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For the treatment of interacting electrons in crystal lattices approximationsbased on the picture of effective sites, coupled in a self-consistent fashion,have proven very useful. Particularly in the presence of strong localcorrelations, a local approach to the problem, combining a powerful method forthe short ranged interactions with the lattice propagation part of thedynamics, determines the quality of results to a large extent. For aconsiderable time the non crossing approximation (NCA) in direct perturbationtheory, an approach originally developed by Keiter for the Anderson impuritymodel, built a standard for the description of the local dynamics ofinteracting electrons. In the last couple of years exact methods like thenumerical renormalization group (NRG) as pioneered by Wilson, have surpassedthis approximation as regarding the description of the low energy regime. Wepresent an improved approximation level of direct perturbation theory forfinite Coulomb repulsion U, the crossing approximation one (CA1) and discussits connections with other generalizations of NCA. CA1 incorporates allprocesses up to fourth order in the hybridization strength V in aself-consistent skeleton expansion, retaining the full energy dependence of thevertex functions. We reconstruct the local approach to the lattice problem fromthe point of view of cumulant perturbation theory in a very general way anddiscuss the proper use of impurity solvers for this purpose. Their reliabilitycan be tested in applications to e.g. the Hubbard model and theAnderson-lattice model. We point out shortcomings of existing impurity solversand improvements gained with CA1 in this context.  This paper is dedicated to the memory of Hellmut Keiter.

Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems