Interplay between crystal structure and magnetism in CeCrB$_4$
CeCrB$_4$ belongs to the family of MTB$_4$ ternary metal borides, some of whose magnetic phases have recently been identified as quantum spin dimers. Employing density functional theory within the GGA+$U$ framework, this work investigates the key ground-state properties of this system, including its ferromagnetic ground state, the mixed-valence nature of cerium, and the charge transfer from the cationic cerium--chromium layers to the anionic boron layers. In particular, we focus on elucidating the bonding mechanism within the structurally embedded Cr-Cr dimers. Specifically, we evaluate a previously proposed interpretation that the localized Cr $3d_{z^2}$ states are molecular-like bonding and antibonding states. The calculated dependence of the $3d_{z^2}$ electronic structure on the intra-dimer distance correlates with the level-splitting characteristics known for classical homonuclear diatomic molecules.