Graph Theory of Chemical Series and Broad Categorization of Clusters

The recent introduction of skeletal numbers has made it much easier to analyze and categorize a wide range of many chemical clusters. In the process, it has been found that a large number of transition metal clusters with and without ligands are capped and do possess closo nuclear clusters. On the basis of the nuclear index, the clusters have been categorized into groups. The categorization of the clusters will greatly assist in promoting deeper understanding and the synthesis of novel clusters and their applications. A simple concept of graph theory of capping clusters has been introduced.


Brief Background
The early research on clusters pointed to the possibility of explaining boranes using numbers (Lipscomb, 1963), Jemmis" MNO numbers (Jemmis, 2008), topological numbers (Teo et al, 1984), and cluster shapes numbers (Fehlner and Halet, 2007) (18-monomer, 34-dimer, 48-trimer, 60-tetrahedron, and so on). The existence of cluster series was also detected and implied through the work of Wade (Wade, 1976), Mingos (Mingos, 1987) and the correlation of cluster symmetries by Rudolph (Rudolph, 1976). This is confirmed by the current work of applying skeletal numbers derived from the 4n series method (Kiremire, 2015b) which clearly indicate the existence of a whole vast universe of clusters linking up naked metallic elements to metal carbonyls (Kiremire, 2016b), metalloboranes (Kiremire, 2016c), Zintyl ions (Kiremire, 2016d), carboranes and other heteroboranes, boranes, and main group element clusters (Kiremire, 2017c). Using the skeletal numbers, a cluster comprising of a mono-skeletal element to a giant one of several hundred skeletal elements can readily be decomposed and represented by a single number now referred to as a cluster number Kor [K(n)parameter] (Kiremire, 2017b). The method can be broken into the following simple steps of analysis summarized in Scheme 1(SC-1). Further simplification is shown in Scheme 2(SC-2). For Os 5 (CO) 16 , K = 5[5]-16(1) = 9. This calculation is based on the observation from series that the skeletal elements act as providers of skeletal number values some of which are utilized by the ligands in a cluster. It is based on the observation that for every ONE electron donated the skeletal number value decreases by 0.5. Thus, 1H( K = -0.5) when acting as a ligand while a CO(K = -1) since it is a 2 electron donor. The K value can be utilized to calculate the q value in the series S = 4n+q. Thus, K = 2n-½ q = 2(5) -1/ q = 9, ½ q = 1 and q = 2. The cluster belongs to the series S = 4n+2 as derived earlier. The use of skeletal numbers to derive the cluster number, K is much faster than using the cluster fragments. The method used in Ex-1(b) has been applied in most of the cluster categorizations in this paper.

The Double Meaning of K(n) Parameter
This concept will be discussed further in this paper. We have seen that K(n) = 9(5) in Os 5 (CO) 16 cluster represents 9 linkages to bind 5 skeletal osmium skeletal elements and in so doing figure is obtained. However, after scrutinizing the series deeper, K also represents the SHORTAGE of the number of the electron pairs needed to ensure that every skeletal element attains the Nobel gas configuration. Let us use this same cluster to demonstrate this concept.

= Capping Os
A short-cut for extracting the K value from a cluster formula is given in SC-3.

K = 80
If we focus on the inner ring (circle) of Figure 4, there are 7 lines linked to the nucleus. This is followed by another 7 linkages to the 7 skeletal elements themselves. Then the internal periphery linkages of one to the next jumping the middle one, gives another 7 linkages. Thus, in total, we have 21 linkages from the inner "circle". The next circle also contributes to another set of 21 linkages. The last circle gives the last 21 linkages. Thus, the gross total of capping linkages is 21(3) = 63. Then the CLOSO NUCLEUS is nine skeletal elements [M9], which gives us K = 2n-1 = 2(9)-1 = 17. This means that the total linkages of the cluster as sketched in the isomeric Figure 4 will be 63+17 = 80. This is an example of one form of representation of the graph of the cluster Pd 30 (CO) 30 L 10 . The cluster Pd 30 (CO) 26 L 10 can be treated in the same manner as Pd 30 (CO) 30 L 10 . Its graph is shown in Figure 5.
In Ex-6, the series method predicts that the platinum golden cluster will have a single nuclear skeletal atom [M1] surrounded by 8 other skeletal elements, Kp =C 8 C[M1]. This has been found to be the case and the Pt is in the nucleus while 8 Au skeletal elements occupy the periphery. This is sketched in Figure 6. Since the cluster is small, one ring of C 8 can readily be utilized to sketch its graph. The number of skeletal linkages = 8x3 + [M1] linkages. Surprisingly, even the one skeletal nuclear element obeys the CLOSO series formula, S = 4n+2 and K = 2n-1 = 2(1)-1 = 1. This means the single skeletal nuclear element is entitled to 1 skeletal linkage. Hence, the TOTAL number of skeletal linkages = 24+1 = 25. What is also interesting is that there are 5 linkages to each periphery Au skeletal element. Since the Au element has a K value of 3.5, then the shortfall on the periphery element is K = 3.5-2.5 = 1. This means each of the 8 periphery Au elements is allowed to accommodate ONE ligand in order for it to obey the 18 electron rule, hence the predominance of [AuL] fragments (Mingos, 1884) in the golden clusters. In summary the graph theory of series involves the construction of skeletal structures in such a way that the skeletal elements (n) are linked in a mathematical precision with the corresponding k skeletal linkages as derived from the cluster formula. In other words, the K(n) parameter of a cluster must obey the "goodness of fit" principle. The concepts that have been revealed in examples Ex-1 to Ex-6 have been applied as closely as possible in the rest of the clusters in this paper.

More Examples to Illustrate the Categorization of Clusters Using 4n Series Method
The examples have been arranged according to nuclearity index. K(n) 1 (2) 2 (2) 3 (2) 4 (2) 3 (2) 1(1) In example 1(N45), Re(K =5.5,see Appendix 2), the hydrogen atom (H•) when regarded as a single electron donor is assigned a K value of -0.5. Similarly, the (:CO) ligand when is regarded as a two electron donor ligand is assigned a value [K = 2(-0.5) = -1]. In case of ligands, the assigned negative K value is directly proportional to the number of electrons donated. For instance a 3 electron donor has a K value of 3[-0.5] = -1.5 while for a cyclopentadienyl ligand, Cp which donates 5 electrons, K = 5[-0.5] = -2.5. The cluster K value is calculated as indicated in N45. This means the 4 rhenium skeletal elements are connected by 4 linkages (bonds) and a reasonable possible ISOMERIC shape is a square. In constructing skeletal shapes, there may be more than one structure. In order to assign tentative ligand distribution to a skeletal element, its k value is utilized. Just as in the CH 4 molecule each H atom is considered to donate an electron to the carbon atom for it to attain an octet rule (noble gas configuration), in the Re 4 skeletal shape (N45), each linkage to Re skeletal element behaves as a hydrogen atom donor and hence it is assigned a K value of -0.5. In the skeletal cluster shape, each skeletal element receives electron donations from the ligands as well as SKELETAL linkages to it. Using this knowledge, we can determine the skeletal linkages AVAILABLE to the external ligands after the linkages to the elements have been taken into account. For instance, in N45 there are 2 linkages to each Re skeletal atom and therefore skeletal linkages available to each skeletal element K1=K2=K3=K4 = 5.5-2(0.5) = 4.5 as indicated in N45. Clearly, a K value of 4.5 implies, in principle, there will be 4CO +1H distributed to each Re skeletal element since a CO ligand is a 2 electron donor and H ligand a 1 electron donor. Essentially, a K value of 1 represents or corresponds to 1 electron pair. The actual structure can be determined from X-ray structural analysis. In this case, one of the skeletal isomers of Re 4 H 4 (CO) 16 is found to have bridged H ligands as sketched in CC-1. The skeletal shape without taking into account ligand rearrangements, has been referred to as a "raw skeletal structure". Once the raw skeletal structure has been constructed and ligands distributed according to skeletal numbers and valences, then it is expected that each skeletal element obeys the noble gas configuration. This can be illustrated by the N45 sketch: Re 1, valence electrons, Ve = 1[Re]+4CO+2[Re-Re]+1[H] = 7+4(2)+2+1=18. We can also use skeletal numbers to verify if the rhenium elements in the cluster obey the 18 electron rule: K = 1[5.5]-4-1-0.5=0. The concepts applied in N45 have been used in the rest of the clusters summarized in the following sections of this paper. A wide range of examples have been analyzed using the skeletal numbers so as to expose the readers to a wide scope of applications of the skeletal numbers to cluster analysis. Note: for every single 1 electron donated by a ligand to a skeletal element, k = -0.5. Also all the skeletal numbers used in the calculations are given in the appedices 1 and 2. K(n) = 5(4)

N46
Cluster interconversions using series Skeletal numbers: Re(K = 5.5), CO(K = -1), (-1)(K = -0. Skeletal numbers: B(K = 2.5, V = 5), H(K = -0.5); K = 5[2.5]-9(0.5) = 8. This means there are 5 skeletal elements linked by 8 lines. In 3 dimensions, such a shape will be a square pyramid. When presented in a 2 dimensional form and the hydrogen ligand atoms are added, the figure N51 (N51 means a cluster with 5 defined skeletal elements, and the last digit refers to the position of such a skeletal sketch -in this case it is the first sketch involving 5 skeletal elements) can be constructed. Deducing from N51, we have 5 skeletal elements [5B] and 9 hydrogen atoms to give a geometrical shape whose skeletal elements hypothetically obey the 8 electron rule by just referring to each of the linkages as a provider of 1 electron to the skeletal element, rather than the normal concept of 2 valence electrons provided by each bond. In orderto get the semblance of the normal 2 electron type of bond, we can assume that the structure N51 undergoes some re-arrangement to give us the structure N52 where some 4 H hydrogen atoms form bridges. Since all the boron skeletal elements in B5H9 are observed to have a H ligand each, then the apex B skeletal element in N52 can be viewed as retaining its linkage of 5 while the 4 basal B skeletal elements have 4 "usual" 2-electron bonds. Figure N51 was referred to as a raw isomeric skeletal structure and N52 as the rearranged isomeric skeletal structure. The raw skeletal structure when done correctly according to series fulfilling the rule of skeletal valences, generally gives a derivation of the cluster formula as is illustrated in N51. The last figure N53 just gives us insight on the general shape of the cluster.  According to the series method, there is one atom in the nucleus which is capped by 5 others.
According to the series symbol Kp= C 5 C[M1], it implies that the cluster has ONE skeletal element in the nucleus with 5 other capping elements. Each of the 5 capping elements has 3 linkages giving a total of 15 linkages. Since the cluster has a K value 16 the nuclear element [M1] has an intrinsic K value of 1, and this is in agreement with the series prediction. According to the series, the nuclear cluster fragment, [M1] obeys the CLOSO series formula S =4n+2 and K =2n-1 = 2(1)-1 =1 in agreement with the above prediction. It has been found easier for clusters with nuclearity index 1-13 (N1-N13) (Kiremire, 2017d) to use the capping symbol as a guide to construct the RAW SKELETAL STRUCTURE of clusters. In this regard, the symbol C 5 acts as a guide to enable us construct a 2-dimensional shape of the cluster. Thus, we can use a 5-membered structure for the capping cluster with a mono-skeletal nucleus; thus the Au 6 L 6 2+ structural frame-work shown in figure N69. There are 5 capping elements and each is connected to the nuclear skeletal element. This utilizes 5 linkages. The 5 periphery skeletal elements are connected to one another giving us another 5 skeletal linkages. Then there are internal alternating linkages which are also 5. This gives us a total of 5+5+5 = 15. This agrees with the capping symbol C 5 which indicates there are 5 capping elements each of which consumes 3 linkages. The calculation of K value from Au 6 L 6 2+ , gives us K=16 for the entire cluster. On the surface, it appears as if there is ONE missing linkage. In order to find it, we must look at the whole capping symbol, Kp = C 5 C[M1]. If C 5 represents 15 linkages, then the missing link must be hidden within the [M1] CLOSO fragment. Since the capping nuclei belong to the CLOSO FAMILY, the mono-skeletal nucleus obeys the series formula S=4n+2 and K =2n-1. Since n =1 for a mono-skeletal fragment, then its K value is given by K =2n-1 = 2(1)-1 = 1.Therefore the grand total of cluster linkages = 5+5+5+1 = 16 which takes into account the one linkage associated with the nucleus.

Distribution of Ligands in Golden Clusters
The number of ligands associated with golden clusters is usually relatively small. Therefore a simple calculation can act as a guide on how the ligands are distributed among all the skeletal elements.
The first approach is to consider the cluster Au 6 L 6 2+ as comprising of Au 6 2+ fragment and 6L. The skeletal linkages of the naked fragment Au 6 2+ subtracting the skeletal linkages of the cluster will give us the linkages remaining to bind the ligands.
The Au 6 2+ fragment has a total of 6[3.5]+2(0.5) = 22 linkages available. According to the calculation, 16 of those are utilized for the internal binding of the skeletal elements. Hence, the remaining linkages are available for the ligands. That is, KL = 22-16= 6, This means there are 6 linkages available for 6 ligands distributed onto 6 gold skeletal elements as shown in N 69.
Alternatively, we can calculate the number of skeletal linkages available for every individual skeletal element as follows: K1= 1[3.5] -5(0.5) =3.5-2.5 =1; K2 is the same as K1 = 1= K3 = K4=K5 =K6. This information indicates that each skeletal element will be assigned to one ligand as indicated in N69. The ideal raw skeletal structure will have a 2-dimentional symmetry of D 5h . The outside circle in N69 is to assist in adding the ligands to the periphery gold atoms. The actual shape has been described as octahedral or edge-sharing bi-tetrahedral (Mingos, 1984). The two dimensional skeletal framework N69 looks like a star. The cluster valence electrons: The cluster valence electrons have been calculated using the series formula (Ve) and the result was compared with the one calculated from the cluster formula, VF. Always, Ve = VF indicating the authenticity of the series formula. Clearly, the skeletal structural frameworks of clusters follow an exact mathematical precision of 4n cluster series.
The series acts as a guide. Although the prediction is 3 skeletal elements capping on another set of three skeletal elements, what is observed is simply two sets of 3 golden skeletal elements as shown in N612. Let us consider the first example G1 S 8 2+ cluster as an illustration. From the appendix 1, sulphur has a K value of 1. The K value of 8 S skeletal elements is simply 8x1 = 8. Also we know from series, that a unit of positive charge (+1) is assigned a K value of 0.5. So for 2+ charge the K value = 2x0.5 = 1. Hence the K value of S 8 2+ = 8+1 = 9. Since there are 8 skeletal elements the K(n) parameter is K(n) = 9(8). This means that the cluster will have 9 linkages and this has been found to be the case (Housecroft & Sharpe, 2005). We also know that K =2n-½ q ; hence 9 = 2(8)-½ q and this gives the value of q = 2[16-9] = 14. Since the cluster series S = 4n+q, this gives us S = 4n+14. We also know that there are a good number of isomers with K = 9 (nine linkages) that can be constructed, but we can only sketch one. This is given in G1 below. Categorization of the cluster based on CLOSO (S = 4n+2) vertical series (∆K = ±3, ∆n =±1) has been developed. Since the cluster belongs to the series S = 4n+14(n = 8), it has to go through the stages 4n+12(n=9)→4n+10(n=10) →4n+8(n= 11)→4n+6(n= 12)→4n+4(n=13)→4n+2(n=14). Clearly, the cluster is 6 steps below the CLOSO step and hence its capping symbol can be expressed as Kp = C -6 C[M14]. Thus, the cluster belongs to the CLUSTER GROUP [M14] on the Rudolph type of numerical categorization (Rudolph, 1976). The symbol [M14] means that S = 4n+2 and n = 14; K = 2n-1 = 2(14)-1 = 27. Hence K(n) = 27(14). The numerical series of this will be 27(14)→24(13)→21(12)→18(11)→15(10)→12(9)→9(8). Clearly, the cluster S 8 2+ , K(n) = 9(8) is numerically 6 steps below the Closo [M14] level. This cluster is graphically linked by 9 lines(linkages). According to the series, is it possible for us to say that cluster linkages are similar or identical to the conventional chemical bonds? The Pt + fragment has a K value is = 4+1(0.5)-7(0.5)-1= 0. Since each of the periphery Au element is assumed to have a zero oxidation state, the positive charge can be assigned to the platinum element. In summary, each of the periphery Au element will bind to a 2 electron donor ligand while the central Pt element will have no ligand. According to this model the ligand distribution is according to simple mathematical precision. The sketch is shown in N82.
Let us consider another example N83 : Au 8 L 7 2+ (L = PPh 3 ): K = 8[3.5]-7+1 = 22;n = 8, K(n) = 22 (8), K = 2n-½ q, 22 = 2(8)-½ q, ½ q = 16-22 = -6, q = -12, S = 4n-12, Kp = C 7 C[M1]. According to the series, the prediction is that this will be a cluster of 7 skeletal elements surrounding one skeletal element that is a member of CLOSO SERIES, S = 4n+2, K = 2n-1 = 2(1)-1 = 1. Thus the closo nucleus has K(n) parameter = 1(1). The rest of the 7 skeletal elements will be capped around it. A capping fragment of the series has a formula S = 4n-2, K = 2n+1; n=1, K = 2(1) + 1 = 3. This means the symbol C 7 with seven cappings involves 7(3) = 21 linkages. This gives us a graphical guide of the diagram, N83. There are 7 linkages to the central closo skeletal nuclear element and 14 linkages on the periphery structure of the diagram giving a total of 21 linkages. Then the [M1] closo nucleus has 1 skeletal linkage (K = 1) in its own right. This makes a grand total of 22 cluster linkages. This provides us with a graphical representation of Au 8 skeletal elements in Au 8 l 7 2+ cluster. On a planar view point, the diagram has D 7h symmetry. How are the ligands distributed? This means that each of the periphery au will bind a 2 electron donor ligand such as pph 3 , while the nuclear au will have no ligands (naked) as sketched in the skeletal graph N83.
The capping symbol Kp =C 2 C[M6] derived from the series predicts a bi-capped octahedral complex and this is what was found (Hughes and Wade, 2000).
Au 8 L 7 2+ : K =22; Au 8 2+ : K = 29, KL = 29-22= 7. There will be 7 linkages available for 7 skeletal elements. The skeletal elements are the periphery ones as indicated in CC-25. This is in agreement with observation and is described as capped centered chair (Konishi). In 2 dimensional form, it forms a nice D 7h symmetry.  ; S = 4n+0. The new series formula when the cluster changes from m = 6 to m = 7 will be given by S = [(4n-4)(n=6) +(4n+0)(n = 1) = 4n-4(n=7)]. Thus, the cluster series formula has remained the same despite n changing by 1. In other words, the series will still remain tri-capped; Kp = C 3 C[Mx], x = 5 -9. This simply means, the capping index will remain the same, that is, same series but the nuclearity index will expand as it is observed (Fehlner & Halet, 2007).
Au 9 L 8 3+ → Au 9 3+ +8L; Au 9 3+ ; K = 9[3.5] +3(0.5) = 33. Some of these linkages will be used for binding the skeletal elements and the rest to bind the ligands. In the calculation above in CC-28, K =25. This means that these are for binding the skeletal elements in the cluster. Therefore those remaining to bind the ligands will be given by KL = 33-25 = 8. As expected, this agrees with the cluster formula. These will be evenly distributed onto the periphery Au skeletal elements as indicated in N95. Note that the nuclear element [M1] obeys the closo series formula S = 4n+2, K = 2n-1= 2(1)-1 = 1. Thus, the closo nuclear skeletal element in its own right has a K value of 1. We can also calculate the number of skeletal linkages available for binding ligand(s). As can be seen from the raw skeletal structure in N95, each peripheral Au skeletal element has 5 linkages. Each linkage behaves as a [H•] ligand donating ONE electron to Au periphery skeletal atom. So we can calculate the respective K value as 1[3.5]-5(0.5) = 1. This shows that each peripheral Au skeletal element will bind ONE ligand as indicated in N95. The cluster of this formula has been described as bi-capped centered or crown centered (Konishi, 2014).

Numerical Periodic Table of Chemical Clusters
The examples of clusters given above cover a range of nuclearity index 1-56. By deriving the K(n) values of many clusters and carefully studying their relationship an interesting pattern was discerned. Firstly, the K(n) values of clusters covered an infinite range of 4n series. That is, S = 4n+q [ q= 2, 4, 6, 8, 10, 12, …..(un-capping series); q = 0, -2,-4, -6, -8, -10, -12, ….(capping series)]. This aspect has been illustrated in Table1. In addition, a grid map shown in Table 2 was obtained. The K(n) values of the clusters derived from the 4n series when carefully analyzed, are found to form an interesting series. A selected range of K(n) values from [M-10] through [M0] to [M14] are given in Table 2. This arrangement is similar to the usual X-axis of x = -10 to x=+10. This array of K(n) values was discovered in earlier work (Kiremire, 2017a) but it is being re-introduced for the purpose of categorizing a vast range of capping chemical clusters.
The horizontal rows show the clusters arranged according to S =4n+q where q is the same while the vertical columns indicate the cluster determinant q varies by ±2. Table 2 appears seemingly complex, but on close observation, it actually commences with one K(n) value. This is demonstrated in Scheme SC-4. Starting with point A of K(n) = 11(6), the vertical movement upwards involves stepwise changes in K value by 3 units; while downward movements involves a decrease of K by the same amount of 3 units stepwise. Thus, the vertical axis can be extended in both directions indefinitely. Hence, new points B, C, D and E in scheme SC-4 are derived from A. Likewise, each of the points B, C, D and E can be handled in the same manner as a resulting in a seemingly complex cluster GRID MAP shown in Table 2. In a way, Table 2 is similar to an array of coordinates of a point P(x, y) except in this case we have the K(n) VALUES where K represents the number of cluster linkages corresponding to x value of the X-axis and n which represents the number of skeletal elements in a cluster corresponding to the y value of the Y axis in the Cartesian coordinate system. Thus, we have the correlation relationship P(x,y)→(K,n) but expressed as K(n). The base line has been chosen to correspond to the values of the horizontal clososeries, S = 4n+2.

 CAPPING CONCEPT FROM SERIES ELABORATED
The capping symbol Kp = C y C[Mx] that has been developed is very useful in explain the capping concept. The letter y (capping index) represents the number of capping skeletal elements outside the cluster nucleus [Mx] (x = nuclear index), represents the number of the skeletal elements in the nucleus of a cluster. If Table 2 were displayed on a larger space, it would cover the [Mx] range x= -10 to x = +14 expressed as [M-10] to [M14]. The construction of the table shows that the un-capped series commence at x= 2 upwards, that is, x≥2. However, there are no un-capped series for x≤1. Therefore, the Rudolph concept of correlation of geometrical structures is more meaningful for clusters x≥2. A section of Table 2 has been numerically demarcated and labeled Table 2C(T-2C). If we focus on the closo base-line, S=4n+2 As a reference, then clusters with genuine skeletal elements in the nucleus commence with [M1]. This means, these will be clusters which have a single element in the nucleus. This is the foundation of golden clusters referred to as torroidal clusters. According to the series in Table 2 . All these clusters belong to the [M1] SERIES, that is, they belong to GROUP [M1] CLUSTERS. As mentioned earlier, these clusters are referred to as torroidal clusters (Mingos, 1984 (6) 14 (7) 8 (5) 5 (4) 20 (9) 17 ( , this corresponds to S = 4n+2 and K(n) = 9(5). The borane closo cluster that corresponds to this point is B 5 H 5 2-. The numerical movement upwards from 9(5), we get 12(6), 15 (7), 18(8), 21(9), and so on. We can refer to these clusters as GROUP 5 clusters, [M5]. Continuing the horizontal movement left-wise, we get [M4], 7(4), S =4n+2; then [M3], 5(3), 4n+2; [M2], 3(2), 4n+2. It has been found that a good range of golden clusters belong to GROUP 2,[M2]. Accordingly, their vertical [M2] numerical series will be; 3(2), 6(3), 9(4), 12(5), and so on. The group 2 clusters lie along the grid-line shown in GREEN in SC-7. These golden clusters which are [M2]-based have been referred to as SPHERICAL while those which are [M1]-based with a single element in the nucleus were referred to as TORROIDAL (Mingos, 1984). A good example of spherical golden cluster is Au 9 L 8 + : K=9[3.5]-8+0.5 =24,K(n) = 24(9), S = 4n-12,Kp = C 7 C[M2]. On the other hand the cluster of the same nuclearity index, Au 9 L 8 3+ : has a K value 25, K(n) = 25(9), S = 4n-14, Kp =C 8 C[M1] is described as torroidal. The transformation from Au 9 L 8 + to Au 9 L 8 3+ results in the increase in K value by 1. The increase in K value of a capping cluster by 1 results in the increase of the corresponding skeletal elements by 1. That additional capping element is extracted from the nucleus. Hence, the [M2] (two skeletal elements in the nucleus) now becomes [M1] (one skeletal element in the nucleus). The group 2 clusters lie along the [M1] orange grid-line in SC-7.

 A SET OF ALL CLUSTERS WHICH POSSESS K(n) VALUES
Let us take K(n) =11(6) as an illustration. With the help of skeletal numbers, the clusters with K(n)=11(6) are readily identified. The Table 4 gives a sample of about 50 clusters identified in this way. Many more can be added to the table. The K(n) parameter acts as a "car-park" in which clusters of n skeletal elements and same K values can be accommodated endlessly. This implies that each K(n) parameter in the CLUSTER TABLE represents a large number of clusters known and unknown. As for the ideal skeletal shapes and their isomers, in the case of K(n)=11(6)→octahedral, 9(5)→trigonal bipyramid, 8(5)→square pyramid and 6(4) →tetrahedral.  IDENTICAL OR SIMILAR GEOMETRY Taking the K(n) =11(6) example, the ideal shape of this parameter is O h symmetry. This is sketched in Figure F-1.

F-1
However, in special cases, K-isomerism occurs and a different shape with the same K value is observed. A good example is H 2 Os 6 (CO) 18 ; K(n) = 11(6) which portrays a capped square pyramid shape as sketched in Figure F -2 (Hughes and Wade, 2000).

 CLUSTER VALENCE ELECTRONS
The calculation of cluster number K has already been explained in the chemical clusters CC-1 to CC-15 as well as in the examples given in Table 2. The skeletal numbers of elements are obtained from the appendices 1 and 2. When the K value and the number of skeletal elements are known, then this gives us the K(n) parameter of the cluster from which the series formula is derived as summarized in the scheme SC-2 (page 14). The valence electrons of the cluster are directly calculated using the same series equation S = 4n+q [Ve = 4n+q] for the main group element clusters or Ve = 14n+q for transition metal clusters]. When the cluster comprises of both main group and transition metal elements, the calculation is adjusted to take into account the presence of transition metal elements due to the isolobal series relationship [S =4n+q ⥈ S =10n+q] observed in previous work (Kiremire, 2015C).

 LIGAND DISTRIBUTIONS
Skeletal numbers are very useful in the distribution of ligands among skeletal elements. This concept has been applied in many examples in this paper. Other examples can be found in literature (Kiremire, 2016b(Kiremire, , 2017.

 DOUBLE MEANING OF K(n)
Ex-3: Let us consider Os 6 (CO) 18 2example; K =6[5]-18-1= 11;K(n) = 11(6), S=4n+2. The parameter K(n) = 11(6) represents an octahedral shape. This means the skeletal elements are linked by 11 lines as in F-1. At the same time, it represents the number of electron pairs that will decompose the cluster into mono-skeletal fragments each of which will obey the 18 electron rule. This is illustrated in equation (i).

+11CO
Os6(CO)29 2-6[Os(CO)5] Os6(CO)30 Thus 6 complexes of Os(CO) 5 are generated each of which obeys the 18 electron rule. This principle applies to even huge clusters. Let us consider Pd 37 (CO) 28 L 12 : K = 108(as calculated above):K(n) =108(37); S=4n-74, Kp =C 35 C[M2]. According to the series method, the giant cluster of 37 skeletal elements is bound by 108 skeletal linkages. It also has a nucleus of two skeletal elements [M2] surrounded by 35 capping skeletal elements. This cluster, according to the series method, requires 108 electron pairs to be dismantled into individual mono-skeletal fragments each of which obeys the 18 electron rule. Since the CO ligand and L each donates 2 electrons to the skeletal elements, they are regarded as being equivalent. For simplicity, let us represent all of them by CO ligands. Then the cluster can be written as Pd 37 (CO) 40 . Since each: CO ligand is 1 electron pair donor, then K=108 will be equivalent to 108 CO ligands. This is illustrated in equation (ii).  BH 5 ]. The fragment BH 5 obeys the 8 electron rule. In general, based on series analysis, the k(n) parameter represents both the number of k linkages that bind a cluster of n skeletal elements on one hand, and a shortage of k electron pairs needed for each of the skeletal elements concerned to attain a noble gas configuration, on the other hand.
 Summary of information derived from k(n) parameter Most of the important information that can be derived from the K(n) parameter is summarized in Figure SC-8. This includes, among others, the type of series, the cluster valence electrons, the possible shape of a cluster, and so on.

The Diagonal Series and the Shrinking Nucleus
The diagonal series of clusters are also very interesting. Let us consider Rh 6 (CO) 16 as an illustration. This cluster as we know has an octahedral symmetry with K(n)=11(6) parameter. It is a family member of the closo series, S=4n+2. Its cluster valence electrons are given by Ve = 14n+2 = 14(6) +2 = 86. If we remove a CO ligand we will get Rh 6 (CO) 15 with a K(n) =12(6). This belongs to the series S =4n+0 with Kp = C 1 C[M5]. Its cluster valence electron content is given by Ve = 14n+0 = 14(6)+0 = 84. The removal of the next CO ligand we will get Rh 6 (CO) 14 with a corresponding K(n) value of 13(6). This belongs to the series S = 4n-2 with Kp = C 2 C[M4]. The cluster valence content will be Ve = 14n-2 = 14(6)-2 = 82. The cluster becomes a bi-capped closo tetrahedron. The next cluster will be Rh 6 (CO) 13 with K(n) = 14(6). The corresponding series for this is S = 4n-4. This cluster will have a capping symbol Kp = C 3 C[M3]. This is a tri-capped closo triangle. The next cluster of the series is Rh 6 (CO) 12 . This has a K(n)parameter of 15(6). It belongs to the series S =4n-6, Kp = C 4 C[M2]. According to the series, the cluster will have 2 of its skeletal elements in the nucleus and 4 capping skeletal elements. The next fragment is Rh 6 (CO) 11 with K =16. Its cluster parameter is K(n) = 16(6). Its cluster series will be S = 4n-8 and Kp = C 5 C[M1]. This is a cluster with a single skeletal element at the nucleus with 5 capping skeletal elements. The entire sequence of K(n) numbers will be: 11 (6) Table 2 and converted into series in Table 5. This movement involving the removal of a co ligand step-wise may be described as stripping series. In the stripping series Kp = C y C[Mx], both the nuclear index x and the capping index y change as we move from one step to the next. We also notice that as y increases, x decreases and vice versa. In the example of the octahedral cluster Rh 6 (CO) 16 , the nuclear index x goes from 6 to 0 where all the 6 rhodium skeletal elements are capped. These changes are diagrammatically expressed in figure SC-17 below. Using the rhodium carbonyl cluster, Rh 6 (CO) 16 as a reference for illustration, the cluster is a member of the CLOSO family, S = 4n+2, K(n) = 12(6) and Kp = C 0 C[M6]. This is a "borderline" cluster with zero capping skeletal element. When a CO ligand is removed via stripping, it becomes Rh 6 (CO) 15 , K = 12(6), Kp = C 1 C[M5]. This represents a mono-capped trigonal bipyramid geometry. The next stripping of a CO ligand creates, Kp = C 2 C[M4] which is a bi-capped tetrahedral cluster. The next stripping gives us Kp = C 3 C[M3] cluster which is a triangular tri-capped cluster. This is followed by a tetra-capped cluster with a two bi-skeletal nuclear cluster, Kp = C 4 C[M2]. Further stripping generates Kp = C 5 C[M1] which is a penta-capped mono-skeletal nuclear cluster. The next stripping creates a cluster Kp = C 6 C[M0] where all the skeletal elements are capping and the nucleus is empty.

Blackholes in the Nucleus and Ghost Skeletal Elements
Further stripping creates clusters with negative nuclear indices. Thus, the cluster number 8 in Table 5, Rh 6 (CO) 9 : K(n) = 18(6), Kp = C 7 C[M-1]; shows that it has 7 capped skeletal elements. However in reality, the cluster has 6 skeletal elements not 7. Hence, we can regard the seventh elements as fictitious and just a creation of the series. We may call such an element created by the series as a hole or a ghost skeletal element. A closer look at Table 5 shows that the capping elements are being taken from the nucleus. When [M0] becomes [M-1], it means the capping fictitious element has been "borrowed" from a nucleus which did not have any at all, hence [Mx], x = -1 or according to series, n = -1 at the nucleus. Since the nucleus is a member of CLOSO family, S = 4n+2 and K = 2n-1 = 2(-1)-1 = -3. This means the cluster nucleus is indebted with -3 cluster linkages. As x in [Mx] becomes x≤0, the number of nuclear skeletal linkages becomes more and more negative. We may regard such nuclei as having BLACK HOLES since they seem to be swallowing skeletal linkages. When all the CO ligands have been removed from Rh6(CO) 16 to become completely naked, cluster number 17 Table 5, Rh 6 : K(n) = 27(6); and Kp = C 16 C[M-10]. We can interpret this to mean that there are 16 capping skeletal elements, 6 genuine ones and 10 ghost ones borrowed from the cluster nucleus which develops a huge black-hole of n = -10 and K = 2n-1 = 2(-10)-1 = -21 skeletal linkages. Can we then regard a cluster with a nucleus possessing a black-hole and ghost capping skeletal elements as a characteristic of a metallic character of a cluster? The nuclear changes in the stripping series of Rh 6 (CO) 16 up to [M0] are illustrated in SC-9. The decomposition fragments of Rh 6 (CO) 16 are shown in Table 5.
On the other hand, the addition of CO ligands to the Rh 6 (CO) 16 cluster reduces the K value from K =11 to finally K =5, the limiting K value when the cluster behaves as a saturated hydrocarbon analogue.
The cluster Pd 23 (CO) 20 L 10 : K(n) =62(23); S=4n-32, Kp = C 17 C[M6]; was chemically converted into Pd 23 (CO) 20 L 8 : K(n)=64(23); S=4n-36, Kp =C 19 C[M4]. The two clusters were found to have different skeletal structures (Mednikov and Dahl, 2010). The former is described to have a Pd 19 pseudo octahedral while the latter has a Pd 15 kernel hexa-capped cubic with 8 capping Pd elements. The transformation can be explained using series in that Pd 23 (CO) 20 L 10 has a K value of 62 with an octahedral nucleus [M6] while the cluster Pd 23 (CO) 20 L 8 has a K value of 64 with a tetrahedral nucleus. According to the series method, the two clusters of the same nuclearity differ by 2 electron donor ligands and hence as expected as in line with the case of rhodium clusters discussed in SC-17, the removal of a ligand from a cluster of the same nuclearity results in the increase of K value and increase in the degree of capping. In the case of capping clusters, the increase in the capping corresponds to the decrease of the number of atoms in the nucleus. An increase in K value by 1 is accompanied by a decrease in the number of nuclear skeletal elements by 1. What was also found to be exciting is the discovery that the series method predicts that the clusters Pt 38 (CO) 44 (Mednikov and Dahl, 2010). The series method can also be useful in rapid categorization of the products encountered in EDESI-MS work (Bucher, 2003). An illustration is given in Table 6.

DIFFERENCES BETWEEN THE 4n SERIES METHOD AND THE SEP METHOD
Arising from the applications of the 4n series method, some salient features of the differences between the method and the existing SEP method have emerged.

 TOOLS OF ANALYSIS OF CLUSTERS
The SEP method utilizes SEP and vertices concept to analyze the clusters while the 4n method utilizes the SKELETAL NUMBERS and the 4n series to analyze and categorize the clusters of skeletal elements from the main group and transition metals.

 CAPPING AND NON-CAPPING CLUSTERS
According to the 4n series method, all clusters comprising of skeletal elements from the MAIN GROUP or TRANSITION METALS are interrelated and can all be analyzed using the skeletal numbers and the 4n series method.
The series S = 4n+q(q≥2) are non-capping types while S = 4n+q(q≤0) represent capping clusters. The SEP method breaks clusters into groups such as 4n, 5n and 6n and finds some clusters which are described as "rule-breakers". On the other hand, there are no rule-breakers in the 4n series method.
 CLUSTER VALENCE ELECTRONS(cve) The series method provides an easy method of calculating the cluster valence electrons directly from the series formula.

 INTER-CONVERSION OF CLUSTERS
The series formula can be used to generate analogous clusters in line with K(n) parameter. For instance, K(n) = 11(6), S = 4n+2(n=6) can be used to derive any appropriate octahedral cluster such as B 6 H 6 2-, Rh 6 (CO) 16 and Os 6 (CO) 18 2-.

 THE K(n) PARAMETER WITH DOUBLE MEANING
The series method gives us a cluster number K which has a double meaning. For instance the oxygen atom has a K value of 1; O(K=1). This means it has the potential of contributing 1 bond in its chemical activity. Since the skeletal number of boron, K= 2.5, its skeletal valence is 5 and this leads us to distribute the H ligands of B 5 H 9 to generate a raw skeletal structure as in F-2 and then propose a rearranged skeletal one in F-3 as is normally observed from x-ray structural analysis.

 BROAD CATEGORIZATION OF CLUSTERS
By analyzing the cluster series, it has been possible to identify 3 broader clusters series, namely, the vertical, diagonal and horizontal ones. With this knowledge, it has now been possible to categorize all the clusters derived from the main group/transition metals according to their respective closo family members, [M0], [M1], [M2], and [M3] and so on. Being able to categorize clusters into groups, means that we have created a system of literally categorizing every cluster comprising of main group/transition metals with the help of a larger type of Rudolph system that operates numerically.

 CATEGORIZATION OF GOLDEN CLUSTERS
The series method has penetrated the golden clusters and showed that they all obey the cluster laws. As much as possible detailed analysis of each cluster regarding consistency with skeletal linkages and ligand distribution are explained for each cluster.

 THE STRIPPING SERIES OF CLUSTERS
The analysis and categorization of stripping series of clusters may assist in improvement of understanding and appreciation of the work by Butcher and his research groups (Butcher, et al, 2003).

 CATALYSIS
The concept can readily be applied in enhancing deeper understanding and selection of cluster catalysts such as the carbonyls. Selected examples of carbonyl tried for catalysis (Gates, 1995) are given in Table7. As can easily be seen from Table7, most of the clusters that have been tried in catalytic work belong to CLUSTER GROUP 5 OR GROUP 6. Furthermore, there is no sample of positive capping index, Kp = C y C( y≥1) on the table list. ) known product (Housecroft & Sharpe, 2005). In the case of the rhodium cluster Rh 6 (CO) 16 , the removal of a CO ligand creates an additional linkage for binding the two fragments, the cluster K value goes from 11 to 23 for the combined 2 fragments (11+1+11=23) according to the series to generate the dimeric cluster, Rh 12 (CO) 31 in the form Rh 12 (CO) 30 2- (Housecroft & Sharpe, 2005). The hypothetical mechanism for the processes is given in P-1 and P-2. The possible extension of rhodium polymerization to produce longer chains is sketched in P-3.

Unique Capping Clusters
Predicting the structures of clusters or unique structures: This may be illustrated by the two examples, Au 6 R 6 Br 2 (CO) 2 2-, R = CF 3 ; K =6[3.5]-6(0.5)-2(0.5)-2(0.5) = 14, n = 6, K(n) = 14(6), S = 4n-4, Kp = C 3 C[M3]. This predicts a cluster of 3 closo skeletal nucleus surrounded by 3 other skeletal elements. According to series approach, the prediction is a closo nucleus of 3 skeletal elements surrounded by 3 capping skeletal elements. However what is observed are two sets of separate trinuclear clusters (Martinez-Salvado et al, 2015). This is sketched in N611. A similar prediction is found in the dinitrogenhexagolden cluster N 2 (AuL) 6 2+ : K = 2[1.5]+6[3.5-1]+1 = 19; n=2+6 =8, K(n) = 19(8), S = 4n-6, Kp = C 4 C[M4]. Again two sets of 4 skeletal elements were observed (Gimeno, 2008). The predicted skeletal sketch and the observed one for N 2 (AuL) 6 2+ are sketched in N85. Another interesting cluster is Ni 12 As 21 3-. The capping series Kp = C 13 C[M20] predicts a nucleus of 20 skeletal elements surrounded by 13 capping skeletal elements. However, what is observed is the other way round; a dome of 20 skeletal elements encapsulating 13 others. The sketch of the cluster is given in N331. The last cluster on this list is the rhenium selenium cluster Re 6 Se 8 I 6 4-: It has a capping symbol Kp = C 9 C[M5]. The prediction here is that the cluster has a trigonal bipyramid nucleus surrounded by 9 capping elements. The x-ray analytical interpretation of the structure is 8 selenium capping elements with an octahedral nucleus. This revelation is quite fascinating as the prediction is 9 skeletal elements by the series and the observed are 8. Since the series acts as a "deaf and dumb" guide, this result is good enough and gives more confidence in understanding the language of the series method deeper. The cluster is sketched in N141.