# Magnetic Energy Landscape of Dimolybdenum Tetraacetate on a Bulk Insulator Surface
## 3 Materials and Methods  
DFT calculations were performed using the publicly available, plane-wave-pseudopotential Quantum ESPRESSO package [73; 74]. The \(xc\) functional was constructed using a generalized gradient approximation (GGA) with the PBE parametrization [59]. In molecule on-surface calculations, van der Waals interactions were included through the Grimme D2 scheme [75]. Ultrasoft pseudopotentials, taken from the PSlibrary [76] were employed for all species. The electronic wavefunctions and charge density were expanded up to a kinetic energy cut off of 80 and 640 Ry, respectively. To make the energy comparisons consistent, all calculations were performed using the simulation cell resulting from the structural optimization of the calcite (10.4) surface. The cell has an almost perfect orthorhombic symmetry with in-plane lattice parameters of 15.13 and 16.21 A (the angle between them being 89.996\({}^{\circ}\)) and a perpendicular axis of 26.0 A. The Brillouin zone was sampled with the \(\Gamma\) point. The calcite (10.4) substrate was modeled with a periodically repeated slab of three layers, allowing a vacuum gap between the adsorbed molecule and the slab replica of 18 A. The force threshold for atomic relaxations was set to \(10^{-3}\) eV/A. Only the molecule and the slab top-layer were allowed to relax. Figure 3: The total energy of the system as a function of the polar angle \(\theta\) between the total magnetic moment and the molecular Mo-Mo axis (the axis of easy magnetization—EMA). The energy is referred to the ground state with the magnetization aligned along the EAM. The two sets of data refer to two values of the azimuthal angle \(\phi\): \(0^{\circ}\) (direction of one acetate arm–black diamonds), and \(45^{\circ}\) (bisecting direction between neighboring arms–blue solid line). To capture the Mo \(d\) electron localization, both DFT \(+U\)[60, 61, 62, 63, 64] and the extended DFT \(+U+V\)[64, 65] Hubbard corrective functionals were adopted. These schemes have been successfully used on a broad variety of cases and proved very effective in improving the structural [77], vibrational [64, 78, 79, 80, 81, 82, 83, 84], and electrochemical [85] properties as well as the phase-stability and transitions of \(d\) and \(f\) electron systems, notably involving magnetism [86, 87, 88, 89]. Like in many of the works cited above the effective Hubbard interactions (i.e., the on-site \(U\) for Mo \(d\) states and inter-site \(V\) between Mo \(d\) and nearest-neighbor O \(p\) states) were determined from first-principles linear-response theory [77] using a density functional perturbation theory (DFPT) implementation [90, 91]. The calculation of \(U\) and \(V\) was done for the molecule in gas phase in a self-consistent way that ensures the consistency of their values with both the electronic structure and the geometry of the system [64, 85, 89, 91]. The obtained \(U\) and \(V\) values were also adopted for the molecule adsorbed on calcite, assuming negligible changes. Hubbard corrections are based on orthogonalized atomic orbitals and the same basis is adopted for the forces calculation (thanks to a recent extension [92]), thus improving the consistency between the molecular geometry and the electronic ground state. Most calculations were performed with fixed KS occupations, possibly with an imbalance between the two spin populations to achieve the desired total magnetization. As mentioned in Section 2.3 the calculation with the SOC required updating the pseudopotential of Mo to a fully relativistic version. In order to ascertain that this pseudopotential provides an equivalent description as the non relativistic one a preliminary set of calculations with the SOC was performed with DFT \(+U\) for the collinear (magnetic moments all along the Mo-Mo direction) AFM and FM8 configurations. These calculations recovered the same results (in particular, the same energy differences) as those shown in Table 3 for the collinear case.