Bismuth Polycations Revisited: Alternative Synthesis and Electronic Structure of Bi 6 Br 7 , and Bonding in Main-Group Polyatomic Ions from a Direct Space Perspective

: A bismuth subbromide, Bi 6 Br 7 , was synthesized in the form of single crystals using the reaction between Bi and Hg 2 Br 2 in a gradient furnace. Its crystal structure was reinvestigated by low-temperature single-crystal X-ray di ﬀ raction ( Pnnm , a = 15.4996(6) Å, b = 23.6435(7) Å, c = 9.0231(2) Å, Z = 8, R 1 = 0.041, wR all = 0.087). Based on the di ﬀ raction data, the structure description was revised as containing Bi 95 + cluster polycations and 1 ∞ [Bi 3 Br 145 − ] ladder-like anions. DFT calculations of band structure showed the compound to be a narrow-gap semiconductor with a band gap of ca. 1.3 eV, with the nature of the compound as ionic salt conﬁrmed by charge density analysis. Direct-space bonding analysis based on the ELF topology and QTAIM partitioning, performed for all known homoatomic bismuth polycations, as well as isoelectronic main-group metal ions, shows patterns of localized pairwise and three-center bonding forming the frameworks of the clusters. In addition to obtaining new data, the use of highly augmented basis sets allowed us to revise and amend several previously made conclusions regarding bonding in such species.


Introduction
From a chemical point of view, bismuth is a pretty remarkable element, which in a textbook sense, has only a couple of stable oxidation states. However, in the real world, bismuth has incredibly rich subvalent chemistry, for which it was coined "The Wonder Metal" by J.D. Corbett [1], one of the pioneers of bismuth cluster studies. And that was before the true diversity of bismuth subvalent compounds was known! Among particularly interesting objects in the field are bismuth cluster polycations, i.e., electron-deficient polyatomic clusters of varying geometry. The first solid compound containing bismuth cluster polycation was bismuth "monochloride" Bi 6 Cl 7 [2], discovered in the early 60s, and the cation was Bi 9 5+ . In the next six decades, other binary and ternary compounds featuring this cation followed [3][4][5][6][7][8][9][10][11][12][13]. A number of other species were also discovered, such as 'classical' Bi 5 3+ [14][15][16][17][18][19] and Bi 8 2+ [13,14,[20][21][22][23][24], and more recent additions Bi 5 + [25,26], Bi 6 2+ [26,27], and Bi 10 4+ , although the latter has not been obtained as a true homoatomic unity and was mostly centered by palladium or platinum, or on rare occasion even by gold [28][29][30][31] (we leave its capped forms outside the scope of this paper as their homoatomic nature is questionable, as well as other heteroatomic bismuth clusters). The compounds that contain these polycations are essentially complex salts, with counterions for bismuth clusters being metal halide anions. They are typically synthesized either from melts or non-aqueous solutions. A comprehensive review of synthetic approaches to polycations of group 15-16 elements is given in [32], with additional information on the use of ionic liquids provided in [33]. From a topological point of view, these bismuth clusters are fairly unique, however, not completely without relatives. Bismuth's lighter group 15 neighbor, antimony, features far less abundant homopolyatomic cation chemistry, although Sb 5 3+ [16] and Sb 8 2+ [17,34] have been reported, both having matching polyhedral bismuth counterparts. And then there are also naked clusters of group 14 elements, Zintl ions E 5 2− and E 9 4− (E = Si, Ge, Sn, Pb) [35], appearing in various intermetallic and coordination compounds, which are isoelectronic and (more or less) isostructural to Bi 5 3+ and Bi 9 5+ , although to maintain the same electron count they obviously need to be anions.
The compounds that feature bismuth cluster polycations turned out to be rather difficult to work with, due to them often being air-and moisture-sensitive, and particularly complex objects for the X-ray crystallography, for both technical and fundamental reasons (e.g., a tendency to show pseudo-symmetry or form disordered arrangements) [22,24]. As a consequence, some of the reported structures, particularly those from earlier times, were plagued by errors. The quality of structure determination was much improved in more recent times. Some of the older structures, like Bi 6 Cl 7 and Bi 8 (AlCl 4 ) 2 [36], were re-determined with improved quality, which in the case of Bi 6 Cl 7 , led to a slight reinterpretation of its anionic arrangement. More recently, in 2013, these two structures were reinvestigated again, based on the XRD data obtained from crystals synthesized in ionic liquids [37]. Bi 6 Br 7 was also recently synthesized using the reaction of bismuth with its tribromide in [BMIm]Cl·2AlCl 3 [37]. Its crystal structure was reinvestigated by low-temperature single-crystal XRD, however, the authors provided no description of structural details.
In our studies, initially focused on optimizing crystal growth technique for another bismuth subbromide, Bi 4 Br 4 , we have come up with an alternative way of producing crystals of Bi 6 Br 7 .
Having obtained good quality single crystals, we investigated them by means of XRD, primarily to see if there were differences with what was already reported on the compound. And since the structure description still stood as it was given in 1978, we decided to update it based on our data. Improved structure description and atom localization gave us an opportunity to reliably study the band structure of the compound and the topology and bonding in polycations.
To put electronic structure and bonding information in proper context, we also reinvestigated bonding in bismuth polycations and related isoelectronic homoatomic main-group clusters within the concept of R. Bader's Quantum theory of atoms in molecules (QTAIM) [38] via topological analysis of the electron localization function (ELF) [39][40][41]. Almost exactly twenty years ago, bonding analysis, based on a combination of natural bond orbital analysis and ELF topology for bismuth polycations that were known at the time, was reported [42], which was the first comprehensive study of this type. However, with bonding not being the main focus of the study, and with ELF being in its relatively early days of practical use, some of the conclusions made within that paper were not entirely solid. Since then, with the rapid progress of the density functional theory (DFT) applications in chemistry and growing number of bonding analysis tools, along with the discovery of new clusters and emergence of more in-depth approaches to the analysis of bonding indicators, the need to revisit and update that material has been long overdue (although selected cations have received better treatment in more recent publications [10,[28][29][30]). With this research, we offer a sufficiently expanded and amended picture of homoatomic bonding in bismuth cluster polycations and their isoelectronic analogues of group 14 and 15 from the point of direct-space analysis.

Synthesis and Primary Characterization
Bismuth metal (powder, 99.98%) and Hg 2 Br 2 (99.8%) were used as starting materials for the synthesis. A batch of 1.15 g of Bi and 2.33 g of Hg 2 Br 2 was placed into a silica tube (210 mm long, Crystals 2020, 10, 940 3 of 15 16 mm in diameter), flame-sealed under vacuum (ca. 10 −2 Torr), and placed into a horizontal gradient furnace, with the temperatures at the opposite ends of the tube being 230 and 265 • C. Chemicals were placed at the hotter end. The experiment was kept running for 6 days, after which it was cooled down to room temperature. Visual inspection of the tube showed two kinds of crystals formed, elongated prismatic (or thick needle-like) ones near the colder end of the tube, and thin needles almost above the starting materials. Both types of crystals were black with a hint of metallic luster; also visible in the tube was liquid mercury.
Energy dispersive X-Ray spectroscopy (EDS) using Jeol/Nikon Neoscope 6100 (U acc = 20 kV, internal calibration, 3-5 points per crystal) showed both the types of crystals to be binary bismuth bromides, with the first type having Bi:Br ratio of 1, and the second of 0.88, which allowed us to identify the compounds as Bi 4 Br 4 and Bi 6 Br 7 , respectively. The latter appeared as well-shaped rectangular crystals up to 1 mm long with a thickness of ca. 0.2-0.3 mm. This appears as a bit of a surprise, since the method involving chemical transport with HgBr 2 [4] or Hg 2 Br 2 [43] was supposed to produce Bi 4 Br 4 , while the other subbromide was described as forming from Bi-BiBr 3 melts [4]. However, the location of Bi 6 Br 7 crystals at the top side of a tube indicated that the gas phase reaction was involved in crystal growth. The binary Bi-Br system is rather unique among the Bi-X systems (X = Cl, Br, I), since it features both salt-like polycationic subhalides like Bi 6 Br 7 and bismuth subhalides with extended bismuth fragments of the Bi 4 Br 4 family, while for X = Cl only polycation-containing phases exist, and for X = I only subhalides with extended structures are found. And the way of separating them was to use different synthetic conditions. Obviously, to stabilize bismuth polycations, suitable anions must be present in the system, which makes synthetic approaches involving BiBr 3 melts or BiBr 3 -containing solutions highly favorable. In our case, we do not introduce extra BiBr 3 in the system, so the anionic part in our case must be formed in the process of redox reactions involving Bi and Hg 2 Br 2 .

X-ray Crystallography
X-ray diffraction data for a suitable single crystal of Bi 6 Br 7 was recorded at 100 K using automated Stoe IPDS diffractometer (graphite monochromator, MoK α -radiation, λ = 0.71073 Å) equipped with Imaging Plate detector. The structure was solved by direct methods and refined against F o 2 using SHELX-2018 package [44]. Data collection and structure refinement details, as well as atomic coordinates and atomic displacement parameters, are given in Tables S1 and S2, respectively, in the Supplementary Materials. Select interatomic distances are given in Table 1. A summary of crystallographic data for the single-crystal experiment is available from CCDC (ref. number 2032218).

Computational Details
Band structure calculations on crystalline Bi 6 Br 7 were performed on the density-functional theory (DFT) level utilizing the Projector Augmented Wave method (PAW) as implemented in the pseudopotential Vienna Ab initio Simulation Package (VASP) code [45,46]. For the modeling, experimentally determined structure was transformed into the P1 space group. The PBE exchangecorrelation functional [47] of the GGA-type was used in the calculations. Spin-orbit coupling Crystals 2020, 10, 940 4 of 15 was accounted for in the calculations. The energy cut-off was set at 500 eV with a 12 × 6 × 12 Monkhorst-Pack [48] k-point mesh used for Brillouin zone sampling. Atomic charges were analyzed according to Bader's QTAIM approach [38].
Isolated cluster calculations were performed on the DFT level using hybrid B3LYP [49,50] exchange-correlation functional as implemented in the Orca 4.2.1 package [51,52]. For the polycations, a combination of def2-QZVPPD [53] basis sets and relativistic effective core potentials (ECP) was used: 60-electron core ECP60MDF [54] for Bi and Pt, 28-electron core ECP28MDF [54] for Sb and Pd. For the geometry optimizations of polyanions, ma-def2-TZVPP [55] basis sets were used, all-electron for Ge, and in combination with ECP28MDF for Sn and ECP60MDF for Pb. For energy calculations for optimized structures, def2-QZVPPD basis sets with respective ECP's were used. Constrained optimization retaining point symmetry group was performed.
The electron localization function (ELF) was calculated and analyzed using the Multiwfn 3.7 package [56]. Structure visualization was done using the VESTA package [57]. Band structures near the Fermi level were plotted using the wxDragon package [58].

Results and Discussion
3.1. Crystal Structure of Bi 6 Br 7 and the Geometry of Bi 9

5+
The crystal structure of Bi 6 Br 7 is shown in Figure 1, and isolated bismuth polycations in Figure 2. exchange-correlation functional [47] of the GGA-type was used in the calculations. Spin-orbit coupling was accounted for in the calculations. The energy cut-off was set at 500 eV with a 12 × 6 × 12 Monkhorst-Pack [48] k-point mesh used for Brillouin zone sampling. Atomic charges were analyzed according to Bader's QTAIM approach [38]. Isolated cluster calculations were performed on the DFT level using hybrid B3LYP [49,50] exchange-correlation functional as implemented in the Orca 4.2.1 package [51,52]. For the polycations, a combination of def2-QZVPPD [53] basis sets and relativistic effective core potentials (ECP) was used: 60-electron core ECP60MDF [54] for Bi and Pt, 28-electron core ECP28MDF [54] for Sb and Pd. For the geometry optimizations of polyanions, ma-def2-TZVPP [55] basis sets were used, all-electron for Ge, and in combination with ECP28MDF for Sn and ECP60MDF for Pb. For energy calculations for optimized structures, def2-QZVPPD basis sets with respective ECP's were used. Constrained optimization retaining point symmetry group was performed.
The electron localization function (ELF) was calculated and analyzed using the Multiwfn 3.7 package [56]. Structure visualization was done using the VESTA package [57]. Band structures near the Fermi level were plotted using the wxDragon package [58].

Crystal Structure of Bi6Br7 and the Geometry of Bi9 5+
The crystal structure of Bi6Br7 is shown in Figure 1, and isolated bismuth polycations in Figure  2.  The experiment at low temperature, with a relatively small but well-shaped and well-reflecting single crystal, produced the results that generally agree with the data published in [37], which exchange-correlation functional [47] of the GGA-type was used in the calculations. Spin-orbit coupling was accounted for in the calculations. The energy cut-off was set at 500 eV with a 12 × 6 × 12 Monkhorst-Pack [48] k-point mesh used for Brillouin zone sampling. Atomic charges were analyzed according to Bader's QTAIM approach [38]. Isolated cluster calculations were performed on the DFT level using hybrid B3LYP [49,50] exchange-correlation functional as implemented in the Orca 4.2.1 package [51,52]. For the polycations, a combination of def2-QZVPPD [53] basis sets and relativistic effective core potentials (ECP) was used: 60-electron core ECP60MDF [54] for Bi and Pt, 28-electron core ECP28MDF [54] for Sb and Pd. For the geometry optimizations of polyanions, ma-def2-TZVPP [55] basis sets were used, all-electron for Ge, and in combination with ECP28MDF for Sn and ECP60MDF for Pb. For energy calculations for optimized structures, def2-QZVPPD basis sets with respective ECP's were used. Constrained optimization retaining point symmetry group was performed.
The electron localization function (ELF) was calculated and analyzed using the Multiwfn 3.7 package [56]. Structure visualization was done using the VESTA package [57]. Band structures near the Fermi level were plotted using the wxDragon package [58].

Crystal Structure of Bi6Br7 and the Geometry of Bi9 5+
The crystal structure of Bi6Br7 is shown in Figure 1, and isolated bismuth polycations in Figure  2.  The experiment at low temperature, with a relatively small but well-shaped and well-reflecting single crystal, produced the results that generally agree with the data published in [37], which The experiment at low temperature, with a relatively small but well-shaped and well-reflecting single crystal, produced the results that generally agree with the data published in [37], which significantly improves atom localization and atomic coordinates as compared to [4]. This leads to interatomic distances precision improved by ca. half an order of magnitude. In [4], the average esd in Bi-Bi bond length is ca. 0.007 Å, while in our refinement, it is ca. 0.0013 Å. Similarly, the average esd in Bi-Br bond length was improved from ca. 0.015 [4] to ca. 0.0026 Å. The results from [37] are very similar to ours. Apparently, different synthetic approaches did not lead to variations in the structure.
The authors of [37] decided not to make direct comparisons to the structure from 1978, citing large temperature differences of analysis, making such comparisons not applicable. We do not believe that in this case, major effects take place, like phase transition, which separate the data obtained at 100 and 298 K. In fact, the comparison of our data with [4] and [37] confirms this is not the case, and the structure is essentially the same. The results of side-by-side comparisons of the crystallographic data from this work, [4], and [37], performed with the help of COMPSTRU routine [59], are given in Supplementary Materials (Tables S3-S11). Therefore we think that the discussion of the structure after redetermination at low temperature is valuable and relevant to room temperature. Such discussion is particularly important since it affects further studies of the electronic structure of the compound. The general description of the structure remains similar to that obtained at room temperature in 1978 [4], especially in the cationic part. Bi 9 5+ polycations in the structure (see Figure 2) are close to the D 3h symmetry (trigonal tricapped prism) and feature Bi-Bi distances in the 3.082-3.302 Å range, while in [4] they are characterized as being inbetween D 3h and C 4v (monocapped square antiprism) symmetries, but better described by D 3h , with the interatomic distances falling into the 3.087-3.320 Å range. Our data show, based on the thermal ellipsoid analysis, that the cations are not prone to the significant orientational disorder and rather well-localized. Following the logic of the authors of [36], who changed the description of the anionic part of Bi 6 Cl 7 from isolated anions to a polymer-like framework, and based on our own analysis of crystallographic data, the description of the anionic part of the structure of Bi 6 Br 7 also needs to be revised as compared to that provided in [4], where it was described as consisting of BiBr 5 2− (Bi-Br = 2.798-3.148 Å) and  Figure 3a). We prefer to describe these polyhedra as edge-sharing, but even if one chooses to discard 3.15 Å as the non-bonding distance (although in [4] it is a part of the bismuth-centered bromine polyhedra, forming one-dimensional chains along the c axis, linked via vertices by dimeric edge-sharing BiBr 6 octahedral units (see Figure 3). Thus, 1 ∞ [Bi 3 Br 14 5− ] forms a ladder-like framework with side rails formed by pentagonal bipyramids and rungs by edge-sharing octahedra (see Figure 3a). The voids left by missing rungs are filled with bismuth polycations, forming two Bi-Br contacts with a length of 3.266 Å. This picture is similar to the one provided in [36] for Bi 6 Cl 7, which is only logical since these two compounds are isotypic. Based on the comparison of our data with that published in [37] for Bi 6 Cl 7 and Bi 6 Br 7 (see Supplementary Materials, Tables S12-S14), we believe that this conclusion is consistent with the data.

Electronic Structure of Bi6Br7
Band structure of Bi6Br7 was evaluated based on the DFT calculations. Figure 4 shows the calculated total (TDOS) and projected (PDOS) density of states (DOS) plot near the Fermi level. According to the calculations, Bi6Br7 is a narrow-gap semiconductor with the band gap value of ca. 1.3 eV. The top of the valence band features major contributions from Bi 6p-states of Bi9 5+ and bromine 4p-states, and a relatively minor contribution from Bi 6p-states from polymeric bromobismuthate anions. Polycationic bismuth 6s pairs are located over 9 eV below the Fermi level and are very well localized, thus confirming their chemical inertness. The bottom of the conduction band is contributed equally by p-states of Bi and Br from the anions, but the major contribution is from 6p-states of Bi in Bi9 5+ . All these states are located in the same energy range, which means that electronic transport can be conducted between bismuth and bromine states in the anionic framework, which explains the predicted narrow-gap semiconductor behavior of the compound. However, based on the contributions from Bi9 5+ 6p-states to the conduction and valence bands, we cannot completely rule out that polycations may participate in the electric conductivity, rather than the latter being facilitated through the anionic framework only. This appears to be surprising for the compound built like an ionic salt, however, looking at the Bi-Br contacts, we observe that the longest inter-anionic distances of 3.112 Å (or 3.149 Å , depending on vertex-or edge-sharing model) are just ca. 0.15 Å (or ca. 0.12 Å ) shorter than the shortest distance from Bi9 5+ to Br (Bi2-Br8 = 3.266 Å ), which might explain how bismuth clusters may theoretically participate in the electronic transfer.

Electronic Structure of Bi 6 Br 7
Band structure of Bi 6 Br 7 was evaluated based on the DFT calculations. Figure 4 shows the calculated total (TDOS) and projected (PDOS) density of states (DOS) plot near the Fermi level. According to the calculations, Bi 6 Br 7 is a narrow-gap semiconductor with the band gap value of ca. 1.3 eV. The top of the valence band features major contributions from Bi 6p-states of Bi 9 5+ and bromine 4p-states, and a relatively minor contribution from Bi 6p-states from polymeric bromobismuthate anions. Polycationic bismuth 6s pairs are located over 9 eV below the Fermi level and are very well localized, thus confirming their chemical inertness. The bottom of the conduction band is contributed equally by p-states of Bi and Br from the anions, but the major contribution is from 6p-states of Bi in Bi 9 5+ . All these states are located in the same energy range, which means that electronic transport can be conducted between bismuth and bromine states in the anionic framework, which explains the predicted narrow-gap semiconductor behavior of the compound. However, based on the contributions from Bi 9 5+ 6p-states to the conduction and valence bands, we cannot completely rule out that polycations may participate in the electric conductivity, rather than the latter being facilitated through the anionic framework only. This appears to be surprising for the compound built like an ionic salt, however, looking at the Bi-Br contacts, we observe that the longest inter-anionic distances of 3.112 Å (or 3.149 Å, depending on vertex-or edge-sharing model) are just ca. 0.15 Å (or ca. 0.12 Å) shorter than the shortest distance from Bi 9 5+ to Br (Bi2-Br8 = 3.266 Å), which might explain how bismuth clusters may theoretically participate in the electronic transfer.

Electronic Structure of Bi6Br7
Band structure of Bi6Br7 was evaluated based on the DFT calculations. Figure 4 shows the calculated total (TDOS) and projected (PDOS) density of states (DOS) plot near the Fermi level. According to the calculations, Bi6Br7 is a narrow-gap semiconductor with the band gap value of ca. 1.3 eV. The top of the valence band features major contributions from Bi 6p-states of Bi9 5+ and bromine 4p-states, and a relatively minor contribution from Bi 6p-states from polymeric bromobismuthate anions. Polycationic bismuth 6s pairs are located over 9 eV below the Fermi level and are very well localized, thus confirming their chemical inertness. The bottom of the conduction band is contributed equally by p-states of Bi and Br from the anions, but the major contribution is from 6p-states of Bi in Bi9 5+ . All these states are located in the same energy range, which means that electronic transport can be conducted between bismuth and bromine states in the anionic framework, which explains the predicted narrow-gap semiconductor behavior of the compound. However, based on the contributions from Bi9 5+ 6p-states to the conduction and valence bands, we cannot completely rule out that polycations may participate in the electric conductivity, rather than the latter being facilitated through the anionic framework only. This appears to be surprising for the compound built like an ionic salt, however, looking at the Bi-Br contacts, we observe that the longest inter-anionic distances of 3.112 Å (or 3.149 Å , depending on vertex-or edge-sharing model) are just ca. 0.15 Å (or ca. 0.12 Å ) shorter than the shortest distance from Bi9 5+ to Br (Bi2-Br8 = 3.266 Å ), which might explain how bismuth clusters may theoretically participate in the electronic transfer.  calculated charge for Bi 9 is ca. +3.7, and its counterion ca. −3.7, instead of +5 and −5, respectively. Yet, these are still very high calculated charges, considering that Bi in the polyanion, which is formally +3, has a calculated charge of less than +1.5. Therefore, we may conclude that the description of this compound as an ionic salt still holds well, and there are obviously strong ionic interactions between the counterparts in the structure.

Bonding in Bismuth Polycations and Isoelectronic Main-Group Metal Clusters: 3D-Aromaticity vs. Localized Bonding
Bonding was investigated by analyzing ELF topology for isolated clusters. ELF has three types of attractors, namely atomic shells, lone pairs, and localized chemical bonds, and it is the latter type that is of special interest to us, since it shows us bonding interactions in the clusters. By integrating electron density over the bonding ELF basins, we can quantify the number of electrons participating in bonding for each particular attractor. Geometries and ELF isosurfaces for bismuth polycations and isoelectronic main-group metal clusters are depicted in Figure 5, and symmetry and bond basin characteristics in Table 2.
Before moving to the analysis of specific clusters, general observation must be made on the ELF basis dependence. This function is generally considered very robust and fairly insensitive to a computational method. This may well be true as far as the level of theory is concerned, however, ELF might be sensitive towards basis sets used, as discussed in [60]. While evaluating computational conditions for the ELF calculations for post-transition elements, we have found out that a basis set well augmented with polarization and diffuse functions is essential to observe all the ELF features. For instance, the use of reliable energy-optimized LANL2DZ Los Alamos valence double-zeta basis with Hay-Wadt ECP [61] or its triple-zeta LANL2TZ version leads to almost no non-atomic centered ELF attractors visible while scanning the localization parameter. The use of more advanced Stuttgart/Dresden family ECPs [62] does not change the picture, as long as they are paired with DZ or TZ quality basis sets with minimum diffuse or polarization functions. We did not specifically study the effect of small core (SC) vs. large core (LC) ECPs on ELF, yet a limited number of tests indicates that 60-electron core ECPs produce better results for Bi and Pb than 78-electron core ones.

Five-vertex Polyhedra
Five vertex polyhedra are represented by isoelectronic Bi 5 3+ , Sb 5 3+ , Ge 5 2− , Sn 5 2− , Pb 5 2− in the shape of trigonal bipyramid, and a single Bi 5 + as a square pyramid. The ELF topology of the latter shows four non-atomic attractors, which are localization domains corresponding to the Bi-Bi disynaptic bond basins in the base of the pyramid (Ω1), appearing at higher localization parameter (η) values, and another four domains (disynaptic Ω2) corresponding to the bonds between the base and the apical atom at lower η values (see Figure 5). The integration of electron density gives us basin populations of ca. 1.8 e for Ω1 and ca. 0.9 e for Ω2. Thus, this cluster has four almost 2c,2e-bonds in its base, and the apical atom is bonded to the base via four 1e-bonds. Quantitative part aside (as there was no basin population analysis done), this is very close to what the authors of [42] observed using ELF. However, the authors were laboring upon the assumption that η = 0.5 means "perfectly delocalized electrons", and thus observing Bi-Bi bonds to apical Bi at η = 0.53, they called them "almost fully delocalized". This rather common misinterpretation of ELF has been corrected since then [63]. All the ELF range from 0 to 1 is eligible for analysis, and, in fact, some bonds, particularly transition metal-based, only appear below η = 0.5, which does not make them perfectly delocalized. By its nature, ELF is only capable of displaying regions of electron localization so that truly delocalized electrons, such as conduction electrons, cannot be seen in ELF. Another possible source of misinterpretation of bonding in these objects is the mix-up between delocalized and multi-center bonds. ELF is perfectly capable of displaying the latter, but has not been intended to show the former. Therefore, multi-center bonds observed in the ELF topology can only be called 'delocalized' in the sense that they are not localized between pairs of atoms or in terms of a small number of electrons localized within bond basins. The results of the NBO analysis for Bi 5 + were better and worse at the same time. Better because they correctly predicted localized bonding between the base and the apex, and worse because they showed only three bonds and broke the symmetry of the cluster. Now we can revise the description of the bonding in Bi 5 + and show the essentially localized nature of bonding in it.   Next in line is the group of trigonal-bipyramidal clusters starting from Bi 5 3+ , whose ELF topology has certain differences as compared to all other isoelectronic clusters. ELF topology for Bi 5 3+ shows two types of localization domains, trisynaptic Ω1 corresponding to the three-center bonding basin over the center of the triangular face (population 0.72 e), and disynaptic Ω2 (population 0.70 e) corresponding to pairwise Bi-Bi interactions along the face, which, however, quickly merges with bismuth lone pair. Thus, Bi 5 3+ is held together by multi-center and pairwise bonding alike. However, no other isoelectronic cluster features Ω2-type basins, and all of them are apparently held together by three-center bonds. These bonds are close to 3c,2e for Sb 5 3+ , but the population of the respective basins in E 5 2− (E = Ge, Sn, Pb) is lower and further decreases with increased atomic number (see Table 2). This might be associated with the anionic nature of these clusters. Bi 5 3+ was also studied by NBO and ELF in [42]. Once again, NBO correctly predicted the presence of three-center bonds but completely broke the symmetry. ELF correctly showed three-center bonds at lower η values, which were called delocalized on that basis. Now, having calculated basin populations, we can agree that, at least, a three-center bond involving 0.7 e can be in a certain sense regarded as less well-localized than the same kind of bond involving 1.7 e. In the same sense, we can observe the decrease in bond basin population from Ge to Pb and take it as a possible indication of decreasing bond localization. Nevertheless, there are proper electron localization domains for all these bonds.

Six-vertex Polyhedra
A single cluster of this type is known; Bi 6 2+ . ELF topological analysis (see Figure 5, Table 2) shows two kinds of bond basins, disynaptic Ω1 (1.45 e) corresponding to pairwise interactions in the equatorial plane, and disynaptic Ω2 (0.95 e), corresponding to the pairwise bonds between apical and equatorial atoms. This picture is very close to what was observed for Bi 5 + . Once again, we have a cluster based on essentially localized two-center interactions. Only alternative geometries for Bi 6 2+ were studied in [42], so there is no material for comparison.
ELF topological analysis shows two kinds of bond basins, Ω1 (1.61 e) corresponding to pairwise bonds in the square bases (see Figure 5), and Ω2 (0.21 e) corresponding to pairwise interactions connecting the bases. Unlike Bi 5 3+ , these basins, although they appear in triangular faces, are not trisynaptic and do not correspond to three-center bonds, but rather disynaptic. Notably, Ω2-type basins have very small populations as compared to Ω1, which indicates that localized bonding between the bases of the prism is much weaker than in the bases. Yet, the fact that there are localized bonds connecting the bases of the prism explains the lack of rotational disorder in the Bi 8 2+ polycation. These localized interplanar bonds were completely missed in the ELF topology in [42], most likely, due to the lack of basis augmentation. Sb 8 2+ shows the same ELF features; two types of two-center bond basins, disynaptic Ω1 (1.52 e) and Ω2 (0.51 e). Notably, interplanar bonds, in this case, involve more electrons, although they are still short even of 2c,1e bond.

Nine-vertex Polyhedra
These include Bi 9 5+ and its isoelectronic analogues E 9 4− (E = Ge, Sn, Pb). Extensive discussion on the subject of nine-vertex cluster symmetry (D 3h vs. C 4v ) has already been made [11,42,64], therefore, we have investigated both trigonal tricapped prism and square antiprism for Bi 9 5+ (as well as experimentally observed C s in Bi 6 Br 7 ) and square antiprisms for E 9 4− .
Bismuth clusters, in the shape of a trigonal tricapped prism (as seen in Figure 5), have two types of bonding basins, trisynaptic Ω1 (1.36 e) over the centers of trigonal bases, corresponding to three-center bonds, and disynaptic Ω2 (1.17 e), corresponding to the two-center bonds connecting the caps with the prism. A similar picture, with slightly less populated basins, and minimal differences in Ω2 populations (see Table 2) due to the breaking of the idealized symmetry of Bi 9 5+ , is observed in Bi 6 Br 7 . Nine-atomic bismuth clusters of the C 4v symmetry show slightly different pattern (see Figure 5) where three-center bonds, making up the sides of the prism (Ω1, 1.36 e), have a smaller corresponding basin population than pairwise bonds connecting the atoms in the base (Ω2, 1.57 e), and the basins corresponding to pairwise bonds connecting the cap (Ω3, 1.02 e) appear to be even less populated. No bonds between the atoms within the top square are observed. Most interesting is the situation with E 9 4− polyanions, where the topology of ELF only reveals pairwise interactions (see Figure 5), and each pairwise bond connecting the squares of the prism has its own basin (Ω2). The population of bond basins in the anions is fairly evenly distributed and slightly decreases from Ge to Pb, as it was observed for E 5 2− .
The description of the ELF for Bi 9 5+ (D 3h ), given in [42], also confirms three-center bonds in the bases of the prism, however, they also consider bonds to caps being multi-center, which our study does not confirm, as we observe disynaptic localization domains connecting caps to rectangular faces. Interestingly, a similar picture was observed for this cation using ELI-D topological analysis [10], yet, based on the fact that the Ω2-type domains merge together in a ring when the localization parameter is decreased, these bonds were interpreted as three-center. However, it is clear from the figure that, before merging, these are domains centered on pairwise interactions, even if they are slightly off the straight line connecting atoms; in our ELF topology, they also merge at η~0.44 (see Figure 5), yet at higher η they are recognizable as single attractors with defined corresponding basins. It is also interesting to note that trigonal faces do not automatically mean three-center bonding in these clusters. To better illustrate the point, which at times can be obscured in 3D ELF plots, we have plotted ELF for triangular faces for several species (see Figure 6). As seen from the figure, Bi 8 2+ (as well as Bi 10 4+ , vide infra) has two bases connected by two-center bonds, while nine-atomic bismuth clusters show three-center bonding for a similar fragment, and Sn 9 4− shows only pairwise interactions. Particularly interesting is that Bi 9 5+ has both three-center and two-center bonds for triangular faces, the former between the squares, and the latter between the cap and the square.
in these clusters. To better illustrate the point, which at times can be obscured in 3D ELF plots, we have plotted ELF for triangular faces for several species (see Figure 6). As seen from the figure, Bi8 2+ (as well as Bi10 4+ , vide infra) has two bases connected by two-center bonds, while nine-atomic bismuth clusters show three-center bonding for a similar fragment, and Sn9 4-shows only pairwise interactions. Particularly interesting is that Bi9 5+ has both three-center and two-center bonds for triangular faces, the former between the squares, and the latter between the cap and the square.

Ten-vertex Polyhedra
We have analyzed ELF for "empty" Bi10 4+ clusters, which have not been practically observed yet, as well as for characterized Pd-and Pt-centered ten-atomic bismuth polyhedra, which the authors of [28][29][30][31] call Pd@Bi10 4+ and Pt@Bi10 4+ . Our ELF analysis of Bi10 4+ (see Figure 5) shows the bonding picture very similar to the one observed for Bi8 2+ , i.e., almost perfect 2c,2e bonds in the bases (Ω1, 1.83 e), and two-center bonds connecting the bases (Ω2, 1.03 e). The latter are tricky to observe unobscured in 3D view, as they merge very quickly with Ω1, so we also provide 2D plot for a better view (see Figure 6). This cluster was investigated in [30] using ELI-D topological analysis, and the authors also observed bonding within five-member rings, however, they have reported no indications of bonding between them, concluding that there is no covalency between the rings. We can now revise that view, as such covalency is observed in our study.

Ten-vertex Polyhedra
We have analyzed ELF for "empty" Bi 10 4+ clusters, which have not been practically observed yet, as well as for characterized Pd-and Pt-centered ten-atomic bismuth polyhedra, which the authors of [28][29][30][31] call Pd@Bi 10 4+ and Pt@Bi 10 4+ . Our ELF analysis of Bi 10 4+ (see Figure 5) shows the bonding picture very similar to the one observed for Bi 8 2+ , i.e., almost perfect 2c,2e bonds in the bases (Ω1, 1.83 e), and two-center bonds connecting the bases (Ω2, 1.03 e). The latter are tricky to observe unobscured in 3D view, as they merge very quickly with Ω1, so we also provide 2D plot for a better view (see Figure 6). This cluster was investigated in [30] using ELI-D topological analysis, and the authors also observed bonding within five-member rings, however, they have reported no indications of bonding between them, concluding that there is no covalency between the rings. We can now revise that view, as such covalency is observed in our study.
Another very interesting question about these clusters is the nature of bonding between Pd or Pt and Bi. The authors of [28][29][30] did not see any indication of interactions between centering atom and bismuth. However, our investigation clearly shows localization domains between Bi and Pt at η~0.33 (Ω3, see Figure 5). These are disynaptic basins with fairly low population of ca. 0.3 e, corresponding to, perhaps, not well-localized, yet still covalent bonds. The fact that they are observed at low η values is quite typical for Pt and Pd in metallic systems [65][66][67][68]. We could not observe the same kind of domains for PdBi 10 4+ in a 3D view (see Figure 5), as we needed to decrease the localization parameter to the extent when the picture was completely cluttered with bismuth shells; however, there were basins similar to Ω3 found analytically, and they are also visible in the cross-section (see Figure 7). They have an even lower population than for Pt-Bi ones, 0.21 e, yet, in our view, this is still an indication of covalency and somewhat localized bonding. Moreover, the tendency for Pd to show less localized bonds than Pt in isotypic metallic systems was previously observed in ELI-D topology on several occasions [65,66]. Thus we can conclude that group 10 metals in both cases perform a more important function than mere fillers, and PdBi 10 4+ and PtBi 10 4+ , based on our direct-space bonding analysis, are most likely to be heteroatomic clusters.
To summarize this section, after providing the description of bonding in a series of main-group clusters, we would like to return to the question of 3D aromaticity and whether the description we provide, based on the picture of localized bonds revealed by ELF topologies, rules it out completely. We would not say so, as we only show that bonding patterns in such clusters are of more localized nature than was assumed before and can be rationalized using localized multi-centered and pairwise bonding interactions. Although previous conclusions regarding delocalization were made based on indirect indications, i.e., the lack of certainly observed bonding attractors, which we now can see, this does not preclude delocalization of a part of electrons. Indeed, relatively low basin populations for supposedly two-center interactions might serve as an indication of such delocalization (how such electrons would participate in bonding is another question). But since ELF only allows us to observe localized electron areas, this kind of indication would, once again, be indirect. cross-section (see Figure 7). They have an even lower population than for Pt-Bi ones, 0.21 e, yet, in our view, this is still an indication of covalency and somewhat localized bonding. Moreover, the tendency for Pd to show less localized bonds than Pt in isotypic metallic systems was previously observed in ELI-D topology on several occasions [65,66]. Thus we can conclude that group 10 metals in both cases perform a more important function than mere fillers, and PdBi10 4+ and PtBi10 4+ , based on our direct-space bonding analysis, are most likely to be heteroatomic clusters. To summarize this section, after providing the description of bonding in a series of main-group clusters, we would like to return to the question of 3D aromaticity and whether the description we provide, based on the picture of localized bonds revealed by ELF topologies, rules it out completely. We would not say so, as we only show that bonding patterns in such clusters are of more localized nature than was assumed before and can be rationalized using localized multi-centered and pairwise bonding interactions. Although previous conclusions regarding delocalization were made based on indirect indications, i.e., the lack of certainly observed bonding attractors, which we now can see, this does not preclude delocalization of a part of electrons. Indeed, relatively low basin populations for supposedly two-center interactions might serve as an indication of such delocalization (how such electrons would participate in bonding is another question). But since ELF only allows us to observe localized electron areas, this kind of indication would, once again, be indirect.

Conclusions
To summarize this paper, an alternative synthetic approach allowed us to obtain good quality crystals of Bi6Br7 and to reinvestigate its crystal structure, and update the description of the structure as containing Bi9 5+ cluster polycations and polymer-like 1 ∞[Bi3Br14 5-] anions. Based on the DFT calculations, the compound was found to be semiconducting with a band gap of ca. 1.3 eV. The compound is best described as complex salt, with polymer-like anionic framework, most likely, responsible for electronic transport. Additionally, band structure and crystallographic analysis indicate a possibility of polycations, embedded in the anionic framework, also contributing to it. Direct-space bonding analysis using ELF for all known bismuth homoatomic polycations and their isoelectronic analogues has shown the pattern of two-and three-center localized bonding in these

Conclusions
To summarize this paper, an alternative synthetic approach allowed us to obtain good quality crystals of Bi 6 Br 7 and to reinvestigate its crystal structure, and update the description of the structure as containing Bi 9 5+ cluster polycations and polymer-like 1 ∞ [Bi 3 Br 14 5− ] anions. Based on the DFT calculations, the compound was found to be semiconducting with a band gap of ca. 1.3 eV. The compound is best described as complex salt, with polymer-like anionic framework, most likely, responsible for electronic transport. Additionally, band structure and crystallographic analysis indicate a possibility of polycations, embedded in the anionic framework, also contributing to it. Direct-space bonding analysis using ELF for all known bismuth homoatomic polycations and their isoelectronic analogues has shown the pattern of two-and three-center localized bonding in these species, rather than a tendency for dominating 3D-aromaticity. We have also found indications of localized bonding between bismuth and group 10 metals in PdBi 10 4+ and PtBi 10 4+ , which characterizes these species as heteroatomic clusters rather than homoatomic bismuth species centered by non-interacting atoms.
Supplementary Materials: The following are available online at http://www.mdpi.com/2073-4352/10/10/940/ s1, Table S1: Crystallographic parameters, X-ray data collection, and refinement details for a single-crystal experiment on Bi 6 Br 7 , Table S2: Fractional atomic coordinates and anisotropic displacement parameters for Bi 6 Br 7 . Tables S3-S14-side-by-side comparisons between the structures of Bi 6 Br 7 from different sources and between the structure of Bi 6 Cl 7 and Bi 6 Br 7 .