  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.
