## 4 Discussion  
The comparison of DFT results with experimental polarized neutron/X-ray diffraction [21; 23] and magnetic Compton scattering [40] data has to be carried out with some caution. After removing the instrumental and thermal effects, diffraction experiments provide the structure factors that are related to the charge/spin density by a Fourier transform. The atomic and orbital contributions to the experimental charge/spin density are obtained from the multipolar model [41] or from the maximum entropy method [42]. However, it is possible to perform a detailed QTAIM analysis on the experimental charge and spin density after a multipolar refinement [43; 44]. The other possibility is to calculate the theoretical structure factors directly from the DFT and perform multipolar refinement on them. Therefore, in this paper, we can perform only a qualitative comparison between the calculated and experimental spin densities. As reported in Ref. [21], the experimental spin density of YTiO\({}_{3}\) shows a four-lobe structure tilted by \(\sim\)45\({}^{\circ}\) degrees with respect to the Ti-O2-O2\({}^{\prime}\) plane. This situation is well reproduced by hybrid functionals (HSE06 and PBE0) as well as by DFT+U with a Hubbard U value larger than 3 eV. Smaller U values and ACBN0 are not sufficient to open an insulating gap, and, as a result, the spin densities show a six-lobe structure with results from a superposition of the \(t_{2g}\) orbitals. Conversely,  
Figure 7: Spin density and VSCC of SrRuO\({}_{3}\). The spin density isosurface is 0.3/au\({}^{3}\). O1 is the apical oxygen, and O2 and O2’ are the basal oxygens. the calculated SrRuO\({}_{3}\) spin density appears to be less sensitive to the DFT method and XC approximation. The spin density is always made of eight lobes centered on Ru and a non-negligible doughnut-shaped region center on O. Overall, its shape compares extremely well with the spin density maps that are reported in Ref. [23] without the color scale. The electronic structure of these materials is better described by hybrid functionals [36] and by ACBN0 to the same extent. The large values of U on the O-2\(p\) orbitals found by ACBN0 have the effect of moving the occupied states (which mainly have an O character) to lower energy and increasing their bandwidth in agreement with photoemission experiments and many-body (i.e., GW) techniques. Large U values for oxygen are found with ACBN0 in a number of perovskites [45]. As discussed in Ref. [45], according to the cRPA calculations and spectroscopy results, the magnitude of U(2p) is expected to be of the same order of U(3d/4d). Based on our past experience with BaBiO\({}_{3}\)[46], we noted that when the states at the Fermi level have a large O(2p) character, applying U on oxygen is more effective than applying it on the transition metal. The result is a slightly more ionic metal-oxygen bond, which allows charge disproportionation in BaBiO\({}_{3}\) (i.e., Ba\({}_{2}\) Bi\({}^{III}\) Bi\({}^{V}\) O\({}_{6}\)). As a corollary, because ACBN0 acts both on the metal and on the oxygen ions, the Hubbard value of the transition metal is diminished. By changing the set of active orbitals (i.e., including O-2\(s\) and metal-\(s\)), we expect changes in the U values. Indeed, by removing oxygen from the active orbitals, the U value on the metal is expected to be large. We added this paragraph to the discussion. Therefore, the inclusion of \(s\) orbitals in \(sp\) elements and the empty 4\(s\) orbital of Ti would constitute a big improvement to ACBN0 [45]. Unfortunately, the current implementation of DFT+U in Quantum Espresso is limited to one angular momentum per ion. Finally, we stress that partitioning of the charge/spin density with methods that do not involve projection over atomic or local orbitals is a very meaningful way to compare different DFT methods among them and with experiments. In this work, we used QTAIM, but other space partitioning methods can be used--for instance, Voronoi or Hirshfeld partitioning [47].