Instabilities in the Iterative Solution of the Hartree-Fock Equations for Excited Electrons

For many excited $d$- and $f$-electron configurations there exist instabilities in the iterative self-consistent solution of the Hartree-Fock (HF) equations which make it practically impossible to obtain HF solutions using standard methods. These instabilities are associated with the sudden contraction of the $d$- or $f$-electron wave function to smaller radii near the beginning of the corresponding transition or rare-earth series. The instabilities have been overcome by new methods of solving the HF problem which involve the temporary relaxation of the normalization condition and the application of techniques which search directly for that integral of the differential equation having the highest degree of self-consistency. Using these new methods, HF solutions have been obtained for excited $d$- and $f$-electron configurations throughout the transition and rare-earth series of elements, and the results of these calculations have been employed to study the nature of the solutions and of the associated instabilities.