Symmetry coordinates of a homonuclear diatomic molecule

Chapter 3. Diatomic Molecules and their Molecular Orbitals 

A no less important point to note is that the six symmetry coordinates comprise a complete set, in terms of which any arbitrary molecular motion can be described. A composite motion like the one in Fig. 3.9 can be constructed by a superposition of symmetry coordinates with suitably chosen phase and amplitude, and is therefore assigned to a reducible representation, the direct sum of its component irreps. It is easy to see that the motion of a single atom also belongs to a reducible representation Displacement of the left-hand X atom parallel to x is clearly a superposition of Tx and the negative phase of Ry, so it belongs to bsu b2g, whereas that of the right-hand atom along z, composed of Tz and transforms as ag 

When the atoms are not identical - and it is immaterial whether the dissymmetry is substantial, as in HCl, or minimal, as in CF Cl - the symmetry of the diatomic molecule is reduced in our hypothetical quadrupolar field to C2y- Their symmetry labels are obtainable from those in Fig. 3.9 by inspection, by comparing Tables 3.1 and 3.2, or by consulting the Correlation Table for D2/1 in Appendix C. Most simply, we consult the Character Table of The three translations transform as ai, bi and 62 and the rotations about x and y as 62 and 61 respectively. There is no molecular rotation about z in a linear molecule, where the nuclei lie on the internuclear axis as in the case of the homonu-clear molecule, the vibration is necessarily totally symmetric (ai), because the molecular symmetry remains 

One can, of course, imagine inducing rotation of electrons about the molecular axis this, does not produce molecular rotation, but amounts to excitation to a higher electronic state. 

Diatomic molecules are necessarily linear, but a triatomic molecule can be either linear like CO2 and HCN or bent like SO2 and H2O. Mulliken s correlation diagram procedure was extended to tri- and tetraatomic molecules by Walsh [1], who promulgated a set of simple but remarkably viable [2, 3, 4] rules for predicting whether or not a molecule will remain linear, from the effect of the departure from linearity on the energy of its occupied molecular orbitals. 

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Figure 3.9. Symmetry coordinates of a homonuclear diatomic molecule (D2/1) (Consult Table 3.1 to obtain the Dooh labels)...

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