The full quantum mechanical treatment of a 5 nm nanoparticle is still not feasible. The modeling of magnetic nanoparticles from the first-principle side is, therefore, normally limited to small clusters of hundreds of atoms [71,72] and cannot take into account to a full extend the spin non-collinearities, their dynamics and temperature. Larger magnetic nanoparticles of 10 nm diameter are normally modeled using the Heisenberg model. An important role of spin non-collinearities  in understanding the magnetic behavior of nanoparticles has been reported using these types of studies. This so-called "atomistic" description [74,75,76,77,78,79] is relied on the Heisenberg-type Hamiltonian and can include spin dynamics and temperature. As a handicap these calculations use phenomenological surface anisotropy models, such as the transverse anisotropy one . One of the most justified models for the surface anisotropy is the widely-used Néel surface anisotropy model [47,74,14] which will be also used in the present thesis.
The challenge, however, is the understanding of not only individual nanoparticles but their ensembles, where the distributions of individual properties and interactions play an important role. This is normally done using the representation of each nanoparticle as one macrospin [80,81].
As for the thin films modeling, although intrinsic surface effects, based on the electronic interactions are localized on few layers, the magnetic exchange correlation length makes the surface anisotropy to influence the magnetic structure down to nanometers distance inside the material and may cause the inhomogeneous magnetization. The correct account for the domain structure belongs to the area of micromagnetism - a "continuous" approximation which has been proven to be very useful, especially in understanding the hysteresis and dynamics of nanostructures with dimensions up to several microns .
Therefore, one of the challenges of theoretical modeling in magnetism is the proper account of its multiscale character. The physics of magnetism in materials involve many length, energy and time scales. And, unfortunately at this moment, a unified description of all these scales on the same footing is impossible.
Our "ideal" multiscale scheme is presented in Fig. 1.8. In the case of thin films the microscopic on-site parameters are calculated from ab-intio calculations limited to several atomic cells. The output of these calculations is the parameterized Heisenberg Hamiltonian which can be used to evaluate temperature-dependent parameters. These are used for large-scale micromagnetic modeling where also the microstructure such as the distribution of grains can be taken into account. In "theory" the atomic-scale defects should be taken into account either on the ab-initio or on the atomistic scale, also the atomistic discretization could be used near defects and the micromagnetic one - far from it, as in Ref. .
For the nanoparticles the "ideal" scheme consists first of the calculation of the nanoparticle structure, basing, for example, on the molecular dynamics with suitably parameterized potentials .
We should note that even this scheme is too ambitious for the present state of art and its development is making the first steps. Also this scheme is too simplified, since as we mentioned above, the surface anisotropy, for example, has multiple ingredients, many of them related to the presence of defects. The precise knowledge of defects on the atomistic scale and its consequences is basically not available.