Cubic interaction parameters for $t_{2g}$ Wannier orbitals

Many-body calculations for multi-orbital systems at present typically employ Slater or Kanamori interactions which implicitly assume a full rotational invariance of the orbitals, whereas the real crystal has a lower symmetry. In cubic symmetry, the low-energy $t_{2g}$ orbitals have an on-site Kanamori interaction, albeit without the constraint $U=U^{\prime }+2J$ implied by spherical symmetry $(U$ is the intra-orbital interaction, $U^{\prime }$ is the interorbital interaction, $J$ is Hund's exchange). Using maximally localized Wannier functions we show that deviations from the standard, spherically symmetric interactions are indeed significant for $5d$ orbitals $(\sim 25\%$ for ${\mathrm{BaOsO}}_3$; $\sim 12\%$ if screening is included) but are less important for $3d$ orbitals $(\sim 6\%$ for ${\mathrm{SrVO}}_3$; $\sim 1\%$ if screened).

Figure 1

(left) In the perovskite $\left(AB{\mathrm{O}}_3\right)$ structure, the oxygen octahedron around the central transition-metal ion breaks spherical symmetry down to cubic. The $B$ ion (large sphere) occupies the center of a cube with $A$ at the corners and O at the face centers. (right) Schematic representation of the low-energy $d_{xy}^{\prime }$ orbital of Eq. (5) [light and dark shading indicates opposite signs of the wave function].

Figure 2

Densities of states and Wannier functions for the $d$-only Wannier orbitals for ${\mathrm{SrVO}}_3$ (top) and ${\mathrm{BaOsO}}_3$ (bottom). On the left, we show the total DOS (dotted line) and the projections on transition-metal $t_{2g}$ (dark) and O (light) states. The shaded area marks the region of integration used to estimate the O-$p$ weight [corresponding to ${\left|b\right|}^2$ in Eq. (5)], see text. On the right, the light (dark) lobes are isosurfaces for the positive (negative) parts of the real-valued Wannier functions. The strong $p-t_{2g}$ hybridization and the antibonding character are plainly visible.

Figure 3

Twelve-orbital Wannier functions for ${\mathrm{SrVO}}_3$ [plotted as in Fig. 2]. (top left) By symmetry, the twelve orbitals are grouped into three equivalent $d$-like orbitals; and two types of $p$-like orbitals, (bottom) three “$p_{\sigma }$” whose symmetry axes point toward their $B$ neighbors, and (top right) six “$p_{\pi }$” pointing away from the $B$ sites. With the O-$p$ states explicitly included, no $p-t_{2g}$ hybridization is seen in these orbitals. Correspondingly, the $d$ and $p_{\pi }$ orbitals are close to their atomic counterparts. Conversely, the $p_{\sigma }$ orbitals, which mediate the $\sigma$ bonding between O-$p$ and $B-e_g$ states, are elongated along their symmetry axis and have large contributions at their $B$ neighbors [26, 27].