Identifying AGNs from X-ray detections-I: Metallicity calibrations in AGNs with X-ray luminosity as the primary input parameter

We present the first semi-empirical strong-line calibrations to determine metallicity in Active Galactic Nuclei (AGNs) that use the directly observable X-ray luminosity (Lx) instead of the dimensionless ionization parameter ($U$). The calibrations are derived from an extensive grid of photoionization models computed with the {\sc Cloudy} code, which are compared with observational data of Seyfert nuclei from the Burst Alert Telescope (BAT) AGN Spectroscopic Survey (BASS). In this first paper, we develop new calibrations for two key optical metallicity diagnostics based on the $N2$ and $O3N2$ indices, which are valid in a metallicity range of $8.0 \lesssim 12 + \log({\rm O/H}) \lesssim 9.1\, {\rm or}\, 0.2 \lesssim (Z/Z_{\odot}) \lesssim 2.6$, with precision of $1\sigma \approx 0.22$ dex ($N2$) and $\approx 0.20$ dex ($O3N2$). We systematically investigate the influence of the AGN spectral index $(\alpha_{ox})$, narrow-line region (NLR) gas density $(N_{\rm e})$, the characteristic peak temperature of the Big Blue Bump $(T_{\rm BB})$, and Lx. We find a strong, opposing secondary dependence on Lx for both indices. We demonstrate that neglecting this parameter overlooks systematic offsets intrinsic to the diagnostics, leading to metallicity errors of up to $\sim 1.0$ dex, particularly for the least and most luminous sources. This framework offers a more precise characterization of chemical enrichment in the NLRs of AGNs by leveraging their intrinsic X-ray emission to mitigate these systematic biases.