Spin-1/2 chains with alternating antiferromagnetic (AFM) and ferromagnetic (FM) couplings have attracted considerable interest due to the topological character of their spin excitations. Here, using density functional theory and density-matrix renormalization-group (DMRG) methods, we have systematically studied the dimerized chain system ${\mathrm{Na}}_2{\mathrm{Cu}}_2{\mathrm{TeO}}_6$ with a $d^9$ electronic configuration. Near the Fermi level, in the nonmagnetic phase the dominant states are mainly contributed by the Cu $3d_{x^2-y^2}$ orbitals highly hybridized with the O $2p$ orbitals, leading to an “effective” single-orbital low-energy model. By calculating the relevant hoping amplitudes, we explain the size and sign of the exchange interactions in ${\mathrm{Na}}_2{\mathrm{Cu}}_2{\mathrm{TeO}}_6$. In addition, a single-orbital Hubbard model is constructed for this dimerized chain system where the quantum fluctuations are taken into account. Both AFM and FM couplings (leading to an $\uparrow \text{−}\downarrow \text{−}\downarrow \text{−}\uparrow$ state) along the chain were found in our DMRG and Lanczos calculations, in agreement with density functional theory and neutron-scattering results. The hole pairing binding energy $\mathrm{\Delta }E$ is predicted to be negative at Hubbard $U\sim 11\mathrm{eV}$, suggesting incipient pairing tendencies.

The van der Waals oxide dichlorides $M\mathrm{O}{X}_2$ ($M=\mathrm{V}$, Ta, Nb, Ru, and Os; $X=\text{halogen}$ element), with different electronic densities, are attracting considerable attention. Ferroelectricity, spin-singlet formation, and orbital-selective Peierls phases were reported in this family with $d^1$ or $d^2$ electronic configurations, all believed to be caused by the strongly anisotropic electronic orbital degree of freedom. Here, using density functional theory and density matrix renormalization group methods, we investigate the electronic and magnetic properties of ${\mathrm{RuOCl}}_2$ and ${\mathrm{OsOCl}}_2$ with $d^4$ electronic configurations. Different from a previous study using ${\mathrm{VOI}}_2$ with $d^1$ configuration, these systems with $4d^4$ or $5d^4$ do not exhibit a ferroelectric instability along the $a$ axis. Due to the fully occupied $d_{xy}$ orbital in ${\mathrm{RuOCl}}_2$ and ${\mathrm{OsOCl}}_2$, the Peierls instability distortion disappears along the $b$ axis, leading to an undistorted $Immm$ phase (No. 71). Furthermore, we observe strongly anisotropic electronic and magnetic structures along the $a$ axis. For this reason, the materials of our focus can be regarded as “effective one-dimensional” systems even when they apparently have a dominant two-dimensional lattice geometry. The large crystal-field splitting energy (between $d_{xz/yz}$ and $d_{xy}$ orbitals) and large hopping between nearest-neighbor Ru and Os atoms suppresses the $J=0$ singlet state in $M{\mathrm{OCl}}_2$ ($M=\mathrm{Ru}$ or Os) with electronic density $n=4$, resulting in a spin-1 system. Moreover, we find staggered antiferromagnetic order with $\pi$ wave vector along the $M$-O chain direction ($a$ axis) while the magnetic coupling along the $b$ axis is weak. Based on Wannier functions from first-principles calculations, we calculated the relevant hopping amplitudes and crystal-field splitting energies of the $t_{2g}$ orbitals for the Os atoms to construct a multiorbital Hubbard model for the $M$-O chains. Staggered AFM with $\uparrow \text{−}\downarrow \text{−}\uparrow \text{−}\downarrow$ spin structure dominates in our density matrix renormalization group calculations, in agreement with density functional theory calculations. Our results for ${\mathrm{RuOCl}}_2$ and ${\mathrm{OsOCl}}_2$ provide guidance to experimentalists and theorists working on this interesting family of oxide dichlorides.