## 3 Results
### SrRuO\({}_{3}\)  
SrRuO\({}_{3}\) is a weakly interacting ferromagnetic metallic perovskite. SrRuO\({}_{3}\) crystallizes in the Pnma structure, but differently from YTiO\({}_{3}\)  the octahedra in SrRuO\({}_{3}\) are less tilted and less distorted. The Sr-O distances are 1.986 A(Sr-O1 apical), 1.986 A  and 1.987 A (Sr-O2 basal). The Sr-O-Sr angle is \(\sim\)161\({}^{\circ}\)[22]. Figure 2: Electronic density of states of YTiO\({}_{3}\) calculated as function of U(Ti-3d) and with ACBN0 with respect to the Fermi level. Figure 4 shows the spin density isolines on the _ac_ plane passing through the Ru ions. The plane containing the Sr and O atoms is almost parallel to the (010) plane. The atomic charges and spin densities integrated in the Bader basins of the charge density are reported in Table 2. Similarly to YTiO\({}_{3}\), the ACBN0 method yielded a relatively small value of U on the transition metal and a large value of U on the oxygen ions: U(Ru-4d) = 2.06 eV and U(O-2p) = 5.08 eV. The calculated atomic charges are far from the formal valence charges (Sr:+2, Ru:+4, O:\(-\)2). The sum of the partial charges, integrated on a real space mesh, differs by \(10^{-4}\) electrons from the total number of electrons per unit formula. The atomic charges of oxygen are smaller than those of YTiO\({}_{3}\). The ACBN0 and HSE06 functionals tend to concentrate both the charge and the spin on the Ru ion, making the system slightly more ionic in character. The total magnetization is close to 2 \(\mu_{B}\) and is compatible with a Ru\({}^{+4}\) ion in the \(t_{2g}^{4}\epsilon_{g}^{0}\) configuration. The atomic magnetic moments on the oxygen atoms are about 0.2 \(\mu_{B}\), one order of magnitude larger than those of YTiO\({}_{3}\). With respect to YTiO\({}_{3}\), the smaller Bader charge and the larger magnetic moment on oxygen can be explained by the larger overlap between the O-2p orbitals and the Ru-4d, which has a larger spatial extent than the Ti-3d orbital. Differently from YTiO\({}_{3}\) the spin density map, atomic charges, and spin populations are nearly insensitive to the value of U for the Ru ion. The spin density map shows only regions of positive spin density. Figure 3: Spin density and valence shell charge concentrations (VSCCs) of YTiO\({}_{3}\). The spin density isosurface is 0.3/au\({}^{3}\). O1 is the apical oxygen, and O2 and O2’ are the basal oxygens. Ti–O2’ is the largest Ti–O interatomic distance. The electronic densities of states of SrRuO\({}_{3}\) are shown in Figure 5. At U = 0, the system is a ferromagnetic metal, and upon increasing U(Ru), the system turns into a half-metal state. with a gap in the spin-up channel opening for U > 2 eV. Consequently, the total magnetic moment reaches the value of 2 \(\mu_{B}\) in the half-metallic state. The system is also half-metal with the ACBN0 method (U(Ru) = 2.06 eV), but the bandwidth is larger than in the DFT+U calculations. The half-metallic character is consistent with previous reports [38], as well as with quasi-particle self-consistent GW (sc-QSGW) calculations [39]. The transition from the metal to the half-metal has important consequences for the structural parameters. In Figure 6, we report the equilibrium volume of SrRuO\({}_{3}\) as a function of U(Ru-4d). For small values of U, the agreement with respect to experiments is rather good. However, as the system becomes half-metallic, the volume increases away from the experimental value. ACBN0 underestimates the equilibrium volume by 1.88%. The spin density and the VSCC are reported in Figure 7. The analysis of the laplacian of the charge density shows that the there are eight VSCC points located \(\sim\)0.44 A from the  
Figure 5: Electronic density of states of SrRuO\({}_{3}\) calculated as function of U(Ru–4d) and with ACBN0 with respect to the Fermi level. Figure 6: Equilibrium volume of SrRuO\({}_{3}\) calculated as a function of the U(Ru-4d) compared with ACBN0 and an experiment [22]. Ru ion.