## 1 Introduction  
Density functional theory (DFT) [1; 2] is the most used computational method for calculating the physical and chemical properties of materials. Its success is due to the possibility of obtaining reasonably accurate results at a convenient computational cost. In principle, DFT is an exact theory, but in practice, approximations to the exchange-correlation (XC) functional must be made. Therefore, the DFT's accuracy and predictive power are strongly dependent on the choice of the XC functionals. Moreover, as it is a mean-field theory, the DFT is not self-interaction free [3], and no XC functional to date is able to exactly cancel the Hartree term. This is in contrast with the Hartree-Fock method, which is also a mean-field theory, but is one-electron self-interaction free [4].  
The problem of the self-interaction is particularly severe for highly localized orbitals, like the \(d\) and \(f\) orbitals. The self-interaction leads to an excessive wavefunction delocalization and vanishing band gaps in many insulating systems. To solve these drawbacks, one should employ generalized DFT methods, such as DFT+U [5], hybrid functionals (i.e., mixing of the DFT and Hartree-Fock methods) [6], DMFT [7], or GW [8; 9]. Nowadays, these methods are widely accepted, as they tremendously improve the description of the electronic properties (i.e., band structures, photoemission spectra) with respect to the plain DFT. Note that most of these methods are not, strictly speaking, fully ab initio, but depend on empirical parameters, like the on-site Hubbard terms U and J.  
One relevant question is if and how these methods improve the description of other measurable quantities, such as the charge and spin density. For instance, it was observed that climbing [PERSON]'s ladder (LDA, GGA, mGGA, and higher rungs) does provide more accurate total energies, but this is not true for the charge density [10] when compared to experiments. There exist other papers in the literature comparing DFT charge density to the CCSD(T) one [11] and to quantitative electron diffraction data [12]. Clearly, one has to devise a method to perform a numerical comparison of charge/spin density [13]. Another relevant question is if it is possible to determine DFT parameters from experiments.  
For instance, the authors of Ref. [14] were able to determine the Hubbard U parameters of NiO from quantitative convergent-beam electron diffraction experiments. All these theoretical and experimental works mainly addressed the charge density, whereas critical discussions of if DFT can accurately reproduce the experimental spin density are rare.  
In this work, we present a quantitative analysis of the theoretical spin density map of two ferromagnetic perovskites, YTiO\({}_{3}\) and SrRuO\({}_{3}\), calculated using two DFT+U methods and one hybrid functional (HSE06) [15]. In the standard DFT+U method [16], the U correction is applied only to the Ti and Ru ions, while in the pseudo-hybrid ACBN0 method [17; 18; 19], the Hubbard U parameters of every ion are determined self-consistently. We used the Quantum Theory of Atoms in Molecules (QTAIM) [20] theory to integrate the spin density in the Bader atomic basins. The advantage of the QTAIM over projection (Mulliken, Lowdin) methods is that it does not depend on the set of atomic orbitals used to project the Bloch wavefunctions. Moreover, the QTAIM can be used to quantitatively analyze the experimental charge and spin densities on the same footing as the theoretical ones.