Categorization of Boranes Into Clan Series

Boranes, despite their instability in nature, can be regarded as hydrocarbon relatives since a [BH] fragment corresponds to a carbon [C] skeletal element in terms of the number of valence electrons. The borane formula which can be expressed as BnHm usually appears in such a way that when (n) is even, then (m) is even and when (n) is odd, (m) is odd as well. Through the study of cluster series, it appears that the cluster number K which represents skeletal linkages is usually a whole number. This inherent characteristic confers unique order within borane clusters with nodal connectivity of 5 and the polyhedral nature of the borane clusters. The orderliness of the borane clusters is reflected by the ease of their categorization into clan series and their readily constructed geometrical isomeric structures. The cluster valence electrons can easily be calculated using one of the six recently discovered fundamental equations.


The Six Equations for Calculating the Cluster Valence Electrons
Some or all the six equations have been used in the analysis of many clusters (Kiremire, 2019b) are being applied here in the analysis of the selected borane clusters. They all give the same result of the cluster valence electrons (VE) as the one calculated from the cluster formula (VF). This underpins the validity of the developed cluster valence equations. They are derived from the series equations and the capping principle of the 4N series method. These equations are: i. K(n); VE=8n-2K ii. S=4n+q; VE=4n+q K=C y +D z ; VE0=2z+2, VEDz=4z+2, y+z=n= the number of skeletal elements.

vi. VE=VE0+2y+2z
The symbol n= the number of skeletal elements in a cluster excluding the ligands, K= skeletal linkages linking up the skeletal elements, S=series equation, q is a numerical variable that defines the type of the cluster, the K value can also be expressed in terms of C y +D z where C and D represent the categorization of a cluster, y is in principle, the number of the capping skeletal elements, and z is the number of the nuclear elements. Knowing the values of n and K of a cluster, the q value of a cluster can be derived from (V). Hence the cluster series equation can readily be obtained. From the series equation, the K value is then expressed in terms of n and q. The Kp value and K* can also be derived from K. The K* is finally expressed in terms of C y and D z which determine the categorization of a cluster. This approach has been applied in the example 1 given above.

Construction of the Skeletal Isomeric Structures
Arising from the experience of sketching isomeric skeletal isomers of clusters, it has been found to be easier and useful to use quasi-circular skeletal structures to construct the isomers. The K(n) parameter has been very helpful. In this regard, a geometrical figure of n sides is selected and K linkages are inserted. Let us consider the following examples as illustrations. Take C 2 B 4 H 6 cluster. The K value for C is 2 and V=4 whereas B has a K value of 2.5 and V=5. The cluster has 11 skeletal linkages and 6 skeletal elements. In order to construct a skeletal isomer, a 6-memberered ring is selected. Then 5 linkages are added in such a way that TRIANGULAR FACES ARE FORMED. Since there are 6 skeletal elements, these can cyclically be linked by 6 linkages out of the 11 total linkages corresponding to K=11. Hence, the remaining 5 linkages are utilized to construct the appropriate triangular faces. The triangles are constructed in such a way that the valences of the constituent skeletal elements are obeyed. This is shown in Figure 9. The same approach was done for B 8 H 14 , B 10 H 14 , C 2 B 10 H 12 and B 12 which are shown in Figures 10-17. The boron skeletal element is quite unique in that in it appears to strictly exert a skeletal valence of 5 in all its hydride clusters and other complexes such as halides, metalloboranes and metallocarboranes (Kiremire, 2017b). According to the 4N series approach, we can regard the bridging hydride structures of boranes as re-arrangements of the clusters so as to achieve more stable conformations. In summary, the construction of the isomeric shapes using the K(n) parameter, can be expressed as K=n+x where K=number of skeletal linkages, n= the number of the skeletal elements and x= the number of linkages inserted within and/or around the n-sided figure in such a way that skeletal TRIANGLES are generated. The parent structures on which linkage lines are drawn are based on Symyx Draw 3.2 program.

Categorization of Borane Clusters
A systematic method of categorization of clusters was recently developed (Kiremire 2019a-c). According to the method, a categorization parameter K*= C y +D z was introduced where y+z=n = normally is number of skeletal elements in the cluster and C = represents the outer capping symbol and D represents the inner capping symbol of the cluster as it was discovered that clusters and chemical elements portray double capping phenomena. The symbol D z is also regarded as representing CLAN series of the clusters (Kiremire, 2019a-c).The analyzed boranes fall within the range D 1 -D 22 cluster clans and are shown in Table 1. The capping is in line with the descending cluster series that Rudolph identified more than 40 years ago (Rudolph, 1976).

The Evolution of Isomeric Polyhedral Geometries
During the analysis of borane clusters including the construction of clusters shapes with nuclearity index of 5 and above, it has been found helpful to utilize ring structures corresponding to the level of the nuclearity index. This means that for example B 5 clusters, a 5-membered ring is used, B 6 clusters, we use a 6-membered ring, B 7 ,7-membered ring, B 8 , 8membered ring and so on. According to this approach, there is a relationship between the cluster skeletal linkages K and the number of skeletal elements, n. This relationship is given between by K=n+x where x=number of skeletal lines or triangles constructed. As this relationship is considered very important, is being repeated here for emphasis. Thus, for K(n)=7(6), 7=6+1 and therefore a 6-membered ring with one triangle constructed inside will fulfill the cluster valence content of all the hydrogen ligands in the cluster. This is the case for B 6 H 16 shown in Figure 18-0. The cluster B 6 H 14 with K(n)=8(6); 8=6+2, we will construct two triangles to be in resonance with the formula B 6 H 14 . This is shown in Figure 19. The procedure goes on for B 6 H 12 , K(n)=9(6); 9=6+3→3 triangles; B 6 H 10 , K(n)=10(6); 10=6+4→4 triangles, B 6 H 8 [B 6 H 6 2-]→K(n)=11(6); 11=6+5→5 triangles, B 6 H 6 ,K(n)=12(6);and 12=6+6→6 triangles. The increasing number of skeletal triangles are given in Figures 20-27. The K(n)=12(6) value for B 6 H 6 is the same as that of C 6 and Os 6 (CO) 18 . Therefore, these clusters have the same skeletal structure. These are shown in Scheme 1.  (6) Linking up the nodal charges to get a closed shape  (6) (6) The same as B6H6.

Closo Capping Series
Os6(CO)18 Os(K=5, V=10) Scheme 1. Equivalent Isomeric graphical mono-capped closo structures of B 6 H 6 , C 6 and Os 6 (CO) 18 Since B 6 H 6 , C 6 and Os 6 (CO) 18 have the same K(n) value, we can regard them as being equivalent SKELETAL ISOMERS. Since the CO ligand is a two-electron donor, it can be considered as using up 2 skeletal linkages of osmium skeletal element. The carbon skeletal element has 4 skeletal linkages which are all utilized in forming the isomeric skeletal structure of C 6 .

The Closo Series: B n H n 2-
The construction of the closo structures is similar to the general one of constructing other non-closo structures. The guidance is simply the K(n) parameter of the cluster. First, an n-sided ring is selected. Then additional linkages are added to the ring in such a way that the total linkages are the same as the numerical number of skeletal linkages K. That is, K=n+x where x represents the number of additional linkages constructed in such a way that triangles are created including the nodal points carrying the negative charges. This procedure is shown in Scheme 2 for the closo B 5 H 5 2-. In closo structures, the points with negative charges are linked by a line sealing off a triangle.

General Descending Series
A wide range of clusters have been categorized into clan series using skeletal numbers. More than 60 borane clusters have been ranging from D 5 to D 22 . Their analysis is given examples 1-51 some of their isomeric graphical shapes in Figures1-54 and are presented in Table 1.

The popular Rudolph Clan Series
Clusters have broadly been categorized (Kiremire,2018). Those which change by ∆K= ±3 and ∆n =±1 are in line with the cluster relationship which was brilliantly identified by Rudolph more than 40 years ago (Rudolph,1976).
The Rudolph clan series behave as if a boron (B) skeletal element is transmuted into a hydrogen(H) ligand at each descending step. Also a closo skeletal structure is obtained when the NODAL points bearing negative charges are linked up to generate an additional triangular face. The changes in the skeletal structure are shown in Scheme 3.

Conventional Capping in Borane Clusters
The proper capping based on 4N series approach is very rare in borane clusters unlike the clusters such as those found in golden and transition metal complexes. According to the sample of clusters analyzed in this paper, only 3 were found. These are B 20 H 16 , and K*=C 3 +D 17 (tri-capped) is given in Figure 66, B 20 H 20 , K*= C 1 +D 19 (mono-capped) Figure 67

Conclusion
The borane clusters have been categorized into clan series. The six newly discovered equations for calculating cluster valence electrons have been demonstrated. The formation of triangles in borane clusters could be associated with the attainment of 5 linkage nodes at borane nodal points and the polyhedral nature of borane clusters. A few rare normal capping borane have been identified. The number of triangles of borane clusters formed using the 4N approach for closo system are the same as nuclearity index of a cluster. The borane formulas follow a well-organized mathematical sequence.