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Ca$_{3}$Ru$_{2}$O$_{7}$ is a fascinating material that displays physicalproperties governed by spin-orbit interactions and structural distortions,showing a wide range of remarkable electronic phenomena. Here, we present adensity-functional-based analysis of the interplay among degrees of freedom,such as magnetism, Coulomb repulsion (Hubbard-U), and structural degrees offreedom, considering two exchange-correlation methods: Local-DensityApproximation (LDA) and Perdew-Burke-Ernzerhof revised for solids (PBEsol). Ourgoal is twofold: first, to provide a brief overview of the current state of theart on this compound underpinning the last proposed theoretical models andexperimental research, and second, to give another view to model the electronicproperties compared with the previous theoretical models. Our findings showthat Ca$_{3}$Ru$_{2}$O$_{7}$ displays several electronic states (metal,semimetal, and narrow insulator) as a function of Hubbard-U while it exhibitsstructural transition depending on the functional. We disentangle the effect ofthe different degrees of freedom involved, clarifying the role ofexchange-correlation in the observed electronic and structural transitions.

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