Crucial role of out-of-plane Sb-$p$ orbitals in Van Hove singularity formation and electronic correlation for superconducting Kagome metal CsV${}_3$Sb${}_5$

First-principles density functional theory calculations are performed to understand the electronic structure and interaction parameters for recently discovered superconducting Kagome metal CsV${}_3$Sb${}_5$. A systematic analysis of the tight-binding parameters based on maximally localized Wannier function method demonstrates that the out-of-plane Sb${}^{\mathrm{o}\mathrm{u}\mathrm{t}}$-$p$ orbital is a key element in complete description of the three Van Hove singularity structures known in this material at $M$ point near the Fermi level. Further, the correlation strengths are also largely determined by Sb${}^{\mathrm{o}\mathrm{u}\mathrm{t}}$-$p$ states. Based on constrained random phase approximation, we find that on-site and inter-site interaction parameter are both significantly affected by the screening effect of Sb${}^{\mathrm{o}\mathrm{u}\mathrm{t}}$-$p$ orbitals. As the role of this previously unnoticed orbital state can be tuned or controlled by out-of-plane lattice parameters, we examine the electronic structure and particularly the evolution of Van Hove singularity points as a function of strain and pressure, which can serve as useful knobs to control the material properties.