Symmetry homonuclear diatomic molecule

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

For practical purposes the rules for diatomic molecules concerning even and odd J reduce to the statement that for homonuclear diatomic molecules the molecular partition function must be divided by two (s = 2), while for heteronuclear diatomic molecules no division is necessary (s = 1). The idea of the symmetry number, s,... 

For this case, the primary change that is observable in the IR spectrum is due to changes in the vibrahonal frequencies of the probe molecule due to modificahons in bond energies. This can lead to changes in bond force constants and the normal mode frequencies of the probe molecule. In some cases, where the symmetry of the molecule is perturbed, un-allowed vibrational modes in the unperturbed molecule can be come allowed and therefore observed. A good example of this effect is with the adsorption of homonuclear diatomic molecules, such as N2 and H2 (see Section 4.5.6.8). 

Problem 10-5. In a homonuclear diatomic molecule, taking the molecular axis as z, the pair of LCAO-MO s tpi = 2p A + PxB and tp2 = 2 PyA + 2 PyB forms a basis for a degenerate irreducible representation of D h, as does the pair 3 = 2pxA PxB and 4 = PxA — PxB Identify the symmetry species of these wave functions. Write down the four-by-four matrices for the direct product representation by examining the effect of the group elements on the products 0i 03, 0i 04, V 2 03) and 02 04- Verify that the characters of the direct product representation are the products of the characters of the individual representations. 

Fig. 4.14 Symmetry of rotational levels of a homonuclear diatomic molecule. The letters s and a refer to the nuclear-interchange symmetry of the wave function with the nuclear-spin factor omitted. The signs + and - refer to the parity of the wave function with respect to inversion of all particles.

As with homonuclear diatomic molecules, the letters s and a are used to give the nuclear-exchange symmetry of / ns for polyatomic molecules. 

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. 

This is the correct expression for the rotational partition function of a heteronuclear diatomic molecule. For a homonuclear diatomic molecule, however, it must be taken into account that the total wave function must be either symmetric or antisymmetric under the interchange of the two identical nuclei symmetric if the nuclei have integral spins or antisymmetric if they have half-integral spins. The effect on Qrot is that it should be replaced by Qrot/u, where a is a symmetry number that represents the number of indistinguishable orientations that the molecule can have (i.e., the number of ways the molecule can be rotated into itself ). Thus, Qrot in Eq. (A.19) should be replaced by Qrot/u, where a = 1 for a heteronuclear diatomic molecule and a = 2... 

Free atoms are spherically symmetrical, which implies conservation of their angular momenta. Quantum-mechanically this means that both Lz and L2 are constants of the motion when V = V(r). The special direction, denoted Z, only becomes meaningful in an orienting field. During a chemical reaction such as the formation of a homonuclear diatomic molecule, which occurs on collisional activation, a local held is induced along the axis of approach. Polarization also happens in reactions between radicals, in which case it is directed along the principal symmetry axes of the activated reactants. When two radicals interact they do so by anti-parallel line-up of their symmetry axes, which ensures that any residual angular momentum is optimally quenched. The proposed sequence of events is conveniently demonstrated by consideration of the interactions between simple hydrocarbon molecules. 

A second useful symmetry operation exists for homonuclear diatomic molecules, namely the permutation of two identical nuclei, P 2. In the same way that E has two possible eigenfunctions 1 in equation (6.206), so there are two possible ways in which the molecular wave function can transform under P 2 ... 

One final symmetry aspect for homonuclear diatomic molecules to be mentioned here is that g/u states can, under some circumstances, be mixed by the nuclear spin part of the molecular Hamiltonian. This mixing, which is explained by Bunker and Jensen [71], has some interesting spectroscopic consequences, particularly in the molecular ion, which are described elsewhere in this book. 

Because of the symmetry of the homonuclear diatomic molecule, every alternate rotational level is missing those that exist have N odd and positive parity, as shown for the first three rotational levels in figure 10.43. The magnetic dipole transitions arise from coupling of the electron spin magnetic moment with the oscillating magnetic field, represented by the interaction term... 

Homonuclear diatomic molecules, such as N2 and O2, have but a single vibrational mode of Vcoh symmetry and are thus transparent to infrared radiation. By contrast, other molecules in the Earth s atmosphere, such as CO2 and CH4, can absorb in the infrared. This is the cause of the greenhouse effect. While almost all of the components of the atmosphere are transparent to the ultraviolet radiation from the Sun. much of the infrared which would be radiated back into space is trapped by the IR-absorbing greenhouse gases. In 1862, John Tyndall referred to infrared radiation As a dam buill across a river causes a local deepening of the slreain, so our atmosphere, thrown as a barrier across the terrestrial rays, produces... 

Diatomic molecules such as CO and NO, formed from atoms of two different elements, are called heteronuclear. We construct MOs for such molecules by following the procedure described earlier, with two changes. First, we use a different set of labels because heteronuclear diatomic molecules lack the inversion symmetry of homonuclear diatomic molecules. We therefore drop the g and u subscripts on the MO labels. Second, we recognize that the AOs on the participating atoms now correspond to different energies. For example, we combine the 2s AO of carbon and the 2s AO of oxygen to produce a bonding MO (without a node). 

The final symmetry operation in this Book is inversion through a centre of symmetry. You met this operation when you were studying homonuclear diatomic molecules, but it is not confined to diatomic molecules. Structure 6.7, for example, has a centre of symmetry in the middle of the benzene ring ... 

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. 

For homonuclear molecules, the g or u symmetry is almost always conserved. Only external electric fields, hyperfine effects (Pique, et al., 1984), and collisions can induce perturbations between g and u states. See Reinhold, et al., (1998) who discuss how several terms that are neglected in the Born-Oppenheimer approximation can give rise to interactions between g and u states in hetero-isotopomers, as in the HD molecule. An additional symmetry will be discussed in Section 3.2.2 parity or, more usefully, the e and / symmetry character of the rotational levels remains well defined for both hetero- and homonuclear diatomic molecules. The matrix elements of Table 3.2 describe direct interactions between basis states. Indirect interactions can also occur and are discussed in Sections 4.2, 4.4.2 and 4.5.1. Even for indirect interactions the A J = 0 and e / perturbation selection rules remain valid (see Section 3.2.2). 

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. 

Since the occurrence of the Raman effect depends on the change in polarizability as vibration occurs, the selection rules are different for the Raman effect than they are for the infrared spectrum. In particular, in molecules with a center of symmetry the totally symmetric vibration is Raman-active, but is forbidden in the infrared since it produces no change in dipole moment. Thus the homonuclear diatomic molecules, H2, O2, N2, show the Raman effect but do not absorb in the infrared. There is also a purely rotational Raman eff ect in these molecules. However, in this case the selection rule is A J = 2. Thus we have for the rotational Stokes lines... 

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