Recent improvements to atomic energy-level data allow, for the first time, accurate predictions to be made for the Fe III line emission strengths in the spectra of luminous, $L_text{bol}=10^{46}-10^{48}$ erg/s, Active Galactic Nuclei. The Fe III emitting gas must be primarily photoionized, consistent with observations of line reverberation. We use CLOUDY models exploring a wide range of parameter space, together with 26,500 rest-frame ultraviolet spectra from the Sloan Digital Sky Survey, to constrain the physical conditions of the line emitting gas. The observed Fe III emission is best accounted for by dense ($n_H=10^{14}$ cm$^{-3}$) gas which is microturbulent, leading to smaller line optical depths and fluorescent excitation. Such high density gas appears to be present in the central regions of the majority of luminous quasars. Using our favoured model, we present theoretical predictions for the relative strengths of the Fe III UV34 $lambdalambda$1895,1914,1926 multiplet. This multiplet is blended with the Si III] $lambda$1892 and C III] $lambda$1909 emission lines and an accurate subtraction of UV34 is essential when using these lines to infer information about the physics of the broad line region in quasars.
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