Double-Hybrid Density Functional Theory Investigation of MgScH_n and MgTiH_n Clusters (n ≤ 18)

Abstract

Transition metal-doped magnesium hydride solids are leading candidates as hydrogen storage materials. Here, a double-hybrid density functional theory method is used for the first time to explore the ground state geometries and electronic properties of small MgScH_n and MgTiH_n (n = 1–18) clusters. It is determined that hydrogen atoms aggregate to the metal core of the cluster up to a saturation limit of MgScH₁₃ and MgTiH₁₄ for each transition metal. Additional hydrogen atoms exist as weakly interacting dissociated H₂ molecules. These saturated clusters containing scandium and titanium contain a large hydrogen mass percent of 15.9% and 16.4%, respectively. A detailed discussion of cluster growth mechanisms, hydrogen dissociation pathways, the effect of each different transition metal, and the cluster stabilities is presented as determined at the DSDPBEP86/6-311++G(3df,3pd) level of theory.

1. Introduction

The world’s dependence for decades on fossil fuels as a primary energy source is leading to severe environmental strains, including air pollution and global warming, and major health issues in our current population [1,2]. In the search for an alternative energy source to help replace fossil fuels, hydrogen is a very promising candidate [3,4,5]. It is light, has a high energy density and, perhaps most important, is a clean energy source producing water as a byproduct when burnt in oxygen. However, H₂ is a gas at ambient temperatures and pressures, and using gases requires special equipment and additional safety and handling considerations. To elevate this, alternative ways of storing hydrogen more safely for use as an energy source are an active focus of discussion and research. Liquefying H₂ requires cryogenic temperatures, which has unique safety and handling concerns of its own. Physisorption of hydrogen on a substrate also often requires extreme cold temperatures, given the inert nature of H₂, and has a limit to the hydrogen coverage that can be accumulated. Hydrogen can be stored as a chemical hydride (e.g., methanol), but this often requires high temperatures and expensive catalysts to extract H₂ by sophisticated chemical reactions. However, storing hydrogen as a solid metal hydride is a viable option as a reversible hydrogen storage material [6,7]. While simple binary metal hydrides often have the form of MH or MH₂ ionic compounds, many other ratios between metal and hydrogen atoms exist for more complex hydrides and solid materials. Here, we define a hydride as any species containing hydrogen, and typically hydrogen carries a net negative charge in binary metal hydrides.

Of the various possible metal hydrides, MgH₂ shows promise, as magnesium has the advantages of being light, non-toxic, abundant, and inexpensive. The largest issue with using magnesium hydrides as hydrogen storage materials is the slow sorption kinetics of hydrogen in the solid [8,9]. This can be partially alleviated by doping the magnesium hydride solids with a transition metal. While much is still unknown about which transition metal, and at which dopant concentration, is most effective, investigations have shown that titanium and similar early transition metal-doped magnesium hydride materials show better sorption kinetics compared to other metal dopants [10,11]. In addition, Mg₇TiH_x solids are stable and have been prepared that release hydrogen at lower temperature requirements than both pure magnesium hydrides and titanium hydrides [12]. Likewise, mixed magnesium-scandium materials have demonstrated good hydrogen uptake and storage capacity [13]. Hence, these transition metals are two of the more promising candidates to improve the hydrogen sorption kinetics in magnesium hydride solid materials. Thus, research is actively being explored on scandium- and titanium-doped magnesium and magnesium hydride clusters for hydrogen storage [14,15,16,17,18,19,20,21].

In order to understand the detailed binding characteristics and hydrogen saturation limits of small cluster systems, we previously studied the structures and properties of various MgScH_n and MgTiH_n clusters using standard density functional theory (DFT) methods [22,23]. Clusters, such as these, are frequently used as models for larger bulk materials and real-world applications [24]. Advances in, and access to, high-performance computing (hpc) technologies now allow for both of these MgMH_n systems (M = Sc, Ti) to be investigated at a higher level of theory, allowing for a more accurate and detailed comparison of the two transition metal dopants and their role in improving hydrogen sorption kinetics. We report the results from these investigations here with the main goals to (1) investigate the precise cluster structures of MgScH_n and MgTiH_n clusters using a double-hybrid DFT method for the first time, (2) determine discrepancies and similarities in the assigned ground state structures at this new higher level of theory, and (3) re-analyze cluster stability, and other electronic properties, determining which sizes are the most stable for each dopant metal. To our knowledge, this is the first report using a double-hybrid DFT method to study transition metal–magnesium-hydrogen cluster systems, and this study additionally provides a novel benchmark study for these advanced computational chemistry methods for use in exploring transition metal-doped magnesium hydride materials.

2. Theoretical Methods

Initial cluster structures that were considered were taken from low-energy isomers reported previously, were created by hand, or were generated by using the unbiased global optimization ABCluster 2.1 program [22,23,25,26,27]. We have previously demonstrated the need to use a global optimization procedure, rather than solely relying on prior experiences, to locate the correct global minima structures of clusters [28]. All candidate isomers were then stringently optimized using the DSDPBEP86 double-hybrid DFT method with Grimme’s D3BJ dispersion corrections coupled with the large 6-311++G(3df,3pd) basis set for all atoms [29,30,31]. This method was chosen for several reasons. For example, this double-hybrid method has been shown to reproduce theoretical predictions nearing the coupled cluster level of theory, while taking the computational time similar to Møller–Plesset levels [29]. It has also accurately reproduced experimental results for a variety of systems [32,33], and logically expands upon the theoretical levels previously employed for these cluster systems (both in terms of method and basis sets). To illustrate the additional computational demand of the higher double-hybrid method employed here (i.e., DSDPBEP86/6-311++G(3df,3pd)), optimization and vibrational frequency calculations on small MgTiH clusters took on average over 11 times longer to complete than at the lower level B3PW91/6-311++G(d,p) level employed previously when both had otherwise identical input files. Although double-hybrid DFT methods have higher computational costs compared to standard and hybrid DFT techniques, they (and DSD-type double-hybrids in particular) approach the accuracy of wavefunction methods in terms of predicting energetics, vibrational frequencies, reactions, and other properties [29,34]. The Gaussian 16, revision C.01, program package was used for all calculations [35]. Vibrational frequencies were calculated to ensure true minima structures were located with no imaginary frequencies, and all reported energy values include zero-point energy corrections. Although various spin states were tested, all reported global minima adopt the lowest possible spin state (i.e., either with spin multiplicity 2s + 1 = 1 or 2). To further understand the cluster’s electronic properties, natural bond orbital analysis was performed using the NBO 7 program [36]. Calculations were performed on the Expanse hpc cluster housed at the San Diego Supercomputing Center (SDSC) through the NSF-ACCESS program [37,38]. The optimized structures and various properties were visualized using the GaussView 6 program [39]. The xyz coordinates for all ground state structures reported here are provided in the Supporting Information for this article.

3. Results and Discussion

3.1. Geometric Structure Determination and Growth Mechansim

3.1.1. MgMH_n (n = 1–6)—Formation of Mg(μ-H)₃M Subunits

The determined ground state structures for all MgMH_n clusters (M = Sc, Ti; n = 1–18) are shown in Figure 1. The triatomic MgScH and MgTiH species form small triangular clusters with the hydrogen atom bridging each transition metal–magnesium bond. These smallest-sized clusters are similar to those predicted previously at the B3PW91/6-311++G(d,p) level of theory [23]. For comparison and reference, the optimized Mg-H, Mg-Sc, and Sc-H distances in MgScH are predicted to be 2.458 Å, 3.205 Å, and 1.794 Å here at the DSDPBEP86/6-311++G(3df,3pd) level, whereas they are 2.423 Å, 3.165 Å, and 1.787 Å at the lower B3PW91/6-311++G(d,p) level. For the small analogous titanium species, the optimized Mg-H, Mg-Ti, and Ti-H distances are predicted to be 2.222 Å, 2.828 Å, and 1.841 Å here at the DSDPBEP86/6-311++G(3df,3pd) level, but were previously 2.130 Å, 2.772 Å, and 1.807 Å at the lower B3PW91/6-311++G(d,p) level of theory. Hence, the optimized distances with the double-hybrid DSDPBEP86 methods are slightly longer than those at the previously reported B3PW91 hybrid DFT level for MgMH₁ species. The Mg-M length using the larger double-hybrid method as the cluster size grows with additional hydrogen atoms is depicted in Figure 2. We note that while the Mg-M distance is larger at the DSDPBEP86 level for some smaller-sized MgMH_n clusters than the B3PW91 method (such as for MgMH₁), the Mg-M distance is almost exclusively shorter at the DSDPBEP86 level for larger clusters. That is, the Mg-M distance is, with few exceptions, shorter in MgMH_n clusters using the double-hybrid DFT method when n > 6 for both transition metals.

MgScH₂ and MgTiH₂ are both related structures with the two hydrogen atoms binding to the transition metal center as individual H atoms, which is a similar structure to that reported previously with the lower B3PW91 method [23]. However, for MgMH₃, different ground state structures are located for MgScH₃ and MgTiH₃. Hence, reported for the first time, this is the smallest size determined with a metal dependence on hydrogen binding to MgMH_n clusters. MgScH₃ contains two hydrogen atoms bound to Sc with one additional hydrogen atom bridging both scandium and magnesium (i.e., bridging the Sc-Mg bond). MgTiH₃ contains one hydrogen atom bound to Ti with two additional hydrogen atoms bridging Ti-Mg. Note that neither of these structures is the same as that found for MgTiH₃ at the B3PW91/6-311++G(d,p) level, which contained three bridge-bound hydrogen atoms. At the DSDPBEP86/6-311++G(3df,3pd) level, this MgTiH₃ isomer is found to lie 9.2 kJ/mol higher in energy than the lowest energy structure shown in Figure 1. Hence, this is also the smallest-sized cluster where a new ground state isomer is located at this higher double-hybrid level of theory, demonstrating the importance of higher-level theoretical predictions of atomic clusters.

MgMH₄ and MgMH₅ ground state structures are similar for both scandium and titanium transition metals, and are similar to the lowest energy geometries found previously for MgTiH_n [23]. For MgMH₅, we note that this is the smallest size with a hydrogen atom bound solely on magnesium. This is also the size where an H-Mg(μ-H)₃M unit first appears for both scandium and titanium. This unit will be prevalent and discussed further in the next section for larger clusters. We also note in Figure 2 that the Mg-M distance has decreased in MgMH₅ for both scandium- and titanium-containing clusters, and that the distance remains small for larger sizes. Hence, this Mg(μ-H)₃M unit (which is present in all larger sizes) aids in decreasing the Mg-M bond distance. MgMH₆ again shows a difference between the two transition metals. Both MgScH₆ and MgTiH₆ have three bridge-bound hydrogen atoms and form a Mg(μ-H)₃M subunit. However, MgScH₆ contains three additional hydrogen atoms bound on the Sc center (one as a hydrogen atom, and two hydrogen atoms as molecular H₂), whereas MgTiH₆ contains only two hydrogens on the Ti center (as individual hydrogen atoms) with the last hydrogen atom bound on Mg. While this is a uniquely different structure for MgScH₆ than reported previously with the lower B3PW91 method, it is the same structure found previously for MgTiH₆ [22,23].

3.1.2. MgMH_n (n = 7–12)—Continued Hydrogen Aggregation on Transition Metal Centers

The located ground state structures for MgScH₇ and MgTiH₇ both contain H-Mg(μ-H)₃M units, with three additional hydrogen atoms bound to the transition metal center. In both cases, the additional three hydrogens are coordinated as an H₂ molecule and a separate H atom to the transition metal center (Figure 1). This is similar to that predicted at the B3PW91 level of theory [22,23]. For MgMH₈, a difference is again noted between ground state structures with Sc and Ti transition metals. Both contain the H-Mg(μ-H)₃M unit, with four additional hydrogens bound to the transition metal atom. However, these four hydrogen atoms bind as two H₂ molecules for Sc, but as a single H₂ molecule and two separate H atoms for Ti, which is similar to that predicted previously at the B3PW91 level of theory [22,23]. Hence, the difference seems indeed to be due to the different transition metals, and is method independent (see additional details in the MgScH_n and MgTiH_n comparison section below).

For MgMH₉—MgMH₁₂, hydrogen atoms continue to aggregate on the Sc and Ti metal centers. For each MgMH_n with an odd number of hydrogen atoms, there is an H-Mg(μ-H)₃M unit with an additional single H atom bound to the transition metal and the remaining hydrogens bound as H₂ units on the transition metal center. For each MgMH_n with an even number of hydrogen atoms, there is an H-Mg(μ-H)₃M unit with all remaining hydrogens bound as H₂ units on the transition metal (i.e., there is not a lone H atom attached to the metal as there is for species containing an odd number of hydrogen atoms). The structures are similar to each other for both transition metals, are similar to the structures predicted at the B3PW91 level of theory [22,23], and do not need additional comment here.

3.1.3. MgMH_n (n = 13–18)—Hydrogen Saturation and Dissociation

The structures for MgScH₁₃ and MgTiH₁₃ continue to build upon the smaller clusters reported in the previous section, and have additional hydrogen atoms coordinating to each transition metal center. For each metal, the lowest energy isomer located at the DSDPBEP86/6-311++G(3df,3pd) method shown in Figure 1 is related to that reported previously at the B3PW91/6-311++G(d,p) level [22,23], with a few small differences described below. The harmonic H₂ stretching frequencies for diatomic hydrogen units in these clusters are provided in Table S2 of the Supporting Information, and show a weakening of the H₂ bond, confirming all of the molecular hydrogens are interacting in the cluster at this size.

For MgScH₁₄, our double-hybrid calculations predict a ground state isomer with a dissociated H₂ molecule. This structure can be seen as MgScH₁₂ with an additional dissociated H₂. The H-H bond lengths for the five H₂ units in MgScH₁₄ are predicted to be 0.7746 Å, 0.7724 Å, 0.7721 Å, 0.7599 Å, and 0.7433 Å, with the last belonging to what we refer to as the dissociated H₂. As our calculations predict the bond length in free H₂ to be 0.7417 Å, four of the five H₂ molecules in MgScH₁₄ show considerably more bond weakening due to the interaction in the cluster than the dissociated H₂. Table S2 in the Supporting Information also shows that four of the five H₂ stretching frequencies in MgScH₁₄ are at considerably lower frequencies than free H₂, whereas one H₂ frequency (belonging to the dissociated H₂) is only red-shifted by <35 cm⁻¹ from that predicted for free H₂. Hence, for species with scandium, MgScH₁₃ is saturated with hydrogen and dissociation begins at the next larger size (i.e., for MgScH₁₄). This is similar to our previous report using a lower level of theory [22]. However, a different report has also indicated that MgScH₁₄ contained all hydrogen atoms bound to Mg or Sc, and these two investigations conflicted on the saturation size where dissociation occurs in MgScH_n clusters [22,25]. We previously reported this species with all bound hydrogens to be 8.6 kJ/mol higher in energy at the B3PW91/6-311++G(d,p) level, and 4.0 kJ/mol higher in energy at the MP2/6-311++G(3df,3pd) level [22]. Here, at the DSDPBEP86/6-311++G(3df,3pd) level, this isomer without a dissociated H₂ is only 0.7 kJ/mol higher in energy than the ground state isomer reported in Figure 1. Regardless, this now demonstrates that even at this higher-level double-hybrid method, hydrogen is predicted to dissociate already for MgScH₁₄. However, the presence of isomers containing a dissociated hydrogen molecule, and an isomer with no dissociated H₂, lying close in energy, further indicates that this is a transitional size in the growth of MgScH_n clusters. It is interesting to note that we also find a low-energy isomer with H₂ dissociated, but coordinated to the H-Mg-Sc backbone, which lies only 0.6 kJ/mol higher in energy than the ground state in Figure 1 at this DSDPBEP86/6-311++G(3df,3pd) level of theory. This isomer has been reported as 1.8 kJ/mol and 0.5 kJ/mol higher in energy at the B3PW91/6-311++G(d,p) and MP2/6-311++G(3df,3pd) levels, respectively [22], and has been proposed as an isomer of particular interest in a possible route for H₂ dissociation in transition metal-doped magnesium hydride clusters and materials. That is, it has been suggested that H₂ dissociation occurs by coordinating to the anionic hydride in the H-Mg-M backbone of the cluster through species resembling this intermediate. The presence of this low-lying isomer here, with a large negatively charged hydride (−0.504 Mulliken charge, −0.528 APT charge, −0.696 NBO charge) further supports this possible dissociation pathway. Regardless, our main point here is that MgScH₁₄ contains a dissociated H₂ molecule.

The ground state isomer for MgTiH₁₄ at this level (Figure 1) contains all hydrogen atoms bound to the metal core, and hydrogen has not dissociated yet for the titanium analog at this size. The H-H bond distances are predicted to be 0.8553 Å, 0.8152 Å, 0.8152 Å, 0.7885 Å, and 0.7885 Å in MgTiH₁₄. These are all longer than the predicted 0.7417 Å distance in free H₂. Note also the large red shift in H₂ stretching frequencies (provided in Table S2 in the Supporting Information) for the diatomic hydrogen units in this cluster. Hence, all of the H₂ molecules show some interaction within the cluster, weakening the H-H bond, and we state that no H₂ is dissociated in the MgTiH₁₄ cluster. This is in agreement with the lower B3PW91/6-311++G(d,p) results [23], and confirms that the structures for MgMH₁₄ (M = Sc, Ti), and hence the saturation limit, are different depending on the transition metal dopant.

The ground state structures of MgMH₁₅ (M = Sc, Ti) are also shown in Figure 1. MgScH₁₅ builds upon MgScH₁₄ by adding a single H atom to the Sc center and still having a dissociated H₂ molecule along the cluster’s side. For MgTiH₁₅, the ground state structure possesses a single dissociated H₂ unit on an analogous MgTiH₁₃ structure. Here, the five H-H bond lengths in MgTiH₁₅ are predicted to be 0.7794 Å, 0.7790 Å, 0.7687 Å, 0.7683 Å, and 0.7427 Å. The last (smallest) H-H length, which is close to the 0.7417 Å distance predicted in free H₂, is the length associated with the hydrogen molecule we are referring to as dissociated. Hence, MgTiH₁₅ is the first size cluster with titanium to have a dissociated H₂ molecule, and MgTiH_n clusters are saturated with hydrogen at MgTiH₁₄. It is interesting to note that the saturation limit in these clusters is indeed transition metal dependent (i.e., MgScH₁₃ and MgTiH₁₄ are the saturation limits for MgScH_n and MgTiH_n clusters, respectively). It is also worth noting here that a prior investigation using the B3PW91 method predicted MgScH₁₅ to be the saturation limit [25], a claim we disputed using the same theoretical method [22]. This study, using a high DSDPBEP86 level of theory, further validates these saturation limits, and the structures reported here are similar to those determined previously at the B3PW91/6-311++G(d,p) level [22,23].

For MgMH₁₆-MgMH₁₈, the ground state structures are determined, which continue to resemble saturated MgScH_n (n = 12, 13) and MgTiH_n (n = 13, 14) with additional H₂ molecules, with small deviations denoted below for each size. These structures are, in general, similar to those reported previously at lower levels of theory [22,23]. However, it is interesting to note that the dissociated H₂ molecules are at different positions around the core cluster compared to previous predictions. This is not entirely unexpected, as the dissociated H₂ molecules are only weakly interacting, and we have already shown that the dissociated H₂ molecule is predicted to be at different positions with the MP2 method for MgScH₁₇ and MgScH₁₈ compared to that predicted at the B3PW91/6-311++G(d,p) level of theory (and the DSDPBEP86 method used here contains some MP2 character). Hence, it appears that the position of weakly bound dissociated H₂ may be more dependent on the method employed, and this study at the DSDPBEP86/6-311++G(3df,3pd) level provides a new higher-level reference for the position and interaction of the H₂ molecule while dissociating from scandium- and titanium-doped magnesium hydride materials. The predicted lowest energy structures in this size range are also depicted in Figure 1, and the coordinates are provided in the Supporting Information. For MgScH₁₈ where the three dissociated H₂ molecules coordinate to the anionic hydride along the Sc-Mg-H backbone, the hydride in the cluster is predicted to have a large negative charge (−0.513 Mulliken charge, −0.541 APT charge, −0.689 NBO charge). This again provides support for a possible route for H₂ dissociation from MgMH_n clusters, with each molecular hydrogen being stabilized by coordinating to anionic hydrides in the cluster. MgTiH₁₈ is a unique cluster in this section which merits additional comments here. For the ten hydrogen atoms bound to the Ti center in MgTiH₁₈, eight of them are bound as (four) H₂ molecules, and the remaining two are bound as individual atoms. This cluster is uniquely different compared to others in this size range in that other sizes have all hydrogens bound as H₂ molecules (for even n), or contain only one single H atom bound to the transition metal (for odd n). That is, MgTiH₁₈ is the only cluster in this size range to have multiple single H atoms bound to the transition metal. All attempts to find a MgTiH₁₈ isomer with five H₂ molecules bound to Ti ended with isomers lying at least 10.4 kJ/mol higher in energy than the ground state structure depicted in Figure 1. It is also interesting to note that in this larger size range, for all sizes of MgMH_n (n > 13), differences in the ground state structures are noted between species containing the Sc and Ti dopant atoms. Only for smaller sizes (i.e., n ≤ 13) are similar structures noted for most, but not all, sizes as described in more detail later in the section dedicated to comparing the two transition metals.

3.2. Cluster Stability, Energetics, and Electronic Properties

The relative stabilities of each cluster size can be analyzed in several ways. Here, we first calculate a binding energy (fragmentation energy) for each ground state species by Equation (1), which compares the energy of each cluster with the sum of the energy of the individual H, Mg, and M (M = Sc or Ti) atoms making up the cluster.

$${\mathrm{E}}_{\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{d}}\left(n\right)={\displaystyle \frac{{\mathrm{E}}_{\mathrm{M}\mathrm{g}}+{\mathrm{E}}_{\mathrm{M}}+{n\mathrm{E}}_{\mathrm{H}}-{\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_n}}{n+2}};\mathrm{M}=\mathrm{S}\mathrm{c},\mathrm{T}\mathrm{i}$$

(1)

Equation (1) is normalized for the increasing size of the cluster by dividing by the total number of atoms (i.e., the number of hydrogen atoms plus 2). It is standard to calculate the binding energy this way in terms of individual atoms and to normalize it by the total number of atoms [14,15,18,19,22,23,25,28,40,41], but we have also provided it in different ways (see further discussion later in this paragraph and in Table S1 of the Supporting Information). A plot of the binding energy calculated in Equation (1) as a function of cluster size for each metal is shown in Figure 3, and a few things merit comment. First, all binding energies are positive, indicating that each cluster is more stable than the individual atoms. Second, an oscillation is observed when there is an odd or even number of hydrogen atoms present in the cluster. We have noted previously that MgMH_n clusters with an odd number of hydrogens tend to have a larger binding energy and be more stable than their counterparts with an even number of hydrogen atoms [23]. Clusters with an odd number of hydrogens present have new M-H interactions (rather than M-(H₂) interactions) occurring, which helps explain this observation. For example, in looking at the structures in Figure 1 for the middle sizes presented in the insert in Figure 3 (i.e., MgMH_n; n = 9–14), we see that those structures for both transition metals that contain an odd number of hydrogen atoms contain a lone M-H bond. That is, aside from the H-Mg(μ-H)₃M unit present in all of these clusters, MgMH₉, MgMH₁₁, and MgMH₁₃ contain an additional two, three and four H₂ molecules, respectively, and a single H atom bound to the transition metal. Clusters with an even number of hydrogen atoms (i.e., MgMH₁₀, MgMH₁₂, and MgMH₁₄) contain an additional two, three and four H₂ molecules, respectively, without any additional H atoms bound to the transition metal center. It is this fundamental difference in the presence of M-H vs. M-(H₂) groups that gives rise to this odd-even oscillation. The numerical values of the binding energies shown in Figure 3 are provided in Table S1 in the Supporting Information. This table also lists the binding energies calculated relative to H₂ molecules (rather than H atoms), as well as the calculated binding energies relative to a mixture of the H atoms and H₂ molecules that compose each individual cluster.

A stabilization energy compares the energy of a given cluster size to related cluster sizes. We calculate a stabilization energy (E_stab1), as depicted in Equation (2), as twice the energy of the given cluster minus the energy of clusters containing one less and one more hydrogen atom.

$${\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}1}\left(n\right)={2\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_n}-{\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_{n-1}}-{\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_{n+1}};\mathrm{M}=\mathrm{S}\mathrm{c},\mathrm{T}\mathrm{i}$$

(2)

A negative E_stab1 indicates more stable clusters relative to their neighboring sizes. Given the odd/even oscillation observed in E_bind, a similar odd/even oscillation is expected for E_stab1. Hence, an E_stab2 is also calculated by Equation (3), which compares the energy of a cluster with those of similar clusters also containing an odd or even number of hydrogen atoms. That is, E_stab2 is calculated as twice the energy of a cluster minus the energy of clusters containing two fewer and two more hydrogen atoms.

$${\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}2}\left(n\right)={2\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_n}-{\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_{n-2}}-{\mathrm{E}}_{{\mathrm{M}\mathrm{g}\mathrm{M}\mathrm{H}}_{n+2}};\mathrm{M}=\mathrm{S}\mathrm{c},\mathrm{T}\mathrm{i}$$

(3)

E_stab1 and E_stab2 are plotted in Figure 4 for both the Sc and Ti metals in MgMH_n clusters. In looking at these plots, a larger negative number suggests a cluster size that is more stable than others for both E_stab1 and E_stab2. We note that smaller clusters often have larger negative stabilization energies. This has been noted previously and is due, in part, to the dramatic change in structure made sometimes by adding a single hydrogen in these small clusters. Perhaps more important are the observed clusters of enhanced stability once the H-Mg(μ-H)₃M unit is formed and remains (i.e., for MgMH_n; n ≥ 7 clusters), and the cluster size range approaching the saturation limit. In this region, the energies are more comparable, but local minima appear present for MgScH₁₂ and MgTiH₁₀ (for species containing an even number of hydrogen atoms), and for MgTiH₁₃ (for species containing an odd number of hydrogen atoms). This particular size region for E_stab2 is discussed in more detail for each transition metal in the next section, which compares both MgScH_n and MgTiH_n clusters.

While these clusters are clearly stable, we note again that a major importance of cluster studies is to provide detailed studies on smaller systems that may be extended to, and provide insight about, larger bulk materials. While the dissociated H₂ molecule(s) in larger clusters here are only slightly perturbed from free H₂ (for example, the harmonic frequencies listed in Table S2 in the Supporting Information are only slightly perturbed in the dissociated H₂ units), the bound clusters are predicted to be stable and likely could be formed in at least gas-phase experiments. The strength of the interaction of molecular hydrogen with the cluster, which has been discussed in terms of H₂ bond elongation and stretching frequency red shifts, is related to dehydrogenation enthalpy and desorption temperature properties. The binding energy (fragmentation energy) in terms of fragmenting H₂ molecules (rather than H atoms) is further analyzed in Table S1 in the Supporting Information.

The predicted partial charges (i.e., Mulliken and Natural charges) and Natural Electron Configurations (NECs) for each magnesium and transition metal atom in MgMH_n clusters are provided in

Table 1. Natural Electron Configuration (NEC), Mulliken charges, and Natural charges for each metal atom in MgMH_n (M = Sc, Ti; n = 1–10) clusters.

H Atoms (n)	M	NEC	Mulliken Charge (Mg/M)	Natural Charge (Mg/M)
1	Sc	Mg:3s1.82 3p0.04 4s0.01 3d0.01	0.207/0.106	0.130/0.457
		Sc:4s1.70 3d0.71 4p0.10 5s0.01 4d0.02
	Ti	Mg:3s1.62 3p0.06 3d0.01	0.213/0.110	0.307/0.442
		Ti:4s1.13 3d2.33 4p0.08 4d0.02
2	Sc	Mg:3s1.81 3p0.09	0.064/0.701	0.094/1.150
		Sc:4s0.91 3d0.81 4p0.11 4d0.01
	Ti	Mg:3s1.68 3p0.05 3d0.01	0.213/0.493	0.263/0.918
		Ti:4s0.50 3d2.42 4p0.13 4d0.02 4f0.01
3	Sc	Mg:3s1.67 3p0.04 3d0.01	0.279/0.851	0.276/1.592
		Sc:4s0.47 3d0.82 4p0.09 4d0.02
	Ti	Mg:3s1.06 3p0.07 3d0.01	0.513/0.474	0.854/1.249
		Ti:4s0.36 3d2.35 4p0.01 4d0.03
4	Sc	Mg:3s1.03 3p0.07 3d0.01	0.527/0.975	0.893/1.691
		Sc:4s0.47 3d0.78 4p0.04 4d0.01
	Ti	Mg:3s1.31 3p0.03 4s0.01	0.559/0.684	0.639/1.094
		Ti:4s0.61 3d2.19 4p0.07 4d0.03 4f0.01
5	Sc	Mg:3s0.48 3p0.02	0.879/0.784	1.484/1.696
		Sc:4s0.37 3d0.88 4p0.03 4d0.02
	Ti	Mg:3s0.48 3p0.02	0.909/0.511	1.485/1.387
		Ti:4s0.38 3d2.16 4p0.03 4d0.03 4f0.01
6	Sc	Mg:3s1.00 3p0.05 3d0.01	0.490/0.770	0.927/1.460
		Sc:4s0.41 3d1.08 4p0.02 4d0.02
	Ti	Mg:3s0.46 3p0.02	0.731/0.605	1.502/1.095
		Ti:4s0.45 3d2.37 4p0.04 4d0.03 4f0.01
7	Sc	Mg:3s0.48 3p0.02	0.819/0.700	1.491/1.550
		Sc:4s0.37 3d1.03 4p0.02 4d0.02
	Ti	Mg:3s0.47 3p0.02	0.839/0.423	1.495/1.125
		Ti:4s0.38 3d2.42 4p0.02 4d0.05 4f0.01
8	Sc	Mg:3s0.47 3p0.02	0.728/0.663	1.499/1.067
		Sc:4s0.27 3d1.62 4p0.02 4d0.02
	Ti	Mg:3s0.45 3p0.02	0.830/0.337	1.514/0.685
		Ti:4s0.40 3d2.83 4p0.02 4d0.05 4f0.01
9	Sc	Mg:3s0.47 3p0.02	0.784/0.673	1.499/1.349
		Sc:4s0.36 3d1.24 4p0.02 4d0.02
	Ti	Mg:3s0.47 3p0.02 3d0.01	0.770/0.229	1.502/0.876
		Ti:4s0.40 3d2.64 4p0.02 4d0.06 4f0.01
10	Sc	Mg:3s0.46 3p0.02	0.758/0.452	1.507/0.839
		Sc:4s0.28 3d1.84 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.46 3p0.02	0.800/0.020	1.513/0.188
		Ti:4s0.36 3d3.38 4p0.01 4d0.04 4f0.01

(for n = 1–10) and

Table 2. Natural Electron Configuration (NEC), Mulliken charges, and Natural charges for each metal atom in MgMH_n (M = Sc, Ti; n = 11–18) clusters.

H Atoms (n)	M	NEC	Mulliken Charge (Mg/M)	Natural Charge (Mg/M)
11	Sc	Mg:3s0.46 3p0.02	0.739/0.603	1.507/1.124
		Sc:4s0.37 3d1.47 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.46 3p0.02 3d0.01	0.717/0.084	1.510/0.426
		Ti:4s0.38 3d3.11 4p0.01 4d0.06 4f0.01
12	Sc	Mg:3s0.46 3p0.02 3d0.01	0.616/0.501	1.510/0.652
		Sc:4s0.29 3d2.02 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.45 3p0.02	0.706/−0.131	1.510/−0.151
		Ti:4s0.36 3d3.72 4p0.01 4d0.04 4f0.01
13	Sc	Mg:3s0.45 3p0.02	0.727/0.331	1.516/0.815
		Sc:4s0.36 3d1.79 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.44 3p0.02 3d0.01	0.622/−0.212	1.524/−0.075
		Ti:4s0.37 3d3.61 4p0.01 4d0.08 4f0.01
14	Sc	Mg:3s0.46 3p0.02 3d0.01	0.582/0.499	1.510/0.650
		Sc:4s0.29 3d2.02 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.44 3p0.02 3d0.01	0.486/−0.298	1.530/−0.774
		Ti:4s0.35 3d4.35 4p0.01 4d0.04 4f0.02
15	Sc	Mg:3s0.45 3p0.02	0.681/0.366	1.516/0.813
		Sc:4s0.36 3d1.79 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.44 3p0.02 3d0.01	0.609/−0.190	1.525/−0.079
		Ti:4s0.37 3d3.61 4p0.01 4d0.08 4f0.01
16	Sc	Mg:3s0.46 3p0.02 3d0.01	0.527/0.528	1.511/0.648
		Sc:4s0.29 3d2.02 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.44 3p0.02 3d0.01	0.445/−0.336	1.531/−0.777
		Ti:4s0.35 3d4.36 4p0.01 4d0.04 4f0.02
17	Sc	Mg:3s0.44 3p0.02 3d0.01	0.563/0.390	1.523/0.484
		Sc:4s0.36 3d2.12 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.44 3p0.02 3d0.01	0.607/−0.253	1.525/−0.087
		Ti:4s0.36 3d3.62 4p0.01 4d0.07 4f0.01
18	Sc	Mg:3s0.45 3p0.02 3d0.01	0.597/0.232	1.513/0.410
		Sc:4s0.29 3d2.26 4p0.01 4d0.02 4f0.01
	Ti	Mg:3s0.43 3p0.02 3d0.01	0.632/−0.748	1.538/−0.737
		Ti:4s0.41 3d4.25 4p0.01 4d0.04 4f0.02

(for n = 11–18). In looking at the charges, we notice first that Natural charges are typically, with a few exceptions, predicted to be larger in magnitude than Mulliken charges for Mg, Sc, and Ti. We also notice that the sign of both the Natural and Mulliken partial charges for all three metals is the same for all sizes. Magnesium and scandium atoms are always predicted to be partially positively charged. Titanium is predicted to be partially positively charged in MgTiH_n clusters for n = 1–11, and then is predicted by both charge schemes to be partially negatively charged for n ≥ 12. This is uniquely different from a previous study at the lower B3PW91 level of theory where Ti was predicted to have a partial negative Mulliken charge in MgTiH₁₀ and MgTiH₁₁ [23]. We also note that Sc in MgScH_n (n = 12–20) has previously been predicted to have a negative Natural charge at a lower B3PW91 level of theory [25], whereas a more recent investigation contradicted this and suggested a small positive partial Natural charge on Sc in this size range [22]. This investigation at the higher double-hybrid DSDPBEP86 level further validates a prediction of a positive Natural charge on Sc in all cluster sizes.

In looking at the Natural Electron Configurations in Table 1 and Table 2, it is noted that the partial positive charges in Mg primarily arise from a loss of electron density in the 3s atomic orbital. The partial charges in Sc and Ti both come about largely from a loss of electron density in the 4s atomic orbital, and a gain of electron density in the 3d atomic orbital. It is interesting to note that the only exception to this for the species studied here is for small MgScH_n (n ≤ 5), where Sc loses electron density from both the 4s and 3d atomic orbitals. Comparing the NECs here to those predicted previously at the lower B3PW91 level, a main difference is in the 3d atomic orbital of the transition metal. While the metal is still predicted to accept electron density in the 3d atomic orbital at the DSDPBEP86 level, it is predicted to accept less electron density than at the lower B3PW91 level [22,23]. The full predicted NECs for Mg, Sc, and Ti for all sizes studied here using the DSDPBEP86 method are provided in Table 1 and Table 2. The highest occupied molecular orbitals (HOMOs) or singly occupied molecular orbitals (SOMOs) for MgMH_n (M = Sc, Ti; n = 13–15) at the transitional size region where dissociation begins to occur are provided as Figure S1 in the Supporting Information. They show that these frontier orbitals are comprised primarily of hydrogen s and transition metal d atomic orbitals. We also note that the frontier orbitals of MgMH₁₃ and MgMH₁₅ are quite similar, showing that the dissociated molecular hydrogen does not play a significant role in the MO. That is, MgMH₁₅, which adopts a similar structure to MgMH₁₃ with an additional dissociated H₂ as described above, has a similar HOMO to MgMH₁₃ (as well as a similar NEC as shown in Table 2).

3.3. MgScH_n and MgTiH_n Comparison

Comparing the geometries of MgScH_n and MgTiH_n (n = 1–18) in Figure 1, we first note similar geometries regardless of the transition metal for n = 1, 2, 4, 5, 7, 9, 10, 11, 12, 13, and 15. MgMH₃ is the first size that shows a transition metal dependence, with MgTiH₃ preferring a structure with two hydrogen atoms bridging Ti-Mg, and MgScH₃ preferring a structure with only one hydrogen atom bridging Sc-Mg. As both metals prefer a structure with no M-Mg bridging hydrogen atoms in MgMH₂, and both metals prefer a structure with two M-Mg bridging hydrogen atoms in MgMH₄, MgMH₃ is evidently a transitional size in the preferred hydrogen location in these clusters. MgMH₆ (M = Sc, Ti) each have a Mg(μ-H)₃M unit, with Sc preferring the three additional hydrogen atoms on Sc, whereas Ti prefers a structure with two hydrogen atoms on Ti, and the remaining hydrogen atom bound solely on magnesium. The MgMH₈ (M = Sc, Ti) clusters have nearly similar geometries with four hydrogen atoms bound solely on the transition metal. However, MgScH₈ adopts a structure where these four hydrogens are present as H₂ molecules, whereas MgTiH₈ adopts a structure with one H₂ molecule and two individual H atoms.

MgMH₁₄ (M = Sc, Ti) is the size with probably the most important structural difference. MgScH₁₄ prefers to have a dissociated H₂, whereas MgTiH₁₄ continues to have H₂ units bound to the metal centers. Hence, this difference sets the saturation limit for the small systems with different transition metals as MgScH₁₃ and MgTiH₁₄, which contain large 15.9% and 16.4% hydrogen by mass, respectively. Both MgMH₁₅ clusters (M = Sc, Ti) are similar with a dissociated H₂ molecule. MgMH_n (n = 16–18, M = Sc, Ti) all change slightly depending on the transition metal, but often differ only in the position of the weakly interacting dissociated H₂ species. These positions are also often different from those predicted previously at the lower B3PW91 method, as described in the geometric structure determination section. Figure 2 shows the Mg-M distance for both scandium and titanium metals as the cluster grows in the number of hydrogen atoms. We have already noted that the distance is larger for smaller-sized clusters and remains fairly constant in MgMH_n (n ≥ 5). In comparing the effect of the transition metal, we note that Mg-Sc distances are longer than Mg-Ti distances in MgMH_n for almost all clusters in the n = 1–18 size range. We also note that for larger MgMH_n species, where n ≥ 13, the odd-even oscillation in Mg-M distances is more pronounced for scandium than it is for titanium. This was also noted at the B3PW91 level, that the Mg-Ti distance changes little for MgTiH_n species where n ≥ 13 [23].

Here, we also discuss the differences in the cluster stability (i.e., binding and relative energy) containing each transition metal in more depth. Figure 5 shows an enlargement of E_stab2 (from Figure 4) for both metals in the MgMH₇–MgMH₁₆ size range. We notice that the most stable MgScH_n clusters predicted by E_stab2, seen as local minima in the plots in Figure 5, are MgScH₁₂ and MgScH₁₃ for clusters with an even and odd number of hydrogens, respectively. For MgTiH_n, MgTiH₁₃ and MgTiH₁₀ are the most stable clusters with an odd and even number of hydrogens with respect to E_stab2, respectively. Although it is clearer here in Figure 5, these sizes are also evident as the most stable species containing an odd and even number of hydrogen atoms in inspecting E_stab1 in Figure 4. It is interesting to note that while MgMH₁₃ is the most stable cluster containing an odd number of hydrogen atoms for both transition metals, the most stable cluster with an even number of hydrogens is different for the two transition metals (i.e., MgScH₁₂ and MgTiH₁₀). We note that these most stable sizes are also close to the saturation limits of MgScH₁₃ and MgTiH₁₄.

Lastly, a comment can be made comparing the partial charges of each transition metal in Table 1 and Table 2. With only a few exceptions, the partial charges for magnesium in MgMH_n are larger (more positive) when the transition metal in the cluster is titanium rather than scandium. In addition, scandium is more positively charged than titanium for all cluster sizes, and titanium is actually predicted for larger clusters to even be partially negatively charged.

4. Conclusions

A double-hybrid DFT approach is used for the first time to explore small MgMH_n (M = Sc and Ti, n = 1–18) clusters, which serve as models to explore the internal bonding and properties of transition metal-doped magnesium hydride clusters. At this higher level of theory than has been previously considered, several differences with regard to the effect and role of the transition metal are noted. It is found that the hydrogen saturation limit in these clusters occurs at MgScH₁₃ and MgTiH₁₄, which have large 15.9% and 16.4% hydrogen by mass, respectively. Additionally, the most stable clusters with an even and odd number of hydrogen atoms are found to be MgScH_x (x = 12 and 13) and MgTiH_y (y = 10 and 13). Overall, the distance between atoms is often larger at this higher level of theory for small clusters, but the Mg-M distance is almost exclusively smaller when utilizing this double-hybrid DFT method for larger clusters that contain more than six hydrogen atoms. For particular sizes, differences in the ground state structures are noted at this high DSDPBEP86/6-311++G(3df,3pd) double-hybrid level of theory.