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We calculate the Kondo temperature ($T_K$) and crystal-field levels ofstrongly correlated multiorbital systems solving the Anderson Impurity Modelwith the finite U Non-Crossing Approximation (UNCA) in its simplest scheme,that is, considering the self energies at lowest order in the 1/N diagrammaticexpansion. We introduced an approximation to the vertex function that includesthe double energy dependence and investigate its effect on the values of $T_K$for simple electronic models. We also analyze the competition between the twospin flip mechanisms, involving virtual transitions to empty and doublyoccupied states, in the determination of the ground state symmetry by includingan extra diagram of higher order in $1/N.$ We finally combine the resultingsimple formalism with {\it ab initio} calculated electronic structures toobtain $T_K$'s, ground states, and crystal field splittings in excellentagreement with experimental results for two particular Ce compounds, namelyCeIn$_3$ and CeSn$_3$.

Kondo temperature and crystal field levels of Ce systems within a finite U Non-Crossing Approximation