Beads , Necklaces , Chains and Strings in Capping Carbonyl Clusters ENOS

The paper attempts to explain at length the close relationship between transition metal carbonyl clusters with main group clusters especially the boranes using the 14n and 4n rules. When the ‘shielding’ electrons are removed from a transition metal carbonyl cluster and becomes ‘naked’, it resembles a corresponding one in the main group elements. A an expanded table of osmium carbonyl clusters was constructed using the capping fragment Os(CO)2(14n-2) and the fragment Os(CO)3 (14n+0). The table reveals the fact that the known series such closo, nido and arachno are part and parcel of a wide range of series especially the capping series 14n+q, where q takes up negative multiple integers of two including 0 such as such = 0, -2,-4, -6, and so on. The linkage between capping series in transition metal carbonyl clusters has also been identified. Apart from the capping series generated in the table, there is another type of series where the skeletal cluster elements remained the same but the number of carbonyl ligands successively decreased. These types of series are referred to as stripping series. Mapping generating functions were also derived which produces any cluster formula or series required. Also the table shows that many clusters form utilizing some of its atoms as closo nucleus around which the larger ones are built and thus forming clusters within larger clusters. The table may be used to categorize a given cluster formula that falls within its range. Otherwise, using the 14n rule or 4n rule can be used for cluster classification. Furthermore, the table indicated that atoms, fragments and molecules can be classified into series. Through this approach of using series, Hoffmann’s importantisolobalrelationship of chemical species can splendidly be explained.Using the 14n rule and 4n rules creates a framework under which chemical species such as atoms, fragments, molecules and ions some of which may appear unrelated from main group elements and transition metal may be grouped together like ‘birds of a feather flock together’ provided the skeletal elements obey octet or eighteen electron rule.


INTRODUCTION
The Wade-Mingos rules have significantly enhanced the understanding and development of cluster chemistry since 1970s 1-2 .Wade's rule related rule by illustrating its application with a few examples.Let us take B 6 H 6 2¯ cluster with a regular octahedral symmetry 3 .If V represents the number of valence electrons of the cluster and B e H the number of electrons involved in the B-H bonds and S e the skeletal electrons, then V = 3x6+6+2 = 26, B e H = 2x6 = 12 and hence S e = 26-12 = 14.Therefore, the number of skeletal electron pairs in this case, S ep = 14/2 = 7.Since n = 6 for B 6 H 6 2¯ cluster, this give us the relationship S ep = n+1 = 6+1.This is the same as,S ep = n+1 which applies to all B n H n 2¯c loso clusters.In actual fact, this represents the closo series of clusters whose sleletal elements obey octet (8) or eighteen (18) electron rule.Let us consider B 5 H 9 cluster; n = 5, V = 3x5+9 = 24, ,B e H =2x5 =10, S e = 24-10 = 14.Hence, S ep = 14/2 = 7.In this example, S ep = n+2 = 5+2 = 7.Thus, B 5 H 9 (B-5)is a nido cluster a derivative of an octahedral cluster (B-6) where B-5 and B-6 represent the 5 and 6 skeletal elements respectively.The B-5 cluster will have an ideal square pyramid shape (C 4v ) 4 as opposed to B-6 octahedral shape(O h ).In addition to S ep = n+2 relationship assisting us to predict the shape B 5 H 9 cluster, it actually represents the nido series of clusters.Let us look at the last example of B 4 H 10 cluster.In this case, n = 4, V = 3x4+10 = 22 and B e H = 2x4 =8.Therefore, S e = 22-8 = 14 and S ep = 14/2 =7.This gives us the series, S ep = n+3 which represents arachno series and B 4 H 10 cluster is a member.The geometrical relationship of arachno to that of the corresponding closo 'parent' may simply be represented as M-4'!M-5'!M-6.Hence, B 4 H 10 is derived from B 6 H 6 2¯o ctahedral symmetry.On the other hand, Mingos related the series to the number of electron count of the cluster 5 .The relationship is as follows: (4n-2) Bicappedcloso, (4n) Monocappedcloso, (4n+2) closo, (4n+4) Nido, (4n+6) Arachno, and (4n+8) Hypho.Both Wade and Mingos relationship were derived from Molecular Orbital Theory 6 .

Universality of the 4n and 14n Rules
In an attempt to find a simpler way to teach Lewis shapes of simple molecules, a fundamental question arose.Is there a simple formula that could link the number of atoms that obey the octet rule with the corresponding number of valence electrons?This was found and has been exceedingly useful in analyzing clusters such as hydrocarbons, boranes, heteroboranes and transition metal carbonyl clusters.Let us consider a few examples to illustrate how the cluster series were simply derived by this approach.Consider a C 2 diatomic molecule, n =2, the number of valence electrons for the two carbon atoms V =4x2 =8.But if the number of carbon atoms, n = 2 is multiplied by 4 gives the same numerical result as the total valence electrons, then S = 4x2 = 4n = 8 = V where S stands for series.If two hydrogen atoms (2H) are hypothetically added to the C 2 molecule, we generate C 2 H 2 molecule.This means that the two added hydrogen atoms have donated an additional 2 electrons giving rise to a total number of 10.If we take 4n as a standard or baseline on which other molecules with skeletal atoms that obey octet rule then C 2 H 2 , n =2, 4n =8, V =10.Therefore S = 4n+2 = V.Hence, C 2 H 2 may be regarded as a member of S= 4n +2 series.How can we relate the number of bonds (k) in the molecule to its series formula?We know that C 2 H 2 has a triple (3) bond, and therefore k = 2n-1 = 2x2-1 =3.Thus, the two hydrogen atoms acted as 'ligands', guest or well-wisher atoms to simply donate the electrons to assist the 'C 2 ' skeletal atoms attain the octet rule.If we continue the process and add two more H atoms to the 'C 2 ' skeletal atoms we get C 2 H 4 ,n =2, 4n = 8 and V=12.Hence S = 4n+4 and k = 2n-2 = 2x2-2 =2.This is in agreement with the double bond found in C 2 H 4 .If we use the same procedure for O 2 molecule, n = 2, 4n = 8, V =6x2 =12.Hence S =4n+4 and k =2n-2 = 2.That is, O 2 as in C 2 H 4 has a double bond although it does not need any donations from the well-wisher atoms.Continuing adding the two hydrogen atoms we get C 2 H 6 .In this case, n =2, 4n = 8 and V= 14 and hence S =4n+6 and k = 2n-3 = 1.This means that C 2 H 6 has one C-C bond.On the other hand,the diatomic molecule F 2 gives us a similar result, n=2, 4n =8 and V= 14 and therefore S = 4n +6, and k =2n-3 = 1 without the support from donor atoms.This agrees with the single bond observed for F 2 .The method can be extended to heteroatomic systems.For instance, NO + , n =2, 4n =8, V = 5+6-1 = 10.Hence, S = 4n+ 2, k = 2n-1 = 3, the bond order of NO + .We can apply the method to skeletal atoms greater than 2. For instance,C 4 H 4 , n=4, 4n =16, V = 20, S = 4n+4, and k = 2n-2 = 2x4-2 = 6.With the k value of 6 , we can sketch a number of isomers.Two of these are sketched as shown in Figure 1.

Construction of a hypothetical table of carbonyl clusters
After analyzing the structure of cluster carbonyl series, it was observed that the capping series vary by a change in k value of 3 from one cluster member to the next involving an Os(CO) 2 (14n-2 series) fragment while in the noncapping (ordinary)series the variation in k value is 2 units from one cluster member to the next involving an Os(CO) 3 (14n+0 series) carbonyl fragment.With this knowledge in mind, a hypothetical table of osmium clusters was constructed.This is given in Table 3.This table can be extended where possible as required.The table is quite fascinating indeed.In order to analyze the characteristics of clusters, an assumption was made that the clusters in this case was that all clusters will comprise of osmium and carbonyl(CO) ligands only.Since the CO ligand donates two electrons, the charged clusters and other ligands are converted into CO ligand equivalent in terms of donating electrons.For instance, Os 6 (CO) 18 2¯ is taken as Os 6 (CO) 19 for the purposes of this analysis.The columns represent the ordinary series in which the expansion involvesOs(CO) 3 (14n+0) fragment which does not influence the change in the type of series for instance, the 14n represent the members of Monocapped series which include, Os 2 (CO) 6 , Os 3 (CO) 9 , Os 4 (CO) 12 , Os 5 (CO) 15 , Os 6 (CO) 18 , Os 7 (CO) 21 , Os 8 (CO) 24 , and Os 9 (CO) 27 .One interesting member is Os(CO) 3 the building block fragment for noncapping series.The rows represent the capping series since the building fragment Os(CO) 2 (14n-2) is a capping fragment containing the change "operator unit of (-2)"while the columns represent the ordinary series such as Hypho (14n+8), Arachno(14n+6), Nido(14n+4), Closo(14n+2), Monocap(14n), Bicap(14n-2), Tricap(14n-4), Tetracap(14n-6) and so on.Let us take the following capping series as seen in Table 3, namely 10 , Os 2 (CO) 10 → Os 3 (CO) 12 →Os 4 (CO) 14 →Os 5 (CO) 16 → Os 6 (CO) 18 .The respective k values are given by (k = 2n-4 = 0, n =2), ( k = 2n-3= 3, n =3), (k = 2n-2 = 6, n =4), ( k = 2n-1, n =5) and ( k = 2n, n =6) which as a set are(0, 3, 6, 9,).This involves in change of k units by 3. The value of k = 0 means that the clusterOs 2 (CO) 10 exists as Os(CO) 5 molecules.The other k values imply the clusters have shapes changing from a triangle (k =3), to a tetrahedral ( k =6), to a trigonalbipyramid ( k= 9) and to a capped trigonalbipyramid (k = 9+ 3 =12).These changes are shown in Figure 3.     Just as the elements of the periodic table have their atomic numbers and principal quantum numbers start with a digit of 1 and faithfully increase step by step with a numerical value of 1, the carbonyl clusters as derived from Table 3 have three main types of series namely, the 'STRIPPING'series (Diagonal in Table 3) which lose 1 CO ligand step by step and have a cluster k value also correspondingly change by 1, the 'CAPPING' series (Horizontal or row in Table 3) with a periodic change of k value by 3 and the 'ORDINARY' series (Vertical or Column in Table 3) which vary with a periodic k value of 2.

Correlation between 14n series of transition metal carbonyl series and 4n series of the main group elements
When the corresponding series are compared, the correlation between the two series becomes vivid.This is illustrated in the Table4 for a few selected series.
On the basis of the relationship between 14n and 4n system, it is possible to suggest a possible formula of the corresponding main group compound.We can also make a correlation with boranes in the case of capped carbonyl clusters.For instance, Os 10 (CO) 26 2¯ is known to be a tetracapped {C 4 C[M-6]} cluster.Whereas the capping fragment was identified as Os(CO) 2 which is a member of 14n-2 series and with electron content of 12, the corresponding capping fragment in boranes may be taken as (+B, -H) equivalent to adding having a content of 2 electrons.The (+B, -H) may be represented as [B, -H] or [B(-)H] in which case for every capping a B atom is added to the cluster while the H atom is subtracted each time.The fragment[B, -H] may be regarded as (4n-2) series corresponding to Os(CO) 2 (14n-2)fragment .If we take Os 6 (CO) . This implies the octahedral cluster of 6 boron atoms will, in theory, be capped with 4 B atoms.In the case of capping boranes, the process involves replacing every H atom in the neutral borane known or unknown by a B atom.This means for B 6 H 6 2¯ = B 6 H 8 we can, in theory, cap it 8 times resulting into a final B 14 cluster.It is fascinating to note that B n (n = 3-15) boron clusters are being studied 15 .

Beads, strings and chains of clusters
The way Table 3 is constructed is like chains or strings of cluster units(molecules) increasing as we move from left to right along the capping series.The clusters can be synthesized by adding atoms, fragments, and molecules to make small or short , medium or large clusters.It imitates the way beads are put together to make short or long necklaces.In a way, the process is like putting chains tother to make long ones or train cabs being assembled.For instance the cluster chain may start with one skeletal atom (M-1) at one end goes up to more than 30 skeletal atoms(M-30) at the other.In principle, such chain could go on indefinitely by adding one skeletal atom at a time.Since moving from left to right involves the addition of Os(CO) 2 (14n-2) fragment, we can generate horizontal capping cluster series starting with Os atom as follows: Os→Os 2 (CO) 2 → Os 3 (CO) 4 → Os 4 (CO) 6 → Os 5 (CO) 8 → Os 6 (CO) 10 → Os 7 (CO) 12 → Os 8 (CO) 14 → Os 9 (CO) 16 → Os 10 (CO) 18 and so on.It has been shown in this illustration how a capping series can be created starting with Os (horizontal row).This is a simple example of one of the rows of the capping series and Os(CO) 2 (14n-2) fragment may be regarded as a capping constant.The same applies to the movement from the top to bottom for an ordinary cluster series such as closo or nido.The series are mathematically precise in terms of the content of valence electrons.For instance the table classifies osmium atom (Os) (see beginning of the Table 3)as a member of 14n-6 (tetracap)series thus, Os'→ Os 2 (CO) 3 → Os 3 (CO) 6 .The corresponding valence electron contentare numerically equal to 8, 22, and 36.This sequence varies by 14 electron units as expected from ordinary carbonyl series.The osmium atom also obeys the series S =14n-6 =V as n = 1 for one osmium skeletal ton S = 14(1)-6 = 8 the valence content of osmium.If we continue to successively add Os(CO) 3 (14n+0)fragment starting with Os we will generate a column of Tetracapped Series(S =14n -6, C p = C 4 C) as follows: Os'→ Os 2 (CO) 3 → Os 3 (CO) 6 → Os 4 (CO) 9 → Os 5 (CO) 12 → Os 6 (CO) 15 → Os 7 (CO) 18 → Os 8 (CO) 21 → Os 9 (CO) 24 → Os 10 (CO) 27 and so on.As can be seen from Table 3,Os 10 (CO) 27 is a member of tetracapped series with its closo nucleus based on octahedral symmetry.We may symbolically express this as C p = C 4 Using other points or 'beads'similar capping and ordinary cluster series can be derived.'The S = 14n-4 (Tricapped)series has the three first members as OsCO→Os 2 (CO) 4 →Os 3 (CO) 7 with the corresponding electron content as (10, 24, 38).As can be seen, they also vary by 14 electron units.If n = 1, then 14n-4 = 14(1)-4 = 10 which corresponds to the electron content of OsCO fragment.Also the Table 3shows that CO molecule can be placed in the category of CLOSO series( 14n+2) as CO→ Os(CO) 4 →Os 2 (CO) 7 and so on.In terms of valence electron content this corresponds to 2→16→30.Again the change from one member to the next is 14 electron units.Indeed the CO obeys the 14n+2 rule since when n = 0, 14n+2 = 14(0)+2 = 2, the number of electrons one CO donates to a cluster.What is also fascinating is that the CO molecule also obeys the S = 4n+ 2 rule for n =2 (V = 10) of the main group element clusters already discussed.This links up nicely the parallel relationship between (14n +2 →4n+2) series.Finally, let us look at the three first members of the Monocap (14n) series namely, Os(CO) 3 →Os 2 (CO) 6 →Os 3 (CO) 9 .The respective electron content is 14'!28'!42 with the series variation of 14 electron units.If n = 1, then S = 14n = 14(1) =14 the valence electron content of Os(CO) 3 fragment.Indeed the Os(CO) 3 fragment is a member of the 14n series.THE DIAGONAL RELATIONSHIP The way the table is arranged reveals yet another interesting relationship of the cluster series.For example consider the following diagonal relationship type of series.Let us begin with Os 4 (CO) 16 and follow its path; Os 4 (CO) 16 →Os 4 (CO) 15 → Os 4 (CO) 14 → Os 4 (CO) 13 → Os 4 (CO) 12 → Os 4 (CO) 11 →Os 4 (CO) 10 → Os 4 (CO) 9 → Os 4 (CO) 8 → Os 4 (CO) 7 → Os 4 (CO) 6 → Os 4 (CO) 5 → Os 4 (CO) 4 → Os 4 (CO) 3 →Os 4 (CO) 2 → Os 4 (CO)'!Os 4 .The stripping series showing a 'naked' cluster such as Os 4 is quite interesting.Can the clusters of the series Os n (n =2, 3, 4, 5, 6, 7, 8, 9, 10..)exist in the absence of supporting ligands?The above type of series may be called STRIPPING SERIES since at every step the CO ligand is stripped off.Some good work has been done that reveals the existence of stripping series [16][17] .

The generating functions of series and Cluster
When these carbonyl cluster series are scrutinized, it becomes clear that they have inherent orderly simple patters.Such patterns can easily be expressed by simple algebraic functions.For instance, the Hypho Series (S =14n+8) can be generated by a simple mapping function, f(n) →(3n+4).Let us apply the mapping function to generate a few clusters as follows: f(0) →3(0) +4 = 4.This means that when there is no skeletal element, therefore we get 4CO ligands as members of the 4n+8 series.Since n = 0, then S = 14n+8 = 0+8 =8, the valence content of 4CO (4x2=8) ligands.When n =1, then f(1) →(3x1+4) = 7.This means we have generated the hypothetical chemical species Os(CO) 7 which has also been obtained in the process of constructing the series table.For n =4, for example, we get f(4) →(3x4+4) = 16.This gives us the cluster member of Os 4 (CO) 16 one of the carbonyl complex of osmium.It also belongs to Hypho Series (S = 14n+8).The generating function for Arachno Series (S = 14n+6) is f Selected generating functions of Nido, Closo, Monocaped and Bicapped series as well as osmium carbonyl clusters produced up to n =12 are presented in Table7.

Detrmining a required hypothetical cluster formular and the simple algebra of clusters
Apart from generating members of the clusters series using generating functions such as those shown above, it is possible to create a simple formula to produce a required or targeted cluster carbonyl complex.Let us consider the following hypothetical examples.Let us take Os 3 (CO) 12 complex as a starting material on which to build other carbonyl clusters.Let us further visualize adding 57 more fragments of the formula Os(CO) 3 .The final cluster will have a molecular formula obtained by the following simple relationship F = Os 3 (CO) 12 + 57 [Os(CO) 3 ] = Os 3 (CO) 12 +Os 57 (CO) 171

The growth of clusters
A close analysis of Table 3 shows that there are two principal fragments responsible for producing osmium carbonyl clusters namely Os(CO) 3 (14n+0) and Os(CO) 2 (14n-2).The repeated addition of Os(CO) 3 (14n+0)fragment produces an increase in cluster size)length without change in the type of series.This is illustrated and summarized in the example shown in Scheme 1.On the other hand, the addition of Os(CO) 2 (14n-2) fragment introduces a cap to the chemical system in question.When a molecule being extended is a member of series below the closo series, the capping goes on till a closo cluster has been attained.After that the capping becomes centered on that closo unit.This is illustrated in Scheme 2 where the capping is based on a trigonalbipyramid unit.As it can clearly be seen, there is no change in the type of series as it starts with Os 3 (CO) 12 (S=14n+6) and ends with Os 60 (CO) 183 (S=14n+6) for the successive addition of 57 Os(CO) 3 (14n+0) fragments.Let us compare the above hypothetical example with a similar one but this time using the transforming fragment Os(CO) 2 (14n-2).The final cluster formed will be given by F = Os 3 (CO) 12 + 57 [Os(CO) 2 ] = Os 3 (CO) 12 +Os 57 (CO) 114 = Os 60 (CO) 126 .This cluster can be categorized as follows, n = 60, 14n =14x60 = 840,V = 732, S = 14n-108.This can be written as S = 14n +54(-2).Hence the capping notation by this method derived from this series is given by C p = C 1 + C 54 = C 55 C [M-5].This symbol implies the Os 3 (CO) 12 starting simple carbonyl cluster was capped 57 times but 2 of those copings were utilized to create a Closo nucleus with a trigonalbipyramid entity symbolized as [M-5].Since the nucleus is of Closo Series its generating mapping function is f(n) '!(3n+1).Hence, [M-5] = f(5)→(3x5+1) = 16.This gives us the valence equivalent cluster of Os 5 (CO) 16 .This nuclear cluster has, in principle, a trigonalbipyramid shape.In other words, Os 60 (CO) 126 is expected to be a huge carbonyl cluster of 60 skeletal atoms but hidden inside is another smaller cluster of a trigonalbipyramid(D 3h )symmetry.In Scheme 2, we can give a possible process of the formation of the giant cluster and the transformation of the cluster series.The addition of Os(CO) 2 (14n-2) fragment decreases the code of the cluster by (-2).For instance, if the molecular system belongs to S = 14n+10 series, it will be converted into [14n+10-2] = 14n+8.The next addition generates 14n+6(hypho series) , then 14n+4(nido), 14n+2(closo), 14n(monocap), 14n-2(bicap).14n-4(tricap), 14n-6(tetracap), and so on.In this way, the concept of cluster formation is more or less similar to that of POLYMERIZATION.An attempt to demonstrate the concept of cluster formation is illustrated in Shemes 1 and 2 as well as in Table 7.
It is quite clear that in the capping process, the 14n parameter remains constant, the change involves only the interactions of the outside digits in the series codes.With this observation, we can do the arithmetical additions of the outside digits to determine the final code of the cluster.In this case, we have to consider the changes of the code additions of (14n+6)+ 57(14n-2).In terms of the code addition, this will be equal to [14n+6+57(-2)]=(14n-108).This is indeed the cluster series code for Os 60 (CO) 126 carbonyl complex.We also know that 14n represents Monocapping.Any extra capping after 14n is represented by (-2).Hence, -108 = 54(-2) represents another 54 cappings.Therefore, we can translate the code 14n-108 to mean 1 cap from 14n and 54 caps from (-108) giving us a total of 55 caps all together.For visual purposes we may represent a huge cluster with a smaller one inside shown in Figure 4.

Classification of carbonyl clusters using the osmium carbonyl cluster table
The above constructed carbonyl table(Table 3) is extremely useful in categorizing a carbonyl cluster into its type of series code.That is, to deduce whether a given cluster is a member of klapo, hypho, arachno, closo, monocapped and so on.If a given cluster comprises of osmium and CO ligands only, the deduction is simple.It is a matter of tracing the formula of that cluster and determine its location in Table 3 and hence the code of the series is just on top of the table.Let us illustrate this idea using the table with the following osmium clusters.Starting with a simple oneOs 3 (CO) 12 and doing a little calculation will quickly show you where it is located.For instance, looking for its lower members in the ordinary series, by removing Os(CO) 3 fragment we get the following: Os 3 (CO) 12 -Os(CO) 3 ®Os 2 (CO) 9 ¯ Os(CO) 3 ®Os(CO) 6 and this places it in the ARACHNO family(14n+6).In the stripping series, the next member on the diagonal is Os 3 (CO) 12 ¯(CO)®Os 3 (CO) 11 .In the capping series, its neighbor will be Os 3 (CO) 12 + Os(CO) 2 '!Os 4 (CO) 14 .Thus, with certainty, Os 3 (CO) 12 is a member of Arachno series.We can also use the 14n rule which is exceedingly useful for the classification of large clusters which may not be in Table 3 as follows, n = 3, 14n = 14x3 = 42, valence electrons V = 3x8+12x2 = 48 and series S = 14n+6 =V (Arachno).This process is summarized in Scheme 3. Take Os 5 (CO) 16 example.We can remove 4Os(CO) 3 = Os 4 (CO) 12 fragments to determine the starting building block of its ordinary series.Thus,Os 5 (CO) 16 ¯Os 4 (CO) 12 = Os(CO) 4 .This places Os 5 (CO) 16 in the Closo series (14n+2).When there is a charge as in Os 6 (CO) 18 2¯ cluster, the two negative charges are converted into its CO equivalent.In this case, (-2) = 1 CO and the cluster becomesOs 6 (CO) 19 .Removal of 5Os(CO) 3 = Os 5 (CO) 15 from Os 6 (CO) 19 gives is the starting unit as Os(CO) 4 .Clearly, Os 6 (CO) 19 (Os 6 (CO) 18 2¯ ) is a member of the closo series and since it has six skeletal atoms Os 6 [M-6], they will take up an O h symmetry.If a cluster has "guest atoms"or ligands other than CO, then the valence electron content of the cluster is converted into its osmium equivalent.For instance H 3 Os 6 (B)(CO) 16 its valence electron value V = 3+6x8+3+16x2 = 86.However the 6 Os atoms have a valence electron content given by Vos = 6x8 = 48.Hence the remaining electrons Vco = 86-48 = 38 will be converted into its CO equivalent ligands as 38/2 = 19.Hence, the osmium equivalent cluster will beOs 6 (CO) 19 (Os 6 (CO) 18 2¯) which is closo and has an O h symmetry.Consider Ir 4 (CO) 12 cluster, V = 4x9+12x2 = 60, Vos = 4x8 =32, Vco = V-Vos =60-32 = 28 and number of CO ligands Nco =Vco/  14 .The starting block in its ordinary series is given by Os 4 (CO) 14 ¯ 3Os(CO) 3 '!Os 4 (CO) 14  Ōs 3 (CO) 9 = Os(CO) 5 .This places Ir 4 (CO) 12 cluster in the Nido family (14n+4).For very large carbonyl clusters, their formulas get out of the range of the table.In that case, the 14n rule explained in our earlier work 18 is very helpful.

Giant python clusters swallowing smaller ones
Take Os 6 (CO) 19 (closooctahedtral cluster) as a starting cluster on which others can be built.If 94 Os(CO) 2 fragments are added to it we will generate a huge cluster of the formula F = Os 6 (CO) 19 + Os 94 (CO) 188 = Os 100 (CO) 207 .Applying our method of classification, 14n = 14x100 = 1400, V = 1214.Hence this cluster belongs to S =14n-186 series.The series may be written as S = 14n+93(-2).This has a capping given by the symbol C p = C 1 +C 93 = C 94 C[M-6].This symbol means that this will be a huge cluster with an [M-6] cluster embedded inside other 94 skeletal atoms.The [M-6] notation is a symbol to represent an octahedral nuclear cluster enclosed in the giant cluster of total 100 skeletal atoms.The cluster series code can also be derived from S =[(14n+2)+94(-2)] =[14n+2-188] =(14n-186) as obtained above.All the fragments have been utilized for capping purposes since the starting cluster is already a closo member.is a unique one as 14 times the number of skeletal atoms gives a numerical result exactly the same as the number of valence electrons of the cluster.It may regarded as a hexa-mer of Os(CO) 3 (14n) fragment.We can also write the series as Os(CO) 3 (14n+0).Since the digit after 14n is 0, the various combinations of the same fragment will generate the same series.Let us generate some of its members up to n =10 to enable us have a feel of this type of series: Os(CO) 3 → Os 2 (CO) 6 → Os 3 (CO) 9 → Os 4 (CO) 12 → Os 5 (CO) 15 → Os 5 (CO) 15 → Os 6 (CO) 18 → Os 7 (CO) 21 → Os 8 (CO) 24 → Os 9 (CO) 27 → Os 10 (CO) 30 .It is also interesting to note that the ratio of n to number of CO s is 1:3.Hence, the series can be expressed as Os n (CO) 3n ={Os(CO) 3 } n .This process is similar to the polymerizationencountered in organic chemistry involving fragments such as CH 2 '!n(CH 2 ) = (CH 2 ) n .Since capping usually refers to a capping on a closo system, in case of Os 6 (CO) 18 cluster, it implies capping will be on M-5 closo system which will be a trigonalbipyramid, Os 5 (CO) 16  which is a member of its capping series.Indeed if we add upthe two chemical species [Os 5 (CO) 16 (14n+2) + Os(CO) 2 (14n-2) = Os 6 (CO) 18 [14n+2+(-2)] = Os 6 (CO) 18 (14n+0) = Os 6 (CO) 18 (14n)] we generate the mono-capped cluster.We have seen 10 members of the 14n series in which Os 6 (CO) 18 belongs.Such members are generated using Os(CO) 3 (14n)fragment.We can also generate some members of its capping series by adding Os(CO) 2 (14n-2) fragments.These are already given in Table 7 in the bicapped series column.

The Capping Symbol, C n C
The desire to understand clusters more stimulated the need to create a symbol to express capping.Consider the following osmium clusters, Os 6 (CO) 18 2¯( Os 10 (CO) 27 ) (Tetracapped octahedral, closo).The respective series are, (14n+2), (14n+0), (14n-2), (14n-4), and (14n-6).After pondering over these series, it was deemed a good idea to use, C 1 to represent mono-capped geometry.Hence, (14n) '!C 1 , (14n-2) = [14n+1(-2)].Since the capping starts at 14n, then any additional multiple of (-2) corresponds to an additional capping.Therefore, The symbol [M-6] represents a closo cluster equivalent to Os 6 (CO) 19 or B 6 H 6 2¯.The cluster Os 6 (CO) 18 has attracted a lot of attention [19][20] probably due to its having the same number of CO ligandsas Os 6 (CO) 18 2¯ with the exception that it does not carry the two negative charges.Its shape is usually described as a monocapped cluster with respect to a trigonalbipyramid or a bicapped cluster with respect to a tetrahedral geometry.If we look at it in terms of series(see Table 3), we find that it is simply a member of mono-capped series (14n).This series [14n+1(-2)]'!C 1 +C 1 = C 2 .This simply means, the (14n-2) series is a Bicaped series (C 2 ).Since these clusters are closo systems, the symbol becomes C 2 C. In this particular set of clusters, the capping is based on an octahedralcloso geometry of six skeletal elements.For further refinement of the introduced symbol of closo six skeletal elements, the symbol [M-6] was added.Thus, Mono-caaped symmetry based on octahedral is represented by 6] , and so on.It is also proposed that a symbol (Mn) could represent capping other than that based on non-closo system.For instance in the case of Os 6 (CO) 18 we could refer to it as monocapped based on [M-5](closo) or bi-capped based on (M-4)(nido).The nido mapping generating function is given by F(n) '!3n+2.For(4) = 3x4+2 = 14.Hence, Os 6 (CO) 18 is bicapped onOs 4 (CO) 14 cluster.The symbol [M-6] has been introduced to represent an octahedral clososystem in this case.We also know that the Generating function of a closo system is given by F(n) = 3n+1.This means that for n = 6, 3n+1 = 3x6+1 = 19 giving us the osmium cluster Os 6 (CO) 19 .Therefore, [M-6] is equivalent to Os 6 (CO) 19 cluster that has valence electron content of 86 for an octahedral geometry.Using the 14n rule or Table 3 above, the cluster Os 10 (CO) 26 2¯ can be shown to belong to the series S =14n-6 = [14n+3(-2)] = C 1 +C 3 = C 4 C[M-6].If the series type of a cluster is determined, then the type of capping (Cp) can be deduced.Using this approach, the capping systems of small to medium to large clusters have readily been derived with amazing results.Consider the following clusters, Ru 8 Pt(CO) 19 2¯; n = 9, 14n =14x9 =126, V = 114, S = 14n-12 = 14n+6(-2).Hence the capping will be given by C p = C 1 +C 6 = C 7 C[M-2].This symbol means that 7 skeletal atoms will be involved in capping while the remaining two will form the closo nucleus.Following the same method, Os 17 (CO) 36  The algebra of series A closer scrutiny and study of the series in Table 3 clearly shows that the clusters series can easily be added and subtracted.This may have some implications on future synthetic work of carbonyl clusters.Let us look at the following illustrations based on the Table.As we have observed, the transition metal carbonyl series are based on 14n baseline.Therefore the digits after the 14n are the deciding factor on the type of product series that is theoretically expected to be formed.This means the digits can be added or subtracted using similar concepts found in algebra.This is illustrated by a few examples given in Table1O.

Isolobal concept
Table 3 also demonstrated that individual atoms, molecules and chemical fragments can be classified into series based on 14n or 4n rules.In order to greatly appreciate Hoffmann's extremely important isolobal concept 12,22 , it is equally important to briefly provide some highlights on how series are derived for re-emphasis as this is a new approach 13,14,18, .This summarized in Scheme 4 below.The classification is based on 4n rule for the main group elements which obey the octet rule and 14n rule for transition metal atoms which obey the 18 electron rule.If the value of 4n is smaller than the valence value V of the atom, molecule or fragment in question, then the difference d between 4n and is added to 'top up' in order to be equal to the value of V. Then the series of the chemical species in question will be given by S = 4n +d = V for main group elements and S = 14n +d for transition metal carbonyl clusters.However, if 4n is greater than V then the difference between 4n and V must be subtracted in order to be equal to the value of the valence electrons V and hence the series will be given by S = 4n-d = V for main group elements and S = 14n-d for the transition metal carbonyl clusters.The borderline case occurs when 4n = V (main group) or 14n=V(transition metal).This is where capping begins in clusters( refer to Scheme 4).For S= 14n+d, these are series which are below mono-capped series such as closo, nido and arachno.For the series 4n-d or 14n-d, all these will be capping series beginning with (14n+0).Let us illustrate how the series of fragments are deduced.Consider, the C 2 molecule discussed earlier, n = 2, 4n = 8 and V = 8.Hence, S = 4n= 4n+0.For the monoatomic carbon, C, also S = 4n.However, for the C 2 H 2 ; n =2, 4n = 8, V= 10 and so S = 4n+2 = V, when the C 2 H 2 is fragmented into 2 CH units each unit carries 5 electrons.Thus, C 2 H 2 (V= 10, S = 4n+2, n =2) → 2CH(V=5, S = 4n+1, n =1).What has happened in terms of series, when the molecule is fragmented, the baseline, 4n or 14n does not change except the digit after it.In short, C 2 H 2 (S=4n+2) →2 CH 2 (S = 4n+1).This in line with the algebra of series concept discussed above.Alternatively, the series of the fragment can be deduced by saying for CH 2 , n =1, 4n = 4 and V = 5 and hence S = 4n+1.Likewise, forC 2 H 4 (S=4n+4, n =2, V =12) →2 CH 2 (S = 4n+2, n =1, V =6).For ethane,C 2 H 6 (S=14n+6) →2 CH 2 (S = 4n+3).For transition metal elements, 14n rule is used.The series for selected fragments from the main group elements and transition metals have been worked out and are presented in Table 11.This includes some fragments derived from Table 3.The table also puts more emphasis on the method of classifying molecules and fragments.

CONCLUSION
As can be deduced from the Table 3, there are three main types of cluster series.The first typeis the one which we may call the ORDINARY series and this includes the familiar ones such as Arachno, Nido and Closo.In this type of series the cluster type remains constant and valence electrons change by 14 units as the chain size or length increases.These utilize the Os(CO) 3 (14n+0) FRAGMENT.Then there are the CAPPING series.These involve the use of Os(CO) 2 (14n-2)fragment as the chain length increases, the cluster type varies by12 electron units.Finally there are the STRIPPING series in which CO ligands are being removed.The cluster type varies but the number of skeletal elements constant.But the valence electrons change by 2. Using the14n and 4n rules, shows that there is close relationship between the transition metal carbonyl clusters and the corresponding ones of the main group elements especially the boranes.Many carbonyl clusters have a tendency of forming clusters centered around nuclear closo systems.The earlier work by Wade, Mingos with a bearing on series and Hoffmann on isolobal concept all based on Molecular Orbital Theory go along way in providing vital the pillars in supportof the validity of the 14n and 4n rules in explaining the carbonyl clusters, boranes, heteroboranes, Zintyl ion clusters, and the isolobal concept.Furthermore, the 18 electron rule and the octet (8) rule give a strong and firm foundation of the 14n and 4n rules in analyzing clusters.

Scheme 1 .
Scheme 1. Process of generating the same series

Scheme 2 .
Scheme 2. Process of generating capping series 2¯( Os 6 (CO) 19 octahedral, closo)'!Os 7 (CO) 21 (Monocapped octahedral, c l o s o ) → O s 8 ( C O ) 2 2 2 ¯( O s 8 ( C O ) 2 3 ) ( B i c a p p e d octahedral, closo)→Os 9 (CO) 24 2¯ (Os 9 (CO) 25 ) (Tricapped octahedral, closo)→ Os 10 (CO) 26 -28) will have a capping given byC p = C 1 +C 14 = C 15 C[M-2].In this case as well the nucleus will have 2 skeletal atoms while the remaining 15 will be capping the cluster the tiny closo nucleus.On the other hand, Ni 38 Pt 6 (CO) 48 6¯; n =44, 14n = 44x14 = 616, V = 542, S = 14n-74, [14n+37(-2)], C p = C 1 +C 37 = C 38 C[M-6].With this code notation derived from series, means that the skeletal cluster of 44 atoms has six of them with a closo octahedral symmetry surrounded by 38 capping atoms.This fascinating result predicted from series is what is observed 21 .Furthermore, the octahedral inner core comprises of platinum atoms only 21 .With series knowledge we have so far, it can it can be shown the symbol C p = C 1 +C 37 = C 38 C[M-6] represents the valence electron content of the cluster as follows:[M-6] = Os 6 (CO) 19 as already proved, C 38 = 38[Os(CO) 2 ] = Os 38 (CO) 76 .Therefore the cluster formula equivalent is given by F = Os 6 (CO) 19 + Os 38 (CO) 76 = Os 44 (CO) 95 .The valence electrons of this cluster = 44x8+95x2 = 542 as calculated forNi 38 Pt 6 (CO) 48 6¯ complex.We can represent this giant cluster of six platinum closo octahedral nucleus by a ketch shown in Figure 6.Using series method, the cluster Pd 23 (CO) 22 L 10 where L = PEt 3 , has a capping symbol C p =C 15 C[M-8] and Au 6 Ni 32 (CO) 44 6¯ also has a capping symbol C p =C 27 C[M-11].Clearly, the sizes of skeletal closonulearatoms varieswidely.In these few examples we have seen the range [M-2] to [M-11].This reminds us of the notorious African pythons which can ambush, grab and swallow animals as small as rabbits and as large as antelopes.It does appear that these huge carbonyl clusters portray an image of pythons.

Table 1 : Characteristic k values of Selected Closo Boranes
If Table 2 were to be expanded, it would be visible that important Rudolph geometrical relationship is a portion of the borane capping series.Let me present some of these series including those that were presented by Rudolph's work by the structural formulas as seen in Table 2. Starting with B 12 H 14 (B 12 H 12 2-)→B 11 H 15 →B 10 H 16 → B 9 H 17 .Then from B 11 H 13 →B 10 H 14 →B 9 H 15 →B 8 H 16 at the end in the table.Final example is starting with the octahedral borane B 6 H 8 (B 6 H 6 2-)→B 5 H 9 (square pyramid)→B 4 H 10 →B 3 H 11 at the end as indicated in the table.If the Rudolph capping series were to be extended, then the next starting cluster for decapping would be B 13 H 15 , the next starting point

Table 2 : GeneratingFunctions of Borane Series
would be B 14 H 16 , the next B 15 H 17 and so on.It should be pointed out that no neutral closoboranes are known except that they occur as di-ionic B n H n 2¯.