# 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  
Standard \(xc\) functionals, as the GGA-PBE, are affected by electronic self-interaction and miss most of the effects of strong correlations. Localized \(d\) or \(f\) electrons are mostly affected \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **NM** & **FM1** & **FM2** & **FM4** & **FM6** & **FM8** \\ \hline E (eV) & 0 & 0.90 & 1.43 & 3.57 & 4.08 & 5.34 \\ M (\(\upmu_{\text{B}}\)) & 0 & 1 & 2 & 4 & 6 & 8 \\ d\({}_{Mo-Mo}\) (Å) & 2.09 & 2.12 & 2.15 & 2.42 & 2.67 & 2.98 \\ n\({}^{\uparrow}\)- n\({}^{\downarrow}\) & 0.0 & 0.45, 0.44 & 0.88, 0.87 & 1.82, 1.81 & 2.74 & 3.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: GGA-PBE results for the MoMo molecule in the gas-phase. For each state, we report the total energy \(E\) (relative to the nonmagnetic ground state energy), the total magnetization \(M\), the Mo-Mo distance d\({}_{Mo-Mo}\) and the magnetization on the \(d\) states of each Mo, n\({}^{\uparrow}\)- n\({}^{\downarrow}\). Double entries in n\({}^{\uparrow}\)- n\({}^{\downarrow}\) refer to the Mo atoms, when they present different magnetizations. In the first row, the numbering at the end of the label refers to the corresponding value of \(M\). by these flaws as they are typically over-delocalized, especially in presence of degeneracy. Among the consequences, one of the most typical is the suppression of magnetism. In order to ascertain whether some results in the previous Section are in fact artifacts of the GGA-PBE functional (in particular, the stable Mo-Mo multiple bond) and whether a magnetic GS can actually be stabilized by alleviating the self-interaction of Mo \(d\) states, we performed calculations with the Hubbard corrected DFT + \(U\) method [60; 61; 62; 63; 64]. A selection of input and output files are available as Supplementary Materials. Specifically we use two different formulations: the traditional DFT + \(U\) with only on-site interactions (\(U\)), and a generalized corrective functional based on the extended Hubbard model containing both on-site (\(U\)) and inter-site (\(V\)) interactions, known as DFT + \(U\) + \(V\)[65; 64]. While mostly employed for crystals, these corrective functionals have also been used to study molecular systems containing TM magnetic centers, for example to assess the equilibrium geometry and dissociation energies [66; 67; 68]; the relative stability of various magnetic configurations and their implications on chemistry [69]; the potential energy surface along relevant reaction pathways [70; 71]; the energetics of low-lying excited states using a Slater-1/2 approach [72]. In many cases the results compare favorably with those of more sophisticated quantum chemical approaches. Here the Hubbard \(U\) is applied to the Mo \(d\) states, while the inter-site \(V\) operates between the Mo \(d\) and O \(p\) states and between Mo \(d\) states. In general, the extended DFT + \(U\) + \(V\) scheme improves the description of electronic localization, especially in presence of significant hybridization between neighbor sites. Therefore, this method should be quite suitable for the system considered here. Quite surprisingly, we found that the two Hubbard corrections give almost equivalent results and will be collectively indicated as \"DFT+Hubbard\". DFT+Hubbard calculations were conducted following a similar scheme as for the GGA-PBE functional. Table 2 reports the Hubbard parameters obtained for the considered states of the system; these quantities were calculated using a self-consistent procedure which involves the re-optimization of both the electronic structure and molecular geometry (see the Materials and Methods section). The Table only shows results for the molecule in gas phase. For the DFT+\(U+V\) calculation on the calcite surface we used the same value obtained in the gas phase. The Hubbard parameters reflect the substantial structural equivalence of the two Mo atoms, except for those magnetic configurations (AFiM2 and FiM4) where they assume quite different magnetic moments.