Ca$_{3}$Ru$_{2}$O$_{7}$: Interplay among degrees of freedom and the role of the exchange-correlation

Ca$_{3}$Ru$_{2}$O$_{7}$ is a fascinating material that displays physical properties governed by spin-orbit interactions and structural distortions, showing a wide range of remarkable electronic phenomena. Here, we present a density-functional-based analysis of the interplay among degrees of freedom, such as magnetism, Coulomb repulsion (Hubbard-U), and lattice degree of freedom, together with clarifying the role of the exchange-correlation effects (LDA v/s PBEsol functional). Our goal is twofold: first, to provide a brief overview of the current state of the art on this compound underpinning to last proposed theoretical models and experimental research, and second, to give another view to model the electronic properties compared with the previous theoretical models, establishing the lower and upper limits of (Hubbard-U). Furthermore, disentangle the effect of the Coulomb repulsion, structure, and functional approximation on the electronic band structure. Our findings show that local-density approximations (LDA) exchange-correlation outperforms the generalized gradient approximations (GGA-PBE); the Hubbard-U term in range ($0.2 \leq U \leq 2.0$ eV) for LDA+U accurately reproduces the structural and electronic properties of Ca$_{3}$Ru$_{2}$O$_{7}$.