Nonunique fraction of Fock exchange for defects in two-dimensional materials

By investigating the vacancy and substitutional defects in monolayer ${\mathrm{WS}}_{2}$ with hybrid functionals, we find that there is no unique amount of Fock exchange concurrently satisfies generalized Koopmans' condition reproduces band-gap band-edge positions. Fixing mixing parameter based on band gap can lead to qualitatively incorrect defect physics two-dimensional materials. Instead, excellent agreement achieved both experiment many-body perturbation theory within $GW$ approximation once parameters are tuned individually for species edges. We show departure from a optimized inherent systems as edges experience reduced screening while localized subject bulklike screening.