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Various defects in ZnO, focused on substitutional N$_O$ and N$_2$ in varioussites, O-site, interstitial and Zn-site are studied using first-principlescalculations with the goal of understanding the electron paramagnetic resonance(EPR) center reported for N$_2$ in ZnO and substitutional N on the O-site. The$g$ tensors are calculated using the gauge including projector augmented wave(GIPAW) method and compared with experiments. The $g$-tensor of the freeN$_2^+$ and N$_2^-$ radicals and their various contributions within the GIPAWtheory are analyzed first to provide a baseline reference for the accuracy ofthe method and for understanding the N$_2$ behavior in ZnO. Previouscontroversies on the site location of N$_2$ in ZnO for this EPR center and onthe shallow or deep nature and donor or acceptor nature of this center areresolved. We find that the N$_2$ on the Zn site is mostly zinc-vacancy like inits spin density and $g$-tensor, while for the O-site, a model with the N$_2$axis lying in-the basal plane and the singly occupied $\pi_g$-orbital along the{\bf c} axis provides good agreement with experiment. For the interstitiallocation, if the N$_2$ is not strongly interacting with the surroundings, nolevels in the gap are found and hence also no possible EPR center. Thecalculated $g$-tensors for N$_O$ and $V_ Zn$ are also found to be in goodagreement with experiment. The effects of different functionals affecting thelocalization of the spin density are shown to affect the $g$-tensor values.

Calculated electron paramagnetic resonance $g$-tensor and hyperfine parameters for zinc vacancy and N related defects in ZnO