The mathematical form of hybrid orbitals

Applying PAl to tpt and y t is not necessary since the same combination as PAl y i will be produced. The two combinations obtained from P are by inspection, linearly independent and a third combination which can be found by applying P to ipti will be a combination of these two. If the combinations are normalized with the assumption that 

Under the 3h point group, an s-orbital belongs to and the pair of p-orbitals p and p belongs to r If these orbitals have been used to construct the three hybrid orbitals (sp type), then we can 

The x and y axes are shown in Fig. 11-5.1. To establish the values of the coefficients alt ait blt and bt it is necessary to investigate the detailed effect of one or more of the transformation operators Og of lh on the pair of combinations. 

Since ip% and —ips have negative y components (y1 has none), we take the negative sign, i.e. 

Eqns (11-5.1), (11-5.4), and (11-5.5) may be brought together in the single matrix equation 

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Transformation properties of atomic orbitals 11-3. Hybrid orbitals for c-bonding systems 11 -i- Hybrid orbitals for >r-bonding systems 11-5. The mathematical form of hybrid orbitals... 

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