## 3 Results
### YTiO\({}_{3}\)
The spin density and the VSCCs are reported in Figure 3. The analysis of the laplacian of the charge density shows that the there are six VSCC points, which are located \(\sim\)0.37 A from the Ti ion, corresponding to the outer 3\(d\) shell of Ti. When U(Ti) is small (less than 2 eV), the VSCCs are arranged as a nearly regular octahedron. The VSCC octahedron is rotated such that two VSCCs are in the O1-Ti-O2 plane, and the remaining four VSCCs are located in the plane tilted by \(\sim\)40\({}^{\circ}\) from the Ti-O2-O2' plane. For small values of U(Ti), the six VSCCs correspond to the six lobes of the spin density, showing that the spin density is given by a superposition of the \(t_{2g}\) orbitals. When the Hubbard U on Ti is larger than 2 eV and the system becomes insulating, the VSCC octahedron changes its shape and becomes highly distorted: Four VSCCs were found to correspond two the four spin density lobes, while the two remaining VSCCs were located close to the Ti-O1 bond. This is reflected by the values of the laplacian (curvature of the charge density): When U(Ti) is small, the laplacian at the VSCC ranges from \(-\)9.3 to \(-\)10.0 au\({}^{-5}\), meaning that the charge density is concentrated nearly equally at the VSCC. When U(Ti) is larger that 2 eV, the laplacian is more negative (\(-\)12.5 au\({}^{-5}\)) at the four VSCCs corresponding to the spin density lobes, while it is less negative (\(-\)8.7 au\({}^{-5}\)) at the two remaining VSCCs. Therefore, in YTiO\({}_{3}\) the curvature of the charge density distribution gives a direct indication of the shape of the spin density. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline Method & **q(Y)** & **q(Ti)** & **q(O1)** & **q(O2)** & **m(Y)** & **m(Ti)** & **m(O1)** & **m(O2)** & **Tot. magn.** \\ \hline U = 0 eV & 2.1101 & 1.9066 & \(-\)1.3325 & \(-\)1.3423 & 0.0594 & 0.7974 & 0.0521 & 0.0456 & 1.0000 \\ U = 1 eV & 2.1131 & 1.9198 & \(-\)1.3387 & \(-\)1.3474 & 0.0549 & 0.8161 & 0.0461 & 0.0414 & 1.0000 \\ U = 2 eV & 2.1182 & 1.9336 & \(-\)1.3448 & \(-\)1.3537 & 0.0457 & 0.8348 & 0.0396 & 0.0399 & 1.0000 \\ U = 3 eV & 2.1220 & 1.9496 & \(-\)1.3510 & \(-\)1.3605 & 0.0376 & 0.8511 & 0.0342 & 0.0386 & 1.0000 \\ U = 4 eV & 2.1243 & 1.9671 & \(-\)1.3574 & \(-\)1.3673 & 0.0316 & 0.8647 & 0.0299 & 0.0369 & 1.0000 \\ U = 5 eV & 2.1255 & 1.9854 & \(-\)1.3636 & \(-\)1.3739 & 0.0272 & 0.8764 & 0.0264 & 0.0350 & 1.0000 \\ ACBN0 & 2.2678 & 2.0992 & \(-\)1.4481 & \(-\)1.4597 & 0.0450 & 0.8208 & 0.0458 & 0.0442 & 1.0000 \\ HSE06 & 2.2343 & 2.0259 & \(-\)1.4127 & \(-\)1.4239 & 0.0273 & 0.8513 & 0.0369 & 0.0422 & 1.0000 \\ \hline expt. PND & N/A & N/A & N/A & N/A & -0.047 & 0.715 & 0.016 & 0.004 & 0.704 \\ PBE0 & N/A & N/A & N/A & N/A & 0.015 & 0.852 & 0.036 & 0.049 & 0.998 \\ \hline \end{tabular}
\end{table} 
Table 1: Quantum Theory of Atoms in Molecules (QTAIM) charges and spin densities of YTiO\({}_{3}\). O1 is the apical oxygen, while O2 is the basal oxygen. Experimental polarized neutron diffraction (PND) results and PBE0+QTAIM results are from Ref. [21]. Unsurprisingly, the small U(Ti)=0.26 eV found by the ACBN0 method yields a spin density with six lobes, and the VSCCs form a nearly regular octahedron. However, the ACBN0 charge density around the Ti atom is more concentrated with respect to the U = 0 case: The laplacian at the six VSCCs ranges from \(-\)9.4 to \(-\)11.2 au\({}^{-5}\), and this is due to the fact that the ACBN0 Bader charge on Ti is \(\sim\)10% larger than that computed with U = 0. In the HSE06 case, the spin density shows four large lobes and two small ones that become visible with an isosurface of 0.2 au\({}^{-3}\). This situation is intermediate between the small and large U value cases. The laplacians at the VSCCs are \(-\)8.9 and \(-\)12.5 au\({}^{-5}\), comparable to the case of U(Ti) > 2 eV.