We present an ab-initio study of the Mn substitution for Ga in GaN using the Heyd-ScuseriaErnzerhof hybrid functional (HSE). Contrary to semi-local functionals, the majority Mn t2 manifold splits into an occupied doublet and an unoccupied singlet well above the Fermi-level resulting in an insulating groundstate, which is further stabilized by a sizeable Jahn-Teller distortion. The predictions are confirmed using GW calculations and are in agreement with experiment. A transition from a localized to a delocalized Mn hole state is predicted from GaN to GaAs.Semiconductor based spintronics aims to develop hybrid devices that could perform all three operations, logic, communications and storage within the same materials technology [1]. Dilute Magnetic Semiconductors (DMSs) represent the most promising materials, and undoubtedly, transition metal doped III-V semiconductors are presently the workhorse for spintronics [2].Ab-initio simulations based on density functional theory have played an important role in investigating the physics of DMSs [3,4]. Nevertheless, the theoretical understanding has been hindered by the well-known deficiencies of the spin-polarized local density approximation (SLDA) and generalized gradient approximation (SGGA) to the exchange-correlation functional [5]: the non-locality of the screened exchange interaction is not taken into account and the electrostatic self-interaction is not entirely compensated. This lack of compensation causes fairly large errors for localized states, e.g. the Mn d states. It destabilizes the orbitals and decreases their binding energy, leading to an over-delocalization of the charge density [6]. Another closely related issue is that the Kohn-Sham gap is usually a factor 2-3 smaller than the fundamental gap of the solid [5]. Whenever the energy position of the defect level with respect to the Valence Band Maximum (VBM) is comparable with the KohnSham gap, e.g. deep acceptor levels introduced by substitutional Mn in GaN (Mn Ga ) [7], the calculation of the thermodynamic transition levels becomes difficult, since all predicted thermodynamic transition levels are strictly bound by the Kohn-Sham one electron gap. Although the underestimation of the one electron gap would even occur for the exact Kohn-Sham functional, discontinuities in the potential upon adding or removing electrons correct for this error [5,8]. For approximate functionals, which lack any such discontinuities-this includes all available semi-local and hybrid functionals -agreement between the Kohn-Sham gap and experimental fundamental gap is a prerequisite for modelling thermodynamic transition levels and band structure related properties [5].Hybrid Hartree-Fock density functionals[9] overcome the two limitations discussed above to a large extend [5]. brid functional[10] to study the Mn impurity in a GaN semiconductor host. Extensive studies of the performance of the HSE functional in solid state systems can be found in Refs. 11,12,13, unequivocally showing that hybrid functionals outperform semi-local ...