# Magnetic Energy Landscape of Dimolybdenum Tetraacetate on a Bulk Insulator Surface
## 2 Results
### Magnetic Anisotropy Energy  
Considering one of the magnetic metastable state (we chose the lowest energy FM8) we now evaluate the magnetic anisotropy energy (MAE), i.e., the energy needed to change the orientation of the total magnetization with respect to the structure of the molecule, from its ground state position (defining the easy magnetization axis, EMA) up to an orthogonal direction (see Figure 2). The energy cost is associated with a finite spin-orbit coupling (SOC), which binds the magnetic moment to the geometrical structure of the molecule. The SOC was included in the calculations by updating the Mo pseudopotential to a fully-relativistic version. We define the orientation of \(M\) via the polar angle \(\theta\), with \(\theta=0\) in correspondence of the Mo-Mo direction, Figure 2a. Given the substantial \(\sigma_{h}\) symmetry of MoMo (only broken by the extremal H atoms) a symmetric behavior for angles \(\theta>90^{\circ}\) is assumed (magnetic bistability).  
Our SOC calculations used a non-collinear magnetic formalism and have been performed with DFT + \(U\) for the isolated module. We assumed the lowest-energy FM configuration, FM8, as the reference state. The MAE is then obtained from the dependence of the total energy on the polar angle \(\theta\) between the total magnetization and the EMA, which, we anticipate, our calculations demonstrated to coincide with the Mo-Mo axis. Examples of input and output files of MAE calculations are available as Supplementary Materials.  
The results are shown in Figure 3, while the technical details of these calculations are discussed in the Material and Methods Section.  
Figure 2: The magnetization of Mo atoms for some of the considered polar angles: \(\theta=0^{\circ}\) (**a**); \(\theta=30^{\circ}\) (**b**); \(\theta=60^{\circ}\) (**c**); \(\theta=90^{\circ}\) (**d**). The Mo-Mo EMA is defined by the direction of magnetizan in (**a**).  
The plot compares two sets of results, that were obtained for two different values of the azimuthal angle \(\phi\) of the magnetization around the EAM. In fact, given the lack of cylindrical symmetry of the molecule around the Mo-Mo axis, with four acetate arms protruding along perpendicular directions in the \(x\)-\(y\) plane (with the EAM taken as \(z\)), we assumed a (light) dependence of the energy on this variable as well. The results shown in the figure, obtained for two values of \(\phi\) corresponding to one of the acetate arms (black diamonds) and with a bisecting direction (solid blue line) show a very minor effect of \(\phi\) on the total energy. The energy cost associated to the deviation of the magnetization vector from its minimum energy direction is thus almost entirely due to the polar angle \(\theta\) it forms with the EMA. The figure shows that the energy barrier is about 9.3 meV. This value is compatible with the one found in the [Mn\({}_{12}\)] compound (5.2 meV) [1] but lower than the MAE observed in single lanthanide ion SSMs [7]. However, to the best of our knowledge, no experimental data exists for the considered system.  
It is important to remark that the MAE of the system is entirely due to (and so, dependent on) the chemical environment of the Mo atoms in the molecule. In fact repeating the same spin-orbit calculations on the _bare_ Mo dimer resulted in a flat energy profile.