3.1. Structural properties

We first optimized the geometries of the bare Ti_mZr_n (n + m ≤ 5) clusters. The most stable structures together with their spin multiplicities and point groups are shown in Fig. 1. Table 2 shows some calculated results obtained from the ground state structures. The TiZr dimer has a bond length of 2.100 Å, which is between the bond lengths of Ti–Ti (1.896 Å) and Zr–Zr (2.271 Å) in the corresponding monoatomic dimers. For the trimers, Ti₃ and Zr₃ are both equicrural triangles and have the same symmetry of C_2v, while Ti₂Zr and Zr₂Ti are of the C_s symmetry, and the distances between the atoms of the two structures are different. When one Ti atom in the Ti₃ cluster is substituted by a Zr atom, the two bonds connecting it are elongated from 2.292 Å to 2.524 Å, and the Ti–Ti bond away from the changed atom is shortened from 2.407 Å to 2.163 Å. By contrast, the Zr₂T cluster has two shorter Ti–Zr bonds and a longer Zr–Zr bond after the Ti atom takes the place of the Zr atom in the Zr₃ cluster. All the tetramers are tetrahedrons. While the pure clusters are of higher symmetries (C_2v for Zr₄ and C_s for Ti₄), the mixed clusters possess the C₁ symmetry. As for the clusters with n + m = 5, the geometries of the most stable structures are all trigonal bipyramids. We also present the structures with energies slightly higher than the minimum-energy ones for comparing. We can see that for the same chemical compositions, the differences of the two most stable structures lie in the atomic arrangements rather than the geometries. Another interesting phenomenon is that to lower the energies of the systems, the Zr atoms tend to occupy the apexes of these trigonal bipyramids, while the Ti atoms are more likely to be located at the middle planes.

From the above, the Ti_mZr_n (n + m ≤ 5) clusters can be seen as deriving from the Zr atoms taking the places of the original Ti atoms in the titanium clusters or vice versa. This may imply that the two metal elements are compatible and substitutable in some cases.

Now we discuss the structural stability, which can usually be described by the binding energy per atom (E_b). The E_b is calculated according to

where E(Ti), E(Zr), and E(Ti_mZr_n) are the total energies of the Ti atom, the Zr atom, and the Ti_mZr_n cluster, respectively. Figure 2 shows the results. It can be seen that for both pure clusters and mixed clusters, the binding energy per atom increases as the cluster size increases, indicating that these clusters become more stable during the growth process. For the clusters of the same size, it is notable that the binding energy increases monotonically with the increase of Zr atoms, and the changes are almost in a linear fashion. The reason is likely to be that the geometric structures of these same-sized clusters are highly similar, and the outer electron configurations of Ti and Zr are 3d²4s² and 4d²5s², respectively, which are also comparable. The lowest and highest values of E_b appear at the pure Ti clusters and Zr clusters, respectively. Thus, doping Zr atoms in Ti clusters may help to stabilize the clusters.

The dissociation energy is another indicator to describe the stability of clusters, which is defined as follows:

The ΔE(Ti_mZr_n−Ti) and ΔE(Ti_mZr_n−Zr) are the energies needed to dissociate a titanium atom and a zirconium atom, respectively, from the Ti_mZr_n cluster. Figure 3 presents the variation of ΔE(Ti_mZr_n−Ti) with the change of the titanium atom number, and figure 4 presents the variation of ΔE(Ti_mZr_n–Zr) with the change of the zirconium atom number. Comparing the two graphs, we can see that for the same cluster, more energy is needed to separate a Zr atom than to separate a Ti atom, which means that Zr atoms take a more important role in the stability of the studied Ti_mZr_n clusters. Seeing Figs. 3 and 4 separately, it is found that as the number of Zr atoms increases, i.e., as the number of Ti atoms decreases, both ΔE(Ti_mZr_n−Ti) and ΔE(Ti_mZr_n−Zr) increase, indicating that the Ti_mZr_n cluster gets more stable when the Ti atoms are replaced by Zr atoms. The rule is exactly consistent with that of the averaged binding energies. For the clusters with different sizes, on the whole, the larger clusters (n + m = 4 and 5) present the higher dissociation energies towards both a Ti atom and a Zr atom when compared with the smaller clusters (n + m = 2 and 3), which indicates that the larger Ti_mZr_n clusters are characterized with the higher stability. In general, in larger clusters, interactions between atoms are stronger and more complicated, and thus the atoms bind with each other closely. As a result, the clusters are likely to possess higher stabilities.

3.2. Electronic properties

The energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), i.e., the HOMO–LUMO gap, reflects the ability of an electron to be excited from an occupied orbital to an unoccupied orbital. Generally speaking, a larger HOMO–LUMO gap is related to enhanced chemical stability. The variation of the HOMO–LUMO gaps of the clusters versus the number of zirconium atoms is plotted in Fig. 5. The HOMO–LUMO gaps for the studied clusters range from 0.211 eV to 0.713 eV, indicating that these clusters have a semiconductive feature. The gap changes markedly with the cluster size, and the clusters with n + m = 5 generally possess lower HOMO–LUMO gaps, which we consider results from their relatively low LUMO energies (see Table 2). In regard of the clusters of the same sizes, we can notice that the minima are always found at the pure zirconium cluster, and the clusters consisting of more Ti atoms are more likely to possess large gaps. Besides, for the bimetallic clusters, apart from Ti₂, the cluster of Ti₂Zr possesses a significantly high HOMO–LUMO gap, indicating that it has better chemical stability.

The ionization potential and electron affinity are another two significant parameters for the evaluation of chemical stability. In the present work, we have obtained the vertical ionization potentials (VIP) and the vertical electron affinities (VEA) of the Ti_mZr_n clusters, which are respectively calculated by

where E(Ti_mZr_n) represents the total energy of the neutral cluster, and and denote the total energies of the cationic and anionic species, respectively, with the same geometries of the corresponding neutral clusters. Generally speaking, when the VIP is larger, more energy should be required to remove an electron from the neutral structure without any structural relaxation, and a larger VEA value means that more energy is released when charging the neutral cluster with an extra electron. The VIP and VEA of the considered clusters are presented in Figs. 6 and 7, respectively. The dimers possess the markedly high VIP, while the tetramers are characterized with the lowest VIP. On the whole, when the numbers of atoms are the same, the clusters consisting of more Ti atoms tend to have the higher VIP. As for their behaviors of VEA, the minima can be found at the dimers. The clusters with n + m = 5 possess the largest VEA, implying that they are of lower chemical stability, which is consistent with the fact that their HOMO–LUMO gaps are small. In fact, it is believed that people can predict VEA from LUMO energy, and in this work, it can be verified from the low LUMO energies of these pentamers. The overall trend for each series is opposite to that of VIP, indicating that for the neutral Ti_mZr_n clusters, which are less likely to lose an electron (larger VIP), it is also more difficult to charge them with an electron (smaller VEA), thus being chemically stable. Apart from the dimers, one can see that Ti₂Zr is less active, which is consistent with the case of the HOMO–LUMO energy gap.

3.3. Chemisorption of H₂ on the Ti_mZr_n (n + m ≤ 5) clusters

One of the main motives of our current work is to investigate the chemical activity of the Ti_mZr_n (n + m ≤ 5) clusters towards molecular hydrogen. We performed an exhaustive search in order to find the minimum-energy structures for the hydrogenated clusters. A hydrogen molecule was introduced to each cluster from various directions and distances, and the stable Ti_mZr_n structures in both the lowest and the second lowest energies were used in this step. The lowest-energy structures after hydrogenation as well as their spin states and point groups are displayed in Fig. 8. We also present the chemisorption energies, the H–H distances, and the net charges derived from the natural population analysis (NPA) in Table 3. We find that in some cases the ground states of Ti_mZr_nH₂ are obtained from the metastable structures of the bare Ti_mZr_n clusters. The H–H distances vary from 2.182 Å to 3.770 Å, which are far longer than the equilibrium bond length of H₂ (0.752 Å). Thus, the molecular hydrogen should be considered as dissociatively chemisorbed onto the considered clusters. Slight structural relaxations of the metallic clusters upon H₂ chemisorption were observed. For n + m = 2, 3, 4, the dissociated H atoms prefer bridge sites of the Ti–Zr clusters, except for the Ti₄H₂ system, in which one H atom is located at the bridge site and the other is on the top of one face. Also, in each dimer-H₂ and trimer-H₂ system, the atoms are not located in the same plane. When it comes to n + m = 5, the H atoms are on the tops of the faces, except for Ti₃Zr₂, in which the H atoms are located at the bridge sites. The top sites are not practical in all the studied clusters as the hydrogen molecule is always pushed away from the clusters and there exist imaginary frequencies in the optimized systems. Thus, at least two metallic atoms in each cluster are involved in the interaction with H atoms.

The activity of H₂ on these clusters can be investigated by calculating the chemisorption energies for the systems according to

where E(H₂) and E(Ti_mZr_nH₂) denote the total energies of H₂ and the hydrogenated Ti_mZr_n cluster, respectively. The chemisorption energies range from 1.289 eV to 2.947 eV, and the variation is clearly shown in Fig. 9. We notice that the pure Zr clusters are more active towards H₂ when compared with the Ti counterparts, characterized by the higher chemisorption energies. Most of the clusters possess chemisorption energies between those of the same-sized pure titanium and zirconium clusters. The dissociative chemisorption of H₂ on a cluster is always accompanied with a charge transfer process. From Table 3, it is clear that the H atoms attract electrons from the host clusters, resulting in the formation of metal hydride. It is found that when the H atoms are located at the bridge sites of the host clusters, more electrons transfer to the two H atoms. For example, for n + m = 5, Ti₃Zr₂ has transfered more charges to the H atoms when compared with the other clusters. As the host clusters become positively charged after hydrogenations, we can compare the chemisorption energies with the vertical ionization potentials of the clusters. In general, except for Ti₃Zr, the trends for ΔE_CE and VIP are opposite; this is because it is harder for the clusters with larger VIP to loose electrons, so they are less likely to transfer electrons to the H atoms, thus showing lower activity towards H₂. This results in the lower chemisorption energies of the clusters with higher VIP.

In Fig. 9, another noteworthy phenomenon is that the maximum of ΔE_CE is found at the Ti₃Zr cluster. To better understand it, we present the HOMO and LUMO distributions of the Ti_mZr_nH₂ clusters with n + m = 4 in Fig. 10. For these clusters, the delocalization for the frontier orbitals is obvious. The s-d hybridization can be observed between H atom and Ti or Zr atom as well as between the TM atoms in the host clusters. For both orbitals, the states between two H atoms and between each H atom and the host cluster are most dense and delocalized for Ti₃ZrH₂, that is, the interaction is strongest within the Ti₃ZrH₂ cluster, which we consider is likely to be the reason behind the markedly large chemisorption energy for Ti₃Zr towards H₂. This also indicates that Ti₃Zr shows high activity with respect to hydrogen, thus promising to be applied as a catalyst in some hydrogenation and dehydrogenation processes.

We finally discuss the magnetic properties of the Ti_mZr_nH₂ clusters. To compare the hydrogenated clusters with the bare clusters, their spin magnetic moments are listed together in Table 4. In most cases, the total magnetic moments of these clusters have decreased after chemisorbing H₂, and there are four Ti_mZr_nH₂ clusters of which the magnetic moments have quenched, implying their electrons are all paired. It is interesting to find that for the pure titanium clusters, the magnetic moments remain the same after hydrogenation. This may be because H₂ has a weak influence on the pure Ti clusters, which can possibly be reflected by their relatively small ΔE_CE values.