3.1. Geometrical structures of PdnAgm clusters
Initially, the lowest energy structures up to four atoms of pure Pd, Ag and bimetallic PdnAgm clusters are determined by optimizing various possible structures at B3P86/Lanl2DZ level of theory and different spin multiplicities. Figure 1 shows the obtained results arranged in the order of decreasing Pd content in the cluster. The lowest energy structures of pure Pd clusters are obtained with triplet spin state apart from the single atom. On the other hand, the ground states of pure Ag and bimetallic PdnAgm clusters are found to be in the singlet and doublet spin multiplicities respectively. It is clear from the figure that the lowest energy structure of pure and bimetallic trimers is an isosceles triangle, while that of pure Pd and Ag tetramers are distorted tetrahedron and rhombus respectively. The calculations also show that the ground state configurations of both Pd3Ag and Pd2Ag2 clusters are similar to that of Pd4, while that of PdAg3 cluster is similar to that of Ag4 indicating that the elemental content dominates the configuration of the PdnAgm tetramer. Similar structural properties have been previously reported for Ag-Pd clusters [3, 52].
Based on the obtained ground state structures of pure and bimetallic clusters, numerous geometries of molecular hydrogen adsorption and dissociation on these clusters were examined at the same level of theory and two spin multiplicities (singlet and triplet or doublet and quartet) in order to predict the lowest energy structures. Figure 2 reveals the optimized ground state geometries (denoted as a) alongside other low-lying isomers (denoted as b-f) of PdnAgm-H2 clusters arranged from the highest to lowest Pd content in the cluster and lowest to highest energy structures. Also, the spin multiplicity, adsorption energy, average bond length, M-M bond length, shortest M-H bond length, H-H bond length, sum of NBO net charges of hydrogen atoms and HOMO-LUMO energy gap of the lowest energy structures are presented in Tables 2-5. It can be noted from these tables that the ground state geometries in all cases are obtained with the lowest spin multiplicity, that is singlet or doublet.
3.2. Molecular hydrogen interaction with PdnAgm clusters
On the Pd atom, molecular hydrogen adsorption takes place without dissociation forming the ground state of Pd-H2 cluster (PdH2_a). The shortest Pd-H, H-H distances, H-Pd-H angle and H-H vibrational frequency in this cluster are 1.661 Å, 0.881 Å, 30.76° and 2764.39 cm-1 respectively. Isomer PdH2-b, in which molecular hydrogen dissociation occurs, is less stable and higher in energy than the ground state by 0.197 eV. This suggest that an energy barrier of the latter value is required for the transformation from the molecular to dissociative adsorption of hydrogen on the Pd atom. Isomer PdH2_c is much less stable than the lowest energy structure and found to be higher in energy by 2.415 eV. The obtained results for PdH2_a and PdH2_b are comparable to previous theoretical findings . On the other extreme, the ground state geometry on the Ag atom (AgH2_a) involves molecular hydrogen dissociation with Ag-H, H-H distances and H-Ag-H angle of 1.660 Å, 2.840 Å and 117.59° respectively. The linear structure (AgH2_b) is less stable and found to be higher in energy than AgH2_a by 0.390 eV. The atomic configuration of Pd is a closed-shell of 4d10, while that of Ag is 4d10 5s1 which means that it has only one s valence electron [22, 24]. This could be the reason of the better reactivity on Ag atom towards hydrogen dissociation.
The interaction behaviour on homonuclear dimers is contrary to that on single atoms. Apparently, H2 prefers to dissociate into two hydrogen atoms at the bridge sites of Pd2 forming the lowest energy structure (Pd2H2_a) with Pd-Pd and shortest Pd-H bond lengths of 2.699 Å and 1.661 Å. This geometry has also been proposed by Ni and Zeng  as the ground state of Pd2H2 cluster with comparable bond lengths to that obtained in this work. Pd2H2_b geometry is higher in energy than the ground state by 0.818 eV. In this geometry, hydrogen molecule binds to the bridge site of the Pd2 dimer with H-H bond distance of 1.036 Å which is longer than the corresponding experimental value 0.746 Å  suggesting more tendency of H2 towards dissociation. Isomer Pd2H2_c is less stable structure than the previous two structures and found to have a higher energy than the ground state by 1.501 eV. This structure involves molecular hydrogen adsorption on top of one Pd atom in line with the axis of Pd2 dimer with Pd-H and H-H bond lengths of 1.681 Å and 0.870 Å which are close to findings in other reports [14, 22, 27]. Similarly, H2 prefers to molecularly adsorb on top of one Ag atom in line with the axis of Ag2 dimer forming the ground state structure (Ag2H2_a) with Ag-Ag, Ag-H and H-H bond lengths of 2.573 Å, 2.519 Å and 0.751 Å respectively. These results suggest that H2 tends to weakly bind to Ag2 dimer and are in harmony with previous observations by Wang et.al . The dissociative form (Ag2H2_b), in which one hydrogen atom is placed at the bridge sites, is less stable geometry and found to be higher in energy than the ground state by 1.425 eV.
On the heteronuclear cluster, molecular adsorption appears to be more favourable than the dissociative adsorption with lower energy when the molecule binds to Pd site. The lowest energy structure is obtained with H2 molecule horizontally placed on the top of Pd atom (PdAgH2_a) with H-Pd-H angle of 28.77°. When H2 molecule is placed on top of Pd atom in line with the axis of the PdAg dimer, another stable structure (PdAgH2_b) is obtained which is higher in energy than the ground state by 0.117 eV. In contrast, placing H2 molecule on top of Ag atom forms a lesser stable structure (PdAgH2_c) with energy higher than the ground state by 0.467 eV. The Ag-H bond length in the latter case is much longer than the Pd-H bond length in the former case which also indicates, as in the case of the Ag2 dimer, that H2 molecule tends to physically adsorb on the PdAg dimer when attached to the Ag site in the bimetallic cluster. A similar behaviour of hydrogen adsorption on Ag2 and PdAg clusters has been stated previously . In the dissociative form, hydrogen atoms prefer binding to the bridge sites of the PdAg cluster (PdAgH2_d) rather than one atom at the bridge site and the other at the Pd or Ag site (PdAgH2_e and PdAgH2_f). These structures are found to be higher in energy than the lowest energy structure by 0.389, 0.398 and 0.609 eV respectively.
For H2 interaction with Pd3 trimer, the dissociation process is more favoured than the molecular adsorption. The ground state structure is obtained with one hydrogen atom located at the bridge site and the other one at the face of the isosceles triangle (Pd3H2_a). The total energy of the next geometrical structures with one hydrogen atom at each face of the triangle (Pd3H2_b) and two hydrogen atoms at the bridge sites (Pd3H2_c) are found to be in close proximity to that of the ground state with a difference of only 0.008 and 0.07 eV respectively. Similar results have been proposed by Ni and Zeng . However, the authors suggest that the structure that involves two hydrogen atoms at the bridge sites is the ground state structure of this system. Isomer Pd3H2_d involves molecular hydrogen adsorption on the top of one Pd atom with much higher energy than the ground state by 1.404 eV. The lowest energy structure on Ag3 cluster is obtained by molecular adsorption on the top of a silver atom with Ag-H and H-H bond lengths of 2.218 and 0.761 Å (Ag3H2_a). The molecular dissociation into two hydrogen atoms at the bridge sites (Ag3H2_b) is less favoured structure and found to be higher in energy than the ground state by 0.226 eV.
The elemental content of the bimetallic trimer controls its behaviour towards hydrogen adsorption. The calculations suggest that H2 molecule prefers to dissociate on Pd2Ag trimer, while the molecular adsorption is more favoured on PdAg2 trimer. Apparently, the adsorption behaviour on the Pd2Ag and PdAg2 is similar to that on pure Pd3 and Ag3 trimers suggesting that doping pure clusters with only one atom from the other constituent does not change their tendency towards adsorbing hydrogen molecule. The lowest energy structure on the Pd2Ag trimer (Pd2AgH2_a) is obtained by placing the two hydrogen atoms at the bridge sites of Pd-Pd. The next isomers (Pd2AgH2_b, Pd2AgH2_c, Pd2AgH2_d and Pd2AgH2_e) are found to be in order higher in energy than the ground state by 0.140, 0.163, 0.175 and 0.455 eV. On the PdAg2 trimer, the ground state is obtained by molecular adsorption at Pd site with Pd-H, H-H bond lengths and H-Pd-H angle of 1.730 Å, 0.840 Å and 28.09° respectively. Calculations have also shown that (PdAg2H2_b, PdAg2H2_c, PdAg2H2_d, PdAg2H2_e and PdAg2H2_f) configurations are less stable than the ground state.
In a similar manner to the behaviour on dimers and trimers, hydrogen molecule favours dissociation on the Pd tetramers. The ground state configuration (Pd4H2_a) is obtained by placing one hydrogen atom at two triangle faces of the Pd4 tetrahedron structure. This structure is more stable than that with two hydrogen atoms placed at the bridge sites (Pd4H2_b) with energy difference of 0.035 eV. Molecular hydrogen adsorption (Pd4H2_c) is also obtained with 0.820 eV energy difference apart from the ground state. The calculations also suggest, only for this configuration, that the triplet spin multiplicity is lower in energy than the singlet spin state by 0.664 eV. On the Ag tetramers, molecular hydrogen dissociation is more preferred with one hydrogen atom at the two-fold bridge sites of the rhombus structure (Ag4H2_a) and shortest Ag-H bond length of 1.816 Å. This is in line with previous observations of atomic hydrogen interaction with Ag4 cluster [9, 24]. The next isomer (Ag4H2_b) involves molecular hydrogen adsorption with total energy higher than that of the ground state by 0.291 eV. Structure (Ag4H2_c) is obtained with the triplet spin state and its energy is higher than (Ag4H2_a) by 1.626 eV.
The adsorption behaviour of hydrogen molecule on the bimetallic PdnAgm tetramers is different from that on the corresponding mono-component tetramers. For Pd3Ag and Pd2Ag2 clusters, the ground states are obtained with molecular adsorption at Pd site in the distorted tetrahedron structure (Pd3AgH2_a and Pd2Ag2H2_a). On the PdAg3 cluster, the ground state is also obtained by molecular adsorption at Pd site in the rhombus structure. This is unlike the behaviour on the corresponding Pd4 and Ag4 clusters where molecular dissociation is more preferred (Pd4H2_a and Ag4H2_a) indicating that doping Pd tetramers with one or two Ag atoms, and doping Ag tetramers with one Pd atom interfere their affinity towards dissociating hydrogen molecule.
Figure 3 illustrates the shortest M-H bond lengths of the ground state geometries of PdnAgm-H2 clusters. Apparently, the bond lengths in the case of Ag single atom, pure Pd and bimetallic PdnAgm clusters are in close proximity and found to be in the range between 1.650-1.758 Å. The Ag-H bond length is longer in pure Ag clusters with significant peaks appearing at Ag2H2 and Ag3H2 clusters. It is also noticeable from the results in the Tables 2-5 that Pd-H bond length is mostly shorter than that of Ag-H which suggests a stronger interaction in the case of Pd-H.
The average Pd-Ag bond lengths (RABL) of the lowest energy structures of PdnAgm clusters before and after hydrogen adsorption are also shown in Figure 4. It is obvious from the figure that the dissociative adsorption increases the RABL of the entire cluster, while, with exception in the case of Pd-Ag cluster, no significant change occurs upon the molecular adsorption indicating that the former type of adsorption weakens the atomic interactions in the cluster. These results are consistent with the suggestion by Zhao et.al that dissociative adsorption of hydrogen greatly changes the geometry of AunPdm cluster .
3.3. Adsorption Energies
The adsorption energy (Eads) of the PdnAgm-H2 cluster system is calculated using the formula [13, 24, 27]:
Eads = EPdnAgm + EH2 – EPdnAgm-H2
where EPdnAgm, EH2, EPdnAgm-H2 are the total energies of the PdnAgm, H2 and hydrogen molecule adsorbed on PdnAgm clusters respectively. A more positive value of Eads is a sign for a stronger bond . The Eads for the ground state geometries of PdnAgm-H2 clusters are displayed in Figure 5. The results reveal that the Eads of pure Pd clusters are higher than the corresponding Ag clusters which is also another evidence for a stronger interaction of hydrogen with Pd atoms as compared to Ag atoms. For the bimetallic clusters, the adsorption energies are between that of pure clusters suggesting that doping Ag clusters with Pd atoms enhances their interaction with hydrogen and inversely doping Pd clusters with Ag atoms weakens their interaction with hydrogen. The data in the figure also suggest that Pd dimer and trimer are the most reactive clusters for hydrogen adsorption, while single Ag atom is the least reactive one. Besides, the values in Tables 2-5 show that the Eads for a given bimetallic cluster and geometry is much greater when the hydrogen atoms are around Pd site. Similar trends have been previously reported for CO and NO interactions with AgnPdm clusters [52, 53]. Based on the adsorption energies presented for the ground states in Tables 2-5, it appears that molecular adsorption is more favourable on the Pd atom and mixed clusters, while the Pdn (2-4) clusters are more appropriate for hydrogen dissociation.
3.4. Charge Transfer
It has recently been shown for small PdnAg clusters that a charge transfer from Ag atom to Pd atoms takes place indicating that Ag atom acts as electron donor and Pd atom acts as an electron acceptor within the clusters . This behaviour can be linked to the fact that Pd is more electronegative than Ag (2.20 and 1.93) resulting in larger electron population and a stronger interaction with the adsorbed molecule at Pd site [52, 53].
The NBO population analysis suggests that the charge distribution within the lowest energy structures of PdnAgm-H2 clusters depends on the nature of hydrogen adsorption. With exception in the case of single Pd atom, one can observe from the data presented in Tables 2-5 that hydrogen atoms possess negative charge in the dissociative adsorption and positive charge in the molecular adsorption. This indicates that hydrogen atoms tend to draw electrons from the metal clusters in the dissociative form, while they tend to donate electrons to the metal clusters in the molecular adsorption form. The charge transfer tendency within the PdnAgm-H2 clusters is similar to the case of H2 interaction with AunPdm clusters . Ni and Zeng  have also indicated for H2 adsorption on Pdn clusters that a charge transfer from Pd atoms to hydrogen takes place in the dissociative form.
3.5. HOMO-LUMO energy gap
The chemical stability and reactivity of nanoclusters can be evaluated by the HOMO-LUMO energy gap (Egap). A larger value of the Egap indicates higher chemical stability and lower chemical reactivity [59, 60]. The Egap of the ground state structures of PdnAgm-H2 clusters alongside the corresponding values of the PdnAgm clusters are represented in Figure 6. The energy gaps of the adsorbed single atoms and dimers are higher than that of the PdnAgm clusters. For the trimers and tetramers, the gaps decrease monotonically with increasing the Ag content in the cluster before hydrogen adsorption and oscillate after the adsorption process. A maximum Egap occurs in the case of Ag-H2 cluster, while the Ag3-H2 cluster exhibits the smallest Egap indicating that it tends to be relatively less stable and more reactive among the examined systems. Besides, the Ag atom has the lowest chemical reactivity which is in harmony with the result in Figure 5 that Ag atom is the least reactive for hydrogen adsorption.
The Egap of the PdnH2 cluster decreases with increasing the Pd content in the cluster, while no similar tendency can be observed for Pdn or Agm and Agm-H2 clusters. Ni and Zeng  have also reported similar findings for PdnH2 and Pdn clusters.
3.6. DOS Analysis
To further analyse the electronic properties of hydrogen interaction with PdnAgm clusters, the electronic density of state (DOS) spectra of the trimers as a representative system of the others is shown in Figure 7 as well as the DOS spectrum of hydrogen molecule in the gas phase. A number of remarkable DOS peaks can be observed in the vicinity of the Fermi level in the energy range 5 to -5 eV. It is clear from the figure that the Fermi level for pure Pd3 and heteronuclear Pd2Ag clusters is less populated. The interaction with H2 increases the density of state near Fermi level of these clusters and that changes their electronic structures due to charge transfer from hydrogen atoms to the cluster. Moreover, the net charges of hydrogen atoms as shown in Table 4 for these clusters are - 0.266 e and - 0.300 e, respectively, implying more electrons are gained by the cluster. This can be ascribed to the decreased M-H bond lengths as shown in Figure 3 which facilitates more amount of charge transfer to the cluster. Also, the large density of state near the Fermi level after the interaction with hydrogen is manifested in lowering of the Homo-Lumo gap as shown in Figure 6 for these cluster. On the contrary, the charge transfer is very small in pure Ag3 and PdAg2 clusters as can be seen from Table 4. Therefore, the Homo-Lumo gap and DOS at the Fermi level are not changing significantly in these clusters as a result of the interaction with hydrogen atoms. Thus, it is interesting to note that molecular hydrogen adsorption is taking place in the latter two clusters with the RH-H values 0.761 and 0.837 Å respectively. On the other hand, the dissociative adsorption is taking place in the former two clusters as shown in Table 4. Similar trends of large change in the Homo-lumo gap and electronic structure during the molecular and dissociative adsorption of hydrogen molecule on AunPdm bimetallic clusters has been reported by Zhao et.al. . Furthermore, higher interaction of hydrogen atoms with the metal clusters occurs when the adsorption energies are high . The obtained adsorption energies for the trimers are in the order Pd3 > Pd2Ag > PdAg2 > Ag3 as shown in Figure 5. Therefore, the electronic structure is more affected in the clusters with large adsorption energies as shown by the density of states for Pd3-H2 and Pd2Ag-H2 clusters in Figure 7.