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Currently, simulations of many-body quantum systems are known to becomputationally too demanding to be solved on classical computers. The mainproblem is that the computation time and memory necessary for performing thecalculations usually grow exponentially with the number of particles $N$. Anefficient approach to simulate many-body quantum systems is the use ofclassical approximation. However, it is known that at least at lowtemperatures, the allowed spin fluctuations in this approach are overestimatedwhat results in enhanced thermal fluctuations. It is therefore timely andimportant to assess the validity of the classical approximation. To this end,in this work, we compare the results of numerical calculations of smallMn$^{3+}$ magnetic clusters in GaN, where the Mn spins are treated classicallywith those where they are treated quantum-mechanically (crystal field model).In the first case, we solve the Landau-Lifshitz-Gilbert (LLG) equation thatdescribes the precessional dynamics of spins represented by classical vectors.On the other hand, in the crystal field model, the state of Mn$^{3+}$ ion($d^4$ configuration with $S=2$, $L=2$) is characterized by the set of orbitaland spin quantum numbers $|m_s,m_L>$. Particular attention is paid to usenumerical parameters that ensure the same single ion magnetic anisotropy inboth classical and quantum approximation. Finally, a detailed comparative studyof magnetization $\mathbf{M}(\mathbf{H}, T)$ as a function of the magneticfield $\mathbf{H}$, temperature $T$, number of ions in a given cluster $N$ andthe strength of super-exchange interaction $J$, obtained from both approacheswill be presented.

Comparative study of magnetic properties of Mn$^{3+}$ magnetic clusters in GaN using classical and quantum mechanical approach