To elucidate the interplay among valence, orbitals, spin, and electronic structure in the mixed-valence compound ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$, we systematically investigate the electronic structure and magnetic properties using density functional theory (DFT) and DFT+dynamical mean-field theory ($\mathrm{DFT}+\mathrm{DMFT}$) methods. We tune the valence by holes and electrons doped ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$. Our results demonstrate that hole doping via Sr substitution on the La site with ${\mathrm{Ni}}^{2+}$ induced the metal-insulator transition. But both K substitution (hole doping) and electron doping retain metallic behaviors. For the magnetic orders, our results show that ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$ has the C-type antiferromagnetic (CAFM, stripe order) ground state, with a gap of 88 meV, which is in agreement with the experimental value of 54–100 meV. Our results also observe strong electronic renormalization in both the Ni-$d_{z^2}$ and Ni-$d_{x^2-y^2}$ orbitals of ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$, similar to the electronic correlation reported in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$. Tuning the Coulomb interaction shows the spin and charge instabilities in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$, which is consistent with the behavior of ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$.

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• Antiferroelectricity
• Charge density waves
• Flat bands
• Magnetic interactions
• Magnetic order
• Spin dynamics
• High-temperature superconductors
• Ruddlesden-Popper compounds
• Strongly correlated systems
• DFT+DMFT
• First-principles calculations
• Methods in magnetism
• Monte Carlo methods