Homonuclear diatomic molecules symmetry orbitals

In general, it is advantageous to use the symmetry elements of a molecule in dealing with the molecular orbitals. For example, consider the symmetry properties of a homonuclear diatomic molecule... 

The symmetry operations E, C, and av (reflection in a plane that contains the axis A-B) are present. All molecules that possess these symmetry properties have the point-group symmetry Coov The orbitals are characterized by symbols similar to those used for a homonuclear diatomic molecule, such as a, n, etc. The character table for CMV is given in Table 2-2. 

You will recall that in homonuclear diatomic molecules, orbitals that were unchanged by inversion through the centre of symmetry were labelled g and those that were changed were labelled u. Orbitals of all molecules with a centre of symmetry can be labelled using g or u subscripts. 

We consider symmetry in Chapter 3, but it is useful at this point to consider the labels that are commonly used to describe the parity of a molecular orbital. A homonuclear diatomic molecule (e.g. H2, CI2) possesses a centre of inversion (centre of symmetry), and the parity of an MO describes the way in which the orbital behaves with respect to this centre of inversion. 

In the Czv point group, the sjunmetry elements of which are conserved during the reaction, we start with two unique bonds of ai S3anmetry. Two identical bonds are formed which must be considered together. Their sum and difference are of i and bi symmetry, respectively. Hence the reaction is forbidden by orbital symmetry. The same conclusion would be drawn for all other reactions of homonuclear diatomic molecules with each other. 

As found in the last section for the function (13.43), the symmetry of the homonuclear diatomic molecule makes the coefficients of the atom-6 orbitals equal to 1 times the corresponding atom-a orbital coefficients ... 

It would be instructive to illustrate the use of these rules. As a first example let us consider the core-hole excited states of homonuclear diatomic molecules. When one electron is removed from the core orbital, the original Da,h symmetry of the wavefunction is lowered to Coov This situation is depicted in the figure below where only the Is core electrons are represented. According to the rules in table 2, the Dooh group can be decomposed into two C v components related by a Ci or Cs operation. The two C >v structures (a) and (b) below ... 

Figure 7-16 (a) Symmetry orbitals for homonuclear diatomic molecules, (b) United-atom AOs... 

Because both electrons reside in the a bonding orbital, an electron configuration of can be used to describe the ground electronic state of H2. (Because H2 is a homonuclear diatomic molecule, we add a subscript g to the label——to indicate the orbital s symmetry property with respect to the center of the molecule. Electrons in the antibonding orbital are labeled a, the u also referring to the orbital s symmetry properties. Symmetry will be discussed in the next chapter.) To emphasize that the electrons in the a orbital derive from Is electrons from H atoms, the more detailed (cTgls) label can also be used. 

When two atoms containing s and p valence orbitals are combined into a homonuclear diatomic molecule a set of molecular orbitals with shapes and symmetry properties as those already described arises. The relative energies of... 

Homonuclear diatomic molecules belong to the point group which includes the inversion operator. Hence the orbitals for a molecule such as N2 have symmetry designations with a or M subscript (gerade or ungerade). 

Consider the correlation diagram for homonuclear diatomic molecules (Fig. 1.10), constructed in the same way as that for Hj. The significance of the non-crossing rule becomes apparent when we consider the details of this diagram. In terms of symmetry the Og 2s molecular orbital could correlate with either the 2s- or the 3s-orbital of the united atom. Similarly the Og 2p molecular orbital could correlate with either the 2s or the 3s united atom orbital. As a consequence of the non-crossing rule, the correlation of the... 

In the case of ethylene the a framework is formed by the carbon sp -orbitals and the rr-bond is formed by the sideways overlap of the remaining two p-orbitals. The two 7r-orbitals have the same symmetry as the ir 2p and 7T 2p orbitals of a homonuclear diatomic molecule (Fig. 1.6), and the sequence of energy levels of these two orbitals is the same (Fig. 1.7). We need to know how such information may be deduced for ethylene and larger conjugated hydrocarbons. In most cases the information required does not provide a searching test of a molecular orbital approximation. Indeed for 7r-orbitals the information can usually be provided by the simple Huckel (1931) molecular orbital method (HMO) which uses the linear combination of atomic orbitals (LCAO), or even by the free electron model (FEM). These methods and the results they give are outlined in the remainder of this chapter. 

Usually the electronic structure of diatomic molecules is discussed in terms of the canonical molecular orbitals. In the case of homonuclear diatomics formed from atoms of the second period, these are the symmetry orbitals 1 og, 1 ou, 2ag,... 

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,... 

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 ... 

More advanced applications of symmetry (not discussed here) involve the behaviour of molecular wavefunctions under symmetry operations. For example in a molecule with a centre of inversion (such as a homonuclear diatomic, see Topic C4). molecular orbitals are classified as u or g (from the German, ungerade and gerade) according to whether or not they change sign under inversion. In... 

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