# 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
In any case, the dependence of the Hubbard couplings on the magnetic configuration turns out to be relatively small (a negative value is obtained for the FM6 state) with a more pronounced decrease only for the FM8 state. Note also that the \(V\) between the two Mo centers is always much smaller than the one between each Mo and the O atoms in its ligands shell. This result is what makes it easy for \(d\) electrons on each Mo to align their spin and give rise to spin-polarized ground states (see the discussion below Table 3). The main results of Hubbard corrected calculations are summarized in Table 3, which illustrates the relative stability of all the meta-stable magnetic states obtained with DFT + \(U\) and DFT + \(U\) + \(V\) for MoMo in the gas phase and with the molecule adsorbed on calcite (denoted DFT + \(U+V_{ads}\)). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & **NM** & **AFM** & **FM2** & **AFiM2** & **FM4** & **FiM4** & **FM6** & **FM8** \\ \hline \multirow{2}{*}{DFT + \(U\)} & \(U_{\text{Mo1}}\) & 3.84 & 3.58 & 3.54 & 4.09 & 3.61 & 3.58 & 3.60 & 3.23 \\  & \(U_{\text{Mo2}}\) & 3.85 & 3.58 & 3.55 & 4.06 & 3.58 & 3.62 & 3.61 & 3.24 \\ \hline \multirow{6}{*}{DFT + \(U\)} & \(U_{\text{Mo1}}\) & 4.53 & 4.20 & 4.06 & 4.09 & 4.20 & 4.34 & 4.16 & 3.80 \\  & \(U_{\text{Mo2}}\) & 4.54 & 4.20 & 4.07 & 4.17 & 4.24 & 4.23 & 4.17 & 3.81 \\ \cline{1-1}  & \(V_{\text{Mo1}-\text{Mo2}}\) & 0.40 & 0.07 & 0.14 & 0.01 & 0.01 & 0.07 & \(-0.18\) & 0.18 \\ \cline{1-1}  & \(V_{\text{Mo1}-\text{O}}\) & 1.31 & 1.20 & 1.16 & 1.16 & 1.19 & 1.27 & 1.16 & 1.08 \\ \cline{1-1}  & \(V_{\text{Mo2}-\text{O}}\) & 1.31 & 1.20 & 1.16 & 1.19 & 1.21 & 1.22 & 1.17 & 1.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Hubbard parameters (in eV) computed self-consistently for various states of MoMo. The scenario that emerges from DFT+Hubbard calculations is radically different from that of GGA-PBE. First of all, the nonmagnetic solution is not the GS state of the system (in fact, it is one of the highest in energy). While still characterized by a zero total magnetization, the GS shows instead an AFM order, with the magnetic moments of the two Mo atoms pointing in opposite directions. This is a first indication that, due to a more pronounced localization of their \(d\) states, the Mo's retain much of their atomic identity. It is important to note that the AFM GS features two almost completely spin-polarized Mo atoms, essentially as if all eight electrons forming the quadruple bond in the NM state were now localized on either Mo atom, depending on their spin. This outcome might seem quite surprising in light of the fact that other AFM could in principle be obtained by breaking one of the four bonds at a time, and localizing the two electrons of opposite spin on either Mo atom. Indeed solutions of this type (intermediate between AFM and NM) could be stabilized using values of \(V_{Mo-Mo}\) in the 1.5-3.5 eV range, much higher than the self-consistent values reported in Table 2. Note that the FM2 state is in fact of this kind, with three remaining Mo-Mo bonds and two electrons (forming the \(\delta\) bond in the NM state) segregated on either Mo with parallel spin. This state features a Mo-Mo distance which is very similar to the experimental value [56]. The stabilization of a NM and of \"intermediate\" AFM states is prevented by the low value of the self-consistent \(V_{Mo-Mo}\) which is not effective in establishing a strong hybridization between Mo centers. It is notable anyway that the molecule does not break apart thanks to the external C-CH\({}_{3}\) linker groups. Whether this picture is an artifact of our linear response-based self-consistent evaluation of the Hubbard parameters, remains to be established. Compared to the GGA-PBE results shown in Table 1, the Mo-Mo distance is almost exactly the same for states of corresponding \(M\), indicating that the geometry of the molecule mostly depends on the number of broken Mo-Mo bonds rather than on the degree of \(d\) states localization.