Parameterizing the AGN radius -- luminosity relation from the Eigenvector 1 viewpoint

The study of the broad-line region (BLR) using reverberation mapping has allowed us to establish an empirical relation between the size of this line emitting region and the continuum luminosity that drives the line emission (i.e. the R$_{\rm BLR}$-L$_{\rm 5100}$ relation). To realize its full potential, the intrinsic scatter in the R$_{\rm BLR}$-L$_{\rm 5100}$ relation needs to be understood better. The Eddington ratio plays a key role in addressing this problem. On the other hand, the Eigenvector 1 schema has helped to reveal an almost clear connection between the Eddington ratio and the strength of the optical FeII emission which has its origin from the BLR. This paper aims to reveal the connection between theoretical entities, like, the ionization parameter (U) and cloud mean density (n$_{\rm H}$) of the BLR, with physical observables obtained directly from the spectra, such as optical FeII strength (R$_{\rm FeII}$) that has immense potential to trace the accretion rate. We utilize the photoionization code CLOUDY and perform a suite of models to reveal the BLR in the U-n parameter space and estimate RFeII. We compare the SEDs for a prototypical Population A and Population B source, I Zw 1 and NGC 5548, respectively, in this study. The results from the photoionization modelling are combined with existing reverberation mapped sources with observed R$_{\rm FeII}$ estimates, allowing us to provide an analytical formulation to tie together the aforementioned quantities. We utilize the comparison of the modelled equivalent widths for the low-ionization emission lines to their observed values to identify the optimal (U,n$_{\rm H}$). The recovery of the correct physical conditions in the BLR suggests that the BLR `sees' a different, filtered ionizing continuum with only a very small fraction (~1-10%) that leads to the line emission in the dustless, low-ionization BLR.