Diatomic molecules symmetry classification

In the point groups of diatomic molecules, the classification of symmetry-adapted orbitals is according to the effect of rotations about the molecular axis. Notice that since diatomic molecules are cylindrical, rotation by any angle about the molecular axis is a symmetry operation. Orbital functions may possess nodal surfaces (surfaces where the wavefunction is zero), and these are planes that include the molecular axis as in the combination of 2p orbitals shown in Figure 10.7. Consider an end-on view of... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

Heteronuclear diatomic molecules are naturally somewhat more complicated than the homonuclear comprehensive comparisons with homonuclear molecules were given by Mulliken [15]. The atomic orbital coefficients in the molecular orbitals ofheteronu-clear diatomic molecules are no longer determined by symmetry alone, and the electrons in the molecular orbitals may be shared equally between atoms, or may be almost localised on one atom. The molecular orbitals can still be classified as a or n, but in the absence of a centre-of-symmetry the g/u classification naturally disappears. Some heteronuclear molecules contain atoms which are sufficiently similar that the molecular orbitals resemble those shown in figure 6.7. In many other cases, however, the atoms are very different. This is particularly the case for hydride systems, like the HC1 molecule,... 

Thus for a diatomic molecule, the vibrational factor is always symmetric with respect to E and so does not play any part in the symmetry classification scheme. 

The molecular orbital model as a linear combination of atomic orbitals introduced in Chapter 4 was extended in Chapter 6 to diatomic molecules and in Chapter 7 to small polyatomic molecules where advantage was taken of symmetry considerations. At the end of Chapter 7, a brief outline was presented of how to proceed quantitatively to apply the theory to any molecule, based on the variational principle and the solution of a secular determinant. In Chapter 9, this basic procedure was applied to molecules whose geometries allow their classification as conjugated tt systems. We now proceed to three additional types of systems, briefly developing firm qualitative or semiquantitative conclusions, once more strongly related to geometric considerations. They are the recently discovered spheroidal carbon cluster molecule, Cgo (ref. 137), the octahedral complexes of transition metals, and the broad class of metals and semi-metals. 

Chemists have adopted from mathematicians a way of classifying molecules according to their symmetry. The classification is based on the idea that if a molecule is sufficiently symmetrical, then an action can be found that will leave the molecule looking the same as it did when you started. You have already met two such actions inversion through a centre of symmetry and rotation about the molecular axis of a diatomic molecule. Actions such as this are called symmetry operations, and a molecule can be placed in a category according to how many such operations you can perform and still leave the molecule looking the same. 

For linear polyatomic molecules, the operator for the axial component of the total electronic orbital angular momentum commutes with the electronic Hamiltonian, and the same term classifications are used as for diatomic molecules we have such possibilities as S, S , 2, ri, and so on. For linear polyatomic molecules with a center of symmetry, the g, u classification is added. 

If a molecule were straightened out into a linear molecule, its symmetry would turn into Pooh, which is the group for homonuclear diatomic and symmetric triatomic molecules. The orbital classification we studied in Chapter 11 would then apply, and we can reclassify the AO s in the triatomic molecule XH2 as follows ... 

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