Continued Hybridization and Localized Bonds

The picture of methane given in the two previous sections is of course entirely different from the conventional one with four localized equivalent C—H bonds linking the atoms together. Our next task is to examine the relation between these two apparently contradictory pictures and to show that they are compatible. 

In the MO description given above, each MO of methane is derived from a single carbon AO and a combination (SO) of hydrogen Is AOs. Let us now try to find a reciprocal description in which each MO is formed from a single hydrogen Is AO and a combination of carbon AOs. 

It is easily seen that the functions 4 are linearly independent, i.e., no one of them can be expressed as a linear combination of the others. Equally, the AOs s, X, y, and z can be expressed as linear functions ti 4 e.g., 

The mathematical form of p AOs is such that any combination of p AOs gives an orbital of the same shape (i.e., another p AO), pointing in some intermediate direction. Each of the orbitals is therefore identical in shape, being drawn from the s AO and p AOs in similar proportions (i.e., 1 3). The shape of each of these hybrid orbitals is indicated in Fig. 1.19(a, b). The combination of the s AO with a p AO leads to a lopsided p-like orbital in which 

The four MOs ij/s, ij/, ij/y, and ij/ [equations (1.26)-(1.29)] represent volumes of orbital space each able to hold two electrons, so, together, the four MOs represent a volume of orbital space able to hold four pairs of electrons. This suggests that we could combine the four MOs and then subdivide their sum into four equal parts in some other way, thus getting an equivalent description of methane. 

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