Self-consistent second-order perturbation treatment of multiplet structures using local-density theory

Various ways of calculating the multiplet structure from a local-density theory are discussed. They are based on a second-order expansion of the total energy in the changes of the occupation numbers from a given reference state. The method is first applied to single determinantal states whose energies are considered as averages over the true multiplet states. In this case, the problem is reduced to the evaluation of a few fundamental parameters which, in general, must be calculated self-consistently. For states which are degenerate by symmetry it is shown that internal consistency imposes relationships between these parameters. These relationships have a unique solution for the $A_1^2{T}_2^2$ configuration in $T_d$ symmetry. This example includes the neutral vacancy in silicon and, as a limiting case, the $s^2{p}^2$ configurations of column-IV atoms. An extension of the method is then proposed to treat directly linear combinations of Slater determinants. This leads to a formalism which closely parallels term-dependent Hartree-Fock theory. Finally, the method is extended to include configuration interaction. A unique solution is again possible for the free carbon atom yielding improved term values.