The Capping Theory of Chemical Clusters Based on 12 N / 14 N Series

The genesis of chemical clusters of the transition and main group elements has been established. The base-line for cluster valence electrons has been demarcated with help of capping series. Using the base-line as a reference, the formulas of fragments and clusters were generated. Also a simple general formula for calculating cluster valence electrons for systems ranging from a single to multi-skeletal element clusters was identified. The concepts of the existence of nuclei in clusters and some having black-holes were well established. The capping principle was extended to the main group and transition elements of the periodic table and a difference between metals and non-metals was discerned. The skeletal elements of clusters whose series have advanced beyond closo baseline level S=4n+2 can be separated into two broad groups, namely those which follow the closo series(nucleus) and those which follow the capping series.


The Genesis of the Chemical Clusters
The capping principle of clusters is well revealed by considering the genesis of the cluster series.Let us illustrate this by focusing on [M6] clan series.The symbol represents 6 skeletal elements which are bound together and belong to the CLOSO family of clusters (Kiremire,2015a).This means that it is associated with the series S=4n+2, K=2n-1=2[6]-1=11, K(n)= 11(6).The cluster valence electrons are given by VE=4n+6= 4[6]+2=26 for the main group elements and VE= 14n+2=14[6]+2=86 for the transition metals.The ideal shape of the fragment is an octahedron as shown in Figure 1.During the analysis of clusters, a K(n) parameter was introduced (Kiremire, 2017a(Kiremire, -f, 2018a-d)-d).The parameter can be transformed into a numerical value of electrons given by VE=18n-2K for transition metal clusters or VE=8n-2K for main group elements.Let us consider an example of K(n)=11( 6 The chemical clusters follow the K(N) sequences very well.In this regard, let us examine the changes in the K(n) parameter when there is a decrease in (K) by 3 units and (n) by 1 unit, that is, ∆K(n) = -3(-1) or simply a change by 3(1).A good starting point is K(n)=11( 6) since this represents an ideal symmetry of an octahedron.Thus, the following K(n) values can readily be generated: 11(6)→8( 5) →5( 4 This means there is a corresponding decrease in cluster valence electron content of 12 at every step.When the K(n) variation of 2(1) decrease is performed starting with K(n)= 11(6), we get the following series: 11(6)→9(5) →7( 4 decrease by ∆[K(N)] =3(1).This means the downward decrease of 12.That is, the cluster valence electrons will decrease step-wise by 12 units.This is a fundamental principle on which the 12N series is based.If we consider the horizontal movement of the series then ∆K(N)=2(1) is utilized.
Hence, the change ∆(VE)=18N-2K=18(1)-2(2)=18-4=14.If we define a focal point as VE(n)=86(6] and decrease vertically by VE(n)= 12(1), we get the series: VE(n)=86(6)→74(5) →62(4) →50(3) →38(2) →26(1) →14(0) with a corresponding change in ∆VE=12 →this belongs to the 12N SERIES, S=12n+14.The numerical constant at the end in this case 14 represents the additional electrons of the baseline when there is no skeletal element (n= 0) involved.Let us now consider the horizontal movement starting with the same point, 86(6) with a decrease of 14(1) step by step, we get: VE(n) series: 86(6) →72(5) →58(4) →44(3) →30(2) →16(1) →2(0) and ∆VE=14 → this forms the basis of the 14N SERIES, S=14n+2.Like the 12N series, the numerical number 2 represents the additional electrons which are associated with n=0.Selected representatives of the 14N series and the 12N series are shown in Table2.The series can be expressed as S=12n+q' and S= 14n+q where q' and q are numerical values obtained when n=0 in the K(n) and VE(n) variations.In the Tables 1and 2, S= 14n+q series form the rows (families) and the S=12n+q series form the columns (clans).The Rudolph concept of correlation of borane clusters (Rudolph, 1976) corresponds to the clan categorization approach of clusters.In order to examine more VE(n=0) values, that is, q' and q capping values, Table 2 was extended to produce Table s 3, 4 and 5.The VE(n) values in Table 2 can also be expressed in the form of Cartesian coordinates indicated in Figure 2. The proposed assignment of clan and family cluster series is illustrated in Figure 3 and relationships between some of the cluster equations developed during the study of cluster series are shown in Scheme 3.    The series general formula S=4n+q for categorizing clusters that was empirically derived (Kiremire, 2016c) can be utilized to determine cluster valence electrons, VE.The cluster valence electrons, VE = 4n+q for the main group clusters and VE =14n+q for transition metal clusters.As can be seen from array of cluster valence electrons in Tables 2-5, the series formula is a consequence of the natural sequence of the cluster valence electrons.Thus, the 14n and q components of the series formula S=14n+q occur naturally from the arrays of the cluster valence electrons.For ease of application, the formula is adjusted to the simpler one S=4n+q for usage in the categorization of clusters of transition metals and main group elements.Both the cluster valence electrons and formulas can readily be derived from the capping principles of clusters.Furthermore, the relationship between VE and q becomes clearer.As can readily be deduced from Table 2 horizontally or vertically, the cluster valence electrons (VE) can be calculated from the relationship VE= q+14n = q'+12n.For instance, VE=2+6[14]=86 and VE=14+6[12]=86.In terms of deducing the type of cluster series it has been found easier to apply q derived horizontally rather than the q' derived vertically.The q' becomes useful in sketching the capping diagrams of clusters as will be illustrated by examples in this paper.We can express the above concepts as follows: VE = q+ VE1+VE2+VE3+……+VEn; and since VE1= VE2= VE3=….. VEn = 14, then VE= q+14n = S(horizontal series) VE=q'+ VE1+VE2+VE3+……+VEn; and since VE1= VE2= VE3=….. VEn = 12, then VE= q'+12n = S(vertical series) A cluster formula can also be derived by transforming the cluster valence electrons into chemical fragments using the relevant cluster series.Let us illustrate this using Rh 6 (CO) 16 as an example;

Types of Capping Phenomena
A given VE(n) parameter is at the intersection of the horizontal (S=14n+q) and vertical(S=12+q') series.There is also diagonal series that could be considered.However, it been found easier to analyze the capping series using the vertical ones, S=12n+q' but categorization and cluster valence determination to use the horizontal series formula S=14n+q or simply, S=4n+q.
Using the 12N series approach to differentiate the types of capping phenomena The capping formula Kp= C y C[Mx](y+x=n) developed earlier (Kiremire,2015b) is quite useful in guiding us to broadly differentiate the different types of capping phenomena.In general, we can classify the capping clusters into 3 broad categories.Other examples can be tested in the same way.

Metallic and Non-Metallic Elements
It has been found that the capping method which has been applied to clusters can equally well be applied to metallic and non-metallic elements from main group and transition elements.A transition element has a set of valence electrons (VE) associated with it and since it is one, then clearly n=1.Hence, it obeys the 14N series.Hence, q=VE-14n formula can readily be applied to them.Let us consider Ti (VE=4, n=1), hence q=4-14[1]= -10 and S=4n-10, K=2n+5, Kp=C 6 C[M-5], Kp'=D 5 +C* 1 .This symbol means, Ti possesses 5 sets of 12 capping electrons in its black-hole nucleus and one capping skeletal element.The capping phenomenon commences at VEC=14n+2 = 14[-5]+2 = -68.This is a low level of deficit -68 e and when the 5 sets of capping dozen electrons are added, there is a balance of deficit of -8e.However, the black-hole is now filled.According to the 12N capping principle, when the only first skeletal fragment of 12 electrons is added on top the net becomes 4 which now becomes the observed valence electrons of Ti[Ar]4s 2 3d 2 .
Other transition metal skeletal elements may be rationalized in the same hypothetical manner.As can be seen from Tables 6 and 7, the transition elements are characterized by valence electron deficit (VE0<0) on capping giving rise to a decreased value of the valence electrons (VE)than the capping set of 12 electrons that were added(Table 6) while in the case of the main group elements the valence electrons VE obtained after the two capping electrons are more than VE0(the number of valence electrons before the capping skeletal element has been added).Thus, the capping principle can be used as a simple qualitative guide to distinguish between metals and non-metals.The metallic elements have VE0<0, non-metallic VE0>0 and VE0=0 may be regarded as a borderline case.The borderline transition metal elements are Zn, Cd and Hg while the corresponding main group elements are Be, Mg, Ca, Sr, Ba and Ra The VE0<0 for group 1 elements, Li, Na, K, Rb, Cs, Fr implying that they may strictly be regarded as being metallic in character.
Table 7.The capping data of naked transition metals Table 8.The capping data for the Main group Elements Kp''=C 1 (Sn)+C 12 (Cu)+D 20 (Sn) This cluster may be regarded as a Zintl-type of matryoshka cluster.The capping formula of the 4N series can only separate the skeletal elements of a capping cluster INTO 2 GROUPS .At this level of its qualitative application, it is unable to go deeper into details.The details can only be elucidated by x-ray crystal structural analysis.In this case, the cluster of 13 skeletal elements is found to comprise of Sn at the center of an icosahedral cluster enclosed in another cluster of 20 Sn skeletal elements (Huang, et al, 2014).
The capping formula Kp' = C 13 +D 20 represents 2 types of cluster series; namely for the outer shell, C y = C 13 : S1 =4n-2(13)=4n-26, K1= Since the nucleus also has capping skeletal elements founded upon base-line cluster valence electron, we could introduce another capping symbol (D) for the elements in the nucleus.The cluster is expressed as a tetra-capped octahedron (Hughe & Wade, 2000).
3. F=Ru 6 Pd 6 (CO) 24 2-:The cluster is observed to be a hexa-capped octahedron (Rossi&Zanello,2011).This what is predicted by the 4N series method.Furthermore, it is found that the inner cluster [M6] comprises of Pd skeletal elements only.This tells us that all the 20 skeletal elements are in the outer shell capping following the series S=4n-2(20)=4n-40 and there is no skeletal element in the nucleus except the 2 electrons.The capping formula has been applied break into 2 groups of skeletal elements of selected clusters.
More examples obtained mainly from literature sources (Felner & Salet, 2007;Dries & NÖth, 2004) are given in Table 9.The extensive reviews by Belyakova and Slovokhotova as well as Mednikov & Dahl provide a wide range of examples (Belyakova & Slovokhotova, 2003;Mednikov & Dahl, 2010) from which the concept of separating the capping elements into two broad groups can be found.

Conclusion
The establishment of the base-line valence electron content VE0 strongly underpins the great significance of categorizing clusters into clans and families.The establishment of VE0 enables to directly calculate the cluster valence electrons VE using either the 12N oe the 14N capping series as well as deriving the formulas of the respective clusters.The construction of capping diagrams of clusters and skeletal elements of transition elements has been well established.
The black-hole concept of capping electrons may qualitatively explain the difference between metals and non-metals.Some clusters especially the giant ones appear to possess back-holes which have been found to be a characteristic of metallic behavior.The capping formula can be utilized as a simple qualitative guide to separate skeletal elements of capping clusters into two groups, the outer shell and the inner shell.The construction of cluster valence electron trees is a good idea of portraying relationships among clusters of the same clan.Cluster series may simply be regarded as some forms of simple arithmetic progressions with suitable common differences such as 14 for 14N and 12 for 12N series.A capping principle of clusters based upon 12N/14N series has been established.

Figure 3 .
Figure 3. Defining clans and gamilies of clusters A): Kp= C y C[Mx], y≥0 and x≥1 The symbol,[Mx] represents nucleus centered clusters.Such clusters have one or more skeletal elements in the nucleus.The nucleus on its own belongs to the CLOSO family which follows the S=4n+2 series.Ideally, the nuclei behave like closo borane fragments[B n H n ] 2-.Examples include [M1] centered with one skeletal element in the nucleus; the corresponding baseline cluster valence electrons are given by VE0= 2n+2=2[1]+2=4; [M2] with 2 skeletal elements in the nucleus and the corresponding VE0=2n+2=2[2]+2=6,[M5] with 5 skeletal elements in the nucleus and corresponding VE0=2n+2=2[5]+2=12 and [M6] with 6 skeletal elements in the nucleus .andcorresponding VE0=2n+2=2[6]+2=14.The capping trees of such clusters with selected examples are given in Figures4-7.The example of the cluster tree in Figure8of main group elements is added for comparison with the corresponding ones from transition elements.

Figure 4 .
Figure 4.A sketch of the capping tree of [M1] cluster clans series

Figure 6 .
Figure 6.A sketch of the capping tree of [M5] cluster clan series Let us consider the following examples as illustrations.These clusters have negative nuclearity index, x= -1,-2,-3,-4 and so on.There is a significant difference say between [M2] and [M-2].In the case of [M2], the cluster has 2 capping skeletal elements each carrying 12 cluster electrons whereas in the case of [M-2], the nucleus has two empty sets of 12 electrons capping in the cluster nucleus, that is, a nucleus without any skeletal elements.A nucleus of negative nuclearity index has been referred to as a black-hole(Kiremire, 2018d) and in this example of [M-2], the negative 2 are also regarded as mini-holes each containing 12 capping electrons.Negative nuclearity index could be regarded as a characteristic of metallic skeletal elements.This concept is further illustrated by the examples in Figures 9-13 for clusters with black-holes and Figures 14-19 for naked skeletal elements for comparison.

Figure 14
Figure 14.A capping diagram for Ti The titanium element not only does it possess a black-hole nucleus, it has a deficit of capping electrons which results into a net of 4 valence electrons.

Table 2 .
The Genesis of cluster valence electrons VE0 for selected [M6] to [M0] clan series Figure 2. A sketch of VE(n) map of selected cluster clan series

Table 3 .
Upward Extension of Table2

Table 4 .
Further extension of Table2

Table 9 .
A collection of clusters showing their categorization into two groups