturbs the conduction band, and these two methods fail for semiconductors whose hole and electron are entangled in the same shell-like regions. In this work, we introduce the occupation-dependent DFT-1/2 method, where conduction band states are not subject to the additional self-energy potential disturbance. This methodology works for difficult cases such as $\text{Li}_2\text{O}_2$, $\text{Cu}_2\text{O}$ and two-dimensional semiconductors. Using a shell-like region for the self-energy potential, and allowing for downscaling of the atomic self-energy potential (with an $A$ < 1 factor), the occupation-dependent shell DFT+$A$-1/2 method yields more accurate conduction band and valence band edge levels for monolayer $\text{MoS}_2$, compared with the computationally demanding hybrid functional approach.