Evolution of the crystal-field splittings in the compounds CeX (X=P, As, Sb, Bi), \\CeY (Y=S, Se, Te) and their alloys CeX$_{1-x}$Y$_{x}$}
The crystal-field splittings of the monopnictides and monochalcogenides of Cerium (CeX and CeY) and their alloys (CeX$_{1-x}$Y$_x$) are calculated by means of an \emph{ab initio} many-body combined technique. The hybridization functions of the 4f states of Cerium with the conduction band for each material are obtained from first principles within the local density approximation (LDA) and are used as input for the Anderson impurity model, which is solved within a multi-orbital Non-Crossing Approximation (NCA). This realistic theoretical approach (LDA-NCA) is able to reproduce the experimental results for the crystal-field splittings of the CeX and CeY series in agreement with previous theoretical calculations. It is also able to describe the non-linear evolution of the splittings in the CeX$_{1-x}$Y$_x$ alloys as a function of x. An analysis of the values of the crystal-field splittings in all the compounds can be done in depth in this contribution, due to a detailed knowledge of the band structure and crystal environment in combination with many-body physics.