A density functional theory study of adsorption and dissociation of H2 molecule on small PdnAgm (n + m ≤ 4) metal clusters

The adsorption and dissociation of hydrogen molecule on small PdnAgm (n + m ≤ 4) clusters is studied employing density functional theory (DFT) calculations. The lowest energy structures of the PdnAgm clusters are initially presented. Based on the obtained structures, various geometries of H2 adsorption and dissociation with these clusters are examined. The cluster size and composition of the PdnAgm clusters are found to control the nature of the adsorption process. The results suggest that molecular adsorption is more favorable on the Pd atom and mixed clusters, while hydrogen dissociation is more favorable on the Pdn (n = 2–4) clusters. The Pd dimer and trimer are found to be the most reactive clusters for hydrogen adsorption, while single Ag atom is the least reactive one. The natural bond orbital (NBO) population analysis indicates that hydrogen atoms tend to draw electrons from the metal clusters in the dissociative form and donate electrons to the metal clusters in the molecular adsorption form. The density of states (DOS) of the PdnAgm trimers are also presented and discussed. The interaction and stabilities of cluster on the graphene support has also been performed.


Introduction
A metal cluster can be defined as aggregates of a small number to hundreds of atoms. This class of materials possesses high surface to volume ratio compared to their bulk counterparts reflecting distinctive physical and chemical properties [1,2]. The geometric structure and size of small transition metal clusters play a key role in controlling their behavior [3]. Consequently, the catalytic properties of small metallic clusters are expected to vary and be sensitive to the cluster size and composition [4]. Adding an impurity atom or alloying can further modify the physical and chemical properties of pure transition metal clusters affecting their catalytic properties [4][5][6]. Typically, the presence of the second component is expected to change the electronic and geometric properties and enhances the catalytic performance of monometallic catalysts [5][6][7]. The enhanced activity of bimetallic catalysts could be attributed to "geometric ensemble effects" where by active sites on the catalyst surface consist of atoms arranged in specific groupings [7].
The study of interactions between transition metal clusters with atoms or small molecules is fundamental to understand catalytic processes. A primary step to the reaction mechanism is the adsorption behavior of the atoms or small molecules on the metal clusters [6,8,9]. Hydrogen is a promising energy carrier that can be produced by an environmentally sustainable manner such as photocatalytic water splitting [10][11][12]. The interactions and behavior of hydrogen on metal surfaces and clusters are of importance to various applications and technologies such as catalytic processes, metal embrittlement, hydrogen storage, and fuel cells [13][14][15][16][17]. In the latter case, for instance, electricity can be generated by harmless electrochemical reactions of hydrogen and oxygen on metal surfaces as catalysts, and the process typically involves the adsorption and diffusion of the constituents. For example, proton-conducting fuel cells employ an acidic media and hydrogen gas as a fuel, and consequently, hydrogen oxidation occurs at the anode producing electrons and protons which migrate to the cathode and recombine with oxygen resulting in water [15,16,18].
The electronic structure of hydrogen atoms and molecules is simple. Therefore, electronic structure methods can be effectively employed to describe their interactions with metal surfaces and provide valuable insights into essential properties of hydrogen-metal systems such as surface reactivity, bond-making, and bond-breaking behaviors [19]. In this regard, a number of theoretical studies have been carried out on the interactions between atomic or molecular hydrogen with pure and/or binary metal clusters [5,6,8,9,14,15,[20][21][22][23][24][25][26][27][28][29][30][31]. It has been largely proposed that the nature of the metallic components, geometry, and size are crucial in controlling the adsorption and dissociation behaviors of hydrogen on the binary cluster surfaces.
A series of Pd-Ag bimetallic catalysts have experimentally been shown to be efficient for methane combustion [32], oxidation of carbon monoxide [33], partial hydrogenation of acetylene [34], dehydrogenation of formic acid [35,36], and fuel cell-related reactions [37][38][39]. Pd-Ag binary alloys have also been acknowledged as efficient and promising materials for hydrogen gas permeation and storing [40][41][42][43] as well as sensing [44]. Additionally, various physiochemical properties of Pd-Ag bimetallic cluster system have been estimated by theoretical calculations [3,[45][46][47]. For instance, a DFT study of small Pd n Ag (n = 1-8) clusters [47] has indicated that the Pd 3-8 Ag clusters possess three-dimensional configurations and their binding energies are nearly similar to that of the corresponding Pd n clusters. The study has also found that stability and reactivity of the ground-state clusters are size-dependent, and that a universal charge transfer takes place from Ag atom to Pd atoms within the clusters. It has also been shown for Ag m -Pd n (m + n = 5) clusters that structures of bimetallic tetramers and pentamers shift from two to three dimensions with the presence of two Pd atoms in the cluster. Besides, stability analysis of the bimetallic tetramers has revealed that Ag 2 Pd 2 is the most stable cluster [3]. Other theoretical studies have focused on the assessment of various chemical processes over Pd-based [48][49][50], Pd-doped and Ag-doped surfaces [51,52]. Furthermore, a recent DFT study by Yang et al. has examined the HCOOH decomposition over pure and Ag-modified Pd nanoclusters [53]. The study suggests that the activity and selectivity of the process can be improved by controlling the size and Ag composition of the nanoclusters.
A particular attention has also been directed towards studying the interactions of small molecules with Pd-Ag binary clusters. Zhao et al. have indicated that a preferential binding of both CO [54] and NO [55] molecules takes place on Pd sites when both elements co-exist in the cluster. The authors have also found that the binding energy in both cases increases with increasing Pd content in the cluster. A study of Ag n PdH (n ≤ 5) clusters [23] has shown that hydrogen atom binds to Ag-Pd bridge site when (n = 1-4) and to top site in the case of Ag 5 Pd.
It has also been verified that hydrogen molecule adsorbs more favorably and dissociates more easily at Pd atom than Ag atom on the PdAg dimer [21,25]. The dissociation behavior of hydrogen molecule on Ag 5 and Pd 5 clusters has been compared by Bartczak and Stawowska to that on Ag 4 Pd and AgPd 5 clusters [8]. The authors have suggested that the molecule dissociates on Pd 5 cluster, while no dissociation occurs on Ag 5 cluster. The study has also concluded that mixing Ag with pure Pd catalyst reduces its ability for hydrogen dissociation, while mixing Pd with pure Ag enhances its ability for the dissociation process. Moreover, Zhang and Yang [31] have shown that, among Pd n Ag (8-n) alloy clusters, Pd 5 Ag 3 and Pd 2 Ag 6 provide promising candidates for hydrogen dissociation and storage processes, respectively. A variety of other DFT studies have focused on the evaluation of the catalytic performance of Pd-Ag clusters for hydrogenation of acetylene [56,57] and CO oxidation [58]. These studies broadly suggest that mixing the two elements enhances the adsorption properties of the reactant molecules on the bimetallic clusters leading to a better performance in comparison with the mono-component clusters.
This work aims to systematically address the adsorption and dissociation behavior of hydrogen molecule on Pd n Ag m (n + m ≤ 4) clusters within the frame work of density functional theory (DFT) at the B3P86/Lanl2DZ level of theory. Initially, the lowest energy structures of Pd n Ag m clusters are presented. Then, a number of structural and electronic properties of most stable structures of Pd n Ag m -H 2 clusters are presented and discussed.

Computational method
All the calculations presented in this work are carried out using Gaussian 09 software [59]. GauusView 05 [60] is used for drawing atomic and molecular structures of Pd n Ag m and Pd n Ag m -H 2 clusters and results visualization. B3P86 exchange correlation functional and Lanl2DZ basis set are used for Pd and Ag atoms, while the 6-311 + + G(d,p) basis set is used for hydrogen atoms. The B3P86/Lanl2DZ level of theory has recently been shown to be reliable for the assessment of small Pd n Ag (n = 1-8) clusters [47]. The reliability of this method for the study of molecular hydrogen adsorption on Pd n Ag m clusters has been initially assessed by calculating the binding energies, bond lengths, and vibrational frequencies of Pd 2 , Ag 2 , PdAg, PdH, AgH, and H 2 dimers at the B3P86/Lanl2DZ level of theory alongside the PW91PW91 and B3LYP methods. The PW91PW91 method has been previously employed for the study of CO and NO bindings on small Ag n Pd m clusters [54,55] and atomic hydrogen interaction with Ag n Pd clusters [23]. In addition, the B3LYP method has been used for the study of molecular hydrogen interaction with various transition metal homonuclear and heteronuclear dimers [21]. The calculated parameters of the dimers are shown in Table 1 alongside other theoretical data from the literature and the corresponding experimental values; the blank values are not available to the best of the authors' knowledge. In comparison with the other two functionals, the results obtained with B3P86 method are largely closer to the experimental values. Besides, the calculated parameters using the B3P86 functional are consistent with the theoretical findings from other published works [3,20,21,[24][25][26][61][62][63]. Hence, the results in the table suggest that the B3P86 is an effective method for the evaluation of hydrogen adsorption on Pd n Ag m clusters.
The study of the H 2 adsorption and dissociation on Pd n Ag m (n + m ≤ 4) clusters is accomplished by, first, optimizing various possible geometries of the clusters to identify the lowest energy structures. Then, a variety of possible configurations of the adsorption and dissociation of hydrogen molecule on the most stable configurations of Pd n Ag m clusters are optimized to identify the lowest energy structures. The most stable geometries of the Pd n Ag m and Pd n Ag m -H 2 clusters are determined by performing structural optimization and vibrational frequency calculations for each structure at different spin multiplicities. Natural bond orbital (NBO) analysis is carried out for the Pd n Ag m -H 2 clusters at the B3P86/Lanl2DZ level of theory to investigate the charge distributions. The DOS analysis is carried out using the GaussSum software [64]. The FWHM (full width at half-maximum, in eV) of the DOS peak in the software is calculated by convolution of DOS and COOP (Crystal Orbital Overlap Population) peaks. Table 1 The calculated bond length, r (Å), binding energy, E b (eV), and vibrational frequency, ω (cm −1 ) of Pd 2 , Ag 2 , Pd-Ag, Pd-H, Ag-H, and H 2 dimers with other theoretical data and experimental values * The density functionals are shown between brackets a [61] b [62] c [21] d [63] e [3] f [25] g [24] h [20] i [26]

Geometrical structures of Pd n Ag m clusters
Initially, the lowest energy structures up to four atoms of pure Pd, Ag, and bimetallic Pd n Ag m 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 Pd n Ag m 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 Pd 3 Ag and Pd 2 Ag 2 clusters are similar to that of Pd 4 , while that of PdAg 3 cluster is similar to that of Ag 4 indicating that the elemental content dominates the configuration of the Pd n Ag m tetramer. Similar structural properties have been previously reported for Ag-Pd clusters [3,54]. 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 Pd n Ag m -H 2 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, 3, 4, and 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. Isomer PdH 2 -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 PdH 2 _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 PdH 2 _a and PdH 2 _b are comparable to previous theoretical findings [24]. On the other extreme, the ground state geometry on the Ag atom (AgH 2 _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 (AgH 2 _b) is less stable and found to be higher in energy than AgH 2 _a by 0.390 eV. The atomic configuration of Pd is a closed-shell of 4d 10 , while that of Ag is 4d 10 5s 1 which means that it has only one s valence electron [24,26]. This could be the reason of the better reactivity on Ag atom towards hydrogen dissociation. The interaction behavior on homonuclear dimers is contrary to that on single atoms. Apparently, H 2 prefers to dissociate into two hydrogen atoms at the bridge sites of Pd 2 forming the lowest energy structure (Pd 2 H 2 _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 [24] as the ground state of Pd 2 H 2 cluster with comparable bond lengths to that obtained in this work. Pd 2 H 2 _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 Pd 2 dimer with H-H bond distance of 1.036 Å which is longer than the corresponding experimental value 0.746 Å [21] suggesting more tendency of H 2 towards dissociation. Isomer Pd 2 H 2 _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 Pd 2 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 [15,24,29]. Similarly, H 2 prefers to molecularly adsorb on top of one Ag atom in line with the axis of Ag 2 dimer forming the ground state structure (Ag 2 H 2 _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 H 2 tends to weakly bind to Ag 2 dimer and are in harmony with previous observations by Wang et al. [21]. The dissociative form (Ag 2 H 2 _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.

Molecular hydrogen interaction with Pd n Ag m clusters
On the heteronuclear cluster, molecular adsorption appears to be more favorable than the dissociative adsorption with lower energy when the molecule binds to Pd site. The lowest energy structure is obtained with H 2 molecule horizontally placed on the top of Pd atom (PdAgH 2 _a) with H-Pd-H angle of 28.77°. When H 2 molecule is placed on top of Pd atom in line with the axis of the PdAg dimer, another stable structure (PdAgH 2 _b) is obtained which is higher in energy than the ground state by 0.117 eV. In contrast, placing H 2 molecule on top of Ag atom forms a lesser stable structure (PdAgH 2 _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 Ag 2 dimer, that H 2 molecule tends to physically adsorb on the PdAg dimer when attached to the Ag site in the bimetallic cluster. A similar behavior of hydrogen adsorption on Ag 2 and PdAg clusters has been stated previously [21]. In the dissociative form, hydrogen atoms prefer binding to the bridge sites of the PdAg cluster (PdAgH 2 _d) rather than one atom at the bridge site and the other at the Pd or Ag site (PdAgH 2 _e and PdAgH 2 _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 H 2 interaction with Pd 3 trimer, the dissociation process is more favored 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 (Pd 3 H 2 _a). The total energy of the next geometrical structures with one hydrogen atom at each face of the triangle (Pd 3 H 2 _b) and two hydrogen atoms at the bridge sites (Pd 3 H 2 _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 [24]. 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 Pd 3 H 2 _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 Ag 3 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 (Ag 3 H 2 _a). The molecular dissociation into two hydrogen atoms at the bridge sites (Ag 3 H 2 _b) is less favored 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 behavior towards hydrogen adsorption. The calculations suggest that H 2 molecule prefers to dissociate on Pd 2 Ag trimer, while the molecular adsorption is more favored on PdAg 2 trimer. Apparently, the adsorption behavior on the Pd 2 Ag and PdAg 2 is similar to that on pure Pd 3 and Ag 3 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 Pd 2 Ag trimer (Pd 2 AgH 2 _a) is obtained by placing the two hydrogen atoms at the bridge sites of Pd-Pd. The next isomers (Pd 2 AgH 2 _b, Pd 2 AgH 2 _c, Pd 2 AgH 2 _d, and Pd 2 AgH 2 _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 PdAg 2 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 (PdAg 2 H 2 _b, PdAg 2 H 2 _c, PdAg 2 H 2 _d, PdAg 2 H 2 _e, and PdAg 2 H 2 _f) configurations are less stable than the ground state.
In a similar manner to the behavior on dimers and trimers, hydrogen molecule favors dissociation on the Pd tetramers. The ground state configuration (Pd 4 H 2 _a) is obtained by placing one hydrogen atom at two triangle faces of the Pd 4 tetrahedron structure. This structure is more stable than that with two hydrogen atoms placed at the bridge sites (Pd 4 H 2 _b) with energy difference of 0.035 eV. Molecular hydrogen adsorption (Pd 4 H 2 _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 twofold bridge sites of the rhombus structure (Ag 4 H 2 _a) and shortest Ag-H bond length of 1.816 Å. This is in line with previous observations of atomic hydrogen interaction with Ag 4 cluster [9,26]. The next isomer (Ag 4 H 2 _b) involves molecular hydrogen adsorption with total energy higher than that of the ground state by 0.291 eV. Structure (Ag 4 H 2 _c) is obtained with the triplet spin state and its energy is higher than (Ag 4 H 2 _a) by 1.626 eV.
The adsorption behavior of hydrogen molecule on the bimetallic Pd n Ag m tetramers is different from that on the corresponding mono-component tetramers. For Pd 3 Ag and Pd 2 Ag 2 clusters, the ground states are obtained with molecular adsorption at Pd site in the distorted tetrahedron structure (Pd 3 AgH 2 _a and Pd 2 Ag 2 H 2 _a). On the PdAg 3 cluster, the ground state is also obtained by molecular adsorption at Pd site in the rhombus structure. This is unlike the behavior on the corresponding Pd 4 and Ag 4 clusters where molecular dissociation is more preferred (Pd 4 H 2 _a and Ag 4 H 2 _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 Pd n Ag m -H 2 clusters. Apparently, the bond lengths in the case of Ag single atom, pure Pd and bimetallic Pd n Ag m clusters are in close proximity and found to be in the range between 1.650 and 1.758 Å. The Ag-H bond length is longer in pure Ag clusters with significant peaks appearing at Ag 2 H 2 and Ag 3 H 2 clusters. It is also noticeable from the results in the Tables 2, 3, 4, and 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 (R ABL ) of the lowest energy structures of Pd n Ag m clusters before and after hydrogen adsorption are also shown in Fig. 4. It is obvious from the figure that the dissociative adsorption increases the R ABL of the

Adsorption energies
The adsorption energy (E ads ) of the Pd n Ag m -H 2 cluster system is calculated using the formula [14,26,29]: where E PdnAgm , E H2 , E PdnAgm-H2 are the total energies of the Pd n Ag m , H 2 and hydrogen molecule adsorbed on Pd n Ag m clusters respectively. A more positive value of E ads is a sign for a stronger bond [29]. The E ads for the ground state geometries of Pd n Ag m -H 2 clusters are displayed in Fig. 5. The results reveal that the E ads 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  Tables 2, 3, 4, and 5 show that the E ads 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 Ag n Pd m clusters [54,55]. Based on the adsorption energies presented for the ground states in Tables 2, 3, 4, and 5, it appears that molecular adsorption is more favorable on the Pd atom and mixed clusters, while the Pd n (2)(3)(4) clusters are more appropriate for hydrogen dissociation.

Charge transfer
It has recently been shown for small Pd n Ag 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 [47]. This behavior 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 [54,55].
The NBO population analysis suggests that the charge distribution within the lowest energy structures of Pd n Ag m -H 2 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, 3, 4, and 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 Pd n Ag m -H 2 clusters is similar to the case of H 2 interaction with Au n Pd m clusters [29]. Ni and Zeng [24] have also indicated for H 2 adsorption on Pd n clusters that a charge transfer from Pd atoms to hydrogen takes place in the dissociative form.

HOMO-LUMO energy gap
The chemical stability and reactivity of nanoclusters can be evaluated by the HOMO-LUMO energy gap (E gap ). A larger value of the E gap indicates higher chemical stability and lower chemical reactivity [61,62]. The E gap of the ground state structures of Pd n Ag m -H 2 clusters alongside the corresponding values of the Pd n Ag m clusters are represented in Fig. 6. The energy gaps of the adsorbed single atoms and dimers are higher than that of the Pd n Ag m 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 E gap occurs in the case of Ag-H 2 cluster, while the Ag 3 -H 2 cluster exhibits the smallest E gap 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 Fig. 5 that Ag atom is the least reactive for hydrogen adsorption.
The E gap of the Pd n H 2 cluster decreases with increasing the Pd content in the cluster, while no similar tendency can be observed for Pd n or Ag m and Ag m -H 2 clusters. Ni and Zeng [24] have also reported similar findings for Pd n H 2 and Pd n clusters.

DOS analysis
To further analyze the electronic properties of hydrogen interaction with Pd n Ag m clusters, the electronic density of Fig. 6 The HOMO-LUMO energy gap (E gap ) of the ground state structures of Pd n Ag m (solid line) and Pd n Ag m -H 2 (dotted line) clusters state (DOS) spectra of the trimers as a representative system of the others is shown in Fig. 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 Pd 3 and heteronuclear Pd 2 Ag clusters is less populated. The interaction with H 2 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 Fig. 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 Fig. 6 for these cluster. On the contrary, the charge transfer is very small in pure Ag 3 and PdAg 2 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 R H-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 Au n Pd m bimetallic clusters has been reported by Zhao et al. [29]. Furthermore, higher interaction of hydrogen atoms with the metal clusters occurs when the adsorption energies are high [31]. The obtained adsorption energies for the trimers are in the order Pd 3 > Pd 2 Ag > PdAg 2 > Ag 3 as shown in Fig. 5. Therefore, the electronic structure is more affected in the clusters with large adsorption energies as shown by the density of states for Pd 3 -H 2 and Pd 2 Ag-H 2 clusters in Fig. 7.

Effects of graphene support
The graphene (C 24 H 12 ) supported sample geometries of Pd 3 and Pd 2 Ag clusters have been examined corresponding to the unsupported clusters in order to obtain a preliminary understanding of their interactions and stabilities on the graphene support. The calculations have been limited to the lowest energy structures of these clusters, and considering placing them at different possible adsorption sites on graphene surface prior to the calculations. Figure 8 illustrates the optimized structures of the free standing defect-free graphene as well as a number of Pd 3 /graphene and Pd 2 Ag/graphene structures. The results suggest that planner alignment in between the trimer plane and graphene is preferred in case of Pd 3 cluster whilst the orthogonal alignment is preferred for Pd 2 Ag cluster. It also noted that Ag atom always prefer to dock at the top site C-atom of the graphene. The observation is in accordance with the reported behavior of Ag atom on the graphene substrate [65]. The stability of the clusters on graphene can be evaluated through the interfacial interaction energy (E int ) and formation energy (E f ) [66]: where E PdnAgm , E graphene , and E PdnAgm/graphene indicate the total energies of unrelaxed Pd n Ag m cluster, the graphene sheet and adsorbed Pd n Ag m cluster on graphene, respectively.
where E Pd and E Ag indicate the total energies of the isolated Pd and Ag atoms, respectively. Table 6 provides the calculated values using the above mentioned equations as well as the HOMO-LUMO energy gap (E gap ) of Pd 3 and Pd 2 Ag clusters supported on graphene. The Pd 3 trimer structures with the lowest energy of formation are always found to arrange parallel to the graphene plane. For both the Pd 2 Ag trimers, the energies of formation are almost similar, however, the interaction E aint = E PdAgm + E graphene − E PdnAgm∕graphene E f = E PdnAgm∕graphene − E graphene − nE pd − mE Ag Fig. 8 The optimized geometries of graphene as well as Pd 3 and Pd 2 Ag sample clusters supported on graphene. Selected bond lengths (Å) are shown for each structure

Pd3_a
Pd3_b energies of the trimer with Ag atom docking on top-site of graphene (Pd 2 Ag_a) is more than the other structure, in which the Ag tom docks at the hollow-site of graphene. The HOMO-LUMO gaps of the graphene supported Pd 3 and Pd 2 Ag clusters are also found to be significantly lower in comparison to that of unsupported clusters. The observed change in the energy gaps of graphene supported clusters suggests that the graphene support enhances the catalytic activity of the clusters [67].

Conclusion
DFT calculations have been carried out in this study to systematically investigate the adsorption and dissociation of hydrogen molecule on small Pd n Ag m (n + m ≤ 4) clusters. The ground state geometries of the Pd n Ag m clusters are initially determined by optimizing various possible structures. Based on the obtained geometries, the lowest energy structures of Pd n Ag m -H 2 clusters have been identified by optimizing several possible configurations of molecular hydrogen adsorption and dissociation on the most stable configurations of Pd n Ag m clusters. The results have revealed that the nature of the adsorption process is controlled by the size and composition of the clusters. Molecular adsorption has been found to be more favorable on the Pd atom and mixed clusters, while hydrogen dissociation is more favorable on the Pd n (n = 2-4) clusters. The adsorption energy calculations have shown that hydrogen interacts more strongly with Pd atoms than Ag atoms. It has also been found that hydrogen interaction with Ag clusters can be enhanced by doping with Pd atoms, while inversely doping Pd clusters with Ag atoms weakens their interaction with hydrogen. The study has revealed that the Pd dimer and trimer are the most reactive clusters for hydrogen adsorption, while single Ag atom is the least reactive one. The NBO population analysis has shown that electrons transfer from the metal clusters to hydrogen atoms in the dissociative form and from hydrogen atoms to the metal clusters in the molecular adsorption form. The DOS analysis of the trimers has shown that Pd 3 cluster possesses higher d density of state close to the Fermi level compared to the other systems resulting in higher adsorption energy. A preliminary understanding of the cluster interactions and stabilities on the graphene support has also been performed. It was found that the graphene supported clusters have lower HOMO-LUMO gaps in comparison to the unsupported ones indicating possible improvement in the catalytic activity of the clusters.