Evolution of Electronic Correlations in the Ruddlesden-Popper Nickelates

We report on optical studies of the Ruddlesden-Popper nickelates La$_{n+1}$Ni$_{n}$O$_{3n+1}$ with $n = 2$ (La$_{3}$Ni$_{2}$O$_{7}$), $n = 3$ (La$_{4}$Ni$_{3}$O$_{10}$) and $n = \infty$ (LaNiO$_{3}$). As the number of the NiO$_{6}$ octahedra layers $n$ grows, the ratio of the kinetic energy determined from the experimental optical conductivity and that from band theory $K_{\text{exp}}/K_{\text{band}}$ increases, suggesting a reduction of electronic correlations. While the strong electronic correlations in the bilayer La$_{3}$Ni$_{2}$O$_{7}$ place it on the verge of the Mott insulating phase, the trilayer La$_{4}$Ni$_{3}$O$_{10}$ and infinite-layer LaNiO$_{3}$ exhibit moderate electronic correlations, falling into the regime of correlated metals. The evolution of the electronic correlations in La$_{n+1}$Ni$_{n}$O$_{3n+1}$ is likely to be dominated by the Ni-$d_{z^2}$ orbital. Our results provide important information for understanding the superconductivity in Ruddlesden-Popper nickelates.