Symmetry-adapted linear combinations

Key point: Symmetry-adapted linear combinations of orbitals are combinations of atomic orbitals that conform to the symmetry of a molecule and are used to construct molecular orbitals of a given symmetry species. A fundamental principle of the MO theory of diatomic molecules (Section 2.8) is that molecular orbitals are constructed from atomic orbitals of the same symmetry. Thus, in a diatomic molecule, an s orbital may have nonzero overlap with another s orbital or with a pz orbital on the second atom (where z is the internuclear direction, Fig. 6.19), but not with a px or py orbital. Formally, whereas the pz orbital of the second atom has the same rotational symmetry as the s orbital of the first atom and the same symmetry with respect to reflection in a mirror plane containing the internuclear axis, the px and py orbitals do not. The restriction that , π, or δ bonds can be formed from atomic orbitals of the same symmetry species stems from the requirement that all components of the molecular orbital must behave identically under transformation if they are to have nonzero overlap. Exactly the same principle applies in polyatomic molecules, where the symmetry considerations may be more complex and require us to use the systematic procedures provided by group theory. The general procedure is to group atomic orbitals, such as the three H1s orbitals of NH₃, together to form combinations of a particular symmetry and then to build molecular orbitals by allowing combinations of the same symmetry on different atoms to overlap, such as an N2s orbital and the appropriate combination of the three H1s orbitals.

Specific combinations of atomic orbitals that are used to build molecular orbitals of a given symmetry are called symmetry-adapted linear combinations (SALCs). A collection of commonly encountered SALCs of orbitals is shown in Resource section 5; it is usually simple to identify the symmetry of a combination of orbitals by comparing it with the diagrams provided there.

Figure 6.19 An s orbital can overlap (a) an s or (b) a pz orbital on a second atom with constructive interference. (c) An s orbital has zero net overlap with a px or py orbital because the constructive interference between the parts of the atomic orbitals with the same sign exactly matches the destructive interference between the parts with opposite signs.

Specific combinations of atomic orbitals that are used to build molecular orbitals of a given symmetry are called symmetry-adapted linear combinations (SALCs). A collection of commonly encountered SALCs of orbitals is shown in Resource section 5; it is usually simple to identify the symmetry of a combination of orbitals by comparing it with the diagrams provided there.

The generation of SALCs of a given symmetry is a task for group theory, as we explain in Section 6.10. However, they often have an intuitively obvious form. For instance, the fully symmetrical A1 SALC of the H1s orbitals of NH3 (Fig. 6.20) is

To verify that this SALC is indeed of symmetry A1 we note that it remains unchanged under the identity E, each C3 rotation, and any of the three vertical reflections, so its char acters are (1,1,1) and hence it spans the fully symmetrical irreducible representation of C_3v. The E SALCs are less obvious, but as we shall see are

Figure 6.20 The (a) A1 and (b) E symmetry adapted linear combinations of H1s orbitals in NH₃.

Figure 6.21 The combination of O2p_x orbitals referred to in Example 6.11.

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مواضيع ذات صلة

Extended silicon oxygen compounds

Compounds of boron with nitrogen

Boron oxygen compounds

Uses of the elements and their compounds

Occurrence and recovery

Boron clusters

The assignment of molecular symmetry from vibrational spectra

Information from the symmetries of normal modes

The exclusion rule

Molecular vibrations

Chiral molecules

Polar molecules