# Magnetic Energy Landscape of Dimolybdenum Tetraacetate on a Bulk Insulator Surface
## 2 Results
### Magnetic States of MoMo in the Gas-Phase
#### 2.1.2 Results from Hubbard-Corrected Functionals
The Hubbard correction has instead a role in destabilizing odd-valued magnetization (OVM) states. In fact, no OVM can be found in Table 3. These solutions could not be stabilized with DFT+Hubbard by constraining \(M\), not even at the lowest odd-value (\(M=1\)\(\mu_{B}\)). A OVM state corresponds to fractional transfers of electrons between the two spin channels, i.e., to fractional occupations of some Mo \(d\) states. Fractional occupations are made energetically unfavorable by the Hubbard correction, which operates on these states and favours integer occupations, compatible with the even-valued magnetization (EVM) states shown in Table 3.  
The energy comparison between the NM and AFM solutions indicates that, at variance with GGA-PBE, the total energy depends not only on the total magnetization (equal to zero in both AFM and NM) but also, and more significantly, on the relative spin orientation of the two Mo atoms. By increasing the total magnetization to \(M=2\)\(\mu_{B}\) and \(M=4\)\(\mu_{B}\), the total energy increases. As evident from the atomic magnetization of each Mo (\(n^{\uparrow}\) \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & **NM** & **AFM** & **FM2** & **AFiM2** & **FM4** & **FiM4** & **FM6** & **FM8** \\ \hline \multirow{4}{*}{DFT + \(U\)} & \(E\) (eV) & 2.43 & 0.0 & 1.72 & 1.51 & 1.59 & 1.78 & 1.22 & 0.99 \\  & \(M\) (\(\mu_{B}\)) & 0 & 0 & 2 & 2 & 4 & 4 & 6 & 8 \\  & \(d_{Mo-Mo}\) (Å) & 2.09 & 2.54 & 2.14 & 2.57 & 2.70 & 2.66 & 2.99 \\  & \(n^{\uparrow}\)-\(n^{\downarrow}\) & 0.0 & 3.28, \(-\)3.28 & 0.94 & 3.38, \(-\)1.5 & 0.53, 3.36 & 3.36, 0.53 & 2.88 & 3.72 \\ \hline \multirow{4}{*}{DFT + \(U\) + \(V\)} & \(E\) (eV) & 2.45 & 0.0 & 2.05 & 1.02 & 2.65 & 1.54 & 1.26 & 0.99 \\  & \(M\) (\(\mu_{B}\)) & 0 & 0 & 2 & 2 & 4 & 4 & 6 & 8 \\  & \(d_{Mo-Mo}\) (Å) & 2.10 & 2.58 & 2.15 & 2.57 & 2.42 & 2.72 & 2.66 & 2.98 \\  & \(n^{\uparrow}\)-\(n^{\downarrow}\) & 0.0 & 3.32, \(-\)3.32 & 0.93 & 3.32, \(-\)1.47 & 1.87, 1.93 & 3.36, 0.49 & 2.86 & 3.69 \\ \hline \multirow{4}{*}{DFT + \(U\) + \(V_{ads}\)} & \(E\) (eV) & 3.48 & 0.0 & 0.94 & & & & 1.29 \\  & \(M\) (\(\mu_{B}\)) & 0 & 0 & 2 & & & & 8 \\ \cline{1-1}  & \(d_{Mo-Mo}\) (Å) & 2.12 & 2.53 & 2.51 & & & & 2.93 \\ \cline{1-1}  & \(n^{\uparrow}\)-\(n^{\downarrow}\) & 0.0 & 3.18, \(-\)3.20 & 3.10, \(-\)1.27 & & & & 3.67, 3.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: DFT+Hubbard results for the MoMo molecule in the gas-phase and as adsorbed on calcite (10.4). DFT + \(U\) + \(V_{ads}\) indicates calculations for MoMo adsorbed on calcite, performed only on lowest-energy magnetic configurations. \(n^{\downarrow}\)), for both \(M=2\)\(\mu_{B}\) and \(M=4\)\(\mu_{B}\), we were able to achieve two kinds of states: a ferromagnetic one (FM2 and FM4), with both atomic moments essentially equal; and a ferrimagnetic state (AFiM2 and FiM4) with very different moments. In fact, while for both \(M\) values the system was initialized with the two spin in opposite directions, only for \(M=2\)\(\mu_{B}\) an \"antiferrimagnetic\" (AFiM2) ground state was stabilized, while for \(M=4\)\(\mu_{B}\) the system preferred a ferrimagnetic (FiM4) configuration. The AFiM2 and FiM4 states are only possible through a reduction of molecular symmetry, that makes the two Mo atoms loose their equivalence. Note that, independent of the value of \(M\), the AFiM2 and FiM4 solutions have energies lower than the FM ones (FM2 and FM4) with the same \(M\). This stems from the fact that when the equivalence of the Mo's is lost, one of them can reach a saturation magnetization (3.36 \(\mu_{B}\)), approaching the [PERSON]'s rule limit (highest \(S_{z}\)). AFiM2 is one of the states whose energy is closest to the GS energy (\(E\approx 1\) eV). Importantly, a comparable energy can be obtained even for FM configurations of higher magnetizations, such as \(M=6\) and \(8\)\(\mu_{B}\) (FM6 and FM8). The total energy of FM6 (\(E\approx 1.3\) eV) is lower than FiM4, while FM8 presents an energy even slightly lower than AFiM2.