American Journal of Chemistry

p-ISSN: 2165-8749    e-ISSN: 2165-8781

2019;  9(2): 33-70

doi:10.5923/j.chemistry.20190902.02

 

The Double Capping Phenomenon of Chemical Clusters

Enos Masheija Rwantale Kiremire

Kironde Road, Kyadondo Block 224, Plot 5387, Kampala, Uganda

Correspondence to: Enos Masheija Rwantale Kiremire, Kironde Road, Kyadondo Block 224, Plot 5387, Kampala, Uganda.

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Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

The application of skeletal numbers has made it far much easier to categorize and analyze a wide range of clusters very quickly according to the 4N series method. It has been revealed that the clusters ranging from mono-to multi-skeletal elements comprise of two shells simply referred to as the inner (D) shell and the outer shell (C). A symbol to represent this duality concept has been introduced as the double capping parameter K* = Cy+Dz, where y+z = n, the number of skeletal elements in a cluster. Depending upon the cluster being analyzed, both C and could represent capping skeletal elements or in some cases, D could refer to capping electrons in dozen pairs. In other cases the C symbol could represent the missing skeletal elements for a cluster in order to achieve a CLOSO (fragment that follows S=4n+2 series) configuration. In order to demonstrate the double capping nature of clusters, about 40 clusters were categorized more than 30 of which were bi-metallic golden clusters. All of them were found to be capped with both C and D comprising of skeletal elements. The D elements are capped on the GENESIS ELECTRONS given by VE0=2z+2 electrons in the inner shell while the C elements are capped on top of the D closo elements. More examples of clusters were readily categorized into CLAN SERIES. A new equation for calculating the cluster valence electrons was derived as VE= 12y+14z+2 from K* = Cy+Dz. The calculated VE value was found to be the same as the one VF calculated directly from the cluster formula. The new equation for calculating cluster valence electrons is on of the six equations which can eaily be applied to calculate the cluster valence electrons of clusters and chemical elements of the periodic table except lanthanides and actinides.

Keywords: Inner shell, Outer shell, Capping parameter, Double capping, Duality, Bi-metallic

Cite this paper: Enos Masheija Rwantale Kiremire, The Double Capping Phenomenon of Chemical Clusters, American Journal of Chemistry, Vol. 9 No. 2, 2019, pp. 33-70. doi: 10.5923/j.chemistry.20190902.02.

Article Outline

1. Introduction
2. Results and Discussion
3. Examples of Selected Clusters
4. Structural Prediction
5. Conclusions

1. Introduction

The unique shapes and intriguing nature of clusters have attracted a lot of scientists to the study of the same (Lipscomb, 1963; Wade, 1971, 1976, Mingos, 1972; Tolman, 1972; Rudolph, 1976; Pauling, 1977; Stone, 1981; Hoffmann, 1982; Teo, 1984; Hughes & Wade, 2000; Belyakova, 2003; Butcher, et al, 2003; Jemmis, 2005; Jemmis, etal, 2001a-b, 2002, 2003, 2008; Grimes, 2003; Goicoechea & Sevov, 2006; Fehlner & Halet, 2007; Mednikov & Dahl, 2010; Welch, 2013) for nearly fifty years. Relatively recently, an interest was ignited to understand the structural pattern of carbonyl clusters and this resulted into the development of a new approach to the categorization of clusters using the 4N series method out of which the skeletal numbers were discovered (Kiremire, 2016a-b; Kiremire, 2017a-e). The skeletal numbers have been found to be exceedingly fast and generally accurate in categorizing clusters and calculating the cluster valence electrons (Kiremire, 2018a-b). The structural prediction is also generally quite good (Kiremire, 2016c). The skeletal numbers have since been successfully applied to the analysis of a wide range of clusters including boranes, metalloboranes, metal carbonyls (Kiremire, 2017c), zintl, matryoshka (Kiremire, 2018c) clusters and the clusters of gold (Kiremire, 2018b). In this paper, the skeletal numbers will be applied to analyze the bimetallic clusters of gold and related clusters and the general features of clusters will be discussed.

2. Results and Discussion

Skeletal numbers
Skeletal numbers were assigned to all the elements of the periodic table except the lanthanides and actinides (Kiremire, 2016a). Their derivation was based upon the valence electrons of the elements. Since main group and transition elements in the periodic table are arranged according to valence electrons, the skeletal numbers are also assigned to them according to their groups.
The range of the skeletal numbers is from K=7.5 to 0. This assignment is given in Table 1.
Table 1. Skeletal numbers of main group and transition metals
     
The skeletal number (K) of an element can be viewed as representing the number of electron pairs that element requires in order to attain the 8 or 18 electron rule for main group or transition elements respectively. Hence the valence V=2K represents the actual number of electrons needed by the respective element so as to attain a noble gas configuration.
THE VARIATION OF THE SKELETAL NUMBER
Let us consider what happens when we move from one skeletal element to the next one. This can be illustrated by considering the changes from Sc(K=7.5) to Zn(K=3) in the first row of transition metals. This is shown in Scheme 1,
Scheme 1. Inter-conversion of skeletal elements
As can be seen, the addition of an electron to a skeletal element results into a reduction of a skeletal number of the element by 0.5 and hence a decrease in the skeletal valence (V=2K). Likewise, the removal of an electron from an element results into an increase in the skeletal number and a corresponding increase in the skeletal valence. This implies that an electron can be assigned a numerical value of -5, that is, e(K=-0,5) and similarly a positive charge (+1) can be assigned a numerical value of +0.5.
The variation of skeletal number can also be done by oxidizing the skeletal element. For instance, the carbon skeletal element C with a K value of 2, valence V=4, can be oxidized to C+ and then C2+ the K value changes from C(K=2, V=4)→C+(K=2.5, V=5) and C2+(K=3, V=6). This variation of K value is reflected in the following golden carbon clusters (Gimeno, 2008)* shown in sketch Figure 1A.
Figure 1A. The structures of C(AuR)4, C(AuR)5+, and C(AuR)62+
LIGANDS HAVE NEGATIVE SKELETAL NUMBERS
By analyzing cluster series, it was observed that addition of a CO ligand to a skeletal element results into a decrease of a cluster skeletal number (Kiremire, 2017c). The decrease is proportional to the number of electrons donated.
For instance, the hydrogen atom acting as a ligand donates one electron hence H(K= -0.5). The carbonyl ligand donates 2 electrons, CO(K= -1) for a cyclopentadienyl ligand C5H5(K= -2.5) and benzene, C6H6(K= -3).
CLUSTER NUMBER, K
A cluster with skeletal elements and ligands has a cluster number, K which can be regarded as a resultant contributions of the skeletal numbers of all the skeletal elements and the ligands in the cluster. If the cluster has no ligands, then the resultant K value will be derived from the constituent skeletal elements. This relationship can be expressed by the equation;
K = KSE + KL,
Where K = cluster number, KSE = contributions from the skeletal elements and KL = contribution from the ligands.
DECOMPOSITION OF A CLUSTER FORMULA INTO A NUMERICAL NUMBER.
Since a cluster from the main group elements or transition metals comprises of skeletal elements and ligands, it can readily be decomposed into a simple numerical figure (K) which is generally a WHOLE NUMBER, This number can be utilized to generate cluster series S=4n+q, the skeletal number formula K=2n- ½ q where n is the number of skeletal elements in the cluster, and q is a variable which acts the determinant of the type of the cluster. Furthermore, the capping formula Kp= CyC[Mx], y+x=n, as well as double capping parameter implies there are two types of skeletal elements can be expressed by the formula K*=Cy+Dz. The symbol y represents the capping elements of the cluster while x represents the number of nuclear elements of the cluster. The value z can be utilized to calculate the genesis cluster valence electrons (VE0=2z+2) of the series when n=0. With the knowledge of the series then, it is possible to derive the cluster valence electrons equation (Kiremire, 2018b) VEn=VE0+12n.
THE SKELETAL NUMBER K AS A STANDARDIZING PARAMETER OF THE SKELETAL ELEMENTS
The skeletal number can also be regarded to act as a standardizing parameter of cluster elements. Thus, all the skeletal elements can be considered to be equivalent and therefore can numerically be added. HOWEVER, THEIR CONTRIBUTIONS TO THE FINAL CLUSTER NUMBER (K) ARE NOT EQUIVALENT UNLESS IF THEY BELONG TO THE SAME PERIODIC GROUP.
THE DOUBLE CAPPING NATURE OF CHEMICAL CLUSTERS, K* = Cy +Dz
The analysis of a wide range of clusters reveals that a cluster intrinsically comprises of two components given by K* =Cy+Dz, where y and z are two separate entities related by y+z=n, the number of skeletal elements in a cluster. In a normal cluster y -represents the capping skeletal elements on top of z -nuclear CLOSO elements. The symbol z has a set of elements which belong to the closo family, S=4n+2 while the capping elements, y obey a separate series given by S=4n-2y. The addition of the two equations yields the final overall cluster equation, S=(4n+2)+(4n-2y)=4n-2y+2. The double capping phenomenon can be illustrated by Figure 1B.
Figure 1B. Illustration of the capping phenomenon in clusters
D6 ILLUSTRATIONS
We can illustrate the double capping nature of chemical clusters by taking the example of D6 clan series. The D6 represents an octahedral fragment with a K value of 11 and hence K(n)=11(6), and S=4n+2 (closo family). By analyzing the K(N) numerical series, it has been found that the transition metal clusters follow the 12N series while the main group element clusters follow the 2N series. But for now, let us focus on the 12N series for the D6 system. As K decreases by 3 and n by 1, there is a corresponding decrease of cluster valence electrons by 12. Further step-wise decreases reaches a stage when n=0 (assigned to be the bottom-line). The corresponding cluster valence electrons(VE) is 14. Since VE0=14 for D6 clan series, then the D6 is related to VE0=14 by a simple equation VE0=2[6]+2 which is generalized to VE0=2z+2 for Dz clan trees (Kiremire, 2018b). The symbol VE0 is introduced to denote the cluster valence electrons when n= 0. A portion of D6 clan tree series is given in Table 2. The table also gives one of the six formulas VE=18n-2K for calculating the cluster valence electrons, and the other five equations are given in Tables 3 and 4. These are:
VE=14n+q
VE=VE0+12n [VE0=2z+2]
VE=VE0+12y+12z
VE=VEDz+12y [VEDz=14z+2 ]
VE=12y+14z+2
In addition, the table also gives the categorization series formula S=4n+q and cluster linkage formula K=2n- ½ q, the capping formula Kp= CyC[Mx] and the categorization formula K*= Cy+Dz. In ‘conventional’ clusters, y represents the number of capping elements and the z the number of elements in the cluster nucleus around which the capping elements form linkages. A selected number of known clusters which belong to D6 cluster clan have also been included. As can be seen from Tables 2-4, the cluster valence electron series follow a smooth arithmetical progression which can be represented by T=a+12n, where T= cluster valence electrons, a= the number of cluster valence electrons when n=0 and n= the number of skeletal elements in a cluster. Since we are thinking of cluster valence electrons, the capping formula becomes VE=VE0+12n. Selected D6 capping process is shown in Scheme 2. As can be seen from Scheme 2, there is some capping before the CLOSO level S=4n+2 is reached which can be expressed as K*= C-y+Dz, whose base is VE0, in this case, VE0(n=0)=14 and the capping after the closo level, K*=Cy+Dz based on D6 closo nucleus. The D6 symbol represents an Oh fragment or a fragment with an ideal symmetry of B6H62-. Thus, there are two phases of capping processes, the one before the closo level and the other after the closo level. When y and z represent genuine skeletal elements, then Dz may be regarded as a cluster nucleus. THIS IS THE BASIS OF WHAT HAS BEEN TERMED AS THE DOUBLE CAPPING PHENOMENON OF CHEMICAL CLUSTERS. All chemical clusters that have been tested including single skeletal elements can be expressed as K*= Cy+Dz (where y or z can have positive or negative numerical values).
Scheme 2. An illustration of the D6 capping process
Table 2. A Sample of the D6 Clan Series
     
Table 3. The D6 Clan Series showing the use of new equations for calculating cluster valence electrons
     
Table 4. Demonstration of more equations for calculating cluster valence electrons using new equations
     

3. Examples of Selected Clusters

Let us consider some selected examples including the bimetallic golden clusters (Ciabatti, 2015) to demonstrate the double capping nature of chemical clusters.
1. Au6L62+:K=6[3.5]-6[1]+2[0.5]=16,n=6
K(n)=16(6)
12-16=-4
S=4n-8
K=2n+4
Kp=C5C[M1]
K*=C5+D1
D1 represents capping of one skeletal element sitting on 4 genesis electrons given by VE0=2z+2=2[1]+2=4.
C5 represents 5 skeletal elements capping upon D1 element in the cluster nucleus.
VEn=VE0+12n=4+12[6]=76; cluster valence electrons based on capping theory.
VF=represents cluster valence electrons calculated from the cluster formula=6[11]+6[2]-2=76.
2. Au8L72+:K=8[3.5]-7[1]+2[0.5]=22,n=8
K(n)=22(8)
16-22=-6
S=4n-12
K=2n+6
Kp=C7C[M1]
K*=C7+D1
VE0=2[1]+2=4
VEn=VE0+12n=4+12[8]=100
VF=8[11]+7[2]-2=100
One skeletal element surrounded by 7
other capping skeletal elements.
The cluster valence electrons from
theory, VEn=VF(from the cluster formula)
3. Au9L83+:K=9[3.5]-8[1]+3[0.5]=25,n=9
K(n)=25(9)
18-25=-7
S=4n-14
K=2n+7
Kp=C8C[M1]
K*=C8+D1
VE0=2[1]+2=4
VEn=VE0+12n=4+12[9]=112
VF=9[11]+8[2]-3=112
One skeletal element surrounded by 8
other capping skeletal elements.
The cluster valence electrons from
theory, VEn=VF(from the cluster formula)
4. Au9L5R3 :K=9[3.5]-5[1]-3[0.5]=25,n=9
K(n)=25(9)
18-25=-7
S=4n-14
K=2n+7
Kp=C8C[M1]
K*=C8+D1
VE0=2[1]+2=4
VEn=VE0+12n=4+12[9]=112
VF=9[11]+5[2]+3[1]=112
One skeletal element surrounded by 8
other capping skeletal elements.
The cluster valence electrons from
theory, VEn=VF(from the cluster formula)
5. Os5(C)(CO)14(AuL)2:K=5[5]-1[2]-14[1]+2[3.5-1]=14
n=5+2=7
K(n)=14(7)
2[7]-14=0
S=4n+0
K=2n+0
Kp=C1C[M6]
K*=C1+D6
A mono-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12(7)=98
VF=5[8]+1[4]+14[2]+2[11+2]=98
Figure 2. Isomeric graphical structure of Os5(C)(CO)14(AuL)2
6. Fe5(C)(CO)14(AuL)2:K=5[5]-1[2]-14[1]+2[3.5-1]=14
n=5+2=7
K(n)=14(7)
2[7]-14=0
S=4n+0
K=2n+0
Kp=C1C[M6]
K*=C1+D6
A mono-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12(7)=98
VF=5[8]+1[4]+14[2]+2[11+2]=98
Figure 3. Isomeric graphical structure of Fe5(C)(CO)14(AuL)2
7. Ru6(B)(CO)17(AuL): K=6[5]-1[1.5]-17[1]+1[3.5-1]=14
n=6+1=7
K(n)=14(7)
2[7]-14=0
S=4n+0
K=2n+0
Kp=C1C[M6]
K*=C1+D6
A mono-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[7]=98
VF=6[8]+1[3]+17[2]+1[11+2]=98
Figure 4. Predicted isomeric structure of Ru6(B)(CO)17(AuL)
The mono-capped octahedron shape is predicted and this is what is observed and the capping element is gold (Ciabatti, 2015).
8. Co5(C)(CO)11(AuL)2-1:K=5[4.5]+1[-2]+11[-1]+2[3.5-1]-1[0.5]=14
n=5+2=7
K(n)=14(7)
2[7]-14=0
S=4n+0
K=2n+0
Kp=C1C[M6]
K* =C1+D6
VE0=2[6]+2=14
VEn=VE0+12n=14+12[7]=98
VF=5[9]+1[4]+11[2]+2[11+2]+1=98
Figure 5. Predicted isomeric structure of Co5(C)(CO)11(AuL)2
9. Co5(C)(CO)11(AuL)3:K=5[4.5]-1[2]-11[1]+3[3.5-1]=17
n=5+3=8
K(n)=17(8)
2[8]-17=-1
S=4n-2
K=2n+1
Kp=C2C[M6]
K*=C2+D6
A bi-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[8]=110
VF=5[9]+1[4]+11[2]+3[11+2]=110
Figure 6. Isomeric graphical structure of Co5(C)(CO)11(AuL)3
10. Ru6(C)(CO)16(AuL)2:K=6[5]-1[2]-16[1]+2[3.5-1]=17
n=6+1=7
K(n)=17(8)
2[8]-17=-1
S=4n-2
K=2n+1
Kp=C2C[M6]
K*=C2+D6
A bi-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n =14+12[8]=110
VF=6[8]+1[4]+16[2]+2[11+2]=110
Figure 7. Predicted isomeric structure of Ru6(C)(CO)16(AuL)2
11. Fe4Au4(CO)164-:K=4[5]+4[3.5]-16[1]-4[0.5]=16
n=4+4=8
K(n)=16(8)
16-16=0
S=4n+0
K=2n+0
Kp=C1C[M7] K*=C1+D7
VE0=2[7]+2=16
A mono-capped pentagonal bipyramid.
VEn=VE0+12n=16+12[8]=112
VF=4[8]+4[11]+16[2]+4=112
Figure 8
12. Ru4Rh2(B)(CO)15(AuL)3:K=4[5]+2[4.5]-1[1.5]-15[1]+3[3.5-1]=20
n=4+2+3=9
K(n)=20(9)
2[9]-20=-2
S=4n-4
K=2n+2
Kp=C3C[M6]
K*=C3+D6
A tri-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[9]=122
VF=4[8]+2[9]+1[3]+15[2]+3[11+2]=122
Figure 9. Predicted isomeric structure of Ru4Rh2(B)(CO)15(AuL)3
13. Os7(CO)20(AuL)2: K=7[5]-20[1]+2[3.5-1=20
n=7+2=9
K(n)=20(9)
2[9]-20=-2
S=4n-4
K=2n+2
Kp=C3C]M6]
K*=C3+D6
A tri-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+2n=14+12[9]=122
VF=7[8]+20[2]+2[11+2]=122
Figure 10. Predicted isomeric skeletal structure of Os7(CO)20(AuL)2
14. Fe4Au5(CO)163-:K=4[5]+5[3.5]-16[1]-3[0.5]=20
n=4+5=9
K(n)=20(9)
2[9]-20=-2
S=4n-4
K=2n+2
Kp=C3C[M6]
K*=C3+D6
A tri-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[9]=122
VE=4[8]+5[11]]+16[2]+3=122
Figure 11. Predicted isomeric skeletal structure of Fe4Au5(CO)163-
15. Co6(C)(CO)12(AuL)4:K=6[4.5]-1[2]-12[1]+4[3.5-1]=23
n=6+4=10
K(n)=23(10)
2[10]-23=-3
S=4n-6
K=2n+3
Kp=C4C[M6]
K*=C4+D6
A tetra-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[10]=134
VF=6[9]+1[4]+12[2]+4[11+2]=134
Figure 12. Predicted isomeric skeletal structure of Co6(C)(CO)12(AuL)4
16. Os8(CO)22(AuL)2: K=8[5]-22[1]+2[3.5-1]=23
n=8+2=10
K(n)=23(10)
2[10]-23=-3
S=4n-6
K=2n+3
Kp=C4C[M6]
K*=C4+D6
A tetra-capped octahedron.
VE0=2[6]+2=14
VEn=VE0=14+12[10]=134
VF=8[8]+22[2]+2[11++2=134
Figure 13. Predicted isomeric skeletal structure of Os8(CO)22(AuL)2
17. Ir6Ru3(CO)21(AuL)-1:K=6[4.5]+3[5]-21[1]+1[3.5-1]-1[0.5]=23
n=6+3+1=10
K(n)=23(10)
2[10]-23=-3
S=4n-6
K=2n+3
Kp=C4C[M6]
K*=C4+D6
A tetra-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[10]=134
VF=6[9]+3[8]+21[2]+1[11+2]+1=134
Figure 14. Predicted isomeric skeletal structure of Ir6Ru3(CO)21(AuL)-1
18. Co10(Au)(C)(CO)24-1:K=10[4.5]+1[3.5]-1[2]-24[1]-1[0.5]=22
n=10+1=11
K(n)=22(11)
2[11]-22=0
S=4n+0
K=2n+0
Kp=C1C[M10]
K*=C1+D10
A mono-capped bi-capped square-antiprism
VE0=2[10]+2=22
VEn=VE0+12n=22+12[11]=154
VF=10[9]+1[11]+1[4]+24[2]+1=154
19. Os9(CO)24(AuL)2: K=9[5]-24[1]+2[3.5-1]=26
n=9+2=11
K(n)=26(11)
2[11]-26=-4
S=4n-8
K=2n+4
Kp=C5C]M6]
K*=C5+D6
Penta-capped octahedron.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[11]=146
VF=9[8]+24[2]+2[11+2]=146
Figure 15. Predicted isomeric skeletal structure of Os9(CO)24(AuL)2
20. Ir7Ru3(CO)23(AuL)-2:K=7[4.5]+3[5]-23[1]-2[0.5]=25
n=7+3+1=11
K(n)=25(11)
2[11]-25=-3
S=4n-6
K=2n+3
Kp=C4C[M7]
K*=C4+D7
Tetra-capped pentagonal bipyramid.
VE0=2[7]+2=16
VEn=VE0+12n=16+12[11]=148
VF=7[9]+3[8]+23[2]+1[11+2]+2=148
Figure 16. Predicted isomeric skeletal structure of Ir7Ru3(CO)23(AuL)-2
21. Os10(C)(CO)24(AuL)-1:K=10[5]-1[2]-24[1]+1[3.5-1]-1[0.5]=26
n=10+1=11
K(n)=26(11)
2[11]-26=-4
S=4n-8
K=2n+4
Kp=C5C[M6]
K*=C5+D6
Penta-capped octahedron
VE0=2[6]+2=14
VEn=VE0+12n=14+12[11]=146
VF=10[8]+1[4]+24[2]+1[11+2]+1=146
Figure 17. Predicted isomeric skeletal structure of Os10(C)(CO)24(AuL)-1
22. Co11C2(CO)23-1:K=11[4.5]-2[2]-23[1]-1[0.5]=22
n=11
K(n)=22(11)
2[11]-22=0
S=4n+0
K=2n+0
Kp=C1C[M10]
K*=C1+D10
Mono-capped bi-capped square-antprism
VE0=2[10]+2=22
VEn=VE0+12n=22+12[11]=154
VF=11[9]+2[4]+23[2]+1=154
23. Os8(CO)20(AuL)4:K=8[5]-20[1]+4[3.5-1]=30
n=8+4=12
K(n)=30(12)
2[12]-30=-6
S=4n-12
K=2n+6
Kp=C7[M5]
K*=C7+D5
Hepta-capped trigonal-bipyramid.
VE0=2[5]+2=12
VEn=VE0+12[12]=156
VF=8[8]+20[2]+4[13]=156
Figure 18. Predicted isomeric skeletal structure of Os8(CO)20(AuL)4
24. Rh12(C2)(CO)23(AuL)-1:K=12[4.5]+2[-2]-23[1]+1[3.5-1]-1[0.5]=29
n=12+1=13
K(n)=29(13)
2[13]-29=-3
S=4n-6
K=2n+3
Kp=C4C[M9]
K*=C4+D9
Penta-capped tri-capped trigonal prism.
VE0=2[9]+2=20
VEn=VE0+12n=20+12[13]=176
VF=12[9]+2[4]+23[2]+1[11+2]+1=176
25. Os10(CO)24(AuL)4:K=10[5]-24[1]+4[3.5-1]=36
n=10+4=14
K(n)=36(14)
2[14]-36=-8
S=4n-16
K=2n+8
Kp=C9C[M5]
K*=C9+D5
Pentagonal bi-pyramd geometry surrounded by 9 capping skeletal elements.
VE0=2[5]+2=12
VEn=VE0+12n=12+12[14]=180
VF=10[8]+24[2]+4[13]=180
Figure 19. Predicted isomeric skeletal structure of Os10(CO)24(AuL)4
26. Ni12Au6(CO)242-: K=12[4]+6[3.5]-24-1=44,
n=12+6=18,
K(n)=44(18)
2[18]-44=-8
S=4n-16
K=2n+8
Kp=C9C[M9]
K*=C9+D9
A tri-capped trigonal prism capped nine times.
K*=Cy+Dz
VE0=2z+2=2[9]+2=20
VEn=VE0+12n=20+12[18]=236
VF=12[10]+6[11]+24[2]+2=236
27. Pt19(CO)24(AuL)3-1:K=19[4]-24[1]+3[3.5-1]-1[0.5]=59
n=19+3=22
K(n)=59(22)
2[22]-59=-15
S=4n-30
K=2n+15
Kp=C16C[M6]
K*=C16+D6
An octahedron capped 16 times.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[22]=278
VF=19[10]+24[2]+3[11+2]+1=278
28. Fe10Au21(CO)405-:K=10[5]21[3.5]-40[1]-5[0.5]=81
n=10+21=31
K(n)=81(31)
2[31]-81=-19
S=4n-38
K+2n+19
Kp=C20C[M11]
K*=C20+D11
An octadecahedron capped 20 times.
VE0=2[11]+2=24
VEn=VE0+12n=24+12[31]=396
VF=10[8]+21[11]+40[2]+5=396
29. Pd28Au4(CO)22L16:K=28[4]+4[3.5]-22[1]-16[1]=88
n=28+4=32
K(n)=88(32)
2[32]-88=-24
S=4n-48
K=2n+24
Kp=C25C[M7]
K*=C25+D7
A pentagonal bipyramid capped 25 times.
VE0=2[7]+2=16
VEn=VE0+12n=16+12(32)=400
VF=28[10]+4[11]+22[2]+16[2]=400
30. Fe12Au22(CO)486-:K=12[5]+22[3.5]-48[1]-6[0.5]=86
n=12+22=34
K(n)=86(34)
2[34]-86=-18
S=4n-36
K=2n+18
Kp=C19C[M15]
K*=C19+D15
VE0=2z+2=2[15]+2=32
VEn=VE0+12n=32+12[34]=440
VF=12[8]+22[11]+48[2]+6=440
31. Ni32Au6(CO)446-:K=32[4]+6[3.5]-44-3=106;
N=32+6=38
K(n)=106(38)
2[38]-106=-26
S=4n-52
K=2n+26
Kp=C27C[M11]
K*=C27+D11
An octadecahedron capped 27 times.
VE0=2z+2=2[11]+2=24
VEn=VE0+12n=24+12[38]=480
VF=32[10]+6[11]+44[2]+6=480
32. Fe14Au28(CO)528-: K=14[5]+28[3.5]-52[1]-8[0.5]=112
n=14+28=42
K(n)=112(42)
2[42]-112=-28
S=4n-56
K=2n+28
Kp=C29C[M13]
K*=C29+D13
A centered icosahedron capped 29 times.
VE0=2[13]+2=28
VEn=VE0+12n=28+12[42]=532
VF=14[8]+28[11]+52[2]+8=532
33. Fe14Au34(CO)506-:K=14[5]+34[3.5]-50[1]-6[0.5]=136
n=14+34=48
K(n)=136(48)
2[48]-136=-40
S=4n-80
K=2n+40
Kp=C41C[M7]
K*=C41+D7
A pentagonal bipyramid capped 41 times.
VE0=2[7]+2=16
VEn=VE0+12n=16+12[48]=592
VF=14[8]+34[11]+50[2]+6=592
34. Co11(C)2(CO)23-1:K=11[4.5]-2[2]-23[1]-1[0.5]=22,n=11
K(n)=22(11)
2[11]-22=0
S=4n+0
Kp=C1C[M10]
K*=C1+D10
A mono-capped bi-capped square anti-prism.
VE0=2[z]+2=2[10]+2=22
VEn=VE0+12n=22+12[11]=154
VF=11[9]+2[4]+23[2]+1=154
The symbol K* = C1+D10 predicts one skeletal element capping other 10 skeletal elements. This is shown in Figure 20 and is in agreement with what was reported (Ciabatti, 2015).
Figure 20. Isomeric skeletal structure of Co11C2(CO)23-1
35. Ru6(B)(CO)17(AuL):K=6[5]-1[1.5]-17(1)+1[3.5-1]=14
n=6+1=7
K(n)=14(7)
2[7]-14=0
S=4n+0
K=2n+0
Kp=C1C[M6]
K*=C1+D6; predicts a mono-capped octahedral symmetry. This is shown in F-2.
VE0=2[6]+2=14
VEn=VE0+12n=14+12[7]=98
VE=6[8]+1[3]+17[2]+1[11+2]=98
Figure 21. Observed isomeric structure of Ru6(B)(CO)17(AuL)
36. Ir6(B)(CO)14(AuL):K =6[4.5]-1[1.5]-14[1]+1[3.5-1]=14
n=6+1=7
K(n)=14(7)
2[7]-14=0
S=4n+0
K=2n+0
Kp=C1C[M6]
K*=C1+D6: predicts a mono-capped octahedral symmetry and this is observed (Ciabatti, 2015).
VE0=2[6]+2=14
VEn=14+12n=14+12[7]=98
VF=6[9]+1[3]+14[2]+1[11+2]=98
Figure 22. Observed isomeric skeletal structure of Ir6(B)(CO)14(AuL)
37. Ru5Pt(C)(CO)15(AuL)2:K=5[5]+1[4]-1[2]-15[1]+2[3.5-1]=17
n=5+1+2=8
K(n)=17(8)
2[8]-17=-1
S=4n-2
K=2n+1
Kp=C2C[M6]
K*=C2+D6; predicts a bi-capped octahedral symmetry and this is observed.**
VE0=2[6]+2=14
VEn=VE0+12n=14+12[8]=110
VF=5[8]+1[10]+1[4]+15[2]+2[11+1]=110
The observed skeletal structure is given in Figure 23.
Figure 23. Observed isomeric skeletal structure of Ru5Pt(C)(CO)15(AuL)2
Figure 24. Isomeric skeletal Ir6(CO)15(AuL)2
Other known isomeric skeletal bimetallic structures of gold are as shown below.
Figure 25. Other observed isomeric bimetallic skeletal structures of K* = C2 + D6
Figure 26. Isomeric skeletal structure of Ru4Rh2(B)(CO)15(AuL)3
Figure 27. Isomeric skeletal structure of Ru6(B)(CO)16(AuL)3
Figure 28. Observed isomeric skeletal structures of Co6(C)(CO)12(AuL)4, L = PPh3
The Concept of a Cluster Nucleus and the Double Capping Phenomenon K* = Cy+Dz
A wide range of bi-metallic gold clusters have been analyzed and categorized
As indicated above, 41 examples have been worked out using skeletal numbers. The categorization parameters K* derived from the calculations using skeletal numbers are provided as illustrations. The clusters have been organized according to the increasing magnitude of the clan parameter Dz, that is, increasing magnitude of z index. The results are summarized in Table 5. The categorization parameter is very useful as the number of skeletal elements involved can be derived, as well as their cluster clan, the series equation and the cluster valence electrons VE can also be calculated. In addition, the cluster geometry can tentatively be predicted. This information is also summarized in Table 5. The double capping nature of clusters is also reflected in Tables 5-6. The cluster clans range from D1 to D20. The matryoshka with K* =C13+D20 (Huang, etal, 2014; King & Zhao, 2006) is a special case. The graphed hypothetical isomeric structures of selected Dz fragments (z=1-15 and 20) are given in Figure 29. According to the 4N series approach, the Dz fragments, in principle, correspond to the closo series BnHn2―(n=1,2,3,4,5,6,7,8,9,etc). They represent the fundamental units around which clan series are based. In other words, they form the nuclei of clan clusters.
Figure 29. Isomeric graphical structures D1-D15 and D20
BRIEF SUMMARY OF THE ANALYSIS OF THE CATEGORIZED CLUSTERS
All the thirty-five bimetallic golden clusters with nuclearity index ranging from 7-42 analyzed were capped capping index ranging from C1 to C29. It was found earlier that golden clusters have a have great tendency towards capping usually centered around D1 or D2 nucleus (Kiremire, 2017f, 2018d). This capping tendency appears to be portrayed in the bimetallic golden clusters as well. The predicted or observed isomeric skeletal structures of selected clusters are given in Figures 2-28. The nuclearity index in the capping bimetallic golden clusters ranged from D5-D15. These results are given in Table 5. According to the 4N series approach, the Dz fragments form the nuclei of the clusters. Selected isomeric graphical skeletal structures of Dz, z=1-15 and z=20 are shown in Figure 29. The formation of clusters can be viewed as a process of adding unitary fragments on to an appropriate CLUSTER VELENCE GENESIS ELECTRONS VE0=2z+2. For instance, z=1, VEO=4. This series will generate the D1 clan series, z=2, VE0=6, will give us the D2 clan series, z=3, VE0=8→D3 clan series, VEO=10→D4, VE0=12→D5, VEO=14→D6 series and so on. It has been observed that clusters or fragments with Dz where z<0, have metallic characteristics. This aspect has been covered mainly for a limited range transition metal clusters (Kiremire, 2018f). In this paper the concept has been applied to more than 200 clusters including many clusters from the main group elements and the skeletal elements of the periodic table excluding lanthanides (4f) and actinides (5f). This wide range of clusters are given in Table 6A. The clan series range from D-6 to D20 and the family series range from C-3 to C50. The negative family capping index (C-y) means that the cluster is below the CLOSO level whereas a positive capping index (Cy) means the cluster family is beyond the CLOSO level. Different cluster clans have been demarcated in the Table 6A with colored highlighters.
Table 6A. Clusters arranged according to clan D (clan) series
     
Table 6B. Matryoshka Clusters
     
SOME CHARACTERISTICS OF CHEMICAL CAPPING SERIES
There are some notable characteristics of the capping chemical clusters which are arranged according to Dz series. Among others, these include the inherent Rudolph (Rudolph, 1976) sequence of series (∆K=3, ∆n=1), the cluster valence electrons, CVE being EVEN NUMBERS except the comparatively few mono-skeletal elements with odd number electrons. If we observe carefully, also Wade-Mingos rules (Wade, 1976; Mingos, 1972) are well accommodated for a wide range of clusters. In the case, of the 4N series method with its skeletal numbers, all from mono-skeletal to multi-skeletal elements can readily be categorized except lanthanides and actinides as demonstrated in Tables 6A and 6B.
MATRYOSHKA AS A SPECIAL CASE OF DOUBLE CAPPING NATURE OF CLUSTERS
The matryoshka clusters belong to the special clusters series S=4n-24, K=2n+12, K= C13C[M20], K* =C13+D20. This symbol predicts that 20 skeletal elements will be surrounded by 13 capping skeletal elements. In actual fact, what is observed is just the opposite (Huang, et al, 2014; Xiang, et al, 2015) and the C13 is found to be a centered icosahedron. A sample matryoshka clusters both hypothetical and known ones which was published earlier is reproduced here comparative purposes in line with double capping concept of clusters (Kiremire, 2018c).
CLUSTER VALENCE ELECTRONS AND CLUSTER NUMBER OCCURING AS EVEN NUMBERS
It has also been observed that in general cluster valence electrons are associated with even numbers and that the cluster number is usually a whole number. Thus, in a cluster the skeletal elements must combine in such a way that the resultant cluster number is a whole number. As is indicated in Table 1, the K value occurs in multiples of 0.5. Let us use boron as an illustration. Boron has a K value of 2.5 and since the cluster number K has to be whole number, that is why in F= BnHm, the fragment Bn when n=even, m must be even and when n=odd, m must have an odd value (Housecroft & Sharpe, 2005). Thus, the boranes have formulas such as B2H4, B2H6, B3H5, B3H7, B3H9, B4H6, B4H8, B4H10, B5H7, B5H9, B5H11, B6H8, B6H10, B6H12, B6H14, B6H16, B7H9, B7H11, B7H13, B7H15, B8H10, B8H12, B8H14, B8H16, B9H11, B9H13, B9H15, B9H17, B10H12, B10H14, B10H16, B11H13, B11H15, B11H17, B12H14, B12H16 and B12H18.
Skeletal numbers and their skeletal valences are so flexible in constructing isomeric graphical structures as illustrated in Figure 30.
Figure 30. Isomeric graphical structures of structures of B6H10

4. Structural Prediction

The 4N series can act as a guide in structural prediction. The details of the geometrical shape can be ascertained by X-ray analysis. The categorization parameter K*=Cy+Dz recently developed can act as a guide for small values of y=0-12 and z=1-13. For example F= Fe6Pd6H(CO)243―:K=6[5]+6[4]-0.5-24-1.5=28, K(n)=28(12), 2[12]-28=-4, S=4n-8, K=2n+4, Kp=C5C[M7], K*=C5+D7, VE0=2z+2=2[7]+2=16, VE12=VE0+12n=16+12[12]=160, F=6[8]+6[10]+1+48+3=160. Since the capping theory gives us CVE=160 which is the same as obtained from the cluster formula, clearly vindicates its validity. According to the 4N series approach, the cluster is a penta-capped pentagonal bippyramid. Another similar example is Ru6Pd6(CO)242―:K=6[5]+6[4]-24-1=29, K(n)=29(12), 2[12]-29=-5, S=4n-10, K=2n+5, Kp=C6C[M6], K*=C6+D6, VE0=2z+2=14, VE12=VE0+12n=14+12[12]=158, VF=6[8]+6[10]+48+2=158. This example also shows that the capping model is correct. The cluster can be described as an octahedral fragment surrounded by six capping elements or the cluster is a hexa-capped octahedron. As for giant clusters with large nuclearity index, we can only describe them in terms of the nuclear size (z) and the number capping elements. A good example is HNi38Pt6(CO)485―; K*=C38+D6, which we can describe as a cluster with an octahedral shaped nucleus surrounded by 38 capping elements. This prediction is what is actually observed (Rossi & Zanello, 2011). What is interesting in this cluster is that all the octahedral fragment comprises of six platinum elements only. This type of predicting the shapes is what is reflected in Tables 5, 6A and 6B.

5. Conclusions

A wide range of clusters have been categorized using skeletal numbers. Categorization of each cluster has been expressed in the form K*=Cy+Dz. It has been proposed that the symbol K* be referred to a categorization parameter. Clearly, arranging chemical clusters according to clan series (Dz) and family series (Cy) is easier and faster. The genesis electrons for any Dz clan tree are given by VE0 =2z+2. The six fundamental equations for calculating cluster valence electrons (CVE) have been demonstrated. Despite the fact that only more than 200 clusters have been categorized, the method indicates that all elements of the periodic table and their clusters except lanthanides and actinides, can readily be arranged according to Dz and Cy series. This tremendously underpins the great insight of Rudolph of more than 40 years ago (Rudolph, 1976).

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