  An approach is presented which allows a self-consistent description of the
fragmentation of single-particle strength for nucleons in finite nuclei
employing the Greens function formalism. The self-energy to be considered in
the Dyson equation for the single-particle Greens function contains all terms
of first (Hartree-Fock) and second order in the residual interaction. It is
demonstrated that the fragmentation of the single-particle strength originating
from the terms of second order can efficiently be described in terms of the
so-called BAGEL approximation. Employing this approximation the self-energy can
be evaluated in a self-consistent way, i.e. the correlations contained in the
Greens function are taken into account for the evaluation of the self-energy.
As an example this scheme is applied to the nucleus $^{16}O$, using a realistic
nucleon nucleon interaction. The effects of the correlations on the occupation
probabilities and the binding energy are evaluated.
