Electronic structure, self-doping, and superconducting instability in the alternating single-layer trilayer stacking nickelates La$_3$Ni$_2$O$_7$

Motivated by the recently proposed alternating single-layer trilayer stacking structure for the nickelate La$_3$Ni$_2$O$_7$, we comprehensively study this system using {\it ab initio} and random-phase approximation techniques. Our analysis unveils similarities between this novel La$_3$Ni$_2$O$_7$ structure and other Ruddlesden-Popper nickelate superconductors, such as a similar charge-transfer gap value and orbital-selective behavior of the $e_g$ orbitals. However, different from other Ruddlesden-Popper nickelate superconductors, we do not observe any obvious reconstruction of the Fermi surface from ambient conditions (Cmmm phase) to high pressures (P4/mmm phase). Pressure primarily increases the bandwidths of the Ni $e_g$ bands, suggesting an enhancement of the itinerant properties of those $e_g$ states. Furthermore, the $d_{3z^2-r^2}$ orbital also has a layer-selective behavior because the antibonding-bonding-nonbonding splitting can only be obtained in the trilayer. In addition, we observe a "self-doping" effect from the trilayer to the single-layer sublattices and this effect will be enhanced by overall electron doping. Moreover, we find a leading $d_{x^2-y^2}$-wave pairing state that is restricted to the single-layer. Because the effective coupling between the single layers is very weak -- due to the non-superconducting trilayer in between -- this suggests that the superconducting transition temperature $T_c$ in this structure should be much lower than in the bilayer structure.