## 3 Results  
### YTiO\({}_{3}\)  
YTiO\({}_{3}\) is a ferromagnetic perovskite with a Curie temperature of 5 K, and it crystallizes in the GdFeO\({}_{3}\) Pnma structure with large TiO\({}_{6}\) octahedra tilting (19\({}^{\circ}\)). The Ti-O-Ti angles are \(\sim\)144\({}^{\circ}\) on the \(ac\) plane and 140\({}^{\circ}\) along the \(b\) axis. The distance between the Ti and apical O along the \(b\) axis is the shortest (2.0167 A), while the Ti-O distances in the \(ac\) plane are alternatively short and long (2.0178 A and 2.0754 A) [21]. Figure 1 shows the spin density isolines on the \(ac\) plane passing through the Ti ions. The oxygen atoms directly bound to the Ti ions are located 0.44 A above and below the plane. The atomic charges and spin densities integrated in the Bader basins of the charge density are reported in Table 1. The effect of the Hubbard U on the Ti ion is to concentrate both the charge and spin density on the metal ions. Interestingly, the ACBN0 method yielded U(Ti-3d) = 0.26 eV and a large value for oxygen, U(O-2p) = 8.31 eV. The calculated atomic charges are far from the formal valence charges (Y:+3, Ti:+3, 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 total spin magnetization per unit formula is 1 \(\mu_{B}\)k with 80-86% of it located on the Ti ion. This situation corresponds to a Ti\({}^{3+}\) ion in the \(t_{2g}^{1}\,e_{g}^{0}\) configuration. Note that the spin density around the oxygen atoms shows a negative shell near the ion, surrounded by a region of positive spin density. The size of this negative spin region is large when the U is applied to the Ti ion, whereas it is reduced with ACBN0. The ACBN0 atomic charges and spins are close to those obtained with the hybrid HSE06 and PBE0 functionals. The PBE0 calculations accompanying the experimental paper were performed with the gaussian basis set code CRYSTAL [34], and the QTAIM analysis was performed with TOPOND [35]. Note that the small magnetic moment on the O1 (apical) ion is initially larger that those of the O2 ions at U = 0, but the situation changes for U > 3 eV. The larger magnetic moments on the basal O2 ions are also suggested by the experimental data, but this is not the case for ACBN0. Figure 1: Isodensity contour of the spin density of YTiO\({}_{3}\) plotted on the \(ac\) plane. The Ti ion is in the center of the cell. The projections of the O2 ions are indicated by open circles (the O ions are 0.44 Å above and below the cut plane). The isovalues are \(\pm 0.001\cdot 2^{n}\) with \(n=0\ldots 7\). The electronic density of states (DOS) is shown in Figure 2. Upon increasing the Hubbard U on the Ti ion, the system undergoes a metal-to-insulator transition. The singly occupied \(t_{2g}\) orbital in the spin-up channel separates from the empty state manifold and moves to a lower energy below the Fermi level. The situation is different for ACBN0: The small value of U on the Ti ion preserves the metallic character, and the large U value on oxygen has the effect of shifting the occupied manifold down by \(\sim\)2 eV (which has a dominant oxygen character) and increasing its band width. Our results are in good agreement with those of Ref. [36; 37], where it was found that a U(Ti) value of 3.7 eV yields a band gap of 2.20 eV. In the same paper, they also calculated a band gap of 2.07 eV with the HSE06 functional, and the manifold of occupied oxygen bands was found 6 eV below the bottom of the upper Hubbard band (UHB). Overall, the HSE DOS of Ref. [36] is more similar to the case of U = 5 eV, except that the the oxygen states are \(-\)5 eV below the left shoulder of the UHB. In ACBN0, the distance between the oxygen states and the UHB is \(\sim\)6 eV, which is similar to HSE. Unfortunately, ACBN0 is not able to split the lower Hubbard band (LHB ) from the UHB.