Antisymmetric behaviour

Fig. 5.13 Antisymmetrical behaviour of the methylene tIch2 MO with respect to rotation about the principal axis...

We will now consider the consequences of these mles in the simple case of FI2. In this molecule both whatever the value of v, and in the ground electronic state, are symmetric to nuclear exchange so we need consider only the behaviour of lAr A - Since / = i for FI, ij/ and therefore i/ r A rnust be antisymmetric to nuclear exchange. It can be shown that, for even values of the rotational quantum number J, ij/ is symmetric (x) to exchange and, for odd values of J, j/ is antisymmetric a) to exchange, as shown in Figure 5.18. 

For centrosymmetric systems with a centre of inversion /, subscripts g (symmetric) and u (antisymmetric) are also used to designate the behaviour with respect to the operation of inversion. The molecule trans-butadiene belongs to the point group Cik (Figure 2.13b). Under this point group the symmetry operations are /, C2Z, and i, and the following symmetry species can be generated ... 

The 3-F symbols are invariant under an even permutation of the columns in the symbol and are multiplied by (—l)r i+r 2+rs under an odd permutation. Thus their behaviour under odd permutations classifies them as symmetrical (even 3-F symbols. Sect. 4 a) or antisymmetrical (odd 3-F symbols. Sect. 4b). If a 3-F symbol belongs to the even or odd class, the exponent A + A + A is an even or odd number, respectively. This classification, which will be discussed on p. 210 and treated in detail in Chapt. Ill, is associated with the relationship... 

Evidently, there are two distinct frequencies, where either the numerator or the denominator of the complex impedance becomes zero. However, the case of zero impedance is determined exclusively by the serial capacity, whereas the parallel determines the frequency of infinite impedance. These two frequencies thus correspond to the resonance frequencies of the two part circuits mentioned above and are also correctly reproduced in the frequency spectrum of the QCM. Observable side resonances (as shown especially in the insert with lower span) can be traced back to mechanical oscillations that differ from the main one one is the result of antisymmetric thickness shear oscillation, the other of a twist oscillation. The ratio of intensity between the desired thickness shear wave and the side resonances is mainly defined by the ratio between electrode diameter and quartz substrate thickness. This is illustrated in Fig. 4, where the damping spectra for both a 10 MHz and a 5 MHz device are given. In both cases the electrode diameter here is 8 mm (the spectra in Fig. 3 were recorded with 4 mm electrode diameter). Evidently, the 5 MHz QCM shows the desired response pattern, where the shear resonance by far dominates the electrical behaviour. The 10 MHz QCM, however, shows very pronounced side resonances. The rather large electrode diameter (compared to the thickness) very strongly favours the occurrence of torsional motions within the substrate, thus reasonable amplitudes are generated for this mode. 

Irreducible representations, or symmetry species, are ways to depict all possible properties (such as molecular orbitals or normal vibrations) of a molecule in terms of their symmetry properties. The symmetry properties are defined in terms of the behaviour of the property when the symmetry elements of the molecular point group are applied. Thus, for instance, the Och-MO of methylene (Fig. 5.12) is symmetrical with respect to rotation about the principal axis (it does not change) whereas the tCch2" MO is antisymmetrical (it changes its phase. Fig. 5.13). 

All possible properties can be described in terms of their behaviour (symmetrical or antisymmetrical) with respect to the symmetry elements of the molecule. The combinations of these behavioural patterns are the irreducible representations. They are classified using a symbolic notation that defines at least some of their behaviours. The meanings of the indi-ividual characters in the names of the irreducible representations are ... 

A nice example of this behaviour is the selective population of A-doublet states of NO in the photodissociation of (CH3)2NNO at two different wavelengths. In this case, it was found that the antisymmetric A-doublet was dominating at one wavelength, whereas the symmetric A-doublet was dominating at the other wavelength. It was concluded that the excitation leads to two different electronically excited states. 

If there would be no electron spin, the Pauli principle would only allow the antisymmetric solution with repulsion, and no homo-polar binding would exist. The existence of homo-polar chemistry appears to rest entirely on the presence of spin in connection with the Pauli principle. This yields an additional degree of freedom (which is not of importance for the energetic behaviour) and the anti-symmetry requirement of the Pauli principle can be satisfied with the spin coordinates. As a result of this, the eigenfunction symmetric in the coordinates of the centre-of-mass with a homo-polar binding is allowed it is only necessary to suppose the spins of the electrons different and anti-symmetrically related. 

A familiar example is the expansion of a function that is symmetric (or antisymmetric) under some operation in terms of basis functions with the same behaviour (see e.g. p. 73). 

Inspection of Eqa (6) for T2(MO) shows that it corresponds to the Heitler-London function obtained when the + of Eqn. (2) is replaced by a - this is a result that we have obtained previously in section 3-5. We may also obtain T3(M0) (= (ionic)) from T3 (HL) by exciting an electron from one atomic orbital into the other and changing the sign of the linear combination. (The sign change is necessary in order to satisfy the spectroscopic rule that an even — odd excitation is allowed, whereas toth even — even and odd —> odd excitations are forbidden. The even and odd characters of T (HL) and T (ionic) refer to the behaviour of the wave-functions with respect to inversion through the centre of symmetry of the molecule. Thus T (HL) and T. (ionic) are synunetric (even) and T (HL) and T (ionic) are antisymmetric (odd).)... 

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