Symmetry group orbital

Consider now the construction of the A i symmetry group orbital of the hydrogen s atomic orbitals in ammonia as an example of the application of the projection operator. (The various kinds of orbitals will be discussed in detail in Chapter 6.) The projection operator for the A irreducible representation in the C3v point group is... 

Ws3 Vs3 Figure 6-23. Projection of the two E symmetry group orbitals of the three H Is ... 

Figure 6-27. Generation of the. 1 i v symmetry group orbitals of benzene.

Figure 6-31. The two E g symmetry group orbitals formed from the carbon 2pz orbitals in benzene.

The summary highlights an important observation, which identifies a general procedure that is particularly easy to apply for the cases of molecules in which a principal rotational axes can be identified [the dihedral groups]. For an orbit of n vertices, it is necessary only to identify the first occurrences of n distinct a group orbitals, which occur in the hierarchical order s-, p-, d-,. .. like, with respect to the central functions. Then the construction of the other group orbitals of the valence set decorations are formed by superimposition of the uj [jTe here] which operation leads to the Fz x Fa symmetry group orbitals, with the remainder identified on concerted local rotation of the uj into the here] and of symmetries... 

In Figure 10.14, for a slab calculation with the extended Hiickel method of Ag, the group orbital projections onto the valence band 6s PDOS are shown for the interaction with an adsorbate with an s-atomic orbital in three-, two-, or onefold coordination. Note the enhancement of density at low energy for the threefold symmetric group orbital compared to that of the twofold symmetry group orbital or the PDOS of the s orbital of a single metal atom. 

The local symmetry of the M —C bond is In this symmetry group, orbitals of the metal and carbon atoms always form a and n bonds (Table 4.1). 

CC and CH bond orbitals but also for the CTL, ami CH3 group orbitals. If the local symmetry elements are preserved in the full molecule, the 7r (or a) local orbitals can combine to give v (or o) molecular orbitals. The reader should, therefore, not be surprised to find, for instance, tt type molecular orbitals in cyclopropane which are delocalized over the CH2 groups. 

The interaction between the three cr localized orbitals is slightly more complex. As we have just shown, it is proper to start by combining orbitals of same energy (the crCIf2 pair), and then to interact the new combinations with the remaining orbitals. The procedure is simple here because, by symmetry, only the in-phase combination of the crCH2 group orbitals mixes with the acc bond orbital (see Fig. 17). The reader will notice that the acc orbital has been placed,... 

In molecules with little or no symmetry, it may still be possible to recognize the main localized-orbital component of certain molecular orbitals. It is then convenient to adopt the label of this localized type as the label of the molecular orbital, even though the molecular symmetry does not coincide with the local symmetry. For instance, in methylenimine again, the 5A orbital is clearly built out of the in-plane 7rc 2 group orbital, with a small NH component. We therefore label the orbital t CU2, although the molecule does not have a vertical symmetry plane. Similarly, the orbitals 7A and 8A of propylene are labeled 7TqH3, tt CU2 (111.49).a Other examples where the local symmetry is sufficiently preserved and only weakly perturbed by the molecular environment are hydrazine (111.34) and methylamine (III.31). In some cases we have omitted the label as no unambiguous classification is possible. 

Table 6-3. Labelling of metal and ligand group orbitals in O symmetry.

Symmetry Metal orbital Ligand group orbital... 

The three-fold degenerate set of p orbitals are labelled tiu t for three-fold, u for odd under inversion through the centre of symmetry). As shown in Fig. 6-6, each metal p orbital matches symmetry with ligand group orbitals comprising just two... 

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