  Under temperature or pressure tuning, tetragonal EuPd$_2$Si$_2$ is known to
undergo a valence transition from nearly divalent to nearly trivalent Eu
accompanied by a volume reduction. Albeit intensive work, its microscopic
origin is still being discussed. Here, we investigate the mechanism of the
valence transition under volume compression by $ab~initio$ density functional
theory (DFT) calculations. Our analysis of the electronic and magnetic
properties of EuPd$_2$Si$_2$ when approaching the valence transition shows an
enhanced $c$-$f$ hybridization between localized Eu 4$f$ states and itinerant
conduction states (Eu 5$d$, Pd 4$d$, and Si 3$p$) where an electronic charge
redistribution takes place. We observe that the change in the electronic
structure is intimately related to the volume reduction where Eu-Pd(Si) bond
lengths shorten and, for the transition to happen, we trace the delicate
balance between electronic bandwidth, crystal field splitting, Coulomb
repulsion, Hund's coupling and spin-orbit coupling. In a next step we compare
and benchmark our DFT results to surface-sensitive photoemission data in which
the mixed-valent properties of EuPd$_2$Si$_2$ are reflected in a simultaneous
observation of divalent and trivalent signals from the Eu $4f$ shell. The study
serves as well to explore the limits of density functional theory and the
choice of exchange correlation functionals to describe such a phenomenon as a
valence transition.
