Planar vs. three-dimensional X₆²⁻, X₂Y₄²⁻, and X₃Y₃²⁻ (X, Y = B, Al, Ga) metal clusters: an analysis of their relative energies through the turn-upside-down approach

Abstract

Despite the fact that B and Al belong to the same group 13 elements, the B₆²⁻ cluster prefers the planar D_2h geometry, whereas Al₆²⁻ favours the O_h structure. In this work, we analyse the origin of the relative stability of D_2h and O_h forms in these clusters by means of energy decomposition analysis based on the turn-upside-down approach. Our results show that what causes the different trends observed is the orbital interaction term, which combined with the electrostatic component do (Al₆²⁻ and Ga₆²⁻) or do not (B₆²⁻) compensate the higher Pauli repulsion of the O_h form. Analysing the orbital interaction term in more detail, we find that the preference of B₆²⁻ for the planar D_2h form has to be attributed to two particular molecular orbital interactions. Our results are in line with a dominant delocalisation force in Al clusters and the preference for more localised bonding in B metal clusters. For mixed clusters, we have found that those with more than two B atoms prefer the planar structure for the same reasons as for B₆²⁻.

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Introduction

The electronic distribution of nanosized molecular clusters can be very different from that of the bulk state.¹ In fact, metals can exhibit isolating behaviour when reduced to small particles. Since the electronic properties of nanoparticles are quite different from those of the bulk, molecular clusters are expected to have a variety of electronic applications, such as single-electron transistors, diodes, and quantum dots.^2–4 The properties of clusters are profoundly affected by the type of bonding they have. For some of these clusters one can expect an intermediate situation between covalent and metallic bonding. As modern technologies evolve towards the nanoscale, it becomes more important to have a more precise understanding of the bonding in these species to better tune their properties.

Among clusters, those made by group 13 atoms are particularly important.⁵ Both B and Al belong to the same group 13, and thus present a similar electronic structure, [He]2s²2p¹ and [Ne]3s²3p¹, respectively. However, when they form small clusters, B clusters adopt a planar conformation as the equilibrium structure;^6–9 whereas Al clusters present a three-dimensional (3D) closed shape.^10–13 The most relevant examples are B₆²⁻ and Al₆²⁻ clusters, which were obtained experimentally as lithium salts in the form of LiB₆⁻ and LiAl₆⁻.^14–16 B₆²⁻ adopts a planar D_2h geometry in its low-lying singlet state, whereas the Al₆²⁻ cluster is octahedral. Both shapes of the metal clusters are kept when lithium salts are formed.

The chemical bonding of B₆²⁻ and Al₆²⁻ has been widely analysed in previous studies.^14,17,18 In particular, Alexandrova et al.¹⁸ highlighted the fact that B₆²⁻ is able to 2s–2p hybridize and to form 2-center–2-electron (2c–2e) B–B covalent localised bonds. On the other hand, 3s–3p hybridisation in the Al₆²⁻ cluster is more difficult due to larger s–p energy separation, which hampers the formation of directional covalent Al–Al bonds.¹⁹ In this case, bonding comes from the combination of radial and tangential p-orbitals that result in extensive delocalisation.²⁰ Indeed, the Al₆²⁻ cluster displays octahedral aromaticity,^14,21 whereas planar D_2h B₆²⁻ is considered σ- and π-antiaromatic.^17,18,22,23 Thus, as pointed out by Alexandrova et al.,^18,24–26 covalent and delocalised bonding shows opposite effects in determining the molecular structure of many clusters. Huynh and Alexandrova analysed the whole series B_nAl_6−n²⁻ (n = 0–6), from B₆²⁻ till Al₆²⁻ by substituting one B by Al each time, concluding that covalent bonding is a resilient effect that governs the cluster shape more than delocalisation does. Indeed, the planar structure of B₆²⁻ persists until n = 5, the reason being the strong tendency to form 2c–2e B–B bonds in case the cluster contains two or more B atoms.¹⁸ Similar results were reported by Fowler and Ugalde in larger clusters of group 13. In particular, these authors found that B₁₃⁺ prefers a planar conformation²⁷ in contrast to Al₁₃⁻,²⁸ which adopts an icosahedral geometry. Interestingly, in closo boranes and substituted related species, like B₆H₆²⁻ or B₁₂I₁₂²⁻, the delocalised 3D structure is preferred. However, successive stripping of iodine in B₁₂I₁₂²⁻ leads to a B₁₂ planar structure with some localised 2c–2e B–B bonds.^29,30 Similarly, for B₆H_n⁻ clusters, the clusters are planar for n ≤ 3 and become tridimensional for n ≥ 4.³¹

As can be seen in Scheme 1, both 2D D_2h planar and 3D O_h geometries for X₆²⁻ (X = B, Al) can be obtained joining the same two X₃⁻ cluster fragments.^14,17 Therefore, X₆²⁻ species in D_2h and O_h geometries are particularly suitable for an energy decomposition analysis (EDA)^32–35 based on the turn-upside-down approach.^36–39 In this approach, two different isomers are formed from the same fragments and the bonding energy is decomposed into different physically meaningful components using an EDA. Differences in the energy components explain the reasons for the higher stability of the most stable isomer. For instance, using this method we provided an explanation of why the cubic isomer of T_d geometry is more stable than the ring structure with D_4h symmetry for (MX)₄ tetramers (X = H, F, Cl, Br, and I) if M is an alkalimetal and the other way round if M belongs to group 11 transition metals.³⁸ Therefore, the application of this type of analysis to B₆²⁻ and Al₆²⁻ clusters will disclose the factors that make the planar D_2h structure more stable for boron and the octahedral one for aluminium. As said before, boron clusters favour localised covalent bonds whereas aluminium clusters prefer a more delocalised bonding. With the present analysis, we aim to provide a more detailed picture of the reasons for the observed differences. The analysis will be first applied to the above referred B₆²⁻ and Al₆²⁻ clusters, and then further complemented with Ga₆²⁻. Finally, X₂Y₄²⁻ and X₃Y₃²⁻ (X, Y = B, Al, Ga) mixed clusters in their distorted D_2h planar and 3D D_4h geometries will also be discussed.

Scheme 1 D _2h and O_h structures of X₆²⁻ can be formed from C_2v X₃⁻ fragments.

Computational methods

All Density Functional Theory (DFT) calculations were performed using the Amsterdam Density Functional (ADF) program.⁴⁰ The molecular orbitals (MOs) were expanded in a large uncontracted set of Slater type orbitals (STOs) of triple-ζ quality for all atoms (TZ2P basis set). The 1s core electrons of boron, 1s–2p of aluminium, and 1s–3p of gallium were treated by the frozen core approximation. Energies and gradients were computed using the local density approximation (Slater exchange and VWN correlation) with non-local corrections for exchange (Becke88) and correlation (Lee–Yang–Parr 1988) included self-consistently (i.e. the BLYP functional). D3(BJ) dispersion corrections by Grimme were also included in the functional (i.e. BLYP-D3(BJ) functional).^41–44 Analytical Hessians were computed to confirm the nature of the located minima at the same level of theory.

Relative energies between the planar and 3D species were also calculated using the Gaussian 09 program⁴⁵ at the coupled cluster level⁴⁶ with single and double excitation (CCSD)⁴⁷ and with triple excitation treated perturbatively (CCSD(T))⁴⁸ using Dunning's correlation consistent augmented triple-ζ (aug-cc-pVTZ)^49,50 at optimised BLYP-D3(BJ)/TZ2P molecular geometries.

The bonding energy corresponding to the formation of X₆²⁻ for both D_2h and O_h symmetries from two anionic quintet tetraradicals, fragment 1 (αααα) + fragment 2 (ββββ) (see Scheme 1), is made up of two major components (eqn (1)):

ΔE = ΔE_dist + ΔE_int (1)

In this formula, the distortion energy ΔE_dist is the amount of energy required to deform the separated tetraradical fragments in their quintet state from their equilibrium structure to the geometry that they acquire in the metal cluster. The interaction energy ΔE_int corresponds to the actual energy change when the prepared fragments are combined to form the overall molecule. It is analysed in the framework of the Kohn–Sham MO model using a Morokuma-type decomposition^32–35 of the bonding energy into electrostatic interaction, exchange (or Pauli) repulsion, orbital interactions, and dispersion forces (eqn (2)).

ΔE_int = ΔV_elstat + ΔE_Pauli + ΔE_oi + ΔE_disp (2)

The term ΔV_elstat corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the prepared (i.e. deformed) fragments and is usually attractive. The Pauli repulsion ΔE_Pauli comprises the destabilizing interactions between occupied MOs. It arises as the energy change associated with going from the superposition of the unperturbed electron densities of the two fragments to the wavefunction Ψ⁰ = NA [Ψ^4α_fragment1·Ψ^4β_fragment2], which properly obeys the Pauli principle through explicit antisymmetrisation (A operator) and renormalisation (N constant) of the product of fragment wavefunctions. It comprises four-electron destabilizing interactions between occupied MOs and is responsible for steric repulsion. The orbital interaction ΔE_oi is the change in energy from Ψ⁰ to the final, fully converged wavefunction Ψ_SCF of the system. The orbital interactions account for charge transfer (i.e., donor–acceptor interactions between occupied orbitals on one fragment with unoccupied orbitals of the other, including the HOMO–LUMO interactions) and polarization (empty – occupied orbital mixing on one fragment due to the presence of another fragment). Finally, the ΔE_disp term takes into account the interactions which are due to dispersion forces.

In bond-energy decomposition,^51–53 open-shell fragments were treated with spin-unrestricted formalism but, for technical reasons, spin-polarisation was not included. This error causes the studied bond to become in the order of a few kcal mol⁻¹ too strong. To facilitate a straightforward comparison, the EDA results were scaled to match exactly the regular bond energies (the correction factor is consistently in the range 0.97–0.98 in all model systems and does therefore not affect trends). A similar scheme based on the same EDA approach was used by Frenking and coworkers^54,55 and by some of us^36,37,56 to estimate the strength of π-cyclic conjugation in typical (anti)aromatic organic compounds and in metallabenzenes and metallacyclopentadienes.

Let us mention here that, as already mentioned in the introduction, some of the analysed metal clusters exist experimentally as lithium salts.^14–16 On the other hand, these dianionic systems are unstable against the ejection of an electron. However, their molecular and electronic structure is very similar to that of their corresponding lithium salts, which justifies the analysis of the chemical bonding of these doubly charged systems, as it is not affected by the presence of a lithium cation.

Finally, the metalloaromaticity⁵⁷ of these clusters was evaluated at the BLYP/aug-cc-pVDZ level of theory with the optimized BLYP-D3(BJ)/TZ2P geometries by means of multicentre electron sharing indices (MCIs).^58–60 MCIs provide a measure of electron sharing among the atoms considered,⁵⁹ in the present case the six atoms that form each of the clusters studied. MCI values have been calculated using the ESI-3D program.^61,62

Results and discussion

We first focus on the homoatomic X₆²⁻ metal clusters with X = B, Al, and Ga. The optimized O_h and D_2h geometries at the BLYP-D3(BJ)/TZ2P level are depicted in Fig. 1 with the main bond lengths and angles. As expected, B–B bond lengths (1.536–1.768 Å) are much shorter than those for Al–Al (2.574–2.912 Å) and Ga–Ga (2.526–2.898 Å). The similar Al–Al and Ga–Ga distances in X₆²⁻ metal clusters (X = Al, Ga) are not unexpected given the similar van der Waals radii of these two elements.⁶³ In addition, the X–X bond length connecting the two equivalent X₃⁻ fragments in O_h clusters is longer than in the D_2h systems.

Fig. 1 Geometries of X₆²⁻ metal clusters analysed with D_2h and O_h symmetries. Distances in Å and angles in degrees.

Table 1 encloses the energy differences between O_h and D_2h clusters. For B₆²⁻D_2h symmetry is more stable than O_h by 67.5 kcal mol⁻¹, the latter not being a minimum.¹⁸ Meanwhile the opposite trend is obtained in the other two metal clusters, for which O_h is lower in energy by 15.8 (Al₆²⁻) and 9.3 kcal mol⁻¹ (Ga₆²⁻) than D_2h structures. These trends are confirmed by higher level CCSD(T)/aug-cc-pVTZ single point energy calculations at the same BLYP-D3(BJ)/TZ2P geometries (values also enclosed in Table 1). The relative energies of B₆²⁻, Al₆²⁻, and Ga₆²⁻ between O_h and D_2h symmetries are now −38.7, +44.8 and +46.6 kcal mol⁻¹, respectively. CCSD(T) values systematically favour O_h as compared to D_2h structures by about 20–30 kcal mol⁻¹. However, the qualitative picture remains the same.

Table 1 Relative energies of clusters between O_h and D_2h symmetries (in kcal mol⁻¹), and the aromatic MCI criterion

Clusters		BLYP-D3(BJ)/TZ2P^a		CCSD(T)/aug-cc-pVTZ^b		MCI^c
		O h	D 2h	O h	D 2h	O h	D 2h
a B2Ga4^2− (D2h) has not been obtained because optimization breaks the symmetry; whereas B3Al3^2− and B3Ga3^2− (Oh) have not been obtained because the strength of the B3 unit causes the systems to be planar and to avoid a 3D geometry. b Single point energy calculations at BLYP-D3(BJ)/TZ2P geometries. c MCI calculated at the BLYP/aug-cc-pVDZ level of theory with the BLYP-D3(BJ)/TZ2P optimized geometries. d Local minima. e One imaginary frequency. f One small imaginary frequency due to numerical integration problems. g Two imaginary frequencies. h Three imaginary frequencies.
X6^2−	B6^2−	67.5^e	0.0^d	38.7	0.0	0.062	−0.052
	Al6^2−	0.0^d	15.8^e	0.0	44.8	0.077	0.068
	Ga6^2−	0.0^f	9.3^f	0.0	46.6	0.083	0.071

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Clusters		BLYP-D3(BJ)/TZ2P^a		CCSD(T)/aug-cc-pVTZ^b		MCI^c
		D 4h	D 2h	D 4h	D 2h	D 4h	D 2h
X2Y4^2−	B2Al4^2−	0.0^g	15.9^g	0.0	34.0	0.032	0.001
	Al2B4^2−	66.9^h	0.0^g	48.7	0.0	0.032	0.023
	Al2Ga4^2−	0.0^d	13.0^h	0.0	43.3	0.077	0.068
	Ga2B4^2−	79.4^g	0.0^g	47.1	0.0	0.047	0.042
	Ga2Al4^2−	0.0^d	14.8^g	0.0	48.2	0.074	0.072

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Clusters		BLYP-D3(BJ)/TZ2P^a		CCSD(T)/aug-cc-pVTZ^b		MCI^c
		D 3h	C 3v	D 3h	C 3v	D 3h	C 3v
X3Y3^2−	Al3Ga3^2−	0.0^d	13.2^h	0.0	45.3	0.078	0.068

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The aromaticity of these X₆²⁻ metal clusters was evaluated by means of the MCI electronic criterion. The six-membered MCIs are enclosed in Table 1. In all cases, the O_h system is more aromatic than the D_2h one, in agreement with the larger electronic delocalisation of the former, as discussed in the Introduction.²¹ MCI values confirm the octahedral aromaticity²¹ of O_h Al₆²⁻ and the antiaromatic character of D_2h B₆²⁻.^17,18,22,23 Interestingly, MCI values point out the clear aromatic character of all 3D clusters that do not contain boron (MCI = 0.074–0.077); whereas mixed B₂Al₄²⁻, Al₂B₄²⁻, and Ga₂B₄²⁻D_4h clusters containing boron atoms are less aromatic (MCI = 0.032–0.047). For planar structures, there are basically two groups of clusters. First, the group formed by B₆²⁻ and B₂Al₄²⁻ has eight valence electrons distributed in two π-MOs and two σ-MOs (vide infra). Therefore, having four π-electrons and four σ-electrons, they are σ- and π-antiaromatic species. Second, the group formed by Al₆²⁻, Ga₆²⁻, Al₂B₄²⁻, Al₂Ga₄²⁻, Ga₂B₄²⁻, and Ga₂B₄²⁻ have eight valence electrons distributed in one π-MO and three σ-MOs (vide infra) and, therefore, they are σ- and π-aromatic species.

With the aim to obtain a deeper insight into the origin of 2D to 3D relative energies an energy decomposition analysis was performed, following the reaction presented in Scheme 1. As pointed out above, both systems can be constructed from two identical X₃⁻ anionic fragments, both in their quintet state in order to form the corresponding new bonds. Three of these bonds are of σ character, two tangential (σ^T) and one radial (σ^R), and one π character (see Fig. 2). It must be pointed out that, very recently, Mercero et al. have proven the multiconfigurational character of some of the lowest-lying electronic states of Al₃⁻.¹⁹ In the case of the quintet state of Al₃⁻, which is the fragment used in our calculations, the authors showed that the electronic configuration of the four valence electrons is also derived from the occupation of two σ-type tangential and one σ-type radial molecular orbitals arising from the 3p_x and 3p_y atomic orbitals, and one π-type orbital arising from the 3p_z ones. This quintet state was found to be dominated by one-single configuration with a coefficient of 0.92 in the multiconfigurational wavefunction.¹⁹ Moreover, the energy difference between the ground state and the quintet state was almost the same when computed at DFT or at the MCSCF levels of theory.¹⁹ This seems to indicate that DFT methods give reasonable results for this quintet state. Finally, the T₁ test⁶⁴ applied to clusters collected in Table 1 was found to be always less than 0.045, thus indicating the relatively low multiconfigurational character of these species. It is commonly accepted that CCSD(T) produces acceptable results for T₁ values as high as 0.055.⁶⁵

Fig. 2 Molecular orbital diagram corresponding to the formation of Al₆²⁻ in D_2h and O_h symmetries from two Al₃⁻ fragments in their quintet state. Energies of the molecular orbitals are enclosed (in eV), as well as the 〈SOMO|SOMO〉 overlaps of the fragments (values in italics). Energies of the fragments obtained from both D_2h (left) and O_h (right) symmetries are also enclosed.

The different terms of the EDA for B₆²⁻, Al₆²⁻, and Ga₆²⁻ clusters are enclosed in Table 2. First we notice that the total bonding energies (ΔE) are much larger for B₆²⁻ than for Al₆²⁻ or Ga₆²⁻. For the former, ΔE are −100.2 (O_h) and −179.5 kcal mol⁻¹ (D_2h), whereas for the two latter are in between −19.0 and −38.1 kcal mol⁻¹. This trend correlates with the shorter B–B bond lengths mentioned above. Table 2 also encloses the relative EDA energies between the two clusters. The B₃⁻ fragment taken from the B₆²⁻ system in its D_2h symmetry is the one that suffers the largest deformation, i.e. the largest change in geometry with respect to the fully relaxed B₃⁻ cluster in the quintet state (ΔE_dist = 12.5 kcal mol⁻¹), whereas the rest of the systems present small values of ΔE_dist (0.0–1.7 kcal mol⁻¹). However, differences in ΔE are not due to distortion energies (indeed ΔE_dist values follow the opposite trend as ΔE), but to interaction energies (ΔE_int).

Table 2 Energy decomposition analysis (EDA) of X₆²⁻ (X = B, Al, and Ga) metal clusters with D_2h and O_h symmetries (in kcal mol⁻¹), from two X₃⁻ fragments in their quintet state, computed at the BLYP-D3(BJ)/TZ2P level

	B6^2−			Al6^2−			Ga6^2−
	D 2h + D2h → D2h	O h + Oh → Oh	Δ(ΔE)	D2h + D2h → D2h	O h + Oh → Oh	Δ(ΔE)	D 2h + D2h → D2h	O h + Oh → Oh	Δ(ΔE)
ΔEint	−192.0	−101.4	−90.6	−20.7	−39.8	19.1	−19.1	−31.0	11.9
ΔEPauli	533.5	735.3	−201.8	225.7	348.0	−122.3	269.6	384.5	−114.9
ΔVelstat	−239.0	−291.9	52.9	−96.3	−166.5	70.2	−138.0	−207.5	69.5
ΔEoi	−483.4	−542.8	59.4	−146.9	−217.4	70.5	−146.7	−203.4	56.7
ΔEdisp	−3.2	−2.1	−1.1	−3.2	−3.9	0.7	−4.0	−4.7	0.6
ΔEdist	12.5	1.3	11.2	0.0	1.7	−1.7	0.1	1.4	−1.3
ΔE	−179.5	−100.2	−79.3	−20.7	−38.1	17.4	−19.0	−29.6	10.6

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Thus, we focus on the decomposition of ΔE_int into ΔE_Pauli, ΔV_elstat, ΔE_oi, and ΔE_disp terms. As a general trend, in all three X₆²⁻ clusters ΔE_Pauli is larger for the O_h than the D_2h cluster (Δ(ΔE_Pauli) = −201.8, −122.3, and −114.9 kcal mol⁻¹ for B₆²⁻, Al₆²⁻, and Ga₆²⁻, respectively), so making it less stable. The overlaps between doubly occupied MOs are larger in the more compact O_h structure that, consequently, has larger ΔE_Pauli. The larger difference in ΔE_Pauli between the O_h and D_2h structures in the case of B₆²⁻ as compared to Al₆²⁻ and Ga₆²⁻ is attributed to the particularly short B–B distances that increase the overlap between doubly occupied MOs of each B₃⁻ fragment. At the same time, the O_h form presents larger (more negative) electrostatic interactions (Δ(ΔV_elstat) = 52.9, 70.2, and 69.5 kcal mol⁻¹ for B₆²⁻, Al₆²⁻, and Ga₆²⁻, respectively). It is usually the case that higher destabilising Pauli repulsions go with larger stabilising electrostatic interactions. The reason has to be found in the fact that both interactions increase in the absolute value when electrons and nuclei are confined in a relatively small space. The electrostatic interaction together with orbital interaction (Δ(ΔE_oi) = 59.4, 70.5, and 56.7 kcal mol⁻¹ for B₆²⁻, Al₆²⁻, and Ga₆²⁻, respectively) terms favour the O_h structure. However, in the case of O_h B₆²⁻, Δ(ΔV_elstat) and Δ(ΔE_oi) cannot compensate Δ(ΔE_Pauli), which causes the D_2h system to be the lowest in energy. The opposite occurs for Al₆²⁻ and Ga₆²⁻. Finally, the dispersion term almost does not affect the relative energies, as the difference in dispersion is only in the order of ca. 1.0 kcal mol⁻¹. Therefore, what causes the different trend observed for B₆²⁻ on one side, and Al₆²⁻ and Ga₆²⁻ on the other side is basically the ΔE_oi term, which combined with the ΔV_elstat component does (Al₆²⁻ and Ga₆²⁻) or does not (B₆²⁻) compensate the higher ΔE_Pauli of the O_h form.

The comparison of the MOs diagrams of B₆²⁻ and Al₆²⁻, built from their X₃⁻ fragments, justify the trends of ΔE_oi (see Fig. 2 and 3). Both D_2h and O_h clusters are built from the same fragments; the only difference is that the two tangential frag^HOMO(σ^T(b₂)) and frag^HOMO−1(σ^T(a₁)) MOs of Al₃⁻ are degenerate when obtained from Al₆²⁻ in its O_h geometry, whereas they are not when generated from the D_2h system, although they still are very close in energy. As discussed from the EDA, O_h is more stable than D_2h because of more stabilizing electrostatic and orbital interactions, which compensate its larger Pauli repulsion. Fig. 2 also encloses the overlaps for the interactions between the four SOMOs of the Al₃⁻ fragments to form the MOs of the metal clusters in both geometries. We take the Al₃⁻ fragments in their quintet states with three unpaired σ- and one unpaired π-electrons, all of them with spin α in one fragment and β in the other. A more negative ΔE_oi in O_h Al₆²⁻ is justified from the larger 〈SOMO|SOMO〉 overlaps, especially for t_2gO^HOMOa_h and O^HOMOb_h (0.360 compared to 0.225 and 0.232 for b_2uD^HOMO_2h and a_gD^HOMO−1_2h, respectively). D_2h only presents a larger overlap for the π fragment orbital (0.251 for b_3uD^HOMO−2_2h and 0.124 for t_2gO^HOMOc_h). Meanwhile both of them have almost the same overlap for the combination of the radial MO (σ^R) fragment (frag^HOMO−2), with 〈SOMO|SOMO〉 = 0.298 and 0.301 for a_gD^HOMO−3_2h and O^HOMO−1_h, respectively. Overall, the higher orbital interaction term of the O_h system can be explained by the larger 〈SOMO|SOMO〉 overlaps of two of the t_2g delocalised molecular orbitals for this cluster (see Fig. 2). The energies of the occupied MOs of Al₆²⁻ formed are higher than those of the Al₃⁻ SOMOs because we move from a monoanionic fragment to a dianionic molecule.

Fig. 3 Molecular orbital diagram corresponding to the formation of B₆²⁻ in D_2h and O_h symmetries from two B₃⁻ fragments in their quintet states. Electrons in red refer to the formation of B₆²⁻ (D_2h) from B₃⁻ fragments in their triplet state. In the triplet state, π(b₁) is doubly occupied, σ^R(a1) and σ^T(b₂) remain singly occupied, and the σ^T(a₁) becomes unoccupied. Energies of the molecular orbitals are enclosed (in eV), as well as the 〈SOMO|SOMO〉 overlaps of the fragments (values in italics). Energies of the fragments obtained from both D_2h (left) and O_h (right) symmetries are also enclosed.

Now it is the turn to visualize the MOs of B₆²⁻. The fragments for B₃⁻ are the same as those for Al₃⁻ (see Fig. 3). However, the first difference appears in the MOs for B₆²⁻ with D_2h symmetry. In this case, it would be more reasonable to build the MOs of this molecule from two triplet (not quintet) B₃⁻ fragments. The reason is the different occupation of the MOs when compared to the D_2h Al₆²⁻ species. In D_2h B₆²⁻, the HOMO corresponds to the antibonding π MO. To reach doubly occupied bonding (b_3uD^HOMO−4_2h) and antibonding (b_2gD^HOMO_2h) π MOs, the π MO (frag^HOMO−3) should be doubly occupied. Furthermore, the tangential σ^T(a₁) frag^HOMO does not participate in any occupied MO of this metal cluster and only generates virtual MOs. Consequently, MOs of B₆²⁻ are better formed from two B₃⁻ fragments in their triplet state (see red electron in Fig. 3). On the other hand, B₆²⁻ with O_h follows the same trend as Al₆²⁻, and in this case the same SOMOs in their quintet state are involved. At this point, it is worth mentioning that, as pointed out by Mercero et al., due to the strong multiconfigurational character of this species, one must be cautious with the electronic configuration, especially for the triplet state, as radial and tangential MOs are very close in energy.¹⁹

To make results comparable, Table 2 gathers the EDA of O_h and D_2h B₆²⁻ from two B₃⁻ fragments in their quintet states. Also in this case ΔE_oi is more favourable for O_h than for D_2h, however, at a lower extent when compared to Al₆²⁻. There are two main reasons for such a decrease of the strength of ΔE_oi in O_h compared to D_2h. First, and more importantly, because the D^HOMO−2_2h formed presents a much larger 〈SOMO|SOMO〉 overlap than t_2gO^HOMO−1_h (0.518 in the former vs. 0.338 in the latter). In particular, this D^HOMO−2_2h MO contributes to the 2c–2e B–B localised bonds that are related to the larger covalent character of this structure. And second, because the π-interaction between the two π SOMO fragments is much larger in the case of D_2h (0.225 vs. 0.059 for D_2h and O_h, respectively). Nevertheless, these two more favourable orbital interactions are not enough to surpass the ΔE_oi term of the O_h cluster. However, as compared to Al₆²⁻, for B₆²⁻ the Δ(ΔE_oi) term favours the O_h system to a less extent and cannot compensate the higher ΔE_Pauli term of the O_h form, thus making the planar geometry to be more stable in this case. This is related to the determinant force of the formed covalent bonding, involving more localised MOs than for Al₆²⁻. Such a larger covalent component in B₆²⁻ is also supported by the covalent character of the interaction between the two fragments calculated as % covalency = (ΔE_oi/(ΔE_oi + ΔV_elstat + ΔE_disp)) × 100. This formula results in B₆²⁻: 65–67% (O_h, D_2h), Al₆²⁻: 56–60% (O_h, D_2h), and Ga₆²⁻: 49–51% (O_h, D_2h); thus confirming again the larger covalency found in B₆²⁻.

Finally, as done usually in the turn-upside-down approach,^{36–39,56,66,67} instead of building X₆²⁻ in O_h symmetry from the corresponding two X₃⁻ fragments obtained from the O_h structure, we can build the O_h system from two X₃⁻ fragments extracted from the X₆²⁻ cluster in D_2h symmetry, and viceversa (see Tables S2–S4 in the ESI‡). The main conclusions remain unaltered and confirm that the D_2h structures suffer a lower Pauli repulsion whereas those of O_h symmetry have more favourable electrostatic and orbital interactions. The interplay between the Pauli repulsion on the one hand and electrostatic and orbital interactions on the other determines the most favorable symmetry in each case.

Just to conclude this section, we must point out that the whole EDA and turn-upside-down analyses were performed with fragments in their quintet state. However, as we commented before this is not the most reasonable way to build B₆²⁻ in D_2h symmetry. Table S5 (ESI‡) contains the EDA for O_h and D_2h B₆²⁻ systems using B₃⁻ fragments in their triplet states. Results show that although the different terms are larger in the absolute value, the trends discussed above are not affected, and the D_2h cluster is favoured mainly because of smaller Pauli repulsions.

Mixed metal clusters

In this section, we analyse the X₂Y₄²⁻ clusters with X, Y = B, Al, Ga and X ≠ Y (see Fig. 4). The relative energies of the planar and 3D forms are also enclosed in Table 1. In all cases, the D_2h system is preferred when the cluster incorporates four B atoms; otherwise the 3D D_4h geometry is the lowest in energy. In particular, the D_2h symmetry is much more stable for Al₂B₄²⁻ and Ga₂B₄²⁻ by 66.9 and 79.4 kcal mol⁻¹, respectively. On the other hand, when B is not the predominant atom, the D_4h cluster is more stable by about 9–16 kcal mol⁻¹. As for the homoatomic metal clusters, at the CCSD(T) level, the same trend is obtained, although the D_4h system is stabilized with respect to the D_2h one by 20–30 kcal mol⁻¹. It is important to note that the D_4h and D_2h systems are not always the most stable for the X₂Y₄²⁻ clusters. For instance, for Al₂B₄²⁻, a C₂ geometry is the most stable form and, for B₂Al₄²⁻, a C_2v structure is the lowest in energy.¹⁸ However, we are not interested here in finding the most stable structure for each cluster but to discuss the reasons why in some cases 2D clusters are preferred over 3D and the other way round. Finally, Al₃Ga₃²⁻ also prefers an O_h geometry by 13.2 kcal mol⁻¹. Unfortunately, this latter relative energy cannot be compared to those of B₃Al₃²⁻ or B₃Ga₃²⁻ because the strength of the localised bonding between three B atoms prevents the optimization of their 3D structures. In this context, it is worth mentioning that Alexandrova and coworkers²⁶ found in X₃Y₃ (X = B, Al, Ga; Y = P, As) clusters that the lighter elements prefer 2D structures, whereas the heavier ones favour 3D geometries.

Fig. 4 Geometries of mixed metal clusters analysed with planar and 3D geometries. Distances in Å and angles in degrees.

The EDA was also performed for this series of six mixed metal clusters (see Table 3) with the aim to further understand the determinant force towards the most stable cluster. For the X₂Y₄²⁻ clusters, the EDA was carried out taken YXY⁻ fragments in their quintet states. For Al₃Ga₃²⁻, the fragments were Al₃⁻ and Ga₃⁻ in the quintet state too. For those systems for which the out-of-plane geometry is the most stable, the combination of more favourable electrostatic and orbital interactions, even though presenting larger Pauli repulsion, gives the explanation to the trend observed. This is the same behaviour already discussed above for both Al₆²⁻ and Ga₆²⁻ systems. On the other hand, when D_2h symmetry is the cluster lower in energy, as for Al₂B₄²⁻ and Ga₂B₄²⁻ metal clusters, even though the D_4h system presents more stable electrostatic interaction, now the orbital interactions in combination with less unfavourable Pauli repulsion favour the D_2h symmetry. This latter behaviour differs from that of B₆²⁻, for which the orbital interactions also favour the O_h symmetry, thus making Pauli repulsion the determinant factor towards the preference for planar D_2h B₆²⁻.

Table 3 Energy decomposition analysis (EDA) of all mixed metal clusters with planar and 3D symmetries (in kcal mol⁻¹), from two fragments at their quintet states, computed at the BLYP-D3(BJ)/TZ2P level

		ΔEint	ΔEPauli	ΔVelstat	ΔEoi	ΔEdisp
B2Al4^2−	D 4h	−52.1	440.1	−202.7	−285.7	−3.9
	D 2h	−40.4	243.4	−98.1	−182.5	−3.3
	ΔE	11.7	−196.7	104.6	103.2	0.6
Al2B4^2−	D 4h	−75.1	584.0	−251.7	−404.1	−3.3
	D 2h	−139.6	556.6	−238.6	−454.5	−3.3
	ΔE	−64.6	−27.4	13.2	−50.4	0.0
Al2Ga4^2−	D 4h	−35.0	381.2	−201.0	−210.6	−4.6
	D 2h	−19.2	283.1	−147.1	−151.4	−3.8
	ΔE	15.8	−98.1	53.8	59.3	0.8
Ga2B4^2−	D 4h	−83.8	590.4	−262.5	−408.2	−3.5
	D 2h	−157.5	540.0	−225.6	−468.6	−3.2
	ΔE	−73.7	−50.4	36.8	−60.4	0.3
Ga2Al4^2−	D 4h	−38.4	370.1	−188.0	−216.1	−4.3
	D 2h	−20.6	218.0	−90.8	−144.4	−3.6
	ΔE	17.8	−152.0	97.3	71.7	0.8
Al3Ga3^2−	D 3h	−36.8	381.0	−197.8	−215.7	−4.2
	C 3v	−20.7	254.4	−122.8	−148.7	−3.6
	ΔE	16.1	−126.6	75.0	67.0	0.6

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Conclusions

In previous studies,¹⁸ the preference of B₆²⁻ for the planar D_2h geometry and of Al₆²⁻ for the 3D O_h one was justified by the inclination for localised covalent bonding in the former cluster and delocalised bonding in the latter. These two effects point in opposite directions. In the present work, we go one-step further by showing that the preference of B₆²⁻ for the planar D_2h form is due to two particular molecular orbital interactions. From one side the D^HOMO−1_2h(b_2u) formed from two tangential SOMO σ^T(b₂) orbitals. This orbital is related to localised covalent bonding, and has a much more important weight in B₆²⁻ than in Al₆²⁻, proving the dominant localised covalent character in the former. And the second determinant interaction is that of π character. In the case of O^HOMO−1_h(t_2g) for B₆²⁻, its formation from two π SOMO orbitals is much less favourable than for Al₆²⁻. This result is in line with a dominant delocalisation force in Al clusters and more localised bonding in B metal clusters. For mixed clusters, we have found that those with more than two B atoms prefer the planar structure for same reasons discussed for B₆²⁻.

Acknowledgements

This work was supported by the Ministerio de Economía y Competitividad (MINECO) of Spain (Project CTQ2014-54306-P) and the Generalitat de Catalunya (project 2014SGR931, Xarxa de Referència en Química Teòrica i Computacional, ICREA Academia 2014 prize for M.S., and grant No. 2014FI_B 00429 to O. E. B.). The EU under the FEDER grant UNGI10-4E-801 (European Fund for Regional Development) has also funded this research. J. P. thanks the National Research School Combination-Catalysis (NRSC-C), and The Netherlands Organization for Scientific Research (NWO/CW and NWO/NCF). The authors are grateful to Dr Ferran Feixas for fruitful discussions.

References

1. Y. Hu, T. J. Wagener, Y. Gao, H. M. Meyer and J. H. Weaver, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 38, 3037–3044.
2. B. Wang, H. Wang, H. Li, C. Zeng, J. G. Hou and X. Xiao, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 63, 035403.
3. V. Torma, T. Reuter, O. Vidoni, M. Schumann, C. Radehaus and G. Schmid, ChemPhysChem, 2001, 2, 546–548.
4. R. N. Barnett, C. L. Cleveland, H. Häkkinen, W. D. Luedtke, C. Yannouleas and U. Landman, Eur. Phys. J. D, 1999, 9, 95–104.
5. S. Aldridge and A. J. Downs, The Group 13 Metals Aluminium, Gallium, Indium and Thallium: Chemical Patterns and Peculiarities, John Wiley & Sons, Ltd, Chichester, 2011.
6. A. N. Alexandrova, K. A. Birch and A. I. Boldyrev, J. Am. Chem. Soc., 2003, 125, 10786–10787.
7. A. N. Alexandrova and A. I. Boldyrev, Inorg. Chem., 2004, 43, 3588–3592.
8. A. N. Alexandrova, A. I. Boldyrev, H.-J. Zhai and L.-S. Wang, Coord. Chem. Rev., 2006, 250, 2811–2866.
9. H.-J. Zhai, A. N. Alexandrova, L.-S. Wang and A. I. Boldyrev, Angew. Chem., Int. Ed., 2003, 42, 6004–6008.
10. A. I. Boldyrev and L.-S. Wang, Chem. Rev., 2005, 105, 3716–3757.
11. A. E. Kuznetsov, K. A. Birch, A. I. Boldyrev, X. Li, H.-J. Zhai and L.-S. Wang, Science, 2003, 300, 622–625.
12. A. E. Kuznetsov, A. I. Boldyrev, H.-J. Zhai, X. Li and L.-S. Wang, J. Am. Chem. Soc., 2002, 124, 11791–11801.
13. X. Li, A. E. Kuznetsov, H.-F. Zhang, A. Boldyrev and L.-S. Wang, Science, 2001, 291, 859–861.
14. A. E. Kuznetsov, A. I. Boldyrev, H.-J. Zhai, X. Li and L.-S. Wang, J. Am. Chem. Soc., 2002, 124, 11791–11801.
15. O. C. Thomas, W.-J. Zheng, T. P. Lippa, S.-J. Xu, S. A. Lyapustina and K. H. Bowen, J. Chem. Phys., 2001, 114, 9895–9900.
16. A. N. Alexandrova, A. I. Boldyrev, H.-J. Zhai and L.-S. Wang, J. Chem. Phys., 2005, 122, 054313.
17. A. N. Alexandrova, A. I. Boldyrev, H.-J. Zhai, L.-S. Wang, E. Steiner and P. W. Fowler, J. Phys. Chem. A, 2003, 107, 1359–1369.
18. M. T. Huynh and A. N. Alexandrova, J. Phys. Chem. Lett., 2011, 2, 2046–2051.
19. J. M. Mercero, E. Matito, F. Ruipérez, I. Infante, X. Lopez and J. M. Ugalde, Chem. – Eur. J., 2015, 21, 9610–9614.
20. C. Corminboeuf, C. S. Wannere, D. Roy, R. B. King and P. v. R. Schleyer, Inorg. Chem., 2006, 45, 214–219.
21. O. El Bakouri, M. Duran, J. Poater, F. Feixas and M. Solà, Phys. Chem. Chem. Phys., 2016 10.1039/c5cp07011b.
22. J. Ma, Z. Li, K. Fan and M. Zhou, Chem. Phys. Lett., 2003, 372, 708–716.
23. L.-M. Yang, J. Wang, Y.-H. Ding and C.-C. Sun, Phys. Chem. Chem. Phys., 2008, 10, 2316–2320.
24. A. N. Alexandrova, Chem. Phys. Lett., 2012, 533, 1–5.
25. A. N. Alexandrova, M. J. Nayhouse, M. T. Huynh, J. L. Kuo, A. V. Melkonian, G. Chavez, N. M. Hernando, M. D. Kowal and C.-P. Liu, Phys. Chem. Chem. Phys., 2012, 14, 14815–14821.
26. A. N. Alexandrova, M. R. Nechay, B. R. Lydon, D. P. Buchan, A. J. Yeh, M.-H. Tai, I. P. Kostrikin and L. Gabrielyan, Chem. Phys. Lett., 2013, 588, 37–42.
27. J. E. Fowler and J. M. Ugalde, J. Phys. Chem. A, 2000, 104, 397–403.
28. J. E. Fowler and J. M. Ugalde, Phys. Rev. A: At., Mol., Opt. Phys., 1998, 58, 383–388.
29. P. Farràs, N. Vankova, L. L. Zeonjuk, J. Warneke, T. Dülcks, T. Heine, C. Viñas, F. Teixidor and D. Gabel, Chem. – Eur. J., 2012, 18, 13208–13212.
30. M. R. Fagiani, L. Liu Zeonjuk, T. K. Esser, D. Gabel, T. Heine, K. R. Asmis and J. Warneke, Chem. Phys. Lett., 2015, 625, 48–52.
31. J. K. Olson and A. I. Boldyrev, J. Phys. Chem. A, 2013, 117, 1614–1620.
32. K. Kitaura and K. Morokuma, Int. J. Quantum Chem., 1976, 10, 325–340.
33. K. Morokuma, Acc. Chem. Res., 1977, 10, 294–300.
34. T. Ziegler and A. Rauk, Theor. Chim. Acta, 1977, 46, 1–10.
35. T. Ziegler and A. Rauk, Inorg. Chem., 1979, 18, 1558–1565.
36. M. El-Hamdi, W. Tiznado, J. Poater and M. Solà, J. Org. Chem., 2011, 76, 8913–8921.
37. M. El-Hamdi, O. El Bakouri El Farri, P. Salvador, B. A. Abdelouahid, M. S. El Begrani, J. Poater and M. Solà, Organometallics, 2013, 32, 4892–4903.
38. M. El-Hamdi, M. Solà, G. Frenking and J. Poater, J. Phys. Chem. A, 2013, 117, 8026–8034.
39. R. Islas, J. Poater, E. Matito and M. Solà, Phys. Chem. Chem. Phys., 2012, 14, 14850–14859.
40. G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. Fonseca Guerra, S. J. A. van Gisbergen, J. G. Snijders and T. Ziegler, J. Comput. Chem., 2001, 22, 931–967.
41. A. D. Becke, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 38, 3098–3100.
42. C. Lee, W. Yang and R. G. Parr, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 37, 785–789.
43. S. Grimme, J. Antony, S. Ehrlich and H. Krieg, J. Chem. Phys., 2010, 132, 154104.
44. S. Grimme, S. Ehrlich and L. Goerigk, J. Comput. Chem., 2011, 32, 1456–1465.
45. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian 09, Revision A.02 ed., Gaussian, Inc., Pittsburgh, PA, 2009.
46. J. Cizek, J. Chem. Phys., 1966, 45, 4256–4266.
47. G. D. Purvis III and R. J. Bartlett, J. Chem. Phys., 1982, 76, 1910–1918.
48. K. Raghavachari, G. W. Trucks, J. A. Pople and M. Head-Gordon, Chem. Phys. Lett., 1989, 157, 479–483.
49. T. H. Dunning Jr., J. Chem. Phys., 1989, 90, 1007–1023.
50. R. A. Kendall, T. H. Dunning Jr. and R. J. Harrison, J. Chem. Phys., 1992, 96, 6796–6806.
51. F. M. Bickelhaupt and E. J. Baerends, in Rev. Comput. Chem., ed. K. B. Lipkowitz and D. B. Boyd, Wiley-VCH, New York, 2000, vol. 15, pp. 1–86.
52. F. M. Bickelhaupt, A. Diefenbach, S. P. de Visser, L. J. de Koning and N. M. M. Nibbering, J. Phys. Chem. A, 1998, 102, 9549–9553.
53. C. Fonseca Guerra, J.-W. Handgraaf, E. J. Baerends and F. M. Bickelhaupt, J. Comput. Chem., 2004, 25, 189–210.
54. I. Fernández and G. Frenking, Faraday Discuss., 2007, 135, 403–421.
55. I. Fernández and G. Frenking, Chem. – Eur. J., 2007, 13, 5873–5884.
56. R. Islas, J. Poater and M. Solà, Organometallics, 2014, 33, 1762–1773.
57. F. Feixas, E. Matito, J. Poater and M. Solà, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2013, 3, 105–122.
58. P. Bultinck, R. Ponec and S. Van Damme, J. Phys. Org. Chem., 2005, 18, 706–718.
59. F. Feixas, E. Matito, J. Poater and M. Solà, Chem. Soc. Rev., 2015, 44, 6434–6451.
60. J. Poater, M. Duran, M. Solà and B. Silvi, Chem. Rev., 2005, 105, 3911–3947.
61. E. Matito, M. Duran and M. Solà, J. Chem. Phys., 2005, 122, 014109.
62. E. Matito, ESI-3D: Electron Sharing Indexes Program for 3D Molecular Space Partitioning, Institute of Computational Chemistry and Catalysis, Girona, 2006, http://iqc.udg.es/~eduard/ESI.
63. S. Alvarez, Dalton Trans., 2013, 42, 8617–8636.
64. T. J. Lee and P. R. Taylor, Int. J. Quantum Chem., Quantum Chem. Symp., 1989, S23, 199–207.
65. J. M. L. Martin, in Energetics of Stable Molecules and Reactive Intermediates, ed. M. S. Minas da Piedade, Kluwer Academic Publishers, Dordrecht, 1999, vol. 535, pp. 373–417.
66. J. Poater, R. Visser, M. Solà and F. M. Bickelhaupt, J. Org. Chem., 2007, 72, 1134–1142.
67. E. Díaz-Cervantes, J. Poater, J. Robles, M. Swart and M. Solà, J. Phys. Chem. A, 2013, 117, 10462–10469.

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Footnotes

† This work is dedicated to Prof. Evert Jan Baerends as a proof of our admiration for his brilliant contributions to chemistry and of our gratitude for helping us to understand chemistry better.

‡ Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp01109h

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