Isolobal Series of Chemical Fragments

Introduction 

 The arrival of Hoffman’s isolobal principle greatly assisted the advancement of chemical knowledge of linking up the inorganic, organic and organometallic fields^1-2.^. A combination of isolobal principle and Wade-Mingos rules^3-4 laid a strong base for cluster chemistry of boranes, carboranes and transition metal carbonyls. It has recently been found that the organic, and many inorganic  and organometallic compounds and clusters follow the 4n and 14n series^5-6. In continuing to examine the application of the series in analyzing chemical structures, it has been discovered that the isolobal principle also follows the 4n and 14n series.  This revelation is presented in this paper.

Results and Discussion   

In the study of osmium carbonyl clusters⁷, it was found that some of the series follow a pattern in which one carbonyl ligand is removed at every step until a given cluster is completely stripped naked. In the Hoffmann’s isolobal principle the initial fragment molecule used is methane, CH₄. The stripping off one H atom creates CH₃ fragment. The next stages produce the fragments CH₂, CH and finally the naked C is the final product. Using the 4n rule of classification^1-2 the CH₄ molecule or fragment belongs to 4n+4 series, while the CH₃  belongs to 4n+3 series, CH₂ is a member of 4n+2 series, CH  is a member of 4n+1 series and C belongs to 4n series. The series give the value of the electron content of the fragment. For example in CH₄ , n = 1, the series value S = 4n+4 = 4×1+4 = 8, CH₃, S = 4n+3, n =1, S = 7, CH₂ , S = 4n+2, n= 1, S = 6, CH, S = 4, and C, S=4. In terms of series, what is found interesting is that these values correspond to the respective valence content of the fragments.  The CH₃ fragment was found to be isolobal with Mn(CO)₅ fragment. Also applying the 14n rule of classification, Mn(CO)₅ fragment belongs to 14n + 3 series. The 14n +3 series and 4n+3 series while they are not identical, they are similar. If n =1 for the fragment Mn(CO)₅, then S = 14n+3 = 14×1+3 =17, the valence content of the fragment (7+5×2=17). When 10 n is subtracted from 14n+3 it becomes 4n+3. This means if 10n = 10×1 = 10 is removed from 17, then 7 is left which is the same as the valence content of CH₃. Thus, the two fragments CH₃ and Mn(CO)₅ are interrelated by series. Accordingly, the fragments are truly  isolobal CH₃ ⥈ Mn(CO)₅. In this paper, 14n + q and 4n+q will be used interchangeably as needed.

Table 1: Classification of Isolobal Fragments into Series  Click here to View table

 

The Recreation of the Periodic Table  

What is fascinating and exciting is that the use of 14n and 4n rule the Periodic Table of Elements and Fragments has been created. For the ordinary elements of the periodic table they have been placed precisely into their usual places while the isolobal fragments have been classified and placed side by side with the ordinary elements into equivalent periodic groups. This table has implications. For example, if CH fragment may be regarded as equivalent to N atom, it implies that the H atom acted as an electron donor to the  the carbon atom(C) making it have  5 valence electrons in the (CH) fragment like the nitrogen atom (N) and hence placing it into group 5 or 15 like the nitrogen atom. In the same manner, in the fragment [Cr(CO)] when the CO ligand donates the two electrons to Cr which is in group 6 of the transition elements it makes it behave as if it is an iron atom(Fe) atom of group 8. On the basis of the series, it appears that the basis of the isolobal principle is the electron content of the sum valence electrons around the skeletal atom. In the case of CH₃ fragment, the carbon has 4 valence electrons and the three hydrogen atoms donate three more electrons making the carbon atom have a surrounding environment of 7 valence electrons. Thus, it acquires the status of a group 7 element such as fluorine. Group 7 has other family members such as Cl, Br, and I. Moving along the period from C the next element is nitrogen atom(N). The addition of 2H atoms converts it NH₂ which also has a total valence content of 7. The NH₂ fragment will also be isolobal to CH₃. Since N is in group 5, other family members namely, PH₂, AsH₂, SbH₂, and BiH₂ will also be isolobal to CH₃. In case of group 17, those elements with carbonyls will also be isolobal to CH₃. Since we are actually comparing  parallel series 14n+3 with 4n+3, if we remove 10n from 14n+3 we will convert the carbonyl series to main group series of 4n+3. Futhermore, since we are dealing with a single skeletal element, then 10n is equivalent to 10×1 = 10. Hence, if we subtract 10 from 17 we will get 7 the valence electron content of CH₃ fragment and group 7 elements.  The carbonyl fragments include the hypothetical  and real ones  such as V(CO)₆, Nb(CO)₆, Ta(CO)₆, Mn(CO)₅, TcCO)₅, Re(CO)₅, Co(CO)₄, Rh(CO)₄, Ir(CO)₄, Cu(CO)₃, Ag(CO)₃ and Au(CO)₃. The fragments from the main group elements and transition metal fragments which are isolobal to CH₂ and CH fragments corresponding to 6 and 5 valence electrons can similarly be identified. Again the removal of 10 electron units from the carbonyl fragments will yield the corresponding 6 and 5 valence electron contents. It should be pointed out that the hydrogen atom can be replaced by an alkyl group, R or aryl group Ar.  The selected isolobal extended family members are indicated in Tables 2 to 4.  In searching for possible isolobal fragments for CH₃ from main group elements, numbers could act as a guide. For example since we are looking for seven valence electrons, we may use sets such as (4+3) this can lead us to group 4 elements combined with 3 hydrogen atoms. Hence, we generate, CH₃, SiH₃, GeH₃, SnH₃, and PbH₃. In the case of transition metals, we want to generate 17 electron systems. The set of numbers we may need for example could include (5+12). This set will give us the carbonyl fragments such as V(CO)₆, Nb(CO)₆, and Ta(CO)₆, while the set (7+10) could give us the carbonyls M(CO)₅, M = Mn, Tc, and Re. The combination (9+8) will generate M(CO)₄ where M = Co, Rh, and Ir. The other possible combination of numbers to give 17 is (8+5+4). This set will generate the fragments such M(Cp)(CO)₂ where M = Fe, Ru and Os and Cp = η⁵-C₅H₅.

Table 2: Selected CH3 Isolobal Family
	GROUP 17				GROUP 7
		14n+3 SERIES				4n+3 SERIES
Cu(CO)3	Co(CO)4	Mn(CO)5	V(CO)6	CH3	CH3	NH2	OH	F
Ag(CO)3	Rh(CO)4	Tc(CO)5	Nb(CO)6		SiH3	PH2	SH	Cl
Au(CO)3	Ir(CO)4	Re(CO)5	Ta(CO)6		GeH3	AsH2	SeH	Br
					SnH3	SbH2	TeH	I
					PbH3	BiH2
11+6 =17	9+8 = 17	7+10 =17	5+12 =17		4+3 =7	5+2 =7	6+1 = 7	7
17-10 =7	17-10 = 7	17-10 =7	17-10 = 7

 

Table 3: Selected CH2  Isolobal Family

	GROUP 16				GROUP 6
	14n+2 SERIES				4n+2 SERIES
Zn(CO)2	Ni(CO)3	Fe(CO)4	Cr(CO)5	CH2	CH2	NH	O
Cd(CO)2	Pd(CO)3	Ru(CO)4	Mo(CO)5		SiH2	PH	S
Hg(CO)2	Pt(CO)3	Os(CO)4	W(CO)5		GeH2	AsH	Se
12+4 =16	10+6 =16	8+8 = 16	6+10 =16		SnH2	SbH	Te
					PbH2	BiH
16-10 =6	16-10 = 6	16-10 = 6	16-10 = 6		4+2 = 6	5+1 = 6	6

 

Table 4: Selected CH  Isolobal Family
	GROUP 15			GROUP 5
	14n+1 SERIES			4n+1 SERIES
Cu(CO)2	Co(CO)3	Mn(CO)4	CH	CH	N
Ag(CO)2	Rh(CO)3	Tc(CO)4		SiH	P
Au(CO)2	Ir(CO)3	Re(CO)4		GeH	As
11+4 = 15	9+6 = 15	7+8 =15		SnH	Sb
15-10 =5	15-10 = 5	15-10 = 5		PbH	Bi
				4+1 = 5	5

 

Table 5: Recreation of The Periodic Table Using Series  Click here to View table

 

Arising from the arranging isolobal fragments and elements according to series as indicated in Table 1, it became apparent that the 14n and 4n series were in fact recreating the Periodic Table of Elements as is currently known. A portion of this is portrayed in Table 5. Since we are considering one skeletal element in this case, n = 1 and this generates the numbers in the first row of the table. These findings are fascinating.

Assembling the Fragments

 As can be seen from Table 1, atoms and fragments can be categorized according to series. In actual fact, what has happened in Table 1, the 14n and 4n series have created or reproduced a Periodic Table of elements and fragments. Just as atoms can be assembled according to valence rules to produce molecules, fragments can do the same.  Scheme 1 gives some few illustrative examples. Fragments can be combined to generate molecular structures. Let us illustrate this concept starting with the simplest fragment CH₃. This fragment belongs to 4n+3 series. Adding two of these fragments is equivalent to 2[CH₃(4n+3)] = C₂H₆ (4n+6). Thus, the multiplication affects the digit component of the series (2×3 = 6) but not the 4n component. In actual fact the 4n represents the number of the valence electrons contributed by the carbon skeletal atoms and the digital part represents the hydrogen atoms contribution to the valence account. In this result, when n =2, the value of the series S= 4n+6 = 4×2+6 = 14, which the sum of the valence electrons of C₂H_6.. This means that in series, the rule is that the the addition or multiplication does not affect the (4n) component of the series. The digit portion is the one to be added or subtracted whatever the case maybe. This makes sense because 4n for n =1 gives us 4 while 4n for n =2 gives us 8. This gives us a simple algebraic rule of adding series. Let us take CH₂ fragments for illustrating the concept  combining series. The fragment CH₂ belongs to 4n+2 series.

Scheme 1: Combination of two Isolobal Fragment  Click here to View scheme

 

The value k =2n-x where = 0, 1, 2,3, 4 and son comes from a simple correlation of the empirical observation of the number of skeletal atoms involves and the corresponding digit of the molecular fragment series formed. Thus, in the case of C₂H₆, we know it has a single C-C bond and 2 skeletal atoms. In order to create a simple mathematical relationship of a single bond with the series S = 4n+6, it is common sense that k = 2n-3. The digit 3 is just one half of 6 and n represents the number of the skeletal atoms. Naturally for N₂ with S = 4n+2, the k value is given by 2k-1 =3, the triple bond of dinitrogen as we know it.  In this way, a simple mathematical relationship between skeletal bonds or linkages and the corresponding number of skeletal atoms is developed.     Let us try one example to illustrate the concept of assembling fragments. We can combine 4 fragments of Ir(CO)₃ as follows.

4[Ir(CO)₃]  = Ir₄(CO)₁₂.

4[14n+1, n = 1]   = 14n+4 (n = 4); k = 2n-2 = 2×4-2 = 6.

4[14X1+1 =15] = 14X4+4 = 60 = V. This the number of valence electrons of the cluster and k =6 is the cluster number

in this case. It also corresponds to the number of linkages of the skeletal atoms which have a tetrahedral geometry (T_d) as sketched in Figure 1. Since the 14n series run parallel to the 4n series, the removal of 10 n electrons from V = 60 will give us the number of valence electrons characteristic to corresponding cluster of the main group elements. In this case 60-10×4 = 20. This number of valence electrons can give us a (CH)₄ , P₄ or (BH)₄ cluster with the ideal T_d symmetry.

 

Isolobal fragments may be considered to contain the same number of valence electrons and hence can be substituted in similar chemical species. A typical example is Ir(CO)_3 0↔CH.  Complete substitution of Ir(CO)₃  by CH will produce a T_d geometry as in Figure 1.  In C₅H₅ = (CH)₅, the CH may be replaced by P to produce P₅ fragment. Therefore C₅H₅^―  C₀↔ P₅^―. This implies that M(C₅H₅)₂ equivalent clusters M(P₅)₂ where M represents a metal atom such as Fe or Ru may be synthesized. The result is depicted in Figure 2.

 

 

Unique Simplification And Structure Prediction Using Isolobal Principle And Series  

 The most fascinating and exciting aspect of isolobal principle in combination with series is the way they can simplify an apparently imposing cluster to a bare simple number from which its structure (shape) can be predicted with high accuracy. Let us consider the carbonyl cluster, Ru₅(C)(CO)₁₅ which has a square pyramid geometry⁸. Applying the isolobal principle and series, we can split up the cluster into simpler mono-skeletal fragments as follows. The splitting of molecule into fragment, the nature of monoskeletal fragments produced does not matter as long as you can readily work out their corresponding fragmental series. Let us represent the molecular formula with F = Ru₅(C)(CO)₁₅  = 5[Ru(CO)₃] + C.  For Ru(CO)₃ fragment; V = 8 +3×2 = 14 valence electrons, and n =1, hence 14n =14. Therefore the fragment belongs to  S = 14n +0 series. The approach is to determine the overall cluster series (SF) of the formula F. If regard the C atom as a donor of its valence electrons and not part of the skeletal cluster fragment, then the overall series of the formula F is obtained in a straight forward manner.    5[Ru(CO)₃] → SF = 5[14n+0] + 4 = 14n +4 where n = 5. Therefore the series SF = 14n +4 tells us that the cluster is a member of the NIDO family. A nido structure is just one step awy from being a closo (the ideal of full or standard) geometry. In this case,an M-5 skeletal fragment is one step away from an ideal M-6 octahedral (closo) symmetry. For the main group elements, this series is the same as that of SF = 4n+4. Furthermore, it also gives us the valence content of the cluster if we substitute n =5 in the formula series. Just like the 4n+4 series, the cluster k value is obtained from k =2n-2 = 2×5-2 = 8. This implies that an M-5 skeletal cluster with k = 8 has a characteristic shape of a square pyramid sketched in Figure 3. The shape is flattened for ease of visibility. The borane cluster B₅H₉ is known to have an ideal square pyramid shape⁶. We can readily determine its series by splitting up into mono-skeletal fragment as we please and deduce the series of each fragment. Then finally sum up the series to get the formula series of B₅H₉. This is illustrated below.

 

Conclusion   

This paper is a climax of intense efforts in an attempt to fully understand and be able to explain structures of clusters of the main group elements and transition metal carbonyls in much simpler way than is currently presented. The work started with the introduction of cluster number (k value ) which led to the establishment of 14n and 4n rules^14-15. It was demonstrated that clusters could easily be categorized by using the 14n and 4n rules^5-6. Further investigations into the nature of carbonyl clusters with respect to capping series led to the revelation that molecules, fragments, atoms could be categorized by 14n and 4n rules⁷. This sparked great interest in applying the 14n and 4n rules to Hoffmann’s isolobal fragments. This has proved a tremendous success as it was discovered that isolobal fragments, molecules were interlinked by 14n and 4n rules and that they could be categorized in the same way as elements in the periodic table. Furthermore, what is interesting was that the placement of fragments and elements resulted in recreating the modern periodic table. It became easier to appreciate why a fragment such as Os(CO)₄  of a transition behaved like CH₂ of the main group elements. The reason being Os(CO)₄ fragment has a CH₂ fragment belongs to the 4n system. If 10 electrons are removed from 16 electrons of the  Os(CO)₄  fragment the remainder is 6 electrons corresponding to those of the CH₂ fragment. It is as if another CH₂ fragment is hiding within Os(CO)₄ fragment. Finally, when the concept of fragments and series are applied to clusters they are easily reduced to simple series from which cluster structure can be predicted in the usual way. What is much more heart- warming and pleasant was that not only does the 14n and 4n rules reproduce Mandeleev’s periodic table of elements, categorize Hoffmann’s fragments into their positions in the periodic table, but it also provides us with the basis for OCTET RULE AND EIGHTEEN ELECTRON RULES that have played vital roles in science. If we look carefully at Table 1 column 18//8, we can discover all those fragments listed there possess an element that obeys either octet rule or 18-electron rule. For instance, HF, OH₂, NH₃,CH_4, and BH₅ obey the octet rule while Ne, Cr(CO)₆, Mo(CO)₆, W(CO)₆, Fe(CO)₅, Ru(CO)₅, Os(CO)₅, Ni(CO)₄, Zn(CO)₃ and  Cd(CO)₃ in principle obey the 18-electron rule. Since atoms, fragments and molecules can be categorized into series and they can also be combined to produce other fragments and molecules and in turn can be split up into fragments, this implied that the corresponding series of fragments can somehow also be maneuvered. Indeed a simple algebra of series has briefly been introduced to simplify the categorization of clusters. In the author’s reflections and meditations, seeing the beauty and smooth flow of using simple numbers to generate these results with no heavy computational calculations, D. Mendeleev(1834-1907), I. Langmuir(1881-1957), G.N. Lewis(1875-1948) and even R. Hoffmann (1937-) would smile broadly at how nature is so wonderful.  This work complements the existing knowledge on clusters whose skeletal elements obey the 18 or 8 electron rules.

Acknowledgement

The author wishes to acknowledge the financial and material resources provided towards this work, the NAMSOV company, Namibia for  its generous funding and my wife Merab Kambamu Kiremire for her tireless encouragement to write this paper.

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