We report a comprehensive theoretical analysis of the Ruddlesden-Popper layered nickelates ${\mathrm{La}}_{m+1}{\mathrm{Ni}}_m{\mathrm{O}}_{3m+1}$ ($m=1$ to 6) under pressure. These materials have recently received significant attention due to the discovery of superconductivity in some nickelates under pressure. Our results suggest that, while these Ruddlesden-Popper layered nickelates display many similarities, they also show noticeable differences. One of the common features of ${\mathrm{La}}_{m+1}{\mathrm{Ni}}_m{\mathrm{O}}_{3m+1}$ is that the electronic states near the Fermi level are mainly contributed by Ni $3d$ orbitals, slightly hybridized with O $2p$ orbitals. The Ni $d_{3z^2-r^2}$ orbitals display bonding-antibonding, or bonding-antibonding-nonbonding, characteristic splittings, depending on the even or odd number of stacking layers $m$. In addition, the ratio of the in-plane interorbital hopping between $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ orbitals and in-plane intraorbital hopping between $d_{x^2-y^2}$ orbitals was found to be large in ${\mathrm{La}}_{m+1}{\mathrm{Ni}}_m{\mathrm{O}}_{3m+1}$ ($m=1$ to 6), and this ratio increases from $m=1$ to $m=6$, suggesting that the in-plane hybridization will increase as the layer number $m$ increases. In contrast to the dominant $s^{\pm }$-wave state driven by spin fluctuations in the bilayer ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ and trilayer ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$, two nearly degenerate $d_{x^2-y^2}$-wave and $s^{\pm }$-wave leading states were obtained in the four-layer stacking ${\mathrm{La}}_5{\mathrm{Ni}}_4{\mathrm{O}}_{13}$ and five-layer stacking ${\mathrm{La}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{16}$. The leading $s^{\pm }$-wave state was recovered in the six-layer material ${\mathrm{La}}_7{\mathrm{Ni}}_6{\mathrm{O}}_{19}$ with slightly higher calculated pairing strength $\lambda$ than that of the $d_{x^2-y^2}$-wave state. All this evidence suggests that both $s^{\pm }$-wave and $d_{x^2-y^2}$-wave channels are strongly competing in the high-order niceklates based on our random-phase approximation calculations. In general, at the level of the random-phase approximation treatment, the superconducting transition temperature $T_c$ decreases in stoichiometric bulk systems from the bilayer ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ to the six-layer ${\mathrm{La}}_7{\mathrm{Ni}}_6{\mathrm{O}}_{19}$, despite the $m$-dependent dominant pairing. Both in-plane and out-of-plane magnetic correlations are found to be quite complex. Within the in-plane direction, we obtained the peak of the magnetic susceptibility at $\mathbf{q}=(0.6\pi ,0.6\pi )$ for ${\mathrm{La}}_5{\mathrm{Ni}}_4{\mathrm{O}}_{13}$ ($m=4$) and ${\mathrm{La}}_7{\mathrm{Ni}}_6{\mathrm{O}}_{19}$ ($m=6$) and at $\mathbf{q}=(0.7\pi ,0.7\pi )$ for ${\mathrm{La}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{16}$ ($m=5$). Along the out-of-plane direction, four layers are coupled as $\downarrow -\uparrow -\uparrow -\downarrow$ in ${\mathrm{La}}_5{\mathrm{Ni}}_4{\mathrm{O}}_{13}$, five layers are coupled as $\uparrow -\uparrow -\downarrow -\uparrow -\uparrow$ in ${\mathrm{La}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{16}$, and six layers are coupled as $\uparrow -\downarrow -\downarrow -\uparrow -\uparrow -\downarrow$ in ${\mathrm{La}}_7{\mathrm{Ni}}_6{\mathrm{O}}_{19}$.

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• Multiband superconductivity
• Pairing mechanisms
• Superconductivity
• High-temperature superconductors
• Nickelates
• Electron-correlation calculations
• First-principles calculations
• Hubbard model
• Random phase approximation