Electronic structure, dimer physics, orbital-selective behavior, and magnetic tendencies in the bilayer nickelate superconductor La$_3$Ni$_2$O$_7$ under pressure

Motivated by the recently reported high-temperature superconductivity in the bilayer La$_3$Ni$_2$O$_7$ (LNO) under pressure, here we comprehensively study this system using {\it ab initio} techniques. The Ni $3d$ orbitals have a large bandwidth at ambient pressure, increasing by $\sim 22\%$ at 29.5 Gpa. Without electronic interactions, the Ni $d_{3z^2-r^2}$ orbitals form a bonding-antibonding molecular orbital state via the O $p_z$ inducing a ``dimer'' lattice in the LNO bilayers. The Fermi surface consists of two-electron sheets with mixed $e_g$ orbitals and a hole pocket defined by the $d_{3z^2-r^2}$ orbital, suggesting a Ni two-orbital minimum model. Different from the infinite-layer nickelate, we obtained a large {\it interorbital} hopping between $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ states in LNO, caused by the ligand ``bridge'' of in-plane O $p_x$ or $p_y$ orbitals connecting those two orbitals, inducing $d-p$ $\sigma$-bonding characteristics. The competition between the intraorbital and interorbital hoppings leads to an interesting dominant spin stripe ($\pi$, 0) order because of bond ferromagnetic tendencies via the recently discussed ``half-empty'' mechanism.