Relativistic Configuration Interaction Energy Differences Between Gd {\sc iv} $4f^7$ $J=7/2$ Levels

This is an ion of tremendous complexity for the $ab~initio$ computationalist, because a large energy matrix is required (well above 20\,000) and a large number of basis functions (over 100\,000 determinants in some cases) is needed to create its elements. Furthermore, $4f$ pair correlation converges rather slowly in angular space \footnote{K. Jankowski $et~al$, Int. J. Quant. Chem. XXVII, 665 (1985)} in contrast, say, to that of $d$ shell electrons. The ion is of potential interest for determination of the electron electric dipole moment and in PbF$_2$:Gd scintillators. We report energy differences between the 50 $4f^7$ $J=7/2$ levels whose average error \footnote{W. C. Martin $et~al$, Atomic Energy Levels - The Rare Earth Elements, NBS, USGPO, Washington, DC (1978)} for the bottom five levels is 1309 cm$^{-1}$, comparable to the best semi-empirical results \footnote{V. A. Dzuba $et~al$, Phys. Rev. A {\bf 66}, 032105 (2002)}. Our values, however, are obtained by dividing the energy matrix up into about 50 pieces. Future work will focus on assembly (and diagonalization) of the full energy matrix.