Computational efficiences for calculating rare earth $f^n$ energies

Recently\footnote{D. R. Beck and E. J. Domeier, Can. J. Phys. Walter Johnson issue, Jan. 2009.}, we have used new computational strategies to obtain wavefunctions and energies for Gd IV $4f^7$ and $4f^65d$ levels. Here we extend one of these techniques to allow efficent inclusion of $4f^2$ pair correlation effects using radial pair energies obtained from much simpler calculations\footnote{e.g. K. Jankowski \textit{et al.}, Int. J. Quant. Chem. \textbf{XXVII}, 665 (1985).} and angular factors which can be simply computed\footnote{D. R. Beck and C. A. Nicolaides, Excited States in Quantum Chemistry, C. A. Nicolaides and D. R. Beck (editors), D. Reidel (1978), p. 105ff.}. This is a re-vitalization of an older idea\footnote{I. Oksuz and O. Sinanoglu, Phys. Rev. \textbf{181}, 54 (1969).}. We display relationships between angular factors involving the exchange of holes and electrons (e.g. $f^6$ vs $f^8$, $f^{13}d$ vs $fd^9$). We apply the results to Tb IV and Gd IV, whose spectra is largely unknown, but which may play a role in MRI medicine as endohedral metallofullerenes (e.g. Gd$_3$N-C$_{80}$\footnote{M. C. Qian and S. N. Khanna, J. Appl. Phys. \textbf{101}, 09E105 (2007).}). Pr III results are in good agreement (910 cm$^{-1}$) with experiment. Pu I $5f^2$ radial pair energies are also presented.