Chemical Bonding and σ-Aromaticity in Charged Molecular Alloys: [Pd2As14]4− and [Au2Sb14]4− Clusters

We report a computational study on the structures and bonding of a charged molecular alloy D 2h [Pd2As14]4− (1), as well as a model D 2h [Au2Sb14]4− (2) cluster. Our effort makes use of an array of quantum chemistry tools: canonical molecular orbital analysis, adaptive natural density partitioning, natural bond orbital analysis, orbital composition analysis, and nucleus independent chemical shift calculations. Both clusters consist of two X7 (X = As, Sb) cages, which are interconnected via a M2 (M = Pd, Au) dumbbell, featuring two distorted square-planar MX4 units. Excluding the Pd/As or Au/Sb lone-pairs, clusters 1 and 2 are 50- and 44-electron systems, respectively, of which 32 electrons are for two-center two-electron (2c-2e) As-As or Sb-Sb σ bonds and an additional 16 electrons in 1 for 2c-2e Pd-As σ bonds. No covalent Pd-Pd or Au-Au bond is present in the systems. Cluster 1 is shown to possess two globally delocalized σ electrons, whereas 2 has two σ sextets (each associated with an AuSb4 fragment). Thus, 1 and 2 conform to the (4n + 2) Hückel rule, for n = 0 and 1, respectively, rendering them σ-aromaticity.

In the past years, we have been interested in exploring the structural, electronic, and bonding properties of gas-phase clusters [23][24][25][26][27] and synthetic cluster compounds 36- 38 . In the present contribution, we shall report on a quantum chemical study on the structure and bonding of a "charged molecular alloy" cluster: D 2h [Pd 2 As 14 ] 4− (1). This cluster was crystallized in the form of [K([2. 2.2]crypt)] 4 [Pd 2 As 14 ]·5en (en = ethylenediamine) 24 , by Eichhorn and coworkers in 2002, which consists of two [As 7 ] cages interconnected via a Pd 2 dumbbell. The Pd centers are coordinated by As in a distorted square-planar fashion, featuring two PdAs 4 fragments. In this bulk compound, the [Pd 2 As 14 ] 4− tetraanion is effectively stabilized by four K + counter-ions. Eichhorn and coworkers proposed the [As 7 ] 5− and Pd 2 6+ building blocks, the formal Pd(III) centers, and an axial Pd-Pd bond in 1. However, there has been no quantum chemical study on this system so far, to the best of our knowledge, and the nature of bonding in the cluster remains elusive.
Herein we have performed a detailed computational study on the [Pd 2 As 14 ] 4− (1) cluster at the PBE0 level 39 of density-functional theory (DFT) 40 and elucidated its nature of chemical bonding using a range of state-of-the-art quantum chemistry tools: canonical molecular orbital (CMO) analysis, adaptive natural density partitioning (AdNDP) 41 , natural bond orbital (NBO) 42 analysis, orbital composition analysis, and nucleus independent chemical shift (NICS) 43 . To further ensure the computational reliability for a naked multiply charged anion, conductor-like polarizable continuum mode (C-PCM) [44][45][46][47] calculation as an alternative method has also been carried out for 1 to take into account the solvation effects, whose results many people believe should be closer to the truth for a synthetic cluster compound in the bulk: a multiply charged anion being stabilized by bulky ligands including counter-ions 24 . The computational data allows an in-depth understanding of the nature of bonding in 1, which turns out to possess two delocalized σ electrons within the square-planar PdAs 4 fragments, rendering σ-aromaticity for the charged molecular alloy according to the (4n + 2) Hückel rule, for n = 0. Based on the findings in [Pd 2 As 14 ] 4− (1), we have further explored the rational design of a model charged molecular alloy, D 2h [Au 2 Sb 14 ] 4− (2). Cluster 2 has two more valence electrons than 1 and yet differs markedly from the latter in terms of chemical bonding. 2 has two square-planar AuSb 4 fragments, each supporting 6σ delocalized electrons (that is, σ sextet), which conform to the (4n + 2) Hückel rule for aromaticity, for n = 1. Clusters 1 and 2 provide new examples for σ-aromaticity, from gas-phase clusters to synthetic solid-phase compounds. It is stressed that 1 and 2 are relatively large cluster systems and bonding analyses are rather challenging; the current level of understanding for 1 and 2 is largely attributed to the powerful AdNDP tool 41 for chemical bonding analyses, which was not possible even a couple of years ago. The concept of σ-aromaticity, in particular σ sextets, was discussed recently in a synthetic [Au 2 Sb 16 ] 4− compound 38 .

Results and Discussion
Cluster Structure of [Pd 2 As 14 ] 4− . We obtained the initial coordinates from crystal data of the Eichhorn paper 24 and fully reoptimized the structure of [Pd 2 As 14 ] 4− at the PBE0/def2-TZVP level. The ultimate cluster structure is D 2h [Pd 2 As 14 ] 4− (1). Calculated bond distances and bond angles are summarized in Fig. 1 and Table 1. The Cartesian coordinates for 1 are presented in Table S1 in the Supplementary Information, along with those of D 2h [Au 2 Sb 14 ] 4− (2). Overall, the calculated bond distances of 1 are highly coherent with the experimental measurements ( Table 1).
The cluster can be viewed as two As 7 cages interconnected by a Pd 2 dimer ( Fig. 1), as described by Eichhorn and coworkers 24 . The Pd centers hold the As 7 cages together via two distorted square-planar PdAs 4 units. The fourteen As atoms in 1 may be classified into three subgroups, that is, As(i): two bridging As atoms at the far left and far right of the cluster; As(ii): four As atoms that are tricoordinated with As; As(iii): eight As atoms linked to Pd centers. For the As-As interactions, the four As(ii) centers are tricoordinated with two As(i) and eight As(iii), forming 12 As-As links. These As-As distances range from 2.38 to 2.46 Å ( Table 1). The two PdAs 4 units are connected via four interlayer As(iii)-As(iii) links, whose distances are 2.43 Å. In total, there are 16 As-As links in 1.
According to the recommended covalent radii by Pyykkö 48 , typical As-As single and As=As double bonds are around 2.42 and 2.28 Å, respectively. Thus, all 16 As-As links are single bonds.
In terms of the Pd-As interactions in square-planar PdAs 4 units, the eight Pd-As links have distances of 2.50 Å, which are to be compared to the recommended distances of Pd-As single (2.41 Å) and Pd=As double (2.31 Å) bonds 48 . Therefore, the Pd-As bonds in 1 are weaker than single bonds; we tentatively assign them as single  25; ∠AsAsAs, 78.00-108. 42. bonds. The peripheral As(iii)-As(iii) links within two PdAs 4 planes are markedly elongated (As12-As16: 3.09 Å; As16-As3: 3.93 Å) with respect to the As-As bonds discussed above, indicating that no direct As-As interactions are present in the periphery.
The calculated Pd-Pd distance is 2.75 Å, which turns out to be markedly longer than a single bond (2.40 Å) 48 . Even larger Pd-Pd distance was reported in the literature: [Pd 2 @Ge 18 ] 4− (2.831 Å) 49 . In the Eichhorn paper 24 , an axial Pd-Pd bond in 1 was explicitly claimed. We believe the Pd-Pd covalent bonding in 1 is relatively minor (see below), and the Pd atoms can be viewed as isolated, single atoms. Note that despite the fact that the Pd centers in 1 are not in Pd(0) configuration, a "dispersion interaction" mechanism, similar to aurophilicity in Au-Au clusters 50,51 , should help stabilize the Pd 2 dumbbell in 1.
Arguably, the open-shell configuration of Pd centers in 1 can facilitate stronger Pd-Pd dispersion interaction as compared to Pd d 10 centers. The calculated Wiberg index for Pd-Pd amounts to 0.250 (Table 1), which is moderate in spite of the absence of conventional Pd-Pd bond in the system. Bader analysis yields a similar Pd-Pd bond index of 0.170.
Multiply charged cluster anions are not uncommon as synthetic compounds, as well as in computational modeling. Tetraanion D 2h [Pd 2 As 14 ] 4− (1) is intrinsically unstable due to intramolecular Coulomb repulsion. Our outer valence green's function (OVGF) calculations for 1 give a negative electron binding energy of −5.37 eV as anticipated, which is routine for a highly charged tetraanion. Nonetheless, we have analyzed the wavefunction stability for tetraanion 1 and the result indicates indeed that the wavefunction is stable under the perturbations considered.
In response to the concern of one referee, we further technically designed a model neutral cluster, C i [Pd 2 As 14 K 4 ] (3), in which [Pd 2 As 14 ] 4− (1) tetraanion is electrostatically stabilized by four K + cations. Geometry optimizations and frequency calculations show that the tetraanion 1 and the model neutral cluster 3 (see Fig. S1 in the Supplementary Information) have remarkably similar structures with respect to that in the synthetic bulk compound (Fig. S2). The structural and electronic integrity of 1 is fully maintained in 3. The structure of 3 is closely relevant to and yet significantly simpler than the [K([2. 2.2]crypt)] 4 [Pd 2 As 14 ]·5en bulk compound 24 , rendering the former cluster a valuable neutral model of the latter compound. Alternatively, the referee's concern of the computational reliability for a bare tetraanion such as 1 can be addressed using the C-PCM calculations [44][45][46][47] , in which the solvation effects (with dielectric constant ε r (ethylendiamine) = 12.9) are considered. The optimized structure of 1 in the C-PCM calculations is presented in Fig. S3.
Overall A close comparison indicates that the computational data for bare tetraanion in set (a), which some people consider to be questionable or unreliable, are actually a faithful reproduction of the crystal data 24 ; the computational bond distances of As-As and As-Pd are typically 0.01-0.03 Å longer than the experimental data, due to Coulomb repulsion. This observation suggests that tetraanion is not a problem for PBE0, at least in terms of structural optimization. The optimized structures in sets (b) and (c) are even closer to the experiment, with typical errors of 0.00-0.02 Å. However, the improvement is very limited, largely because the computational data of set (a), that is, bare tetraanion at PBE0 without the solvation effects, appear to be excellent. In all cases, the optimized As-As distances deviate very slightly from each other and can all be assigned as single bonds.
All valence electrons are bound in 3 ( Fig. S1b) and the OVGF calculations give a positive ionization potential of 5.43 eV, which is in contrast to the above-mentioned negative value for 1 (Fig. S1a). This difference between 3 and 1 is due to electrostatic stabilization in 3, between four K + cations and tetraanion 1, as pointed out above. Alternatively, with C-PCM calculations ( Fig. S3) to account for solvation effects, all occupied CMOs (that is, the Kohn-Sham orbitals) of 1 become highly negative in energy eigenvalues. Figure S4 depicts the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) of 1 (with solvation effects) and 3, as well as additional low-lying CMOs down to HOMO-4. These frontier CMOs are the same except that their energy order changes slightly due to the perturbation of K + cations in 3, as anticipated. Furthermore, we note explicitly that the calculated CMOs of 1 with and without solvation effects are identical with each other in shapes and in energy order, except that those with solvation effects are shifted to deeper energies. Pd(1/2)-As(iii) As(i)-As(ii) As(ii)-As(iii)  (1) may be classified into three types. As(i): two bridging As atoms; As(ii): four As atoms that are tricoordinated with As; As(iii): eight As atoms linked to the Pd atoms. b Only four As(iii)-As(iii) bonds are present in 1, which link the two square-planar PdAs 4 units (Fig. 1). c Experimental data from ref. 24. d Charge per Pd or As atom.
The above comparative data between 1 (without or with solvation effects) and 3 suggest that the present results for D 2h [Pd 2 As 14 ] 4− (1) should generally be considered reliable for the state-of-the-art quantum chemistry. The four K + counter-ions in 3 are associated with the Pd centers and two As(i) bridging atoms, which carry the majority of four extra electrons in 1 (see below for details). Indeed, the K + counter-ions in the bulk compound occupy similar positions with respect to Pd and As(i) atoms, with slight deviations for two out of eight K + ions (Fig. S2) due to steric effects of the bulky ligands. Therefore, we conclude that cluster 3 (Fig. S1b) is a good and simplified neutral model for the bulk compound, whereas bare tetraanion 1 (Fig. S1a) and tetraanion 1 with solvation effects (Fig. S3) are also reliable computational structures. All four sets of computational and experimental structures actually represent the same chemical entity.
Chemical Bonding in D 2h [Pd 2 As 14 ] 4− : Pd-Pd Bonding and σ-Aromaticity. Considering the Pd 4d 10 and As 4s 2 4p 3 configurations, D 2h [Pd 2 As 14 ] 4− (1) possesses 94 valence electrons in total (including 4 extra charges). It is a relatively large "charged molecular alloy" cluster, and thus an in-depth chemical bonding analysis is difficult. However, the structural description on the basis of bond distances, as outlined in the previous section, allows simplification of the task, because there exist clearly 16 As-As and 8 Pd-As single bonds in 1, which consume 48 electrons. Furthermore, according to our recent work on [Sb 3 Au 3 Sb 3 ] 3− all-metal sandwich 36, 37 , the As 4s 2 electrons in 1 are anticipated to well behave as lone-pairs (14 of them, consuming 28 electrons). The above-mentioned single bonds and lone-pairs use 76 electrons out of 94 in 1, whose corresponding CMOs are depicted in Fig. 2a and b, as well as in Fig. S5b.
Technically, the 14 CMOs for As 4s 2 lone-pairs (Fig. S5b) consist of 7 pairs of constructive/destructive combination between the two As 7 cages (left versus right), which are readily transformed to two sets of 7 orbitals, each for one As 7 cage. For example, HOMO-46 and HOMO-45 are responsible for one completely bonding 7c-2e σ orbital on the left As 7 cage and one on the right. Of these 7 orbitals in the As 7 cage, one is for the lone-pair on As(i), and the remaining 6 are again three pairs of constructive/destructive combination between two As 3 triangles (front versus rear). The three orbitals for each As 3 are in a typical bonding/nonbonding/antibonding combination, eventually leading to three As 4s 2 lone-pairs. For As-As and Pd-As single bonds, we only take the Pd-As bonds as an example (Fig. 2b). Here, HOMO-2, HOMO-10, HOMO-28, and HOMO-32 involve substantial As components, whereas the remaining four CMOs have significant Pd 4d components. Thus, it can be roughly considered that two Pd centers contribute 8 electrons for Pd-As bonding, with the eight As(iii) centers matching the remaining half, which is an ideal case for eight Pd-As single bonds within two PdAs 4 units.
While the above CMO analyses can be difficult to comprehend, the AdNDP method developed by Boldyrev and coworkers 41 offers an alternative, straightforward way to describe such bonding elements. AdNDP is an extension of NBO analysis. It represents the electronic structure of a molecular system in terms of n-center two-electron (nc-2e) bonds, with the value of n ranging from one to the total number of atoms in the molecule. Therefore, AdNDP analysis recovers not only the classical Lewis elements (lone-pairs and 2c-2e bonds), but also delocalized nc-2e bonds. Indeed, the AdNDP data for cluster 1 elegantly recover fourteen As 4s 2 lone-pairs (Fig. 3b), sixteen 2c-2e As-As σ bonds (Fig. 3d), and eight 2c-2e Pd-As σ bonds (Fig. 3e).
Of the remaining 18 electrons in 1, the 4 extra charges are situated on two As(i) and two Pd centers (Table 1), which carry a charge of −0.602 and −0.671 |e|, respectively. To be precise, the charge on an As(i) center smears moderately along the As8-As6-As14 or As7-As5-As13 chains (Fig. 1), which collectively carry a charge of −0.924 |e|, nearly ideal for a single charge. Likewise, the charge on a Pd center smears moderately over a PdAs 4 unit, collectively carrying a charge of −1.079 |e|. From Fig. S5a, one can identify 6 CMOs that are largely Pd 4d based, which are either lone-pairs or nonbonding. Specifically, HOMO-19/HOMO-13 are primarily the constructive/ destructive combination of Pd d yz atomic orbitals (AOs) between two Pd centers; these can be transformed to two Pd d yz lone-pairs. Likewise, HOMO-18/HOMO-14 are readily transformed to two Pd d z 2 lone-pairs. On the other hand, HOMO-15 and HOMO-16 are constructive/destructive combination of Pd − d x y 2 2 or d xz AOs; these are nonbonding in nature between the Pd centers but cannot be transformed to Pd 4d lone-pairs (because their corresponding destructive/constructive CMOs are used for Pd-As σ bonds). In short, these CMOs correspond to four Pd 4d lone-pairs and two pairs of nonbonding Pd 4d electrons. The above 6 CMOs are presented as six Pd 4d lone-pairs in the AdNDP analysis (Fig. 3a); the latter is an effective and approximate description only.
The "missing" complementary Pd 4d based orbitals (four of them) are involved in the Pd-As bonding (Fig. 2b). Furthermore, two As 4p lone-pairs on As(i) are identified in the CMOs (Fig. S5c) and AdNDP data (Fig. 3c). In short, these lone-pairs or nonbonding CMOs further consume 16 electrons. As a consequence, 92 electrons in 1 (out of 94 in total) can be faithfully described as classical Lewis elements: Pd 4d, As 4s, and As 4p lone-pairs (or nonbonding CMOs); 2c-2e As-As single bonds; and 2c-2e Pd-As single bonds.
As for the nature of the HOMO (Fig. 2c), on each PdAs 4 unit it is completely delocalized and completely bonding, due to radial overlaps of As p and Pd d/s/p AOs; see Table 2. The Pd and As components in the HOMO are roughly 55% versus 45%, indicating highly covalent Pd-As bonding. In effect, the HOMO may be cut in halves so that each PdAs 4 unit has a delocalized 5c-1e σ bond. The shape of such 5c-1e σ bond (half bond, to be exact) is consistent with σ-aromaticity, although its electron-counting is only half of that from the (4n + 2) Hückel rule, with n = 0. Effectively, a 5c-1e σ half bond within a PdAs 4 unit in 1 can be considered to be 50% as aromatic as a 5c-2e σ bond. Thus, cluster 1 possesses σ aromaticity. We stress that this conclusion is reached entirely on the bases of the CMO and AdNDP analyses.
NICS calculations are indeed in line with the idea of σ-aromaticity in 1 (Table S2), whose NICS(1) and NICS(1) zz values are −23.31 and −16.43 ppm, respectively, at 1 Å above the PdAs 4 fragment. It is known that electron density affects the result of NICS analysis. We thus also calculated the corresponding NICS(1) and NICS(1) zz values for the model neutral complex 3, which are −49.85 and −43.46 ppm, respectively, at 1 Å above the PdAs 4 fragment. The NICS values for both tetraanion 1 and neutral 3 clusters are highly negative, consistent with σ-aromaticity. It is emphasized here that the CMO analyses and electron counting are the most fundamental tools in elucidating aromaticity of a molecular system; complementary analyses (such as NICS calculations) only offer additional or independent support for the assessment. Since NICS as a criterion of aromaticity has been documented to fail in a number of cases (in particular in metal clusters), we are inclined to state that the correlation between the NICS values and the assessment of aromaticity in this case can be "a mere coincidence" 52 , as one referee suggests. Indeed, we are not claiming σ aromaticity in this system because it has negative NICS values; rather σ aromaticity is established through extensive CMO and AdNDP analyses, as stated above.
Apart from NICS, we also performed calculations using the quantum theory of atoms-in-molecules (QTAIM) 53 . Here, the Multiwfn 54 program is used to generate the para-delocalization index (PDI 55 ; As3-As7-As9-As12-As14-As16) and the multicenter bond aromaticity index (MCI) 56 . The PDI and MCI data for tetraanion 1 and model neutral cluster 3 are presented in Table S2, which are compared with those of [Au 2 Sb 14 ] 4− (2) and benzene (C 6 H 6 ). It is shown that: (a) PDI π and MCI π of species 1-3 are zero, consistent with the nature that these species do not have π aromaticity; (b) The PDI σ and MCI σ values are 0.030-0.121 and 0.271-0.315, respectively, consistent with σ aromaticity. Note that these PDI σ /MCI σ values are to be compared to the PDI π (0.093) and MCI π (0.383) of benzene, because species 1-3 are σ aromatic and benzene has π aromaticity.
In summary, excluding the 44 lone-pairs (or nonbonding electrons), tetraanion 1 is a 50-electron system. Of these, the 2c-2e σ bonds consume 48. Notably, the Pd and As(iii) centers participate evenly in Pd-As 2c-2e  σ bonds, with the Pd 2 dumbbell contributing 8 electrons and the eight As(iii) centers collectively matching the remaining 8. The only delocalized CMO uses the remaining 2 electrons, which are equally split in between two PdAs 4 units, rendering σ-aromaticity for 1. Our analyses indicate that no Pd-Pd covalent bond is present in 1, in contrast to the claim in ref. 24.
Owing to the fact that the Pd 2 dumbbell in 1 has six Pd 4d based nonbonding CMOs or lone-pairs ( Fig. S5a) and that each Pd center carries one extra charge, the oxidation state can be formally viewed as Pd(III), as proposed by Eichhorn and coworkers. However, this is only an oversimplified ionic picture, in which cluster 1 is described as two [As 7 ] 5− cages linked via a Pd 2 6+ dumbbell 24 . The actual bonding in 1 deviates fundamentally from the above picture and indeed the eight Pd-As bonds appear to be highly covalent, with half versus half contributions from Pd 2 and As 8 in the cube (Fig. 2b), which is consistent with their closeness in electronegativity (Pd: 2.20; As: 2.18 [Au 2 Sb 14 ] 4− Cluster as a σ-Aromatic Model "Charged Molecular Alloy". The intriguing bonding and σ-aromaticity in D 2h [Pd 2 As 14 ] 4− (1) have stimulated us to "design" additional charged molecular alloys. One example is the D 2h [Au 2 Sb 14 ] 4− (2) cluster, which differs from 1 by two electrons (Au 5d 10 6s 1 versus Pd 4d 10 ). Structure-wise, the D 2h structure of [Au 2 Sb 14 ] 4− is a true minimum on the potential energy surface and can be considered as a model cluster (Fig. 4a). However, the bonding of 1 and 2 differs markedly.
D 2h [Au 2 Sb 14 ] 4− (2) has 96 valence electrons. Excluding the Au 5d 10 , Sb 5s 2 , two bridging Sb 5p lone-pairs, which use 52 electrons (Figs 4b and S6), cluster 2 can be viewed as a 44-electron system. Of these, the Sb-Sb 2c-2e σ single bonds consume 32 electrons, whose corresponding CMOs are shown in Fig. S7a and AdNDP elements in Fig. 4b. Not surprisingly, these sixteen CMOs show one-to-one correspondence to those of 1 (Fig. S7a versus  Fig. 2a). The extra charges in 2 are distributed on the two bridging Sb centers and, to a lesser extent, two Au centers (Table S3), which are similar to 1 except that the charges around Au seems to be more evenly distributed over the AuSb 4 units. In fact, the HOMO and HOMO-1 of 2 are based on Sb(iii) and Sb(i) centers, suggesting that the net charge on Au is probably due to intramolecular charge transfer (rather than the extra charges), which is in line with the difference in electronegativity between Au (2.54) and Sb (2.05). The peripheral Sb-Sb links within AuSb 4 are elongated (Sb12-Sb16: 3.76 Å; Sb16-Sb3: 3.94 Å) with respect to Sb-Sb single bonds, and thus there is no direct Sb-Sb covalent bonding in the periphery. Furthermore, no classical 2c-2e Au-Sb single bonds are present in 2 and the Au-Au interaction is minimal, the latter showing a "magic" distance of 3.11 Å, which is typical for aurophilicity 50,51 and in line with the calculated Wiberg index (0.186; Table S3). In contrast to 1, cluster 2 has 12 delocalized electrons.
Consistent with the bonding elements discussed above, the calculated bond distances of Sb-Sb and Au-Sb are 2.77-2.88 and 2.73 Å, respectively, which are compared to their recommended values 48 of single bonds: 2.80 and 2.64 Å. Clearly, Sb-Sb single bonds are well defined, whereas Au-Sb bond is elongated with respect to single bond, indicating a delocalized bonding system in the AuSb 4 units.
The 12 delocalized electrons in cluster 2 occupy six CMOs as depicted in Fig. S7b. Their corresponding AdNDP elements are shown in Fig. 4b, bottom row. Thus, each square-planar AuSb 4 unit in 2 supports a completely bonding 5c-2e σ bond, as well a pair of partially bonding ones. The three σ bonds define a σ sextet and follow the (4n + 2) electron-counting of the Hückel rule, which is closely analogous to the π sextet in benzene; see comparisons in Fig. 5. It is thus imperative to claim σ-aromaticity for D 2h [Au 2 Sb 14 ] 4− (2), with n = 1. Indeed, the calculated NICS values for 2 are negative (Table S2).
We believe σ-aromaticity is governing the stability of D 2h [Au 2 Sb 14 ] 4− (2), because otherwise there would be neither Sb-Sb nor Au-Sb bonding within the two AuSb 4 fragments. We have recently described a synthetic [Au 2 Sb 16 ] 4− compound 38 , which is highly distorted with quasi-C 2 point group symmetry, also featuring two σ sextets. In carefully designed and controlled syntheses, we anticipate the D 2h σ-aromatic [Au 2 Sb 14 ] 4− (2) complex to be made as bulk materials in laboratory.
Lastly, we briefly comment on the oxidation state of Au in 2. Here, the two-pairs of extra charges occupy the Sb(i) and Sb(iii) sites; see HOMO-1 and HOMO (Figs S6 and S7). The Au center does not form 2c-2e bonds with Sb, nor does it bound covalently with the other Au center. Thus, the Au centers in 2 should be considered as Au(0) and cluster 2 is best described as two [Sb 7 ] 2− linked via Au 2 0 . While [E 7 ] 3− (E = P, As, Sb) 21 are known as naked cluster anions, the [As 7 ] − or [Sb 7 ] 2− cages in 1 and 2 do not appear odd. Our recent work 36,37 shows that all-metal aromatic [Sb 3 Au 3 Sb 3 ] 3− sandwich complex has two [Sb 3 ] 1.5− ligands, although [Sb 3 ] 3− is routine in solution syntheses.

Concluding Remarks
In conclusion, we have presented a density-functional theory study on the structures and chemical bonding of a synthetic "charged molecular alloy", D 2h [Pd 2 As 14 ] 4− (1), as well as those of a D 2h [Au 2 Sb 14 ] 4− (2) model cluster. Both 1 and 2 possess two distorted square-planar MX 4 units that are virtually parallel to each other, and a dumbbell M 2 interconnects two X 7 cages to form the charged molecular alloys. While 1 and 2 differ only for two valence electrons, their nature of bonding differs markedly. There are only two delocalized σ electrons in cluster 1, conforming to the electron-counting for σ-aromaticity. In contrast, cluster 2 possesses two delocalized σ sextets (each being situated on an AuSb 4 unit), which render σ-aromaticity for 2. No covalent Pd-Pd or Au-Au bonding is observed in 1 and 2. The Pd and Au centers in 1 and 2 clusters are virtually isolated, single atoms in nature 50, 51 , with specific oxidation states, which make them interesting model systems for the so-called "single atom catalysis" 57 . The designer σ-aromatic D 2h [Au 2 Sb 14 ] 4− (2) cluster also invites forth-coming synthetic effort.
Methods Section. Cluster structure of [Pd 2 As 14 ] 4− (1) is optimized using density-functional theory (DFT) 40 at the PBE0/def2-TZVP level 39,58 , whose performance has been tested recently in relevant compound systems 36,37 . We also constructed and optimized the structure of [Au 2 Sb 14 ] 4− (2) at the same level of theory. To elucidate chemical bonding in the systems, the CMO and AdNDP 41 analyses were performed and the results were visualized using the Molekel program 59 . NBO analysis 42 was carried out to obtain the natural atomic charges and Wiberg bond indices. NICS 43 calculations were performed to assess the nature of aromaticity in the systems. The AdNDP analyses were performed using the AdNDP program 41 and all other calculations and analyses were carried out using the Gaussian 09 software package 60 . Orbital composition analysis was performed using the Multiwfn program 54 .