# Magnetic Energy Landscape of Dimolybdenum Tetraacetate on a Bulk Insulator Surface
## 2 Results
### Magnetic States of MoMo in the Gas-Phase  
#### 2.1.1 Generalized-Gradient Approximation (GGA-PBE) Results  
In this Section, we present DFT calculations performed with spin-resolved generalized-gradient approximation (\(\sigma\)-GGA) for the exchange-correlation (\(xc\)) functional, with the PBE prescription [59]. A selection of input and output files are available as Supplementary Materials. Using this functional, independent of the starting value of the Mo's spin polarization, we always obtain a nonmagnetic (NM) ground state, with vanishing magnetic moments on each Mo center (i.e., an equal distribution of \(d\) electrons in the two spin channels). This  
Figure 1: The structure of the dimolybdenum tetraacetate (MoMo) (**a**) in the gas phase and (**b**) as adsorbed on the three-layers slab model of the calcite (10.4) surface. The chemical species are indicated at the center of each atomic ball in (**a**). result is consistent with the formation of a quadruple bond between Mo centers; in fact, a Lowdin population analysis of their \(d\) states reveals that four of them are approximately half-occupied for each spin in each Mo, which reflects the pairing of the electrons spin on the bonding states being formed. Consistent with the available literature [55] the \(d\) states involved are: the \(z^{2}\) (\(\sigma\) bond), the \(xz\) and \(yz\) (\(\pi\) bonds) and the \(xy\) (\(\delta\) bond), where the \(z\) axis is taken along the Mo-Mo axis. However, the well known tendency of approximate \(xc\) functionals to over-delocalize \(d\) electrons, might impair the stabilization of magnetic ground states that could result from the prevalence of the exchange interaction on atomic orbitals (first Hund's rule). To further probe the system's behavior, as an attempt to force a magnetic configuration, we imposed an imbalance between the two spin populations, by fixing the occupations of the Kohn-Sham (KS) spin states during the electronic self-consistent solution. In this way, several magnetic states could be achieved, with different values of the total magnetization \(M\). During the procedure, we relaxed the atomic positions for each value of \(M\). Table 1 reports the total energy \(E\) (relative to the nonmagnetic GS) for all the achieved states, along with the total magnetization M, the distance between the Mo centers and their atomic magnetization (while \(M\) is the unbalance between the number of electrons in the two spin channels and has an integer value in a system with gap, n\({}^{\uparrow}\)-n\({}^{\downarrow}\) are obtained from projecting KS states on atomic \(d\) orbitals, thus yielding fractional numbers. Note, however, that the sum of the atomic magnetization approaches the value of \(M\)). As evident from the table, the Mo-Mo distance in the NM GS is in perfect agreement with the experimental value of 2.09 A [56]. The table also illustrates that our approach allowed us to achieve some \(M>0\) ferromagnetic (FM) states. As revealed by a Lowdin analysis of \(d\) states populations, these FM states, of increasing energy, are obtained by promoting electrons from the Mo-Mo bonding to the corresponding anti-bonding states of progressively higher energy. This gradual disruption of the Mo-Mo bonds implies the corresponding increase of the Mo-Mo distance without, however, compromising the integrity of the molecule, which is still held together by the external C-CH\({}_{3}\) groups. From a technical standpoint, the FM states were obtained through constraining the KS occupations and by re-optimizing the molecular structure each time. However, with the exception of the \(M=6\)\(\upmu_{\text{B}}\) state, they are all excited states; in fact they did not result in stable configurations and a structural relaxation with no constraint on the occupations evolves the system back to its non-magnetic GS. This suggests that the self-consistent solution of constrained KS equations did not succeed in creating a self-consistent potential able to stabilize the corresponding \(M\) state.