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

Despite the fact that B and Al belong to the same group 13 elements, the B6 2 cluster prefers the planar D2h geometry, whereas Al6 2 favours the Oh structure. In this work, we analyse the origin of the relative stability of D2h and Oh 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 (Al6 2 and Ga6 2 ) or do not (B6 2 ) compensate the higher Pauli repulsion of the Oh form. Analysing the orbital interaction term in more detail, we find that the preference of B6 2 for the planar D2h 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 B6 2 .


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.3][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. 5Both B and Al belong to the same group 13, and thus present a similar electronic structure, [He]2s 2 2p 1 and [Ne]3s 2 3p 1 , respectively.5][16] B 6 2À adopts a planar D 2h geometry in its low-lying singlet state, whereas the Al 6 2À cluster is octahedral.Both shapes of the metal clusters are kept when lithium salts are formed.The chemical bonding of B 6 2À and Al 6 2À has been widely analysed in previous studies. 14,17,18In particular, Alexandrova et al. 18 highlighted the fact that B 6 2À 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 Al 6

2À
cluster is more difficult due to larger s-p energy separation, which hampers the formation of directional covalent Al-Al bonds. 19In this case, bonding comes from the combination of radial and tangential p-orbitals that result in extensive delocalisation. 20Indeed, the Al 6 2À cluster displays octahedral aromaticity, 14,21 whereas planar D 2h B 6 2À is considered sand p-antiaromatic. 17,18,22,23Thus, as pointed out by Alexandrova et al., 18,[24][25][26] covalent and delocalised bonding shows opposite effects in determining the molecular structure of many clusters.Huynh and Alexandrova analysed the whole series B n Al 6Àn 2À (n = 0-6), from B 6 2À till Al 6 2À 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 B 6 2À 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. 18Similar results were reported by Fowler and Ugalde in larger clusters of group 13.In particular, these authors found that B 13 + prefers a planar conformation 27 in contrast to Al 13 À , 28 which adopts an icosahedral geometry.Interestingly, in closo boranes and substituted related species, like B 6 H 6 2À or B 12 I 12 2À , the delocalised 3D structure is preferred.However, successive stripping of iodine in B 12 I 12 2À leads to a B 12 planar structure with some localised 2c-2e B-B bonds. 29,30imilarly, for B 6 H n À clusters, the clusters are planar for n r 3 and become tridimensional for n Z 4. 31 As can be seen in Scheme 1, both 2D D 2h planar and 3D O h geometries for X 6 2À (X = B, Al) can be obtained joining the same two X 3 À cluster fragments. 14,177][38][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. 38Therefore, the application of this type of analysis to B 6 2À and Al 6 2À 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 B 6 2À and Al 6 2À clusters, and then further complemented with Ga 6 2À .Finally, X 2 Y 4 2À and X 3 Y 3 2À (X, Y = B, Al, Ga) mixed clusters in their distorted D 2h planar and 3D D 4h geometries will also be discussed.

Computational methods
All Density Functional Theory (DFT) calculations were performed using the Amsterdam Density Functional (ADF) program. 40The molecular orbitals (MOs) were expanded in a large uncontracted set of Slater type orbitals (STOs) of triple-z 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.][43][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 program 45 at the coupled cluster level 46 with single and double excitation (CCSD) 47 and with triple excitation treated perturbatively (CCSD(T)) 48 using Dunning's correlation consistent augmented triple-z (aug-cc-pVTZ) 49,50 at optimised BLYP-D3(BJ)/TZ2P molecular geometries.
The bonding energy corresponding to the formation of X 6

2À
for both D 2h and O h symmetries from two anionic quintet tetraradicals, fragment 1 (aaaa) + fragment 2 (bbbb) (see Scheme 1), is made up of two major components (eqn (1)): In this formula, the distortion energy DE 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 DE int corresponds to the actual energy change when the prepared fragments are combined to form the overall molecule.
The term DV elstat corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the prepared (i.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 DE disp term takes into account the interactions which are due to dispersion forces.
In bond-energy decomposition, [51][52][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 À1 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 p-cyclic conjugation in typical (anti)aromatic organic compounds and in metallabenzenes and metallacyclopentadienes.
5][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.
9][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 X 6 2À 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 (1.536-1.768Å) are much shorter than those for Al-Al (2.574-2.912Å) and Ga-Ga (2.526-2.898Å).The similar Al-Al and Ga-Ga distances in X 6 2À metal clusters (X = Al, Ga) are not unexpected given the similar van der Waals radii of these two elements. 63In addition, the X-X bond length connecting the two equivalent X ) 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 Relative energies of clusters between O h and D 2h symmetries (in kcal mol À1 ), and the aromatic MCI criterion to D 2h structures by about 20-30 kcal mol À1 .However, the qualitative picture remains the same.
The aromaticity of these X 6 2À 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. 21CI values confirm the octahedral aromaticity 21  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 X 3 À anionic fragments, both in their quintet state in order to form the corresponding new bonds.Three of these bonds are of s character, two tangential (s T ) and one radial (s R ), and one p 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 Al 3

À
. 19 In the case of the quintet state of Al 3 À , 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 s-type tangential and one s-type radial molecular orbitals arising from the 3p x and 3p y atomic orbitals, and one p-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. 19Moreover, 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. 19This seems to indicate that DFT methods give reasonable results for this quintet state.
Finally, the T 1 test 64 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 T 1 values as high as 0.055. 65he different terms of the EDA for B    , 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 inter and O HOMOÀ1 h , respectively.Overall, the higher orbital interaction term of the O h system can be explained by the larger hSOMO|SOMOi overlaps of two of the t 2g delocalised molecular orbitals for this cluster (see Fig. 2).The energies of the occupied MOs of Al 6 2À formed are higher than those of the Al 3 À SOMOs because we move from a monoanionic fragment to a dianionic molecule.Now it is the turn to visualize the MOs of B 6 2À .The fragments for B 3 À are the same as those for Al 3 À (see Fig. 3).However, the first difference appears in the MOs for B 6 2À with D 2h symmetry.
In this case, it would be more reasonable to build the MOs of this molecule from two triplet (not quintet) ) p MOs, the p MO (frag HOMOÀ3 ) should be doubly occupied.Furthermore, the tangential s T (a 1 ) frag HOMO does not participate in any occupied MO of this metal cluster and only generates virtual MOs.Consequently, MOs of B 6 2À are better formed from two B 3 À fragments in their triplet state (see red electron in Fig. 3).On the other hand, B 6 2À with O h follows the same trend as Al 6 2À , 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. 19o make results comparable, Table 2     Finally, as done usually in the turn-upside-down approach, [36][37][38][39]56,66,67  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 B 6 2À in D 2h symmetry.Table S5

Mixed metal clusters
In this section, we analyse the X 2 Y 4 2À clusters with X, Y = B, Al, Ga and X a 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 Al  lowest in energy. 18However, 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, Al 3 Ga 3 2À also prefers an O h geometry by 13.2 kcal mol À1 .Unfortunately, this latter relative energy cannot be compared to those of B 3 Al 3 2À or B 3 Ga 3

2À
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 coworkers 26 found in X 3 Y 3 (X = B, Al, Ga; Y = P, As) clusters that the lighter elements prefer 2D structures, whereas the heavier ones favour 3D geometries.
The EDA was also performed for this series of six mixed metal clusters (see Table 3

Conclusions
In previous studies, 18    .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 B 6 2À .
Scheme 1 D 2h and O h structures of X 62À can be formed from C 2v X 3 À

Fig. 1
Fig. 1 Geometries of X 6 2À metal clusters analysed with D 2h and O h symmetries.Distances in Å and angles in degrees.

a B 2
Ga 4 2À (D 2h ) has not been obtained because optimization breaks the symmetry; whereas B 3 Al 3 2À and B 3 Ga 3 2À (O h ) have not been obtained because the strength of the B 3 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 2h ), whereas for the two latter are in between À19.0 and À38.1 kcal mol À1 .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 B 3 À fragment taken from the B 6 2À 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 B 3 À cluster in the quintet state (DE dist = 12.5 kcal mol À1 ), whereas the rest of the systems present small values of DE dist (0.0-1.7 kcal mol À1 ).However, differences in DE are not due to distortion energies (indeed DE dist values follow the opposite trend as DE), but to interaction energies (DE int ).

Fig. 2
Fig. 2 Molecular orbital diagram corresponding to the formation of Al 6 2À in D 2h and O h symmetries from two Al 3 À fragments in their quintet state.Energies of the molecular orbitals are enclosed (in eV), as well as the hSOMO|SOMOi 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.

6 2À
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 (s T (b 2 )) and frag HOMOÀ1 (s T (a 1 )) MOs of Al 3 À are degenerate when obtained from Al 6 2À 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 Al 3 À fragments to form the MOs of the metal clusters in both geometries.

B 3 À
fragments.The reason is the different occupation of the MOs when compared to the D 2h Al 6 2À species.In D 2h B 6 2À , the HOMO corresponds to the antibonding p MO.To reach doubly occupied bonding (b 3u D HOMOÀ4 2h ) and antibonding (b 2g D HOMO 2h much larger hSOMO|SOMOi overlap than t 2g O HOMOÀ1 h (0.518 in the former vs. 0.338 in the latter).In particular, this D HOMOÀ2 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 p-interaction between the two p 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 DE oi term of the O h cluster.However, as compared to Al 6 2À , for B 6 2À the D(DE oi ) term favours the O h system to a less extent and cannot compensate the higher DE 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 Al 6 2À .Such a larger covalent component in B 6 2À is also supported by the covalent character of the interaction between the two fragments calculated as % covalency = (DE oi /(DE oi + DV elstat + DE disp )) Â 100.This formula results in B 6 2À : 65-67% (O h , D 2h ), Al 6 2À : 56-60% (O h , D 2h ), and Ga 6 2À : 49-51% (O h , D 2h ); thus confirming again the larger covalency found in B 6 2À .
(ESI ‡) contains the EDA for O h and D 2h B 6 2À systems using B 3 À 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.

Fig. 3
Fig. 3 Molecular orbital diagram corresponding to the formation of B 6 2À in D 2h and O h symmetries from two B 3 À fragments in their quintet states.Electrons in red refer to the formation of B 6 2À (D 2h ) from B 3 À fragments in their triplet state.In the triplet state, p(b 1 ) is doubly occupied, s R (a1) and s T (b 2 )remain singly occupied, and the s T (a 1 ) becomes unoccupied.Energies of the molecular orbitals are enclosed (in eV), as well as the hSOMO|SOMOi 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.

3 2À, the fragments were Al 3 À and Ga 3 À
) with the aim to further understand the determinant force towards the most stable cluster.For the X 2 Y 4 2À clusters, the EDA was carried out taken YXY À fragments in their quintet For Al 3 Ga 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 Al 6 2À and Ga 6 2À systems.On the other hand, when D 2h symmetry is the cluster lower in energy, as for Al 2 B 4 2À and Ga 2 B 4 2À 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 B 6 2À , 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 B 6 2À .
the preference of B 6 2À for the planar D 2h geometry and of Al 6 2À 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 B 6 2À for the planar D 2h form is due to two particular molecular orbital interactions.From one side the D HOMOÀ1 2h (b 2u ) formed from two tangential SOMO s T (b 2 ) orbitals.This orbital is related to localised covalent

Fig. 4
Fig. 4 Geometries of mixed metal clusters analysed with planar and 3D geometries.Distances in Å and angles in degrees.
3 À fragments in O h clusters is longer than in the D 2h systems.Table 1 encloses the energy differences between O h and D 2h clusters.For B 6 2À D 2h symmetry is more stable than O h by 67.5 kcal mol À1 , the latter not being a minimum.
18Meanwhile the opposite trend is obtained in the other two metal clusters, for which O h is lower in energy by15.8(Al 6 2À ) and 9.3 kcal mol À1 (Ga 6 2À

Table 1 )
h and D 2h symmetries are now À38.7, +44.8 and +46.6 kcal mol À1 , respectively.CCSD(T) values systematically favour O h as compared Thus, we focus on the decomposition of DE int into DE Pauli , DV elstat , DE oi , and DE disp terms.As a general trend, in all three X 6 2À clusters DE Pauli is larger for the O h than the D 2h cluster À fragment.At the same time, the O h form presents larger (more negative) electrostatic interactions (D(DV elstat ) = 52.9,70.2, and 69.5 kcal mol À1 for B 6 2À , Al 6 2À , and Ga 6 2À

Table 2
Energy decomposition analysis (EDA) of X 6 2À (X = B, Al, and Ga) metal clusters with D 2h and O h symmetries (in kcal mol À1 ), from two 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.
, although the D 4h system is stabilized with respect to the D 2h one by 20-30 kcal mol À1 .It is important to note that the D 4h and D 2h systems are not always the most stable for the X 2 Y 4 2À clusters.For instance, for Al 2 B 4

Table 3
Energy decomposition analysis (EDA) of all mixed metal clusters with planar and 3D symmetries (in kcal mol À1 ), from two fragments at their quintet states, computed at the BLYP-D3(BJ)/TZ2P level dominant localised covalent character in the former.And the second determinant interaction is that of p character.In the case of O HOMOÀ1