We compare optical and hard X-ray identifications of AGNs using a uniformly selected (above a flux limit of f_2-8 keV = 3.5e-15 erg/cm2/s) and highly optically spectroscopically complete ( > 80% for f_2-8 keV > 1e-14 erg/cm2/s and > 60% below) 2-8 keV sample observed in three Chandra fields (CLANS, CLASXS, and the CDF-N). We find that empirical emission-line ratio diagnostic diagrams misidentify 20-50% of the X-ray selected AGNs that can be put on these diagrams as star formers, depending on which division is used. We confirm that there is a large (2 orders in magnitude) dispersion in the log ratio of the [OIII]5007A to hard X-ray luminosities for the non-broad line AGNs, even after applying reddening corrections to the [OIII] luminosities. We find that the dispersion is similar for the broad-line AGNs, where there is not expected to be much X-ray absorption from an obscuring torus around the AGN nor much obscuration from the galaxy along the line-of-sight if the AGN is aligned with the galaxy. We postulate that the X-ray selected AGNs that are misidentified by the diagnostic diagrams have low [OIII] luminosities due to the complexity of the structure of the narrow-line region, which causes many ionizing photons from the AGN not to be absorbed. This would mean that the [OIII] luminosity can only be used to predict the X-ray luminosity to within a factor of ~3 (one sigma). Despite selection effects, we show that the shapes and normalizations of the [OIII] and transformed hard X-ray luminosity functions show reasonable agreement, suggesting that the [OIII] samples are not finding substantially more AGNs at low redshifts than hard X-ray samples.
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