Pressure induced redistribution of oxygen hole states in La$_{4}$Ni$_{3}$O$_{10}$

Using density functional calculations and multi-orbital, multi-atom cluster exact diagonalization that includes local exchange and Coulomb interactions, we explored the local low-energy electronic states of trilayer La$_4$Ni$_3$O$_{10}$ via a minimal Ni$_3$O$_{14}$ cluster. We find that, at ambient pressure, starting with all three Ni being nominally 2+ valence, one of the two extra holes is localized in the central NiO$_2$ layer forming a Zhang-Rice singlet (ZRS) with $d_{x^2-y^2}$ orbital. The other hole mainly occupies the antibonding combination of the two interplane O $p_z$ orbitals and thereby hybridizes with an out-of-plane three-spin-polaron (3SP) formed by the $d_{z^2}$ orbitals of three NiO$_2$ layers. In this way, the in-plane spin orientation alternation is carried by the $d_{x^2-y^2}$ orbitals of two outer layers with interlayer antiferromagnetic correlation. Simultaneously, the central layer is insulating with negligible magnetic moment. At high pressure, however, the two extra holes are concentrated on one of two outer layers and the inner layer separately forming the ZRS with $d_{x^2-y^2}$ orbitals or in-plane 3SP with neighboring cluster. We highlight the similarities between the bilayer La$_3$Ni$_2$O$_7$ and trilayer La$_4$Ni$_3$O$_{10}$ via possible charge and spin ordered states suggested by our cluster results.