Polyhedral Dicobaltadithiaboranes and Dicobaltdiselenaboranes as Examples of Bimetallic Nido Structures without Bridging Hydrogens

The geometries and energetics of the n-vertex polyhedral dicobaltadithiaboranes and dicobaltadiselenaboranes Cp2Co2E2Bn−4Hn−4 (E = S, Se; n = 8 to 12) have been investigated via the density functional theory. Most of the lowest-energy structures in these systems are generated from the (n + 1)-vertex most spherical closo deltahedra by removal of a single vertex, leading to a tetragonal, pentagonal, or hexagonal face depending on the degree of the vertex removed. In all of these low-energy structures, the chalcogen atoms are located at the vertices of the non-triangular face. Alternatively, the central polyhedron in most of the 12-vertex structures can be derived from a Co2E2B8 icosahedron with adjacent chalcogen (E) vertices by breaking the E–E edge and 1 or more E–B edges to create a hexagonal face. Examples of the arachno polyhedra with two tetragonal and/or pentagonal faces derived from the removal of two vertices from isocloso deltahedra were found among the set of lowest-energy Cp2Co2E2Bn−4Hn−4 (E = S, Se; n = 8 and 12) structures.


Introduction
The structures of the polyhedral borane dianions B n H n 2− as well as those of the isoelectronic carborane monoanions CB n−1 H n − and neutral dicarbaboranes C 2 B n−2 H n are based on the most spherical deltahedra, which are known as the closo deltahedra ( Figure 1) [1,2]. Except for the 13-vertex systems, the most spherical deltahedra have exclusively triangular faces with vertex degrees as nearly similar as possible. The 11-and 13-vertex systems contain 1 and 2 degree-6 vertices, respectively, whereas the other closo polyhedra have exclusively degree-4 and -5 vertices, for which the degree of a vertex is defined as the number of edges meeting at the vertex in question. The chemical bonding in these polyhedral boranes and carboranes is based on 2n + 2 skeletal electrons for an n-vertex system, with each BH and CH vertex contributing 2 and 3 skeletal electrons, respectively, after providing an electron for the external B-H or C-H bond. This is a key aspect of the Wade-Mingos rules [3][4][5] relating polyhedral geometry to skeletal electron count. A reasonable chemical bonding model for these deltahedra consists of a resonance hybrid of canonical structures containing a single n-center bond composed of orbitals from each of the n-vertex atoms overlapping at the center of the polyhedron supplemented by n two-center, two-electron bonds on the surface of the polyhedron [6]. This structural model accounts for the 2n + 2 skeletal electrons in the stable closo deltahedral borane structures. The delocalization implicit in this bonding model, particularly the presence of the multicenter core bond, is consistent with the interpretation of these species as threedimensional aromatic systems [7,8]. The research groups of Hawthorne [9] and Grimes [10] were the first to boron vertices in these polyhedral boranes could be replaced by transitio The cyclopentadienylcobalt unit (CpCo) was especially useful for this purp robustness of the linkage between the cyclopentadienyl ring and the cobalt more, a CpCo vertex is a donor of two skeletal electrons like a BH vertex replace a BH unit in the polyhedral structures. The research of the Hawth focused on metallaboranes that can be obtained from decaborane (B10H14) a whereas the research of the Grimes group [10] studied metallaboranes obtai taborane (B5H9) as a precursor. The chemistry of polyhedral boranes and t metal derivatives remains of interest even after approximately half a cen the discovery of the original metallaboranes with the possibilities of applic icine [11,12] and catalysis [13].
Further development of metallaborane chemistry, especially by Kenne workers, by using a variety of metal vertices, particularly those containin third-row transition metals, resulted in the discovery of alternative so-call tahedra in 9-and 10-vertex metallaborane structures with a degree-6 verte atom ( Figure 2) [14][15][16][17]. The 11-vertex closo deltahedron can also be an isoclo since it necessarily has a degree 6 vertex. A reasonable chemical bonding m vertex isocloso metallaborane deltahedron consists exclusively of n 3-center, 2e) bonds in n of the 2n − 4 faces of the deltahedron without the n-center co in the closo chemical bonding [18]. Thus, the isocloso metallaboranes are 2n + 2 skeletal electron systems. The research groups of Hawthorne [9] and Grimes [10] were the first to show that the boron vertices in these polyhedral boranes could be replaced by transition metal units. The cyclopentadienylcobalt unit (CpCo) was especially useful for this purpose due to the robustness of the linkage between the cyclopentadienyl ring and the cobalt atom. Furthermore, a CpCo vertex is a donor of two skeletal electrons like a BH vertex so that it can replace a BH unit in the polyhedral structures. The research of the Hawthorne group [9] focused on metallaboranes that can be obtained from decaborane (B 10 H 14 ) as a precursor, whereas the research of the Grimes group [10] studied metallaboranes obtained from pentaborane (B 5 H 9 ) as a precursor. The chemistry of polyhedral boranes and their transition metal derivatives remains of interest even after approximately half a century following the discovery of the original metallaboranes with the possibilities of applications in medicine [11,12] and catalysis [13].
Further development of metallaborane chemistry, especially by Kennedy and his co-workers, by using a variety of metal vertices, particularly those containing secondand third-row transition metals, resulted in the discovery of alternative so-called isocloso deltahedra in 9-and 10-vertex metallaborane structures with a degree-6 vertex for the metal atom ( Figure 2) [14][15][16][17]. The 11-vertex closo deltahedron can also be an isocloso deltahedron since it necessarily has a degree 6 vertex. A reasonable chemical bonding model for an n-vertex isocloso metallaborane deltahedron consists exclusively of n 3-center, 2-electron (3c-2e) bonds in n of the 2n − 4 faces of the deltahedron without the n-center core bond found in the closo chemical bonding [18]. Thus, the isocloso metallaboranes are 2n rather than 2n + 2 skeletal electron systems.
Molecules 2023, 28, x FOR PEER REVIEW Figure 1. The most spherical closo deltahedra having 9 to 13 vertices from which the nido s discussed in this paper are generated. Vertices of degrees 4, 5, and 6 are indicated in red, b green, respectively, in Figures 1, 2, and 8.
The research groups of Hawthorne [9] and Grimes [10] were the first to show boron vertices in these polyhedral boranes could be replaced by transition met The cyclopentadienylcobalt unit (CpCo) was especially useful for this purpose du robustness of the linkage between the cyclopentadienyl ring and the cobalt atom. more, a CpCo vertex is a donor of two skeletal electrons like a BH vertex so th replace a BH unit in the polyhedral structures. The research of the Hawthorne g focused on metallaboranes that can be obtained from decaborane (B10H14) as a pr whereas the research of the Grimes group [10] studied metallaboranes obtained fr taborane (B5H9) as a precursor. The chemistry of polyhedral boranes and their tr metal derivatives remains of interest even after approximately half a century fo the discovery of the original metallaboranes with the possibilities of applications icine [11,12] and catalysis [13].
Further development of metallaborane chemistry, especially by Kennedy an workers, by using a variety of metal vertices, particularly those containing seco third-row transition metals, resulted in the discovery of alternative so-called isoc tahedra in 9-and 10-vertex metallaborane structures with a degree-6 vertex for th atom ( Figure 2) [14][15][16][17]. The 11-vertex closo deltahedron can also be an isocloso delta since it necessarily has a degree 6 vertex. A reasonable chemical bonding model f vertex isocloso metallaborane deltahedron consists exclusively of n 3-center, 2-elec 2e) bonds in n of the 2n − 4 faces of the deltahedron without the n-center core bon in the closo chemical bonding [18]. Thus, the isocloso metallaboranes are 2n rather + 2 skeletal electron systems. Electron-richer polyhedral borane derivatives with n-vertices are obtained by ing a vertex from an (n + 1)-vertex closo or isocloso deltahedron to give a so-called nido polyhedron. Such a vertex removal process creates a smaller polyhedron wit triangular face at the site of vertex removal. This non-triangular face can be regar Electron-richer polyhedral borane derivatives with n-vertices are obtained by removing a vertex from an (n + 1)-vertex closo or isocloso deltahedron to give a so-called n-vertex nido polyhedron. Such a vertex removal process creates a smaller polyhedron with a nontriangular face at the site of vertex removal. This non-triangular face can be regarded as a "hole" in an otherwise triangulated polyhedral surface. Williams [2] introduced a convenient notation for nido polyhedra as ni-n h , where n is the number of vertices, and h is the number of vertices in the non-triangular face as indicated by a Roman numeral. For example, by using this notation, the square pyramid can be designated as a ni-5 IV polyhedron. Since the original (n + 1)-vertex closo deltahedron has 2(n + 1) + 2 skeletal electrons, the n-vertex nido polyhedron derived from it has 2n + 4 skeletal electrons.
The discovery of neutral binary boranes B n H n+4 (n = 2, 5, 6, 8, 9, 10) exhibiting nido structures, of which B 10 H 14 is the most stable [19,20], predates that of the more stable closo boranes by going back to the original boron hydride syntheses of Stock. The structures of B 5 H 9 and B 6 H 10 are derived by the removal of a degree-4 vertex from an octahedron and a degree-5 vertex from a pentagonal bipyramid, respectively, thereby generating a square and a pentagonal face, respectively ( Figure 3). The structures of the larger B n H n+4 (n = 8,9,10) boranes are derived by removal of the unique degree-6 vertex from the corresponding (n + 1) vertex isocloso deltahedron. The polyhedral frameworks of all of the B n H n+4 boranes (n = 5, 6,8,9,10) all have n BH vertices with the four "extra" hydrogen atoms bridging the B-B edges surrounding the non-triangular face. A relatively large non-triangular face such as a hexagonal hole versus a pentagonal or tetragonal face provides more space for the four hydrogen atoms bridging the hole B-B edges.
Molecules 2023, 28, x FOR PEER REVIEW 3 number of vertices in the non-triangular face as indicated by a Roman numeral. Fo ample, by using this notation, the square pyramid can be designated as a ni-5IV po dron. Since the original (n + 1)-vertex closo deltahedron has 2(n + 1) + 2 skeletal elect the n-vertex nido polyhedron derived from it has 2n + 4 skeletal electrons.
The discovery of neutral binary boranes BnHn+4 (n = 2, 5, 6, 8, 9, 10) exhibiting structures, of which B10H14 is the most stable [19,20], predates that of the more stable boranes by going back to the original boron hydride syntheses of Stock. The structur B5H9 and B6H10 are derived by the removal of a degree-4 vertex from an octahedron a degree-5 vertex from a pentagonal bipyramid, respectively, thereby generating a sq and a pentagonal face, respectively ( Figure 3). The structures of the larger BnHn+4 (n = 10) boranes are derived by removal of the unique degree-6 vertex from the correspon (n + 1) vertex isocloso deltahedron. The polyhedral frameworks of all of the BnHn+4 bor (n = 5, 6,8,9,10) all have n BH vertices with the four "extra" hydrogen atoms bridgin B-B edges surrounding the non-triangular face. A relatively large non-triangular face as a hexagonal hole versus a pentagonal or tetragonal face provides more space fo four hydrogen atoms bridging the hole B-B edges. A topic of interest is the design of viable n-vertex nido borane structures that hav necessary 2n + 4 skeletal electrons without the need for bridging hydrogen atoms. T carbaborane structures of the type C4Bn−4Hn provide the required 2n + 4 skeletal elect In this connection, species of the type C4Bn−4Rn (R = H or alkyl) are known experimen to have such structures with 6 [21,22], 8 [23,24], 10 [25][26][27][28], 11 [29], and 12 [30][31][32][33][34][35][36] ver Furthermore, the structures and energetics of the tetracarbaboranes have been studie using modern density functional theory methods [37].
Generating a nido polyhedral borane with the requisite 2n + 4 skeletal electron using electron-richer carbon atoms in CH vertices to replace boron atoms in BH ver requires four such carbon heteroatoms in the carborane polyhedron. Using bare sulf selenium vertex atoms as heteroatoms in polyhedral borane structures to provide th A topic of interest is the design of viable n-vertex nido borane structures that have the necessary 2n + 4 skeletal electrons without the need for bridging hydrogen atoms. Tetracarbaborane structures of the type C 4 B n−4 H n provide the required 2n + 4 skeletal electrons. In this connection, species of the type C 4 B n−4 R n (R = H or alkyl) are known experimentally to have such structures with 6 [21,22], 8 [23,24], 10 [25][26][27][28], 11 [29], and 12 [30][31][32][33][34][35][36] vertices. Furthermore, the structures and energetics of the tetracarbaboranes have been studied by using modern density functional theory methods [37].
Generating a nido polyhedral borane with the requisite 2n + 4 skeletal electrons by using electron-richer carbon atoms in CH vertices to replace boron atoms in BH vertices requires four such carbon heteroatoms in the carborane polyhedron. Using bare sulfur or selenium vertex atoms as heteroatoms in polyhedral borane structures to provide the 2n + 4 skeletal electrons for nido geometry requires only two heteroatoms in an E 2 B n−2 H n−2 (E = S, Se) structure since bare sulfur or selenium vertices are each donors of four skeletal electrons after diverting 2 of their 6 valence electrons to an external lone pair. In this connection, the 11-vertex nido-B 9 H 9 E 2 (E = S, Se) structures have been synthesized and shown via X-ray crystallography to have a central polyhedron obtained via the removal of one vertex from an icosahedron [38].
Dithiaboranes and diselenaboranes with one or two CpCo vertices have been synthesized and structurally characterized via X-ray crystallography ( Figure 4). The 12-vertex cobaltadiselenaborane CpCoSe 2 B 9 H 9 has been synthesized and shown to have a ni-13 VI structure derived via the removal of a degree-6 vertex from a docosahedron, which is the 13-vertex closo deltahedron [35]. This structure can also be derived from a central CoSe 2 B 9 icosahedron with a Se-Se edge by breaking the Se-Se edge and an adjacent Se-B edge to generate an SeCoSeBB pentagonal face, leaving a degree-3 selenium vertex. This structure is shown by density functional theory to be 1 of the 3 lowest-energy structures lying within 1 kcal/mol of each other [39]. Dithiaboranes and diselenaboranes with one or two CpCo vertices have been synthe sized and structurally characterized via X-ray crystallography ( Figure 4). The 12-verte cobaltadiselenaborane CpCoSe2B9H9 has been synthesized and shown to have a ni-13VI structure derived via the removal of a degree-6 vertex from a docosahedron, which is th 13-vertex closo deltahedron [35]. This structure can also be derived from a central CoSe2B icosahedron with a Se-Se edge by breaking the Se-Se edge and an adjacent Se-B edge t generate an SeCoSeBB pentagonal face, leaving a degree-3 selenium vertex. This structur is shown by density functional theory to be 1 of the 3 lowest-energy structures lyin within 1 kcal/mol of each other [39]. In total, 3 isomeric structures of the 11-vertex dicobaltadithiaborane Cp*2Co2S2B7H with 2 pentamethylcyclopentadienylcobalt vertices (Cp*Co) have been isolated by Kan and Sneddon [40] from the mixture that was obtained from the reaction between LiCp* NaS2B7H8, and CoCl2 ( Figure 5). Attempts to obtain definitive structural information on these species via X-ray crystallography were prevented by disorder problems. However the poor X-ray data were sufficient to indicate the relative positions of the cobalt atom and 11-vertex Co2S2B7 geometries obtained via removal from a vertex from a central icosahe dron. Furthermore, 1 of the 3 structures appears to be an ultimate pyrolysis product at 30 °C. The structures were assigned on the basis of the 11 B NMR spectra. In addition, trac quantities of a 10-vertex Cp*2Co2S2B6H6 structure were obtained, which was suggested t have a decaborane-like structure on the basis of its 11 B NMR. Figure 5. The 11-vertex frameworks suggested for the 3 Cp*2Co2S2B7H7 isomers isolated by Kan and Sneddon from the reaction between LiCp*, NaS2B7H9, and CoCl2.
In view of the difficulty in obtaining definitive structural information on the dico baltadithiaboranes, we have undertaken density functional theory studies on the complet In total, 3 isomeric structures of the 11-vertex dicobaltadithiaborane Cp* 2 Co 2 S 2 B 7 H 7 with 2 pentamethylcyclopentadienylcobalt vertices (Cp*Co) have been isolated by Kang and Sneddon [40] from the mixture that was obtained from the reaction between LiCp*, NaS 2 B 7 H 8 , and CoCl 2 ( Figure 5). Attempts to obtain definitive structural information on these species via X-ray crystallography were prevented by disorder problems. However, the poor X-ray data were sufficient to indicate the relative positions of the cobalt atom and 11vertex Co 2 S 2 B 7 geometries obtained via removal from a vertex from a central icosahedron. Furthermore, 1 of the 3 structures appears to be an ultimate pyrolysis product at 300 • C. The structures were assigned on the basis of the 11 B NMR spectra. In addition, trace quantities of a 10-vertex Cp* 2 Co 2 S 2 B 6 H 6 structure were obtained, which was suggested to have a decaborane-like structure on the basis of its 11  Dithiaboranes and diselenaboranes with one or two CpCo vertices have been synthe sized and structurally characterized via X-ray crystallography ( Figure 4). The 12-vertex cobaltadiselenaborane CpCoSe2B9H9 has been synthesized and shown to have a ni-13VI structure derived via the removal of a degree-6 vertex from a docosahedron, which is the 13-vertex closo deltahedron [35]. This structure can also be derived from a central CoSe2B icosahedron with a Se-Se edge by breaking the Se-Se edge and an adjacent Se-B edge to generate an SeCoSeBB pentagonal face, leaving a degree-3 selenium vertex. This structure is shown by density functional theory to be 1 of the 3 lowest-energy structures lying within 1 kcal/mol of each other [39]. In total, 3 isomeric structures of the 11-vertex dicobaltadithiaborane Cp*2Co2S2B7H with 2 pentamethylcyclopentadienylcobalt vertices (Cp*Co) have been isolated by Kang and Sneddon [40] from the mixture that was obtained from the reaction between LiCp* NaS2B7H8, and CoCl2 ( Figure 5). Attempts to obtain definitive structural information on these species via X-ray crystallography were prevented by disorder problems. However the poor X-ray data were sufficient to indicate the relative positions of the cobalt atom and 11-vertex Co2S2B7 geometries obtained via removal from a vertex from a central icosahe dron. Furthermore, 1 of the 3 structures appears to be an ultimate pyrolysis product at 300 °C. The structures were assigned on the basis of the 11 B NMR spectra. In addition, trace quantities of a 10-vertex Cp*2Co2S2B6H6 structure were obtained, which was suggested to have a decaborane-like structure on the basis of its 11 B NMR. In view of the difficulty in obtaining definitive structural information on the dico baltadithiaboranes, we have undertaken density functional theory studies on the complete series of Cp2Co2E2Bn−4Hn−4 (E = S, Se; n = 8, 9, 10, 11, 12) systems with unsubstituted cyclo In view of the difficulty in obtaining definitive structural information on the dicobaltadithiaboranes, we have undertaken density functional theory studies on the complete series of Cp 2 Co 2 E 2 B n−4 H n−4 (E = S, Se; n = 8, 9, 10, 11, 12) systems with unsubstituted cyclopentadienyl rings. In addition, we have included the Cp* 2 Co 2 S 2 B 7 H 7 system with pentamethylcyclopentadienyl (Cp*) rings in this study for comparison with the reported experimental data. This theoretical study extends the previous study [36] of the dicobaltadiselenaboranes by introducing a second cobalt atom into the metallaborane polyhedron. This provides an opportunity to assess preferences for a pair of cobalt atoms in these structures to occupy adjacent bonding positions or to be as far removed from each other as possible or something in between. The dicobaltadithiaboranes and dicobaltadiselena boranes are of interest in representing the first examples of nido structures without bridging hydrogen atoms containing two transition metal vertices in the underlying polyhedron.

Results and Discussion
2.1. The 11-Vertex Systems Cp 2 Co 2 E 2 B 7 H 7 (E = S, Se) In total, 8 structures were found for each of the 11-vertex Cp 2 Co 2 E 2 B 7 H 7 (E = S, Se) systems within 15 kcal/mol of the lowest-energy structure ( Figure 6 and Table 1). The central ni-11 V polyhedron in all of the eight structures can be derived from an icosahedron via the removal of one vertex, leaving a pentagonal face that is similar to the dicarbollide anion C 2 B 9 H 12 − that was originally used by Hawthorne and co-workers [9] to synthesize a variety of transition metal complexes having a central MC 2 B 9 icosahedron. For both Cp 2 Co 2 E 2 B 7 H 7 (E = S, Se) systems, the lowest-energy structures B7Co2E2-1 (E = S, Se) have both cobalt atoms and both chalcogen atoms at the surface of the pentagonal face and, thus, at degree-4 vertices. This is the most prevalent example of generating an n-vertex nido polyhedral borane by the removal of a vertex from an (n + 1)-vertex closo deltahedral borane. Furthermore, in each of the 8 low-energy Cp 2 Co 2 E 2 B 7 H 7 structures, both chalcogen atoms are located at the degree-4 pentagonal face vertices rather than at interior degree-5 vertices. This is consistent with the preference of chalcogen atoms for lower degree vertices in polyhedral selena-and thiaboranes.

The 11-Vertex Systems Cp2Co2E2B7H7 (E = S, Se)
In total, 8 structures were found for each of the 11-vertex Cp2Co2E2B7H7 (E = S, Se) systems within 15 kcal/mol of the lowest-energy structure ( Figure 6 and Table 1). The central ni-11V polyhedron in all of the eight structures can be derived from an icosahedron via the removal of one vertex, leaving a pentagonal face that is similar to the dicarbollide anion C2B9H12 − that was originally used by Hawthorne and co-workers [9] to synthesize a variety of transition metal complexes having a central MC2B9 icosahedron. For both Cp2Co2E2B7H7 (E = S, Se) systems, the lowest-energy structures B7Co2E2-1 (E = S, Se) have both cobalt atoms and both chalcogen atoms at the surface of the pentagonal face and, thus, at degree-4 vertices. This is the most prevalent example of generating an n-vertex nido polyhedral borane by the removal of a vertex from an (n + 1)-vertex closo deltahedral borane. Furthermore, in each of the 8 low-energy Cp2Co2E2B7H7 structures, both chalcogen atoms are located at the degree-4 pentagonal face vertices rather than at interior degree-5 vertices. This is consistent with the preference of chalcogen atoms for lower degree vertices in polyhedral selena-and thiaboranes. Three isomeric pentamethylcyclopentadienyl Cp*2Co2S2B7H7 complexes have isolated by Kang and Sneddon in small quantities from the reaction between NaS2B7H8, and CoCl2 which they designated by the Roman numerals III, IV, and V i paper ( Figure 5) [37]. Extensive disorder prevented complete X-ray structure dete tions on these molecules beyond location of the cobalt atoms. On the basis of 11 B NM 2D 11 B-11 B COSY NMR, Kang and Sneddon assigned structures analogous to B7C B7Co2S-2, and B7Co2S-3 to III, IV, and V, respectively. We found that complete su tion of hydrogen atoms with methyl groups did not affect the relative energy orde the 3 lowest-energy structures with the relative energies of the Cp* structures B7Co B7Co2S-2*, and B7Co2S-3* being 0.0, 0.6, and 6.7 kcal/mol, respectively. What is s is the observation that the Kang/Sneddon isomer III, which has the assigned str B7Co2S2-1* and has both cobalt atoms as well as both sulfur atoms located on penta face vertices, is converted ultimately upon heating to 300°C to the Kang/Sneddon i V ( Figure 5), which was assigned the higher-energy structure B7Co2S2-3*, as it ha cobalt atoms located at degree-5 interior vertices. This is contrary to expectation b pyrolysis, particularly to a temperature as high as 300°C, would be expected to lower-energy isomer rather than a higher-energy isomer. Our theoretical studie some doubt about the structure assignments of III, IV, and V that were given by Kan Sneddon in their 1988 study [37]. Note that the predicted Co … Co distances in B7C B7Co2S-2, and B7Co2S-3 are all approximately 3.8 to 3.9 Å, so the determination o distances by an otherwise incomplete X-ray crystallography study on an extensive ordered system would not be sufficient to distinguish between these three structure improvements in X-ray crystallography methodology in the 35 years since the Kang don report of the 3 Cp*2Co2S2B7H7 isomers might provide a resolution to this dilem    Kang and Sneddon in small quantities from the reaction between LiCp*, NaS 2 B 7 H 8 , and CoCl 2 which they designated by the Roman numerals III, IV, and V in their paper ( Figure 5) [37]. Extensive disorder prevented complete X-ray structure determinations on these molecules beyond location of the cobalt atoms. On the basis of 11 B NMR and 2D 11 B-11 B COSY NMR, Kang and Sneddon assigned structures analogous to B7Co2S-1, B7Co2S-2, and B7Co2S-3 to III, IV, and V, respectively. We found that complete substitution of hydrogen atoms with methyl groups did not affect the relative energy ordering of the 3 lowest-energy structures with the relative energies of the Cp* structures B7Co2S-1*, B7Co2S-2*, and B7Co2S-3* being 0.0, 0.6, and 6.7 kcal/mol, respectively. What is strange is the observation that the Kang/Sneddon isomer III, which has the assigned structure B7Co2S2-1* and has both cobalt atoms as well as both sulfur atoms located on pentagonal face vertices, is converted ultimately upon heating to 300 • C to the Kang/Sneddon isomer V ( Figure 5), which was assigned the higher-energy structure B7Co2S2-3*, as it has both cobalt atoms located at degree-5 interior vertices. This is contrary to expectation because pyrolysis, particularly to a temperature as high as 300 • C, would be expected to give a lowerenergy isomer rather than a higher-energy isomer. Our theoretical studies cast some doubt about the structure assignments of III, IV, and V that were given by Kang and Sneddon in their 1988 study [37]. Note that the predicted Co . . . Co distances in B7Co2S-1, B7Co2S-2, and B7Co2S-3 are all approximately 3.8 to 3.9 Å, so the determination of these distances by an otherwise incomplete X-ray crystallography study on an extensively disordered system would not be sufficient to distinguish between these three structures. The improvements in X-ray crystallography methodology in the 35 years since the Kang/Sneddon report of the 3 Cp* 2 Co 2 S 2 B 7 H 7 isomers might provide a resolution to this dilemma.

The 12-Vertex Systems Cp 2 Co 2 E 2 B 8 H 8 (E = S, Se)
In total, 9 structures were found for the 12-vertex Cp 2 Co 2 E 2 B 8 H 8 (E = S, Se) systems up to 9 kcal/mol (E = S) and 12 kcal/mol (E = Se) in energy (Figure 7 and Table 2). These structures are of three types. The polyhedra for the lowest-energy Cp 2 Co 2 S 2 B 8 H 8 structure B8Co2S2-1, as well as those for 4 of the 5 next higher-energy structures B8Co2S2-2, B8Co2S2-3, B8Co2S2-5, and B8Co2S2-6, lying 3.0, 3.1, 3.8, and 5.6 kcal/mol in energy above B8Co2S2-1, respectively, can be derived from a Co 2 E 2 B 8 icosahedron with an S-S edge by breaking the S-S edge and at least one S-B edge to create typically a gaping, bent hexagonal face. These five structures can be divided in two categories. In B8Co2S2-1, B8Co2S2-5, and B8Co2S2-6, the cobalt atoms are located in meta (non-adjacent, nonantipodal) positions of the original octahedron. However, in B8Co2S2-2 and B8Co2S2-4, the cobalt atoms are located in para (antipodal) positions in the original icosahedron. In total, 9 structures were found for the 12-vertex Cp2Co2E2B8H8 (E = S, Se) systems up to 9 kcal/mol (E = S) and 12 kcal/mol (E = Se) in energy (Figure 7 and Table 2). These structures are of three types. The polyhedra for the lowest-energy Cp2Co2S2B8H8 structure B8Co2S2-1, as well as those for 4 of the 5 next higher-energy structures B8Co2S2-2, B8Co2S2-3, B8Co2S2-5, and B8Co2S2-6, lying 3.0, 3.1, 3.8, and 5.6 kcal/mol in energy above B8Co2S2-1, respectively, can be derived from a Co2E2B8 icosahedron with an S-S edge by breaking the S-S edge and at least one S-B edge to create typically a gaping, bent hexagonal face. These five structures can be divided in two categories. In B8Co2S2-1, B8Co2S2-5, and B8Co2S2-6, the cobalt atoms are located in meta (non-adjacent, non-antipodal) positions of the original octahedron. However, in B8Co2S2-2 and B8Co2S2-4, the cobalt atoms are located in para (antipodal) positions in the original icosahedron.    Substituting selenium for sulfur in the 12-vertex system to give Cp 2 Co 2 Se 2 B 8 H 8 derivatives leads to a different energy ordering of the 9 lowest-energy structures ( Table 2). The lowest-energy Cp 2 Co 2 Se 2 B 8 H 8 structure B8Co2Se2-1 and the higher-energy structure B8Co2Se2-4, which lies 4.0 kcal/mol higher in energy, are exceptional among the complete series of Cp 2 Co 2 E 2 B n−4 H n−4 (n = 8 to 12) structures in exhibiting ideal C 2v symmetry, whereas all of the remaining structures of this type have the lower-symmetry point groups of C 1 , C 2 , or C s . These 2 structures are ni-12 IV structures derived from the 13-vertex closo deltahedron, namely, the docosahedron (Figure 1), by the removal of the unique degree-4 vertex, thereby creating a tetragonal face with alternating degree-5 and degree-4 vertices. In both B8Co2Se2-1 and B8Co2Se-4, the tetragonal face has alternating cobalt and sulfur vertices. The Cp 2 Co 2 S 2 B 8 H 8 structures corresponding to B8Co2Se2-1 and B8Co2Se2-4 are B8Co2S-9 and B8Co2S-8, respectively, which lie 9.0 and 7.9 kcal/mol in energy above B8Co2S-1.

The Cp2Co2E2Bn−4Hn−4 (E = S, Se) Systems Having 8 to 10 Vertices
The central Co2E2B4 polyhedra in 7 of the 8 lowest-energy structure Cp2Co2E2B4H4 (E = S, Se) systems ( Figure 9 and Table 3) are all derived b a vertex from the closo 9-vertex deltahedron, namely, the tricapped trigon 1). Removal of a degree-4 vertex from the tricapped, trigonal prism giv trigonal prism, which is found in the lowest-energy structures B4Co2E2 well as the higher-energy structures B4C2S2-2 and B4Co2Se2-5, which l energy above the lowest-energy structures (Table 3). In the lowest-e  The D 6d bicapped hexagonal antiprism as the 14-vertex closo deltahedron and a 14-vertex isocloso deltahedron derived from it via a diamond-square-diamond process that is the polyhedron found in the experimentally known Cp* 2 Ru 2 C 2 B 10 H 12 . The 12-vertex arachno Cp 2 Co 2 E 2 B 8 H 8 (E = S, Se) structures B8Co2S2-3, B8Co2S2-7, B8Co2Se2-5, and B8Co2eS2-9 are obtained from this 14-vertex isocloso deltahedron by the removal of a degree-4 vertex and a degree-5 vertex, which are so situated that the resulting tetragonal and pentagonal faces share an edge.

The Cp 2 Co 2 E 2 B n−4 H n−4 (E = S, Se) Systems Having 8 to 10 Vertices
The central Co 2 E 2 B 4 polyhedra in 7 of the 8 lowest-energy structures of the 8-vertex Cp 2 Co 2 E 2 B 4 H 4 (E = S, Se) systems ( Figure 9 and Table 3) are all derived by the removal of a vertex from the closo 9-vertex deltahedron, namely, the tricapped trigonal prism (Figure 1). Removal of a degree-4 vertex from the tricapped, trigonal prism gives the bicapped trigonal prism, which is found in the lowest-energy structures B4Co2E2-1 (E = S, Se), as well as the higher-energy structures B4C2S2-2 and B4Co2Se2-5, which lie~6 kcal/mol in energy above the lowest-energy structures (Table 3 1 degree-4 vertex (Figure 8). This 14-vertex deltahedron is closely related to the 14-vertex polyhedron in experimentally known Cp*2Ru2C2B10H12 by lengthening an edge connecting a degree-6 vertex with a degree-5 vertex [41,42]. Figure 8. The D6d bicapped hexagonal antiprism as the 14-vertex closo deltahedron and a 14-vertex isocloso deltahedron derived from it via a diamond-square-diamond process that is the polyhedron found in the experimentally known Cp*2Ru2C2B10H12. The 12-vertex arachno Cp2Co2E2B8H8 (E = S, Se) structures B8Co2S2-3, B8Co2S2-7, B8Co2Se2-5, and B8Co2eS2-9 are obtained from this 14-vertex isocloso deltahedron by the removal of a degree-4 vertex and a degree-5 vertex, which are so situated that the resulting tetragonal and pentagonal faces share an edge.
Of the 8 lowest-energy Cp 2 Co 2 E 2 B 4 H 4 (E = S, Se) structures, B4Co2S2-8 and B4Co2Se2-7, which lie at 10.3 and 14.9 kcal/mol in energy, respectively, above the corresponding B4Co2E2-1 structure, are not derived by removing a vertex from a tricapped trigonal prism (Figure 9 and Table 3). Instead, the central Co 2 S 2 B 4 polyhedron in these structures is generated via the removal of the 2 degree-4 vertices that are bridged by the unique degree-6 vertex from the 10-vertex isocloso deltahedron of ideal C 3v symmetry. This leads to an arachno 8-vertex ar-8 IV,IV structure with 2 tetragonal faces sharing a cobalt atom and a sulfur atom. The process of removing 2 degree-4 vertices from the 10-vertex isocloso deltahedron to give the 8-vertex Cp 2 Co 2 E 2 B 4 H 4 structures B4Co2S2-8 and B4Co2Se2-7 is analogous to the process of removing a degree-4 and a degree-5 vertex from a 14-vertex isocloso 14-vertex deltahedron (Figure 8) to give the 12-vertex ar-14 IV,V Cp 2 Co 2 E 2 B 8 H 8 structures B8Co2S23 and B8Co2S27 (Figure 7) that are discussed above.
The central polyhedra of the 6 lowest-energy 9-vertex Cp 2 Co 2 E 2 B 5 H 5 (E = S, Se) structures ( Figure 10 and Table 4) are all derived from the 10-vertex closo deltahedron, namely, the bicapped square antiprism, by removing either a degree-4 vertex or a degree-5 vertex. Whether a degree-4 vertex is removed to give a capped square antiprism or a degree-5 vertex is removed to give a ni-9 V structure with a pentagonal face makes relatively little difference in energy since the 6 structures lying within 7 kcal/mol of the lowest-energy structures B5Co2E2-1 (E = S, Se) include 2 representatives of the former type and 4 representatives of the latter type. Both chalcogen vertices are always located at a non-triangular face in all of the low-energy structures.
Molecules 2023, 28, x FOR PEER REVIEW 11 of 16 Of the 8 lowest-energy Cp2Co2E2B4H4 (E = S, Se) structures, B4Co2S2-8 and B4Co2Se2-7, which lie at 10.3 and 14.9 kcal/mol in energy, respectively, above the corresponding B4Co2E2-1 structure, are not derived by removing a vertex from a tricapped trigonal prism (Figure 9 and Table 3). Instead, the central Co2S2B4 polyhedron in these structures is generated via the removal of the 2 degree-4 vertices that are bridged by the unique degree-6 vertex from the 10-vertex isocloso deltahedron of ideal C3v symmetry. This leads to an arachno 8-vertex ar-8IV,IV structure with 2 tetragonal faces sharing a cobalt atom and a sulfur atom. The process of removing 2 degree-4 vertices from the 10-vertex isocloso deltahedron to give the 8-vertex Cp2Co2E2B4H4 structures B4Co2S2-8 and B4Co2Se2-7 is analogous to the process of removing a degree-4 and a degree-5 vertex from a 14-vertex isocloso 14-vertex deltahedron (Figure 8) to give the 12-vertex ar-14IV,V Cp2Co2E2B8H8 structures B8Co2S23 and B8Co2S27 (Figure 7) that are discussed above.
The central polyhedra of the 6 lowest-energy 9-vertex Cp2Co2E2B5H5 (E = S, Se) structures ( Figure 10 and Table 4) are all derived from the 10-vertex closo deltahedron, namely, the bicapped square antiprism, by removing either a degree-4 vertex or a degree-5 vertex. Whether a degree-4 vertex is removed to give a capped square antiprism or a degree-5 vertex is removed to give a ni-9V structure with a pentagonal face makes relatively little difference in energy since the 6 structures lying within 7 kcal/mol of the lowest-energy structures B5Co2E2-1 (E = S, Se) include 2 representatives of the former type and 4 representatives of the latter type. Both chalcogen vertices are always located at a non-triangular face in all of the low-energy structures.    The potential energy surfaces for the 10-vertex Cp 2 Co 2 E 2 B 6 H 6 (E = S, Se) systems are the simplest of all, with only 3 structures lying within 16 kcal/mol (E = S) or 21 kcal/mol (E = Se) of the lowest-energy structures B6Co2E2-1 ( Figure 11 and Table 5). The central Co 2 E 2 B 6 framework of all three structures is that of the very stable decaborane, B 10 H 14 , which is obtained via the removal of the unique degree-6 vertex of the closo 11-vertex deltahedron (sometimes called by the confusing name of "edge-coalesced icosahedron") to create a bent hexagonal face. The structures B6Co2E2-1, in which the hexagonal face has only boron and sulfur atoms with the sulfur atoms in opposite ("para") positions, is favored energetically over the next lowest-energy structures B6Co2E2-1 by significant margins of 11 kcal/mol (E = S) and~12 kcal/mol (E = Se). The potential energy surfaces for the 10-vertex Cp2Co2E2B6H6 (E = S, Se) systems are the simplest of all, with only 3 structures lying within 16 kcal/mol (E = S) or 21 kcal/mol (E = Se) of the lowest-energy structures B6Co2E2-1 ( Figure 11 and Table 5). The central Co2E2B6 framework of all three structures is that of the very stable decaborane, B10H14, which is obtained via the removal of the unique degree-6 vertex of the closo 11-vertex deltahedron (sometimes called by the confusing name of "edge-coalesced icosahedron") to create a bent hexagonal face. The structures B6Co2E2-1, in which the hexagonal face has only boron and sulfur atoms with the sulfur atoms in opposite ("para") positions, is favored energetically over the next lowest-energy structures B6Co2E2-1 by significant margins of ~11 kcal/mol (E = S) and ~12 kcal/mol (E = Se).

Theoretical Methods
The chemical models that were investigated in this study are based on various BnHn 2− polyhedra, for which a systematic substitution of 2 BH vertices with 2 CpCo units, followed by the substitution of 2 BH vertices with 2 chalcogen atoms (sulfur or selenium) led to the generation of a total of 5389 different starting structures for each of the Cp2Co2S2Bn-4Hn-4 and Cp2Co2Se2Bn-4Hn-4 systems (n = 8 to 12) (see the Supporting Information).
Full geometry optimizations were carried out on all systems by using the PBE0 DFT functional [43], coupled with the def2-TZVP basis set [44], as implemented in the Gaussian Figure 11. The three lowest-energy Cp 2 Co 2 E 2 B 6 H 6 (E = S, Se) structures are oriented so that the bent hexagonal face is at the top. Table 5. Relative energies (kcal/mol) and geometries of the lowest-energy 10-vertex Cp 2 Co 2 E 2 B 6 H 6 (E = S, Se) structures. In all 3 structures, the central Co 2 E 2 B 6 polyhedron has the same geometry as the B 10 polyhedron in decaborane-14 with a hexagonal face.

Theoretical Methods
The chemical models that were investigated in this study are based on various B n H n 2− polyhedra, for which a systematic substitution of 2 BH vertices with 2 CpCo units, followed by the substitution of 2 BH vertices with 2 chalcogen atoms (sulfur or selenium) led to the generation of a total of 5389 different starting structures for each of the Cp 2 Co 2 S 2 B n−4 H n−4 and Cp 2 Co 2 Se 2 B n−4 H n−4 systems (n = 8 to 12) (see the Supporting Information).
The polyhedral dicobaltadithiaborane and dicobaltadiselenaborane structures Cp 2 Co 2 E 2 B n4 H n−4 (E = S, Se; n = 8 to 12) are designated as B(n−4)Co2E2-x throughout the text, where n is the total number of polyhedral vertices, and x is the relative ordering of the structure on the energy scale. Only the lowest-energy and, thus, potentially chemically significant structures are considered in detail in this paper. More comprehensive structural information including higher-energy structures, connectivity information not readily seen in the figures, and orbital energies and HOMO-LUMO gaps are provided in the Supporting Information.

Summary
The central Co 2 E 2 B n−4 polyhedra in the low-energy structures of the n-vertex polyhedral dicobaltadithiaboranes and dicobaltadiselenaboranes Cp 2 Co 2 E 2 B n−4 H n−4 (E = S, Se; n = 8 to 12) in general are generated from the (n + 1)-vertex most spherical closo deltahedra via the removal of a single vertex, leading to a tetragonal, pentagonal, or hexagonal face, depending on the degree of the vertex removed. In all of these low-energy structures, both chalcogen atoms are located on the non-triangular face vertices, reflecting the energetic preference of chalcogens for lower degree vertices. For the 8-and 9-vertex systems, the structures obtained via the removal of a degree-4 or degree-5 vertex from the corresponding (n + 1)-vertex closo deltahedra, namely, the tricapped trigonal prism and the bicapped square antiprism, have similar energies. The low-energy structures for the 10-vertex Cp 2 Co 2 E 2 B 6 H 6 (E = S, Se) systems all have the framework of the most stable B n H n+4 borane, namely, B 10 H 14 with a bent hexagonal face. The lowest-energy of these 10-vertex Cp 2 Co 2 E 2 B 6 H 6 structures by significant margins exceeding 10 kcal/mol has only boron and both sulfur atoms located at the 6 hexagonal face vertices. The central polyhedra of all of the 11-vertex Cp 2 Co 2 E 2 B 7 H 7 structures are similar to the polyhedron of the dicarbollide anion C 2 B 9 H 12 − in being generated by loss of a vertex from a regular icosahedron to generate a pentagonal face.
In principle, the central polyhedra in most of the low-energy 12-vertex Cp 2 Co 2 E 2 B 8 H 8 (E = S, Se) structures can be derived via the removal of a vertex from the 13-vertex closo deltahedron, namely, the docosahedron. However, the central polyhedron in most of the 12-vertex structures can also be derived from a Co 2 E 2 B 8 icosahedron with adjacent chalcogen vertices by breaking the E-E edge and 1 or more E-B edges to create a hexagonal hole.
Two examples of arachno polyhedra were found among the set of lowest-energy Cp 2 Co 2 E 2 B n−4 H n−4 (E = S, Se; n = 8 to 12) structures. The central polyhedron of one structure within 15 kcal/mol of the lowest-energy structure in each of the 8-vertex Cp 2 Co 2 E 2 B 4 H 4 (E = S, Se) systems is an arachno polyhedron with 2 tetragonal faces sharing an edge that is derived from the 10-vertex isocloso deltahedron via the removal of the 2 degree-4 vertices bridged by the unique degree-6 vertex. In addition, 2 of the structures in each of the 12-vertex Cp 2 Co 2 E 2 B 8 H 8 (E = S, Se) systems are ar12 IV,V structures that are de-rived from the 14-vertex isocloso deltahedron that is found in the experimentally known Cp* 2 Ru 2 C 2 B 10 H 12 by removing the unique degree-4 vertex as well as a degree-5 vertex.
Supplementary Materials: The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/molecules28072988/s1, Initial structures, distance, and energy ranking tables, orbital energies and HOMO/LUMO gaps, complete Gaussian09 reference; concatenated .xyz file containing the Cartesian coordinates of the lowest-energy optimized structure. Table S1A: Initial 8-vertex starting structures. Table S1B: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 S 2 B 4 H 4 structures. Table S1C: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 Se 2 gB 4 H 4 structures. Table S2A: Initial 9-vertex starting structures. Table S2B: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 S 2 B 5 H 5 structures. Table S2C: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 Se 5 B 5 H 5 structures. Table S3A: Initial 10-vertex starting structures. Table S3B: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 S 2 B 6 H 6 structures. Table S3C: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 Se 2 B 6 H 6 structures. Table S4A: Initial 11-vertex starting structures. Table S4B: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 S 2 B 7 H 7 structures. Table S4C: Distance matrices and energy rankings for the lowest energy Cp 2 Co 2 Se 2 B 7 H 7 structures.  Table S6A: Distance matrices and energy rankings for the lowest energy permethylated Cp* 2 Co 2 S 2 B 7 H 7 structures. Table S6B: Distance matrices and energy rankings for the lowest energy permethylated Cp* 2 Co 2 Se 2 B 7 H 7 structures. Table S7: Orbital energies and HOMO-LUMO gaps for the lowest Cp 2 Co 2 S 2 B n−4 H n−4 (n = 8 to 12) structures. Table S8: Orbital energies and HOMO-LUMO gaps for the lowest Cp 2 Co 2 Se 2 B n−4 H n−4 (n = 8 to 12) structures.

Funding:
The computational facilities were provided by the Babeş-Bolyai University under project POC/398/1/1/124155, which is co-financed by the European Regional Development Fund (ERDF) through the Competitiveness Operational Program for Romania 2014-2020.

Data Availability Statement:
The data presented in this study are available in supplementary material.