  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 structural degrees of
freedom, considering two exchange-correlation methods: Local-Density
Approximation (LDA) and Perdew-Burke-Ernzerhof revised for solids (PBEsol). Our
goal is twofold: first, to provide a brief overview of the current state of the
art on this compound underpinning the last proposed theoretical models and
experimental research, and second, to give another view to model the electronic
properties compared with the previous theoretical models. Our findings show
that Ca$_{3}$Ru$_{2}$O$_{7}$ displays several electronic states (metal,
semimetal, and narrow insulator) as a function of Hubbard-U while it exhibits
structural transition depending on the functional. We disentangle the effect of
the different degrees of freedom involved, clarifying the role of
exchange-correlation in the observed electronic and structural transitions.
