Atomic orbitals redundancy

The use of hybrid atomic orbitals in qualitative valence theory has, in the past, rested on two points (i) an empirical justification of their use involving the concept of the valence state of an atom and (ii) a simple linear transformation technique for the construction of the explicit forms of the orbitals. In this section we show that both of these points can be replaced. The justification can be replaced by a derivation and this derivation can be used to suggest variational forms which render the linear transformation technique redundant. 

The electrostatic interactions in a molecule are determined by the structure of the density matrix D. In constructing D from an atomic orbital or atomic spinor basis, we incorporate a lot of redundant information in the Gaussian overlap charge distributions, since most of the electron density is concentrated near the nuclei. One should therefore try to transform D into a block diagonal form in which each dense block corresponds to a one-centre density so that the... 

Fig. 8.31. The benzene molecule. The hybridization concept allows us to link the actual geometiy of a molecule with its electronic structure (al. The sp hybrids of the six carbon atoms form the six o CC bonds, and the structure is planar. Each caibon atom thus uses two out of its three s[7 hybrids the third one lying in the same plane protrudes toward a hydrogen atom and forms the a CH bond. In this way, each caibon atom uses its three valence electrons. The fourth one resides on the 2p orbital that is perpendicular to the molecular plane. The six 2p orbitals form six rr molecular orbitals, out of which three are doubly occupied and three are empty (b). The doubly occupied ones are shown in panel (b). The (fio of the lowest energy is an all-in-phase linear combination rf the 2p atomic orbitals (only their upper lobes are shown). The and

Three common labeling conventions are used in Fig. 7-17. A level may be labeled with reference to the separated atom AOs to which it correlates. The separated atom AO symbol is placed to the right of the MO symmetry symbol (e.g., a 2s). Note the absence of square brackets, which we used to symbolize the SO (2s -I- 2s5>. The symbol (7g2s means the MO of ag symmetry that correlates with 2s AOs at / = oo. An alternative label indicates the united-atom orbital with which the MO correlates. Here the AO label is placed to the left of the symmetry symbol (e.g., 3punited-atom notation are redundant and are often omitted. However, they are helpful in drawing correlation diagrams. Finally, the MOs may be simply numbered in their energy order within each symmetry type, as mentioned earlier (e.g., 2af). 

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