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

We calculate the Kondo temperature ($T_K$) and crystal-field levels of strongly correlated multiorbital systems solving the Anderson Impurity Model with the finite U Non-Crossing Approximation (UNCA) in its simplest scheme, that is, considering the self energies at lowest order in the 1/N diagrammatic expansion. We introduced an approximation to the vertex function that includes the double energy dependence and investigate its effect on the values of $T_K$ for simple electronic models. We also analyze the competition between the two spin flip mechanisms, involving virtual transitions to empty and doubly occupied states, in the determination of the ground state symmetry by including an extra diagram of higher order in $1/N.$ We finally combine the resulting simple formalism with {it ab initio} calculated electronic structures to obtain $T_K$s, ground states, and crystal field splittings in excellent agreement with experimental results for two particular Ce compounds, namely CeIn$_3$ and CeSn$_3$.