# Relative Stability of Small Silver, Platinum, and Palladium Doped Gold Cluster Cations
## 2 Methods
### Theoretical Calculations  
Density functional theory calculations were performed with the ORCA 4.0.1 software package [41]. The lowest energy structures of the clusters were adopted as follows. For the Ag doped clusters, the lowest-energy isomers calculated in Reference [42] were assumed. Experimental characterization of the structures of mixed Au-Ag clusters have been performed up to size \(N=5\) using ion mobility experiments [43], photodissociation spectroscopy measurements [14], and far-infrared multiphoton dissociation studies [44; 45]. The structures assigned in these studies agree with those in Reference [42]. For the case of Pt doping, we are not aware of any experimental investigation characterizing the structure of these clusters, nor of a theoretical study searching systematically for low-lying isomers of the cationic species. A search for low-energy structures was performed by manually constructing different initial geometries. Finally, with regards to Pd doping, a thorough theoretical investigation of the Au\({}_{N}\)+ and PdAu\({}_{N-1}\)+ (\(N=2\)-\(20\)) clusters was presented in our previous work [35]. We adopt the structures found there. All geometries were optimized with the LC-BLYP exchange-correlation functional and the extensive Def2-TZVPP basis set [46]. Long-range corrections in Generalized Gradient Approximation (GGA) functionals have been shown to perform well in Pd and Ag doped Au clusters [14; 33; 35]. In addition, other DFT functionals have been tested, including the GGA PBE, meta-GGA TPSS, and the hybrid B3 LYP. For each cluster those other functionals yielded a similar energetic ordering of the considered isomers as the LC-BLYP functional. Additionally, the even larger Def2-QZVPP basis set was checked, showing no significant difference. For these optimizations, the effective core potential Def2-ECP was used, replacing 60 core electrons for Au and Pt, and 28 core electrons for Ag and Pd. For the Ag and Pd doped clusters only the lowest possible spin configurations were calculated, as previous studies showed these have the lowest energy. For the Pt doped clusters, different spin configurations were computed for each cluster size. Also, in this case the lowest possible spin states were found to be the lowest in energy. Vibrational frequencies were calculated for all the structures at the same level of theory to corroborate that the geometries corresponded to true minima on the potential energy surface and not to transition states. Relativistic effects are known to be important for heavy atoms, such as Au and Pt [47]. The cluster sizes investigated here are small enough to allow the implicit inclusion of relativistic effects by performing single point calculations on the optimized structures using the relativistic Hamiltonian within the zero-order regular approximation (ZORA) [48] and the all-electron ZORA-def2-TZVPP basis set [49]. The size-to-size stability of the clusters was analyzed computationally by the dissociation energies \(D_{N}\), defined as  
\[D_{N}=E\big{(}A_{N-m}{}^{+}\big{)}+E(A_{m})-E\big{(}A_{N}{}^{+}\big{)}, \tag{1}\]  
where \(E(A_{N}{}^{+})\), \(E(A_{N-m}{}^{+})\), and \(E(A_{m})\) correspond to the total energy of the corresponding cationic clusters and of the emitted neutral fragment. Many theoretical studies make use of another computed quantity when analyzing the relative stability of clusters, namely the second energy difference \(\Delta^{2}E_{N}\)[35; 40; 50; 51]. This quantity compares the total energy of size \(N\) with that of \(N+1\) plus \(N-1\). Interpretation of experimental data using computed \(\Delta^{2}E_{N}\) values, however, requires that the fragmentation channel is always the same, irrespective of the cluster size, or that at least one channel dominates the decay, for example the neutral monomer emission.