A Hypothetical Model for the Formation of Transition Metal Carbonyl Clusters Based upon 4 n Series Skeletal Numbers

Skeletal numbers of elements have been introduced as derivatives of the 4n series method. They are based on the number of valence electrons present in the skeletal element. They are extremely useful in deducing possible shapes of skeletal elements in molecules or clusters especially the small to medium ones. For large skeletal clusters, the skeletal numbers may simply be regarded as identity numbers. In carbonyl clusters, they can be used as a guide to facilitate the distribution of the ligands such as CO, H and charges onto the skeletal atoms. A naked skeletal cluster may be viewed as a reservoir for skeletal linkages which get utilized when ligands or electrons get bound to it. The sum of linkages used up by the ligands bound to a skeletal fragment and the remaining cluster skeletal numbers is equal to the number of the skeletal linkages present in the original „naked parent‟ skeletal cluster. The skeletal numbers can be used as a quick way of testing whether or not a skeletal atom obeys the 8-or 18-electron rules.


Introduction
The recently developed 4n series method has been found to analyze and categorize atoms, molecules, fragments and clusters (Kiremire, 2016a(Kiremire, , 2016b(Kiremire, , 2016c)).The method highly complements Wade-Mingos rules which have been in existence for more than forty years (Wade, 1991;Mingos, 1972Mingos, , 1984Mingos, , 1991)).Other methods for dealing with electron counting in clusters have been devised (Lipscomb, 1976;King, 1986aKing, , 1986b;;Jensen, 1978;Teo, et al, 1984;Wales, 2005;Wheeler and Hoffmann, 1986;Jemmis, et al, 2000Jemmis, et al, , 2001aJemmis, et al, , 2001b)).However on closer scrutiny of the 4n series method, it has become apparent that we could go further and assign skeletal linkage numbers to elements and ligands which greatly simplifies the prediction of structures of molecules and clusters.The skeletal numbers are especially very helpful in assigning a specific number of carbonyl ligands to skeletal metal elements and hence generating carbonyl cluster isomers.Furthermore, it makes it easier to deduce whether or not clusters obey the 18-electron rule.The observation of behavior of k values in the hydrocarbons involving the addition of hydrogen atoms to skeletal carbon fragments (Kiremire, 2016e) have triggered the need to observe the behavior of k values by adding the carbonyl (CO) ligands to transition metal skeletal fragments.The impact of this work has been to introduce the concept of assigning skeletal numbers to the atoms of the main group elements and the transition metals.

Assignment of Skeletal Linkages (K Values) to Elements
The procedure for categorization and structural prediction of fragments, molecules and clusters using the 4n series method is now well established (Kiremire, 2016a(Kiremire, , 2016b(Kiremire, , 2016c)).What is more interesting and exciting is that on closer scrutiny of the 4n series method is that skeletal elements and ligands can actually be assigned skeletal k values.For instance, the single carbon atom 1[C], with valence electron content of four [4], S=1[4+0]=4n+0(n=1), and k=2n(n=1)=2.Hence, a carbon atom is assigned a k value of 2. For the nitrogen atom, N with valence electron content of 5, S=4n+1 (n=1) and k=2n-0.5=1.5 while boron, B with valence electrons 3, S=4n-1(n=1) and k =2n+0.5=2.5.The assigned k values of the main group elements are given in Table 1.In the case of transition metals, the S = 14+q is taken as equivalent (Kiremire, 2015a;Hoffmann, 1982) to S = 4n+q.The k values of transition metals are given in Table 2. Furthermore, the k values of naked metal clusters from 1 to 10 for first row transition metals are given in Table 3.The addition of a hydrogen atom to a carbon atom produces the fragment [CH] which has 5 valence electrons like a nitrogen atom [N].Hence its series is given by S = 4n+1 and the k value will also be given by k=2n-0.5=2(1)-0.5=1.5.But the carbon atom [C] belongs to S =4n+0 with k=2n+0=2(1)+0=2.This means that a simple operation of C(k =2) + H → CH (k=1.5)results in the decrease of k value by 0.5.Hence, it makes sense if we could assign a value of k=-0.5 to a hydrogen atom (H) ligand.Structure of benzenehexacarbonitrile BC-1 The chemical fragments C 2 , CN + , BN, and CB -have been shown to possess quadruple bonds (Shaik, et al, 2012) by high level computations although the concept is still controversial.However the skeletal numbers derived from the 4n series approach agree with their results.Also most of the bond orders of chemical fragments obtained from molecular orbital energy level diagrams (Housecroft, et al, 2005) agree with the k values obtained from skeletal numbers of atoms.For the diatomic fragments, the k value obtained is simply the same as the bond order.

Fe Skeletal Element
Let us illustrate this by successive addition of CO ligands to Fe skeletal element.This summed up in Scheme 1.What happens to the k value when the CO ligands are step-by-step added to a single metal atom?Let us use Fe (S=4n-6) atom again as our illustration.The Fe atom belongs to the series S=4n-6, k=2n+3=2(1)+3=5.Addition of the first∶CO ligand, we get the Fe(CO) fragment.This can be expressed by a simple equation Fe + CO→ Fe(CO).Since in 4n series, we are dealing with valence electron content, the:CO ligand contributes two more electrons to the series S =4n-6 +2→S=4n-4.This means that we get the fragment Fe(CO) which belongs to the series S = 4n-4 and k=2n+2 =2(1)+2=4.Thus, the k value of Fe (k=5) has decreased to Fe(CO)(k=4).Further addition of: CO ligand, we get another fragment Fe(CO) 2 (S=4n-2, k=2n+1=3).The next fragment becomes Fe(CO) 3 (S=4n+0, k=2n=2).This will be followed by Fe(CO) 4 (S=4n+2, k=2n-1=2(1)-1=1).The last fragment will be Fe(CO) 5 (S=4n+4, k=2n-2=2(1)-2=0).Clearly the k value of 5 for the Fe skeletal atom implies the number of the skeletal coordinate bonds or electron pairs to be received from the ligands to form coordinate bonds so as to enable the Fe skeletal atom attain the 18-electron rule.Since the Fe atom has 8 valence electrons, it makes sense that it requires additional 5 pairs (10 electrons) electrons so as to obey the 18-electron rule.We know that the addition of one CO ligand decreases the skeletal k value by 1, therefore we could assign a value of k=-1 to the CO ligand.In this way, the CO ligand or any other ligands may be regarded as "neutralizing agents" of the cluster skeletal bonds or linkages of the original naked parent skeletal fragment.Each electron provided by a ligand or a charge, neutralizes the skeletal linkages by k value of 0.5 as deduced from the 4n series.That is, a single electron reduces the skeletal value of a cluster by 0.5.This implies that we could as well assign a hydrogen atom (H) ligand, a value of k=-0.5.

The Following Observations Are Noted Regarding the Hypothetical Model
 For every addition of 1 [:CO] ligand to a fixed naked metallic fragment, the skeletal k value decreases by 1.
 The cluster series last digit (determinant) increases by 2 due to the 2 electrons donated by the CO ligand.
 The capping of the series decreases more and more.
 The capping ends at S = 4n+0 which represents mono-capped series.The series also represents the carbon clusters, C n ; n=1→C 1 , n=2→C 2 , n=3→ C 3 and so on.
 Complete fragmentation of the naked cluster takes place when k = 0 and all the initial skeletal linkages have been consumed by the CO ligands.For example in the case of the Cr 2 (K=12) skeletal fragment, if we add 12CO ligands we get Cr 2 (CO) 12 ,then k value of the cluster attains the value of k=0 and Cr 2 (CO) 12 and hence the cluster decomposes into two fragments as follows, Cr 2 (CO) 12 → 2[Cr(CO) 6 ].
 Also a very important observation is made, that is, the sum of the ligands on the cluster (corresponding to the utilized k values) and the cluster linkages (corresponding to the unutilized k values) present is equal to the original k value of the parent naked metallic cluster.This point is discussed in more details under the heading Fundamental Principle.

Limits of the Carbonyl Cluster Series
The carbonyl cluster complexes may be regarded as being formed by adding the carbonyl ligands stepwise to a metallic cluster fragment.This may be expressed by a simple equation below.
The naked skeletal fragment, M x possesses a fixed number of skeletal linkages which can readily be determined by 4n series based on its valence electrons.These linkages are neutralized one by one for every addition of the CO ligand.If we represent the series as S = 4n+q, then when q≤0 the "metallic character" increases as the CO ligands are removed, the "metallic character" of the fragment decreases with the addition of the CO ligands until the series becomes 4n+0.This series represents the carbon cluster family, F=C n (n=1, 2,3,4,5, 6,…).The large carbon clusters such as C 60 (fullerenes) and C 70 are members of the carbon clusters which belong to the series S=4n+0.The 4n+0 series is the borderline which may be regarded as the highest level of the "hydrocarbon" series but also as the beginning of the "metallic" series.It is interesting to note that carbon has vast industrial applications due its unique properties.The "hydrocarbon" type series can be expressed by the series S=4n+q (q≥0).We can also regard the series range such as S=4n+2, 4n+4, 4n+6, and so on as indicating the increase in the "hydrocarbon character" of the carbonyl clusters despite the inclusion of the metallic skeletal elements.Let us consider the following changes in metallic fragments of chromium.
Cr→S=4n-8, k=2n+4 (n=1)=6, Cr [S=4n-8, k=6]+CO(k=-1)→Cr(CO)[S=4n-6,k=5]→Cr(CO) ]+CO(k=-1).Thus, the cluster series S=4n-8, 4n-6, 4n-4, 4n-2 and 4n+0 associated with Cr fragment may be regarded as having some type of "metallic character".On the other hand the series S = 4n+0, 4n+2, and 4n+4 which correspond to the fragments C, CH 2 and CH 4 may be regarded as having some type of "hydrocarbon character".Hence, the S= 4n+0 series is the borderline between metallic type and hydrocarbon type of series.In fact, S = 4n+0 is refers to the clusters or fragments which are referred to as being mono-capped (Kiremire, 2015a.Other series of fragments may be interpreted in the same way.A good example to illustrate the hydrocarbon character of series is [H 5 Re 6 (CO) 24 ] -complex.Using the skeletal numbers, the k value of the cluster is given by k=6(5.5)+5(-0.5)+24(-1)+1(-0.5)=33-2.5-24-0.5=6.Since for the series S=4n+q, k=2n-q/2, then k=6=2n-q/2(n=6); 6=2(6)-q/2, q=12.Hence, S=4n+12(n=6)→C 6 H 12 .This implies that the shape of [H 5 Re 6 (CO) 24 ] -complex will be similar to one of the isomers of C 6 H 12 .This is found to be the case (Housecroft and Sharpe, 2005 and the ideal shapes are sketched in Figure 1.The reverse of the hypothetical formation of carbonyl cluster may be represented as: M x (CO) y →M x + yCO.This involves the removal of CO ligands from the original cluster.In this manner, the cluster may be viewed as going from a hydrocarbon-type to a more metallic type of cluster fragments.Ideally, this should give rise to the recovery of naked parent metallic fragment M x .The corresponding cluster k values of the fragment should be on the increase.Very interesting extensive work on the metal carbonyls that involves stripping off the CO ligands has been done by several research groups (Butcher, et al, 2002(Butcher, et al, , 2003;;Crawford, et al, 2006;Critchley, et al, 1999;Dyson, et al, 2001;Henderson, et al, 1998Henderson, et al, , 2009)).When the structure of the series is carefully analyzed, the hydrocarbon chain type terminates or reaches a saturation point or limit that corresponds to the corresponding hydrocarbon alkane series F=C n H 2n+2 .Hence, for M 1 →CH 4 →4n+4, and if M=Fe, then the complex will be Fe(CO) 5 .For M    Although the series and the skeletal numbers predict that each of the Re skeletal atoms except one should have a hydrogen atom, the structural determination indicates all the hydrogen atoms are bridging (Miessler, et al, 2014) as observed in borane clusters.

The Conservation of Cluster Skeletal Linkage Content Principle
A naked transition metal element possesses an inherent number of skeletal linkages by virtue of its valence electrons.
When the skeletal element reacts with suitable ligands such as CO and H some or all the linkages are used up, the fragment develops a tendency towards the attainment of the ultimate 18-electron rule.The 18-electron rule implies the maximization of all the atomic orbitals of the orbital set [s(1),p(3)and d( 5)] (Pauling,1977).Since 1[:CO] neutralizes 1 skeletal k unit, it is proposed that we assign it a value of k=-1.We also know that it donates 2 electrons, and so every ligand donor of 2 electrons may similarly be assigned a k value of -1 and for 1 electron donor such as H and Cl or a unit negative charge ligands are correspondingly assigned a numerical value of k=-0.5.As can be seen from Table 9, |k L |+k S =k T where k L represents the used up k-values of the initial skeletal fragment, k S = the skeletal linkages still available and k T = the original skeletal linkages of the parent naked fragment.This result is very important because if we know the value of k T and |k L |, then we can deduce the value of k S and hence use it as a guiding tool in designing and predicting the shape of the carbonyl cluster.This important principle is hereby expressed in an equation form.This relationship is well illustrated in Table 9.

Possible Shapes of the Skeletal Linkages
The main concept is that a skeletal atom possesses naturally inherent linkages as deduced by 4n series.When a ligand is attached to the skeletal atom, it utilizes or neutralizes some of those linkages depending upon the number of electrons that ligand donates to the skeletal element.For every electron donated by the ligand or a negative charge, 0.5 of the linkage is utilized or removed from the skeletal element.Since a:CO ligand donates 2 electrons, it utilizes or removes 1 skeletal value from the element.On this basis, a CO ligand is assigned a value of k=-1.Similarly we can assign H• atom a value of k=-0.5 and one negative charge, k=-0.5.The implications of this is that for 1 Fe(k=5) when combined with 5CO[ k=5x(-1)=-5] ligands, the net k value of the cluster becomes zero (k = 0).Hence Fe(CO) 5 , k =0.The k values for transition elements may be regarded as the number of electron pairs needed for the element to obey the 18 electron rule.Hence Sc, k=7.5 pairs=7.5 x2=15 electrons required for it to obey the 18 electron rule.Accordingly, the other transition metal elements in the same period will require the following electrons, Ti(k=7), 14 electrons; V(k=6.5),13; Cr(k=6), 12; Mn(k=5.5),11;Fe(k=5), 10; Co(k=4.5),9; Ni(k=4), 8; Cu(k=3.5),7 and Zn(k=3),6.We can also tentatively assign possible shapes of the skeletal linkages to individual skeletal atoms.This is shown in Figure 2.For complexes with two or more skeletal elements, the skeletal numbers can be used to determine the k value of the cluster and hence the possible skeletal shape.Take the example of Re 4 (CO) 16 ; Re(k=5.5),CO(k=-1), 1charge(k=-0.2).Therefore, the total skeletal linkages of the Re atoms=4(5.5)=22.These have to be "neutralized" by the ligands and the charge present.The remaining ones will constitute the skeletal bonds or linkages which are remaining to bind the skeletal elements.Hence k value of the complex will be given by k=22+16(-1)+2(-0.5)=22-17=5.The possible skeletal ideal shape of one of its isomers is shown in the labeled example 1(Ex-1) below.A possible ideal isomer shape is a square or rectangle with a diagonal.Using the labeled diagram as a basis, we can also use the skeletal numbers to deduce the possible number of CO ligands needed to complete the remaining skeletal linkages on the skeletal atom so as to enable it fulfill the 18-electron rule.The skeletal linkages available are given the labels k1 to k4.From the sketch, the atom labeled 1, has two bonds connected to it.This means it is receiving one electron donation from each of the bonds linked to it.These two electrons will neutralize a k value by 2(-0.5)=-1.Hence k1 =5.5-1=4.5.This means that atom 1 will have 4.5 CO ligands.In essence, there will be 4 CO ligands and the fractional component will represent one of the negative charges.Other k values, k2=4, k3=k1=4.5 and k4=4 were similarly calculated.In this way a possible isomer of the cluster can be sketched.This is also shown in Ex-1 below.More examples (Ex-2 to Ex-4) are well explained and provided.As the use skeletal numbers as a concept to predict possible shapes of clusters is being introduced for the first time, more well explained examples have been worked out.These are given in Schemes 4-16.

K-Isomerism of Clusters
Let us take the cluster Rh 6 (CO) 16 as an illustration.The k value for the octahedral cluster is 11.Using skeletal numbers tentative distribution of carbonyl ligands on skeletal rhodium atoms can be sketched.Some of the selected isomers are given in Figure 3.The calculated k values on each rhodium atoms indicate the number of carbonyl ligands that can be accommodated according to 4n series approach using the skeletal numbers.Isomer-1 has already been given in F-9b.It is given here for comparison purposes.If the carbon atom is taken as a skeletal atom, then kS = 4(5)+1( 2)-12-1 =22-13 = 9.This means that the 5 skeletal atoms are linked by 9 lines.The sketch appears as in F-7b.The tentative distribution of carbonyl ligands is shown in F-7c.We have learnt that the series S = 4n+q has a corresponding k value given by k=2n-q/2.Since there are 5 skeletal elements if we include the carbon atom, then n=5 and k=9.Hence, k=9=2(5)-q/2; q/2=10-9=1.Therefore, q=2.
Hence the cluster series S=4n+2.This is CLOSO cluster.
The modified skeletal shape will be as in shown in F-8b.The dotted line is not used in the calculation since we are using k =11 for an Oh symmetry.
Final sketch of one of the possible isomer is given in F-9b.
Scheme 18.Using skeletal numbers to derive series and valence electrons

Conclusion
A skeletal transition metal atom possesses inherent skeletal linkages.The linkages are derived from the valence electrons of the element.They correspond to the number of pairs of electrons needed to enable the metal atom obey the eighteen electron rule.The k-values derived are as follows: Group 3, Sc family, k=7.5;Group 4, Ti family, k=7.0;Group 5, V family, k=6.5;Group 6, Cr family, k=6; Group 7, Mn family, k=5.5;Group 8, Fe family, k=5, Group 9, Co family, k=4.5, Group 10, Ni family, k=4, Group 11, Cu family, k=3.5 and Group 12, Zn family, k=3.Ligands have been assigned negative k values as deduced from the 4n series.It is proposed that a single electron donor be assigned a k value of -0.5 and a two electron donor k=-1.The use of skeletal numbers greatly facilitates the categorization of simple to medium large clusters in a simple manner.Furthermore, it is possible to predict the shapes of some clusters.The skeletal numbers can also be utilized as a guide to assigning the ligands and charges to specific skeletal elements of clusters.The method makes the testing of the 18-electron rule, the understanding of some catalytic processes and the isolobal principle much easier.The skeletal values which have now been introduced for the first time in chemistry and the atoms of the main group and transition metal elements can be arranged into groups based on k values.Nearly 80 clusters of different types have been analyzed using skeletal numbers to demonstrate the ease and flexibility of applying skeletal numbers.This paper introduces a fundamental principle of viewing a naked skeletal cluster of elements as being a reservoir of inherent skeletal linkages which are subject to change when it is gets bound to electron donor ligands.The observed linkages or bonds are just remnants of those skeletal linkages which were not utilized by the ligands.This could be viewed as a form of conservation of skeletal cluster linkages.

Figure 2 .
Figure 2. Proposed tentative shapes of the skeletal linkages of first row transition metals 2.7.2 Shapes of Clusters Scheme 6. Derivation of Fe2(CO)9 structure using skeletal numbers Scheme 10.Derivation of the structure of Fe4(C)(CO)12 2-using skeletal numbers Scheme 11.Transforming the k value of a given cluster into series Scheme 12. Derivation of structure of Fe5(C)(CO)15 cluster using skeletal numbers Scheme 14. Derivation of structure of Mo2(Cp)2(CO)4 using skeletal numbers

Table 1 .
Skeletal Values of the Main Group Elements

Table 3 .
Skeletal Values of Selected Naked Skeletal Clusters of First Row Transition Metals

Table 4 .
The k Values of Cluster Fragments Generated by adding Carbonyl Ligands to Selected

Table 5 .
The k Values of Cluster Fragments Generated by adding Carbonyl Ligands to Selected Bi-skeletal Transition x (x>2) Systems M 3 Systems are treated in the same way as in M 2 systems.

Table 6 .
The k Values of Selected Transition Metal Carbonyl Fragments (M x , x = 3-6)

Table 7 .
The k Values Generated by Adding CO ligands to a Large Naked Metallic Fragment

Table 8 .
The Capping Series obtained from Stripping Pd 23 (CO) 46 Cluster

Table 9 .
Examples to Illustrate the Principle of Conservation of Skeletal Value Content of the Naked Parent Cluster Fragment

Table 10 .
Derivation of 4n Series and Valence Electron Counts Using the k-Values of Clusters Cluster k Value q value Series (S) Category n value Borane equivalent Valence Electrons Fe 4 (C)(CO) 12