## 2 Theoretical Studies on Heteroatom-Mediated Metal-Support Interactions
### From Single Metal Atoms to Metal Clusters  
The single atom systems described above represent just ideal situations, which can be used as a model for the interpretation and the prediction of the more realistic metal clusters or metal nanoparticles. In multi-atom systems, additional interactions can be established between adsorbed atoms that are topologically close (adatom-adatom interactions). A general relationship approximating the binding energy (BE) of metal nanoparticles or clusters with carbon surfaces has been proposed by [PERSON] [66], who proposed a linear dependency described by the equation: BE = an + b, where (n) is the number of metal atoms in the cluster, and (a) is an index of surface unsaturation. The intercept (b) accounts for the interactions between the first-layer metal atoms and the carbon surface, and it is strictly related to the electron and structural properties of the metal-support interface. The metal-support interactions between carbon and metal NPs have been disclosed by density functional theory (DFT) supported by experimental investigation. [PERSON] et al. examined the adsorption of several different metal clusters (Pt\({}_{6}\), Pt\({}_{10}\), Pt\({}_{32}\), Pt\({}_{2}\) Ru\({}_{4}\), and Pt\({}_{4}\)Ru\({}_{6}\)) on pristine, boron (B), nitrogen (N), and B,N-co-doped graphene surfaces [57]. The trend in binding energies reproduced the predicted one for the monoatomic counterparts (Pt and Ru atoms): the greatest interaction involved metal clusters and boron-doped graphene, while a weak interaction was produced on pristine graphene. Monitoring the minimum distance between metal clusters and graphene as well as the bond lengths in the metal clusters and the graphene models before and after adsorption provided precious details on atom displacement and surface rearrangement induced by the interaction between metal atoms and carbon atoms at the surface. In general, if the interaction is stronger, the minimum distance will be smaller. Calculated cluster-surface distances reflected the binding energy trend and therefore the interaction order. The higher stability of metal clusters when adsorbed on boron-doped surfaces compared to pristine graphene was confirmed also by AIMD simulations, which predicted lower MSDs for Pt and Ru clusters on B-doped graphene compared to the ones immobilized on pristine graphene. Moreover, as expected, the diffusion of clusters depended on the size of the clusters: the larger the cluster, the lower the diffusion. Concerning N-doping, analogously to single atoms, metal clusters interacted preferentially with pyridinic nitrogen species [67]. The strong covalent chemical adsorption between modified surfaces and metal clusters affected also the density of states of the metal, carbon, and heteroatoms, and the charge distribution. As an instance, the total density of states (TDOS) of Pd in Pd@PdO clusters is deeply altered after adsorption on N-doped graphene, and simultaneously also, the TDOS of C and N atoms are changed after interacting with Pd@PdO shell clusters [68]. Consequently, the electron density was also affected by the adsorption. Electron density difference (EDD) diagrams for Pd@PdO-N-graphene revealed an electron density enhancement around an N atom associated with an electron density depletion around a C atom, indicating that C-N is weakened due to the interaction with Pd atoms. In addition, according to the Mulliken population analysis, an electron transfer process occurred from Pd cluster to support. However, these variations are not observed when Pd@PdO clusters are immobilized onto pristine graphene. This highlights the central role of heteroatoms in activating the carbon atom of the support and mediating the interaction with the metal clusters. QUAMBO (QUAsi-atomic Minimal Basis set Orbitals) charge analysis can be alternatively used to gain theoretical evidence for charge transfer [69]. [PERSON] et al.