# A Holistic View of the Interactions between Electron-Deficient Systems: Clustering of Beryllium and Magnesium Hydrides and Halides

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}and MgX

_{2}(X = H, F, Cl) have been studied in the gas phase using B3LYP and M06-2X DFT methods and the G4 ab initio composite procedure. To obtain some insight into their structure, stability, and bonding characteristics, we have used two different energy decomposition formalisms, namely MBIE and LMO-EDA, in parallel with the analysis of the electron density with the help of QTAIM, ELF, NCIPLOT, and AdNDP approaches. Some interesting differences are already observed in the dimers, where the stability sequence observed for the hydrides differs entirely from that of the fluorides and chlorides. Trimers also show some peculiarities associated with the presence of compact trigonal cyclic structures that compete in stability with the more conventional hexagonal and linear forms. As observed for dimers, the stability of the trimers changes significantly from hydrides to fluorides or chlorides. Although some of these clusters were previously explored in the literature, the novelty of this work is to provide a holistic approach to the entire series of compounds by using chemical bonding tools, allowing us to understand the stability trends in detail and providing insights for a significant number of new, unexplored structures.

## 1. Introduction

_{3}monomers share two hydrogen atoms. As a result of the formation of this quite singular bond, three centers share a pair of electrons as in any typical two-center covalent bond [1,2]. These bonding arrangements, usually named three-center two-electron (3c-2e) bonds, are also found in dialane, the corresponding aluminum hydride dimer. On top of these singular bonding patterns, electron-deficient compounds often present elusive aggregates. A typical example is the corresponding aluminum hydride, whose dimer was predicted to be stable by ab initio calculations [3,4] in the late 1980s but would not be characterized experimentally for the first time until the beginning of the XXI century [5]. In contrast to the stability of diborane, diborane halides, namely B

_{2}F

_{6}and B

_{2}Cl

_{6}, which were supposedly not stable in the gas phase [6], were very recently found to be weakly bound, as revealed by high-level ab initio calculations. The low stabilization enthalpies are due to the fact that interaction between monomers is mainly dispersion [7]. In the same paper, a systematic study of dimers and trimers involving BX

_{3}and AlX

_{3}(X = H, F, Cl) in the gas phase showed that besides dispersion, the rehybridization of the electron-deficient element and its ability to reach pentacoordination are key factors to understand their structure, stability, and bonding characteristics. These findings prompted us to explore the behavior of similar dimers and trimers involving electron-deficient elements from group IIA of the periodic table. Accordingly, in the present paper, we present a theoretical examination of dimers and trimers involving BeX

_{2}and MgX

_{2}(X = H, F, Cl) in the gas phase.

_{2}and its dimer have received a lot of attention. The first two studies we are aware of were on Be

_{2}H

_{4}and Be

_{3}H

_{6}[8] at the Hartree–Fock level, with an estimation of electron correlation effects, and on Be

_{2}H

_{4}and Mg

_{2}H

_{4}using non-empirical approaches, the latter one including also the mixed BeMgH

_{4}dimer [9]. A rather complete survey of the dimers, trimers, and tetramers of BeH

_{2}and MgH

_{2}using high-level ab initio calculations was published in 2005 [10]. More recently, an analysis of the properties of Be

_{2}H

_{4}and other beryllium hydride oligomers and their spectral characteristics have been reported [11], but the Be

_{3}H

_{6}trimer was not included in this survey [11]. In 2011, a study of the interactions of BeH

_{2}and Mg

_{2}H

_{4}with H

_{2}included information on the structure and energetics of these two clusters, revealing their ability for hydrogen storage purposes [12]. The Raman spectrum of Be

_{2}H

_{4}(D

_{2h}) was predicted [13] using a new code from variational configuration interaction theory to allow the calculation of such spectra in a pure ab initio fashion [14]. The magnesium hydride dimer was characterized by matrix infrared spectroscopy, which led to the conclusion that Mg

_{2}H

_{4}is a dibridged molecule analogous to dialane [15]. Some attention was also paid to polymers of BeH

_{2}[16,17]. The information is scarcer for the BeF

_{2}, MgF

_{2,}BeCl

_{2}, and MgCl

_{2}dimers. It can be reduced to Hartree–Fock calculations on Be

_{2}F

_{4}, Mg

_{2}F

_{4}, and BeF

_{2}MgF

_{2}clusters [18], calculations based on polarized models for MgF

_{2}and MgCl

_{2}[19], SCF calculations on the dimers of BeF

_{2}, BeCl

_{2}, MgF

_{2}, and MgCl

_{2}[20], and electron diffraction experiments on the MgCl

_{2}dimer [21]. For the specific case of the BeCl

_{2}dimer, gas-phase electron diffraction (GED) and mass spectrometric (MS) experiments allowed us to obtain its structure assuming a D

_{2h}symmetry, and its force constants and frequencies were also estimated [22]. Some complexes involving Be

_{2}Cl

_{4}and phosphorus-containing compounds have been very recently synthesized and structurally characterized, evidencing the Lewis acid behavior of beryllium in the dimer [23]. However, we are not aware of theoretical studies on trimers involving BeX

_{2}(X = H, F, Cl) derivatives, with the only exception of the BeH

_{2}trimer from [8,10]. The magnesium fluorides and chlorides have instead received most of the attention. To our knowledge, the first survey on the structure and energetics of (MgF

_{2})

_{n}clusters up to n = 24 was published in 1995 [24]. Years later, the structure of MgF

_{2}clusters, from dimers to decamers, was studied [25] using a stochastic optimization technique, namely a genetic algorithm (GA) in association with density functional theory. The structures of Mg

_{3}Cl

_{6}clusters, optimized at the MP2/6-311G* level, were reported in a paper that presents a systematic study of 168 isomers of (MgCl

_{2})

_{n}, where n = 1–4 [26]. It is worth noting that, with the exception of the BeMgH

_{4}[9] and BeMgF

_{4}[18] dimers, there is no information on mixed dimers or trimers involving BeX

_{2}and MgX

_{2}(X = H, F, Cl) compounds. The aim of this paper is to present a systematic theoretical study not only of the corresponding homodimers and homotrimers but also of the mixed dimers and trimers. Although, as indicated above, previous studies have been reported for some of these systems, most of them were focused exclusively on structures and energetics. In our study, we will focus our attention on the bonding characteristics of these clusters to precisely explain their structures and relative stabilities. We wonder what changes are observed in the bonding of the dimers when the two metals involved are not identical and how the nature of the substituent may affect bonding and relative stability trends. A similar scenario arises when dealing with trimers. What are the effects on bonding and stability when the dimer interacts with a monomer of the same nature as those forming the dimer? What occurs if the third component is a monomer different from those forming the dimer? How do these effects depend on the nature of the substituent? These are typical questions we will try to answer in this publication.

## 2. Results and Discussion

^{−1}, improving the agreement with the G4 outcomes. However, the effects of including dispersion on the geometry of the cluster are negligibly small, to the point that if the B3LYP + D3BJ geometry is used in the standard G4 formalism (that does not include dispersion corrections in the geometry optimization procedure), the changes in the final G4 energy are typically smaller than 0.4 kJ·mol

^{−1}. Hence, the conclusion is that the G4 formalism is reliable for investigating this type of complexes, in contrast to the results obtained when dealing with B and/or Al derivatives, where the dispersion effects in some of the complexes are a key factor for a proper description [7]. Additionally, in a few cases that will be commented on later, the G4 formalism predicts stationary points that, if obtained, including dispersion corrections, would collapse to the global minimum. In view of this, for the sake of simplicity, we decided to make our discussion using the M06-2X calculated values, knowing that the general conclusions are fully in line with the G4 values. All investigated clusters are closed-shell systems with stable wavefunctions.

#### 2.1. Dimers

^{−1}lower than the B3LYP ones.

_{total}are practically equal for the three dimers, the stabilizing two-center energy contributions, Δ

^{2}E(AB), are not, being the largest one that of the BeBeH

_{4}binary complex. In contrast, the monomer distortion E

_{R}in the BeBeH

_{4}cluster is almost twice that of MgMgH

_{4}. Indeed, the largest contribution to this term comes from the rehybridization undergone by the metal from sp to sp

^{n}(n ≈ 2), but this energy cost is much larger for BeH

_{2}than for MgH

_{2}(80 vs. 50 kJ·mol

^{−1}) [27]. Therefore, although the attractive two-center term is larger for Be

_{2}H

_{4}, the larger monomer deformation energy compensates for the difference. The rehybridization cost also explains the trends for the halides. In this case, the attractive two-center contributions are rather similar, reflecting the electrostatic character of the interaction, but again, the rehybridization cost is much higher for BeF

_{2}than for MgF

_{2}. Accordingly, the stabilization energy of the MgMgF

_{4}cluster is larger than that of the BeBeF

_{4}analog. Similar arguments explain the trends for the chlorides.

_{2}subunit practically does not change when going from the BeX

_{2}-BeX

_{2}dimer to the BeX

_{2}-MgX

_{2}one, and the same can be said as far as the MgX

_{2}subunit is concerned. The same conclusion is reached when looking at the characteristics and populations of the ELF basins, the components of the MBIE decomposition analysis, or the Wiberg bond indexes [28] (see Table S3 of the Supplementary Materials).

#### 2.2. Homotrimers

_{2}are shown in the first row of Figure 3. The most stable one corresponds to a linear aggregate, whereas the second is a cyclic structure labeled

**A**.

_{2}monomer to the Be

_{2}H

_{4}dimer, in the first case along the Be-Be axis and in the second case perpendicular to it. Accordingly, they present a very different bonding pattern, though the energy gap between them is rather small (9 kJ·mol

^{−1}). This is the result, as we will discuss below, of subtle differences between the different energy components. If the stabilization enthalpy of the dimer (see Figure 1) is compared with those of the two trimers (see Figure 3), it is evident that the trimerization is followed by some kind of cooperativity since both trimers’ stabilization energies are more than twice the stabilization energy of the dimer. As shown in Figure 3, the central Be atom is tetracoordinated, and according to both the AdNDP and ELF analyses, it is involved in two Be-H-Be bonds with each of the terminal Be atoms. Note, however, that the electron densities at the corresponding BCPs are greater than in the dimer, indicating stronger bonding interactions. Consistently, the ELF analysis finds trisynaptic basins in the trimer, similar to those in the dimer (see second row of Figure 3). These trisynaptic basins also have a population very close to 2 e but within a smaller volume (122 vs. 129 au

^{3}), whereas the volume of the disynaptic Be-H basins of both terminal groups remains unchanged. This contraction of the trisynaptic basins in the trimer is reflected in a shortening (0.03 Å) of the distance between the central Be atom and the terminal ones with respect to the dimer, reflecting a reinforcement of the interaction. This is also coherent with the MBIE partition energy shown in Table 2 compared to that in Table 1. The distortion energy of the terminal Be atoms are equal in the dimer and the trimer, whereas that of the central Be atom becomes about 40 kJ·mol

^{−1}greater as beryllium undergoes a change of hybridization from sp

^{2}to sp

^{3}to become tetracoordinated. Consistently, the two-center contributions in the trimer are almost identical to those in the dimer, but the additional three-center term leads to its enhanced stabilization.

**A**is rather different. In this case, the three Be bonds are tetracoordinated, and non-covalent H···H interactions between the negatively charged hydrogens are also detected. The presence of these interactions implies a certain increase in the dispersion contributions to the stabilization energy, which in the cyclic trimer are significantly larger (−198 kJ·mol

^{−1}) than in the linear cluster (−148 kJ·mol

^{−1}). NCIPLOT shows (see third row of Figure 3) that, due to these non-covalent interactions, there is a strongly attractive and quite homogeneous interstitial density between the metals and the hydrogens, in line with the concentration of BCPs found in this area when using AIM. It can also be observed that the dimer subunit is, in this case, significantly distorted, with curved Be-H-Be bond paths, whereas the distortion of the third BeH

_{2}subunit is very small, with an almost linear arrangement. This subunit also appears connected to the dimer through the hydrogen atoms of the former. The AdNDP description provides some interesting additional information, showing that the connectivity between 1Be (see numbering in Figure 3) and the dimer subunit takes place through 1Be-2H-4Be and 1Be-3H-7Be 3c-2e bonds and through 1Be-5H-7Be-8H and 1Be-4Be-5H-8H 4c-2e bonds (both kinds depicted in the second row of Figure 3). The ELF description is not strictly identical since all the basins are trisynaptic, though the ones involving 4Be-5(8)H-7Be are more compact (volume 97.5 au

^{3}) than those involving 1Be-2H-4Be and 1Be-3H-7Be (volume 141.4 au

^{3}). The MBIE decomposition analysis shows (see Table 2) that the two-center term between the Be atoms of the dimer subunit is smaller in absolute value than in the linear trimer (−202.1 vs. −262.0 kJ·mol

^{−1}), but the interaction of these two Be atoms with the third Be atom is more than double in cycle

**A**(−66.8 vs. −30.9 kJ·mol

^{−1}), reflecting the formation of the aforementioned 3c-2e bonds. Still, the overall attractive components in

**A**are 47 kJ·mol

^{−1}above the linear ones. Nevertheless, this difference reduces to only 9 kJ·mol

^{−1}in the stabilization energy due to the E

_{R}deformation energies. Indeed, the E

_{R}distortion values of the dimer subunit of cluster

**A**are higher than in the same subunits of the linear complex (87 vs. kJ·mol

^{−1}). In contrast, the E

_{R}value for the 1Be is much smaller (8 kJ·mol

^{−1}) than that of the central Be atom of the linear trimer (102 kJ·mol

^{−1}). Accordingly, the overall destabilization energies in the linear isomer are 38 kJ·mol

^{−1}greater than in the cyclic one, reducing the gap between their stabilization energies in this amount.

_{2}trimers, the scenario, as illustrated in Figure 4, is a little more complicated, with five (only four shown in the figure) low-energy conformers instead of two. Note that the linear structure is still the global minimum.

**A**. This structure is followed in stability by another cyclic structure,

**B**, that, as with the previous one, can be seen as the result of the interaction of a MgH

_{2}monomer with the MgH

_{2}-MgH

_{2}dimer. Cycles

**A**and

**B**are distinguished by the cis or trans arrangement of hydrogens 6 and 9 in the dimer, with the result that in

**B,**the 1Mg atom is only tricoordinated. At this point, it should be noted, as mentioned previously, that for the BeH

_{2}trimers, only

**A**was found to be stable, as any attempt to find

**B**led to the linear trimer. The third cyclic isomer in terms of stability is a planar hexagonal structure, though another non-planar conformer (not shown in Figure 4) was found to be also a local minimum, but 7 kJ·mol

^{−1}less stable. The bonding analysis discussed above for the BeH

_{2}trimers (linear and

**A)**can be extended to the MgH

_{2}ones, and for similar reasons, again, the linear tautomer is slightly more stable than the trimer

**A**. Specifically, the linear MgMgMgH

_{6}trimer is 20 kJ·mol

^{−1}less stable than its Be-containing analogue. For this structure, in line with what was discussed for the dimers, the Δ

^{2}E attractive interactions and the E

_{R}repulsive ones are smaller in MgMgMgH

_{6}than in the BeBeBeH

_{6}analog (see Table 2), resulting in a smaller stabilizing enthalpy in the former. A comparison of Figure 3 and Figure 4 shows that the structure of cluster

**A**for Mg is less compact than the homologous Be-containing isomer due to the longer interatomic distances (see Table S4 of the Supplementary Materials). Consistently, some of the non-covalent interactions are weaker. NCIPLOT shows that, due to its smaller compactness, the attractive homogeneous interstitial density between the Mg atoms and the hydrogens is less strongly attractive than in the Be analog. Going from cluster

**A**to

**B**, as expected, the overall attractive contributions (Δ

^{2}E and Δ

^{3}E) decrease by about 6 kJ·mol

^{−1}, whereas the E

_{R}terms increase by about the same amount, explaining why complex

**B**is only 12 kJ·mol

^{−1}less stable than conformer

**A**. The lower stability of the hexagonal cycle just reflects the decrease in the Δ

^{2}E attractive terms because no 3c-2e bonds are formed in this case, which is only partially compensated by an increase of the Δ

^{3}E contribution due to the hexagonal arrangement of this complex.

_{2}trimers, the linear arrangement is the global minimum, followed by the hexagonal cycle, only 5 kJ·mol

^{−1}less stable, and by cycle

**B,**which is 50 kJ·mol

^{−1}further less stable. In this case, cycle

**A**is a stationary point with one imaginary frequency. The same clusters are found when Be is replaced by Mg, but in this case, all of them are local minima of the potential energy surface. However, the energy trends are totally different, with cycle

**A**being the second most stable after the linear trimer. As in the dimers, the dominant electrostatic character in fluorides renders the linear trimer more stable than the homologous containing BeH

_{2}(−350.3 vs. −301.3 kJ·mol

^{−1}). Conversely, since this dominant electrostatic character prevents the formation of 3c-2e bonds, cycle

**A**is found to be significantly less stable than its homologous hydride (−214.6 vs. −277.1 kJ·mol

^{−1}). Cycle

**A**is indeed much less stable than the linear trimer due to its compactness, which results in a close vicinity of F atoms reflected by the repulsive terms in the LMO-EDA results of Table S5. The consequence is that for the fluorides,

**A**is a TS that leads to the linear cluster. The MBIE analysis (see Table 2) also shows that the hexagonal cycle is marginally less stable than the linear trimer. This is caused by the very small difference between the attractive (Δ

^{2}E + Δ

^{3}E) and the repulsive (E

_{R}) terms, which is slightly greater (4 kJ·mol

^{−1}) than in the former, so those species can be considered practically degenerate. The main difference when dealing with the Mg-containing systems is the larger size of Mg and the much longer bonds, which contribute to significantly stabilizing cycles

**A**and

**B**. These cycles are less compact, closing the gap with respect to the linear global minimum. It should be mentioned that although the MgF

_{2}trimers we found coincide with those in the literature [25], this is not the case as far as their stabilization energies are concerned since our G4 and M06-2X calculations both predict a different stability order (see Table S6 of the Supplementary Materials), likely due to the effect of dispersion contributions only included in our calculations.

_{2}homotrimers, the hexagonal trimer is planar, whereas in the corresponding chloride, it is not. On top of that, the fluoride is 5 kJ·mol

^{−1}less stable than the linear conformer, whereas for the chloride, this gap becomes ten times larger. Also, for Be trimers, cycle

**A**is found to be a TS and the least stable stationary point, showing once more the significative effect of the repulsion in these compact systems when the substituent is voluminous as Cl. Conversely, for Mg, where these interactions are much weaker due to the much larger interatomic distances (see Table S4 of the Supplementary Materials), cycle

**A**is not only a minimum but close in energy to the global minimum.

#### 2.3. Heterotrimers

_{2}Be-BeX

_{2}, its interaction can be exclusively with MgX

_{2}, leading to a unique BeX

_{2}BeX

_{2}MgX

_{2}arrangement. However, if the interaction involves the heterodimer, X

_{2}Be-MgX

_{2}, the interaction can take place with any of the two monomers. The interaction with BeX

_{2}will yield the same trimer as before if the attachment takes place on the BeX

_{2}or to a new BeX

_{2}MgX

_{2}BeX

_{2}conformer if this attachment takes place on the MgX

_{2}side. If the monomer involved is MgX

_{2}, two new clusters, MgX

_{2}BeX

_{2}MgX

_{2}and BeX

_{2}BeX

_{2}MgX

_{2}, would be produced. All these possibilities, with their corresponding stabilization enthalpies, are shown in Figure 6. As expected from our discussion on the dimers, fluorides are significantly more stable than hydrides, whereas chlorides are only slightly more stable or unstable than hydrides, depending on the case. A second conspicuous fact is that this stability depends on the nature of the central atom and that the stabilities observed for the hydrides reverse on going to fluorides and chlorides. Indeed, for the hydrides, the most stable linear clusters of each kind are those in which the central atom is Be, whereas in fluorides and chlorides, the most stable are systematically those in which the central atom is Mg.

_{R}is almost twice when the central atom is Be instead of Mg. This effect is more than compensated by the Δ

^{2}E terms, more negative in the first case, but mainly by the Δ

^{3}E term, more than three times larger in the former than in the latter thanks to stronger 3c-2e bonds in H

_{2}Be-BeH

_{2}than in BeH

_{2}-MgH

_{2}. When moving to the halides, the Δ

^{3}E contribution is marginal because, in these systems, no 3c-2e bonds are formed. It only remains, as a key factor, the much higher E

_{R}value when the central atom is Be, leading to less stable clusters than those where the central atom is Mg.

^{−1}is less stable than the corresponding linear trimers (see Figure S5 of the Supplementary Materials). More interesting are the cycles similar to the clusters

**A**and

**B**described in the homotrimers section. When dealing with heterotrimers, the different cycles that can be envisaged amount to five instead of two, as shown in Figure 7.

**C**and

**D**result from the interaction of a homodimer with a different monomer. They differ in the cis (

**C**) or trans (

**D**) arrangement of the terminal substituents of the homodimer. On the other hand, cycles

**E**,

**F,**and

**G**arise from the interaction between a heterodimer and a certain monomer. Again, if the substituents of the dimer are cis, cycle

**E**is formed. If the substituents are trans, there are two possibilities, cycles

**F**and

**G**. The important matter is that in some specific cases, these cycles become the global minimum of the potential energy surface. Let us start with the hydrides. The molecular graphs of these clusters are shown in Table 3, together with their stabilization enthalpies and their relative enthalpies with respect to the corresponding linear trimers in Figure 6.

**G**does not exist because it collapses to the linear global minimum. The second and most important is that some of them are rather stable, to the point that one within each family is predicted to be the global minimum of the potential energy surface (negative relative stabilities in red). This is a result of subtle differences, related again essentially to the distortion energy of the monomers and the Δ

^{3}E terms. In cycle

**E**(see Table S8 of the Supplementary Materials), the E

_{R}contributions are smaller than in the linear trimer because the BeH

_{2}moiety at the top of the cycle is almost linear. As expected, the three Δ

^{2}E in cycle

**E**are negative, whereas in the linear trimer, only those of the central unit with the other two are negative. The overall balance is still favorable to the linear trimer, but this is compensated by a larger Δ

^{3}E contribution, which renders cycle

**E**the most stable. For the case of the BeMgMgH

_{6}heterotrimers, the situation is slightly different. Now, the E

_{R}contributions are very similar in both cycle

**C**and the linear trimer (see Table S8 of the Supplementary Materials) because of a larger distortion of the Be derivative at the top in cycle

**C**. Since the Δ

^{2}E contributions are globally similar, even though in the linear trimer, as expected, only two are significantly large, the enhanced stability of cycle

**C**comes essentially from the Δ

^{3}E term. Hence, once more, the two key factors are the E

_{R}and the Δ

^{3}E components, and when the former does not contribute significantly, it is only the Δ

^{3}E term that is behind the stability differences.

**C**–

**G**(see Table S9 of the Supplementary Materials). Indeed, neither in the fluorides nor in the chlorides do the cyclic trimers compete in stability with the linear ones. This reflects the fact already mentioned in previous sections: the dominant electrostatic character of the interactions avoids the formation of 3c-2e bonds and, accordingly, since these cycles are rather compact structures, the repulsive interactions between the substituents increase significantly, and the Δ

^{3}E contributions become highly unstabilizing (see Table S10 of the Supplementary Materials).

## 3. Computational Details

_{2,}MgH

_{2}, MgF

_{2}, and MgCl

_{2}trimers, our most stable structures are the same as those previously reported in the literature [10,24,25,26].

^{−1}.

_{R}(i), which measures the energy associated with the monomer distortion when it is part of the trimer, is the difference between E

_{m}(i), the energy of the i-monomer in its equilibrium geometry, and E(i), the energy of the i-monomer within the geometry of the ABC complex. Δ

^{2}E(ij) and Δ

^{3}E(ABC) are the two- and three-body interaction energies computed at the corresponding geometries in the complex. For the second objective, we have employed the LMO-EDA [38] decomposition analysis based on the generalized Kohn–Sham (GKS) and localized molecular orbitals, which permits us to write the total interaction energy as the sum of electrostatic, exchange, repulsion, polarization, and dispersion contributions (Equation (5)).

_{int}= E

_{elec}+ E

_{exc}+ E

_{rep}+ E

_{pol}+ E

_{disp}

## 4. Conclusions

**A**can compete in stability with the linear structure. As the MBIE showed, in line with observations through topological tools, not only the strength of binding interactions must be taken into account but also the distortion involved to form the structures, together with the presence of non-covalent interactions in cycle

**A**that have to be properly included in the theoretical treatment. Instead, beryllium fluoride clusters prefer the linear conformation, avoiding the compactness of cyclic structures, whereas magnesium, with larger bonds, presents cyclic structures

**A**and

**B**closer in energy to the linear one.

_{6}and BeMgMgH

_{6}hydrides, the compact cycles, as

**C**and

**E**, become the global minima, respectively. However, halogen heterotrimers cannot compete in their cyclic forms with the linear ones in the absence of multicenter two-electron bonds and the predominance of three-body unstabilizing terms. As a final comment, we would like to point out that the study of the interactions between the BeCl

_{2}dimers with phosphorus-containing compounds reported in ref. [23] suggests that the clusters investigated here may exhibit a rather interesting reactivity, whose study could be a good benchmark to analyze it in terms of HOMO-LUMO interactions, as proposed in a very recent publication [47].

## Supplementary Materials

_{4}, MgMgX

_{4}and BeMgX

_{4}(X = H, F, Cl) dimers. Figure S3. Molecular graphs of the homo and heterodimers involving BeX

_{2}and MgX

_{2}(X = H, F, Cl) monomers, showing the electron density, its Laplacian and the energy density at the bond critical points (BCPs). Table S2. AdNPD orbital list for the BeBeX

_{4}, MgMgX

_{4}and BeMgX

_{4}dimers. Table S3. Wiberg bond indexes for Be

_{2}X

_{4}, Mg

_{2}X

_{4}, BeMgX

_{4}(X = H, F, Cl). Table S4. Interatomic distances in the cycles

**A**of Be

_{2}H

_{4}and Mg

_{2}H

_{4}. Table S5. LMO-EDA analysis for the BeBeBeF

_{6}trimers. Figure S4. Bond paths and stabilization enthalpies for the BeCl

_{2}and MgCl

_{2}homotrimers. Table S6. Relative stabilities for the MgF

_{2}trimers obtained by different theoretical approaches. Table S7. MBIE analysis of linear heterotrimer complexes formed by BeX

_{2}and MgX

_{2}(X = H, F, Cl). Figure S5. Bond paths for the stable hexagonal heterotrimers BeCl

_{2}and MgCl

_{2}homotrimers, showing their stabilization and relative enthalpies with respect to the corresponding linear heterotrimer in kJ·mol

^{-1}. Table S8. MBIE analysis of BeBeMgH

_{6}and BeMgMgH

_{6}heterotrimer complexes. Table S9. Molecular graphs of cycles

**C**-

**G**of BeX

_{2}BX

_{2}MgX

_{2}and BeX

_{2}MgX

_{2}Mgx

_{2}(X = F, Cl) and their stabilization enthalpies. Table S10. MBIE analysis of heterotrimer complexes for fluorides and chlorides.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References and Note

- Lipscomb, W.N. Advances in Theoretical Studies of Boron Hydrides and Carboranes. In Boron Hydride Chemistry; Muetterties, E.L., Ed.; Academic Press: Itaca, NY, USA, 1975; pp. 39–78. [Google Scholar]
- Mayer, I. Bond orders in 3-center bonds—An analytical investigation into the electronic-structure of diborane and the 3-center 4-electron bonds of hypervalent sulfur. J. Mol. Struct. Theochem
**1989**, 55, 43–52. [Google Scholar] [CrossRef] - Liang, C.X.; Davy, R.D.; Schaefer, H.F. Infrared-spectra of the unknown dialane (Al
_{2}H_{6}) and recently observed digallane (Ga_{2}H_{6}) molecules. Chem. Phys. Lett.**1989**, 159, 393–398. [Google Scholar] [CrossRef] - Lammertsma, K.; Leszczynski, J. Ab initio study on dialane(6) and digallane(6). J. Phys. Chem.
**1990**, 94, 2806–2809. [Google Scholar] [CrossRef] - Andrews, L.; Wang, X.F. The infrared spectrum of Al
_{2}H_{6}in solid hydrogen. Science**2003**, 299, 2049–2052. [Google Scholar] [CrossRef] - Nori-Shargh, D.; Yahyaei, H.; Mousavi, S.N.; Maasoomi, A.; Kayi, H. Natural bond orbital, nuclear magnetic resonance analysis and hybrid-density functional theory study of sigma-aromaticity in Al
_{2}F_{6}, Al_{2}Cl_{6}, Al_{2}Br_{6}and Al_{2}I_{6}. J. Mol. Model.**2013**, 19, 2549–2557. [Google Scholar] [CrossRef] [PubMed] - Mó, O.; Montero-Campillo, M.M.; Yáñez, M.; Alkorta, I.; Elguero, J. Dispersion, Rehybridization, and Pentacoordination: Keys to Understand Clustering of Boron and Aluminum Hydrides and Halides. J. Phys. Chem. A
**2023**, 127, 5860–5871. [Google Scholar] [CrossRef] - Ahlrichs, R. Ab initio Calculations on Small Hydrides Including Electron Correlation. Theoret. Chim. Acta
**1970**, 17, 348–361. [Google Scholar] [CrossRef] - Kirillov, Y.B.; Boldyrev, A.I.; Klimenko, N.M.; Charkin, O.P. An ab initio calculation of the structure and stability of the complex hydrides MgBeH- and Mg
_{2}H-. J. Struct. Chem.**1983**, 24, 134–136. [Google Scholar] [CrossRef] - Chen, Y.L.; Huang, C.H.; Hu, W.P. Theoretical study on the small clusters of LiH, NaH, BeH
_{2}, and MgH_{2}. J. Phys. Chem. A**2005**, 109, 9627–9636. [Google Scholar] [CrossRef] [PubMed] - Lingam, C.B.; Babu, K.R.; Tewari, S.P.; Vaitheeswaran, G. Quantum chemical studies on beryllium hydride oligomers. Comput. Theor. Chem.
**2011**, 963, 371–377. [Google Scholar] [CrossRef] - Alkorta, I.; Elguero, J.; Solimannejad, M.; Grabowski, S.J. Dihydrogen Bonding vs. Metal-σ Interaction in Complexes between H
_{2}and Metal Hydride. J. Phys. Chem. A**2011**, 115, 201–210. [Google Scholar] [CrossRef] - Erfort, S.; Tschope, M.; Rauhut, G. Efficient and automated quantum chemical calculation of rovibrational nonresonant Raman spectra. J. Chem. Phys.
**2022**, 156, 124102. [Google Scholar] [CrossRef] - Erfort, S.; Tschoepe, M.; Rauhut, G. Toward a fully automated calculation of rovibrational infrared intensities for semi-rigid polyatomic molecules. J. Chem. Phys.
**2020**, 152, 244104. [Google Scholar] [CrossRef] [PubMed] - Wang, X.F.; Andrews, L. Infrared spectra of magnesium hydride molecules, complexes, and solid magnesium dihydride. J. Phys. Chem. A
**2004**, 108, 11511–11520. [Google Scholar] [CrossRef] - Abdurahman, A. Ab initio studies of static dipole polarizability of the polymeric beryllium hydride chain. J. Phys. Chem. A
**2003**, 107, 11547–11552. [Google Scholar] [CrossRef] - Wang, X.F.; Andrews, L. One-dimensional BeH
_{2}polymers: Infrared spectra and theoretical calculations. Inorg. Chem.**2005**, 44, 610–614. [Google Scholar] [CrossRef] - Ramondo, F.; Bencivenni, L.; Spoliti, M. Ab initio study on the Be
_{2}F_{4}, Mg_{2}F_{4}dimers, on the mixed dimers BeMgF_{4}and LiNaF_{2}and on the Li_{2}BeF_{4}, LiBCl_{4}and LiAlCl_{4}ion-pairs. J. Mol. Struct. Theochem**1992**, 96, 171–184. [Google Scholar] [CrossRef] - Gigli, G. On the structure of the alkaline earth dihalides dimers. J. Chem. Phys.
**1990**, 93, 5224–5233. [Google Scholar] [CrossRef] - Ystenes, B.K. Quantum chemical studies of molecular difluorides and dichlorides of beryllium and magnesium. Spectrochim. Acta A Mol. Biomol. Spectrosc.
**1998**, 54, 855–868. [Google Scholar] [CrossRef] - Molnar, J.; Marsden, C.J.; Hargittai, M. Molecular-structures and force-fields of monomeric and dimeric magnesium dichloride from electron-diffraction and quantum-chemical calculations. J. Phys. Chem.
**1995**, 99, 9062–9071. [Google Scholar] [CrossRef] - Girichev, A.G.; Giricheva, N.I.; Vogt, N.; Girichev, G.V.; Vogt, J. Structural investigation of molecules in the vapour over beryllium dichloride using electron diffraction and mass spectrometric data. J. Mol. Struct.
**1996**, 384, 175–182. [Google Scholar] [CrossRef] - Buchner, M.R.; Spang, N.; Ivlev, S.I. Hydrolysis and oxidation products of phosphine adducts to beryllium chloride. Z. Naturforsch. B J. Chem. Sci.
**2022**, 77, 381–390. [Google Scholar] [CrossRef] - Eichkorn, K.; Schneider, U.; Ahlrichs, R. An ab-initio investigation of structure and energetics of clusters Mg
_{n}Cl_{2n}. J. Chem. Phys.**1995**, 102, 7557–7563. [Google Scholar] [CrossRef] - Neogi, S.G.; Chaudhury, P. Structure, electronic properties and vibrational spectra of (MgF
_{2})n clusters through a combination of genetic algorithm and DFT-based approach. Mol. Phys.**2015**, 113, 3729–3739. [Google Scholar] [CrossRef] - Luhtanen, T.N.P.; Linnolahti, M.; Laine, A.; Pakkanen, T.A. Structural characteristics of small magnesium dichloride clusters: A systematic theoretical study. J. Phys. Chem. B
**2004**, 108, 3989–3995. [Google Scholar] [CrossRef] - Values calculated at the M06-2X/aug-cc-pVTZ level of theory (this work).
- Wiberg, K.B. Application of pople-santry-segal cndo method to cyclopropylcarbinyl and cyclobutyl cation and to bicyclobutane. Tetrahedron
**1968**, 24, 1083–1088. [Google Scholar] [CrossRef] - Curtiss, L.A.; Redfern, P.C.; Raghavachari, K. Gaussian-4 theory. J. Chem. Phys.
**2007**, 126, 084108. [Google Scholar] [CrossRef] - Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the Damping Function in Dispersion Corrected Density Functional Theory. J. Comp. Chem.
**2011**, 32, 1456–1465. [Google Scholar] [CrossRef] - Grimme, S.; Hansen, A.; Brandenburg, J.G.; Bannwarth, C. Dispersion-Corrected Mean-Field Electronic Structure Methods. Chem. Rev.
**2016**, 116, 5105–5154. [Google Scholar] [CrossRef] - Walker, M.; Harvey, A.J.A.; Sen, A.; Dessent, C.E.H. Performance of M06, M06-2X, and M06-HF Density Functionals for Conformationally Flexible Anionic Clusters: M06 Functionals Perform Better than B3LYP for a Model System with Dispersion and Ionic Hydrogen-Bonding Interactions. J. Phys. Chem. A
**2013**, 117, 12590–12600. [Google Scholar] [CrossRef] - Castro-Alvarez, A.; Cameros, H.; Sanchez, D.; Vilarrasa, J. Importance of the Electron Correlation and Dispersion Corrections in Calculations Involving Enamines, Hemiaminals, and Aminals. Comparison of B3LYP, M06-2X, MP2, and CCSD Results with Experimental Data. J. Org. Chem.
**2015**, 80, 11977–11985. [Google Scholar] [CrossRef] [PubMed] - Lopez-Lopez, J.A.; Ayala, R. Assessment of the performance of commonly used DFT functionals vs. MP2 in the study of IL-Water, IL-Ethanol and IL-(H
_{2}O)(3) clusters. J. Mol. Liq.**2016**, 220, 970–982. [Google Scholar] [CrossRef] - Stortz, C.A.; Sarotti, A.M. Exhaustive exploration of the conformational landscape of mono- and disubstituted five-membered rings by DFT and MP2 calculations. RSC Adv.
**2019**, 9, 24134–24145. [Google Scholar] [CrossRef] - Hankins, D.; Moskowitz, J.W.; Stillinger, F.H. Water molecule interactions. J. Chem. Phys.
**1970**, 53, 4544–4554. [Google Scholar] [CrossRef] - Xantheas, S.S. Ab-initio studies of cyclic water clusters (H
_{2}O)_{n}, n=1-6. 2. Analysis of many-body interactions. J. Chem. Phys.**1994**, 100, 7523–7534. [Google Scholar] [CrossRef] - Su, P.F.; Jiang, Z.; Chen, Z.C.; Wu, W. Energy Decomposition Scheme Based on the Generalized Kohn-Sham Scheme. J. Phys. Chem. A
**2014**, 118, 2531–2542. [Google Scholar] [CrossRef] - Schmidt, M.W.; Baldridge, K.K.; Boatz, J.A.; Elbert, S.T.; Gordon, M.S.; Jensen, J.H.; Koseki, S.; Matsunaga, N.; Nguyen, K.A.; Su, S.J.; et al. General atomic and molecular electronic-structure system. J. Comp. Chem.
**1993**, 14, 1347–1363. [Google Scholar] [CrossRef] - Bader, R.F.W. Atoms in Molecules. A Quantum Theory; Clarendon Press: Oxford, UK, 1990; pp. 1–456. [Google Scholar]
- Keith, T.A. AIMAll, version 19.10.12; TK Gristmill Software: Overland Parks, KS, USA, 2019. Available online: aim.tkgristmill.com (accessed on 1 January 2023).
- Reed, A.E.; Weinhold, F. Natural localized molecular-orbitals. J. Chem. Phys.
**1985**, 83, 1736–1740. [Google Scholar] [CrossRef] - Glendening, E.D.; Badenhoop, J.K.; Reed, A.E.; Carpenter, J.E.; Bohmann, J.A.; Morales, C.M.; Weinhold, F. NBO 5.G; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, USA, 2004. [Google Scholar]
- Tkachenko, N.V.; Boldyrev, A.I. Chemical bonding analysis of excited states using the adaptive natural density partitioning method. Phys. Chem. Chem. Phys.
**2019**, 21, 9590–9596. [Google Scholar] [CrossRef] [PubMed] - Boto, R.A.; Peccati, F.; Laplaza, R.; Quan, C.; Carbone, A.; Piquemal, J.P.; Maday, Y.; Contreras-García, J. NCIPLOT4: Fast, robust and quantitative analysis of non-covalent interactions. J. Chem. Theory Comp.
**2020**, 16, 4150–4158. [Google Scholar] [CrossRef] [PubMed] - Savin, A.; Nesper, R.; Wengert, S.; Fassler, T.F. ELF: The electron localization function. Angew. Chem. Int. Edit.
**1997**, 36, 1809–1832. [Google Scholar] [CrossRef] - Santos, L.D.; Ramalho, T.C.; Hamlin, T.A.; Bickelhaupt, F.M. Intermolecular Covalent Interactions: Nature and Directionality. Chem. Eur. J.
**2023**, 29, e202203791. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Molecular graphs of the homo and heterodimers involving BeX

_{2}and MgX

_{2}(X = H, F, Cl) monomers. The electron densities at the bond critical points (BCPs) are in a.u. The numbers in magenta are the stabilization enthalpies in kJ·mol

^{−1}. Atomic colors code: Be (yellow), Mg (orange), H (white), F (blue), Cl (green).

**Figure 2.**ELF (0.8) disynaptic and trisynaptic basins for the hydride, fluoride, and chloride homodimers and heterodimers. Basins involving H atoms are colored in yellow, and core basins involving Mg appear in dark blue. For the halides, disynaptic basins between the metal atom and fluorine atom appear in green, while lone pairs belonging to halogen atoms are colored in red. Populations are shown in atomic units (e).

**Figure 3.**Molecular graphs and ELF plots of the homotrimers involving BeH

_{2}. The electron densities at the bond critical points (BCPs) are in a.u. The numbers in magenta are the stabilization enthalpies in kJ·mol

^{−1}. The second row shows the ELF (0.8) trisynaptic basins and their populations, as well as the 3c-2e and 4c-2e orbitals involved in the bonding of cycle

**A**obtained by means of the AdNPD approach. In the third row, the 3D representation of non-covalent interactions obtained with NCIPLOT (s = 0.3); color code: red (strongly repulsive), green (weakly attractive and weakly repulsive), blue (strongly attractive). A less stable hexagonal trimer (not shown in the figure) is also a local minimum of the potential energy surface.

**Figure 4.**Molecular graphs of the homotrimers involving MgH

_{2}. The electron densities at the bond critical points (BCPs) are in a.u. The numbers in magenta are the stabilization enthalpies in kJ·mol

^{−1}. For cluster

**A,**the 3D representation of non-covalent interactions obtained with NCIPLOT (s = 0.3); color code: red (strongly repulsive), green (weakly attractive and weakly repulsive), blue (strongly attractive), is included.

**Figure 5.**Molecular graphs of the homotrimers involving BeF

_{2}and MgF

_{2}. The numbers in magenta are the stabilization enthalpies in kJ·mol

^{−1}.

**Figure 6.**Molecular graphs for the linear heterotrimers, showing their stabilization enthalpies in kJ·mol

^{−1}.

**Figure 7.**Different kinds of cyclic heterotrimers formed by the interaction of a dimer with a monomer that approaches the former in a direction perpendicular to the dimer axis.

**Table 1.**MBIE analysis of the binary complexes formed by BeX

_{2}and MgX

_{2}(X = H, F, Cl). All values in kJ·mol

^{−1}.

Binary Complex | E_{R}(A)
| E_{R}(B)
| Δ^{2}E(AB) | E_{total} |
---|---|---|---|---|

Hydrides | ||||

BeBeH_{4} | 58.7 | 58.7 | −261.4 | −144.1 |

MgMgH_{4} | 36.3 | 36.3 | −216.6 | −143.9 |

BeMgH_{4} | 66.3 | 31.6 | −243.2 | −45.2 |

Fluorides | ||||

BeBeF_{4} | 84.1 | 84.1 | −340.8 | −172.6 |

MgMgF_{4} | 42.3 | 42.3 | −353.1 | −268.5 |

BeMgF_{4} | 85.8 | 42.6 | −357.5 | −229.1 |

Chlorides | ||||

BeBeCl_{4} | 80.5 | 80.5 | −269.8 | −108.9 |

MgMgCl_{4} | 44.2 | 44.2 | −273.6 | −185.3 |

BeMgCl_{4} | 88.0 | 40.0 | −274.8 | −146.8 |

**Table 2.**MBIE analysis of the homotrimers formed by BeX

_{2}and MgX

_{2}(X = H, F). All values in kJ·mol

^{−1}.

Ternary Complex | E_{R}(A)
| E_{R}(B)
| E_{R}(C)
| Δ^{2}E(AB) | Δ^{2}E(AC) | Δ^{2}E(BC) | Δ^{3}E(ABC) | E_{total} |
---|---|---|---|---|---|---|---|---|

BeBeBeH_{6} (linear) | 59.2 | 102.6 | 59.2 | −262.0 | 7.8 | −262.0 | −30.9 | −326.2 |

BeBeBeH_{6} (cyclic, A) | 87.4 | 8.0 | 86.7 | −202.1 | −115.3 | −116.0 | −66.8 | −318.1 |

BeBeBeH_{6} (hexagonal) | 82.9 | 82.9 | 82.9 | −140.1 | −140.1 | −140.1 | −57.0 | −228.7 |

MgMgMgH_{6} (linear) | 37.2 | 66.8 | 37.2 | −215.2 | 0.3 | −215.2 | −10.4 | −299.3 |

MgMgMgH_{6} (cyclic, A) | 58.4 | 4.2 | 58.2 | −185.5 | −99.6 | −99.9 | −30.6 | −294.8 |

MgMgMgH_{6} (cyclic, B) | 28.1 | 44.5 | 54.7 | −124.6 | −54.0 | −188.6 | −42.0 | −281.9 |

MgMgMgH_{6} (hexagonal) | 42.9 | 42.9 | 42.9 | −100.1 | −100.1 | −100.1 | −99. 8 | −271.4 |

MgMgMgH_{6} (hexagonal,non-planar) | 51.6 | 46.7 | 28.5 | −177.8 | −77.2 | −67.8 | −69.3 | −265.3 |

BeBeBeF_{6} (linear) | 84.0 | 167.3 | 84.0 | −345.2 | −2.1 | −345.2 | −2.0 | −359.3 |

BeBeBeF_{6} (hexagonal) | 95.1 | 95.1 | 95.1 | −175.1 | −175.1 | −175.1 | −114.4 | −354.4 |

BeBeBeF_{6} (cyclic, A) | 162.2 | 29.1 | 162.4 | −358.3 | −146.2 | −146.5 | 80.1 | −217.3 |

BeBeBeF_{6} (cyclic, B) | 78.2 | 124.8 | 181.4 | −139.7 | −168.9 | −338.5 | −40.3 | −303.0 |

MgMgMgF_{6} (linear) | 42.4 | 82.1 | 42.5 | −356.3 | 0.3 | −356.3 | 1.2 | −546.1 |

MgMgMgF_{6} (cyclic, A) | 92.3 | 16.0 | 92.4 | −351.9 | −214.4 | −214.4 | 48.2 | −531.6 |

MgMgMgF_{6} (cyclic, B) | 39.5 | 56.2 | 63.1 | −250.2 | −133.0 | −307.4 | 6.8 | −525.0 |

MgMgMgF_{6} (hexagonal) | 47.5 | 47.6 | 47.6 | −188.5 | −188.5 | −188.5 | −65.8 | −488.8 |

**Table 3.**Molecular graphs of cycles

**C**–

**G**of BeH

_{2}BH

_{2}MgH

_{2}and BeH

_{2}MgH

_{2}MgH

_{2}and their stabilization enthalpies (bold numbers). In blue, their relative stabilities with respect to the corresponding linear trimer are also provided. In red, the two cases in which the cycle is the global minimum. All values in kJ·mol

^{−1}.

BeBeMgH_{6} | BeMgMgH_{6} | |||
---|---|---|---|---|

Molecular graph | Stab. enth./ Rel. Stab. | Molecular graph | Stab. enth./ Rel. Stab. | |

Trimers | BeBeMg | BeMgMg | ||

C | −253.655.5 | −319.5−8.2 | ||

D | −299.89.3 | −248.462.9 | ||

E | −313.7−4.6 | −271.340.0 | ||

F | −221.787.4 | −208.1103.2 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mó, O.; Montero-Campillo, M.M.; Yáñez, M.; Alkorta, I.; Elguero, J.
A Holistic View of the Interactions between Electron-Deficient Systems: Clustering of Beryllium and Magnesium Hydrides and Halides. *Molecules* **2023**, *28*, 7507.
https://doi.org/10.3390/molecules28227507

**AMA Style**

Mó O, Montero-Campillo MM, Yáñez M, Alkorta I, Elguero J.
A Holistic View of the Interactions between Electron-Deficient Systems: Clustering of Beryllium and Magnesium Hydrides and Halides. *Molecules*. 2023; 28(22):7507.
https://doi.org/10.3390/molecules28227507

**Chicago/Turabian Style**

Mó, Otilia, M. Merced Montero-Campillo, Manuel Yáñez, Ibon Alkorta, and José Elguero.
2023. "A Holistic View of the Interactions between Electron-Deficient Systems: Clustering of Beryllium and Magnesium Hydrides and Halides" *Molecules* 28, no. 22: 7507.
https://doi.org/10.3390/molecules28227507