Centre of inversion

The symmetry operation i is the operation of inversion through the inversion centre. 

This equation can be interpreted also as implying that, if we carry out a C operation followed by a operation, the result is the same as carrying out an operation. (The convention is to write Bio mean carry out operation B first and A second in the case of C and the order does not matter, but we shall come across examples where it does.) 

From the definition of, it follows that (7 = 51 i = 0082, since a and i are taken as separate symmetry elements the symbols 5i and 82 are never used. 

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

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If a molecule has a centre of inversion (or centre of symmetry), i, reflection of each nucleus through the centre of the molecule to an equal distance on the opposite side of the centre produces a configuration indistinguishable from the initial one. Figure 4.4 shows s-trans-buta-1,3-diene (the x refers to trans about a nominally single bond) and sulphur hexafluoride, both of which have inversion centres. 

We have seen in Section 4.1.4 that = n and that S2 = i, so we can immediately exclude from chirality any molecule having a plane of symmetry or a centre of inversion. The condition that a chiral molecule may not have a plane of symmetry or a centre of inversion is sufficient in nearly all cases to decide whether a molecule is chiral. We have to go to a rather unusual molecule, such as the tetrafluorospiropentane, shown in Figure 4.8, to find a case where there is no a or i element of symmetry but there is a higher-fold S element. In this molecule the two three-membered carbon rings are mutually perpendicular, and the pairs of fluorine atoms on each end of the molecule are trans to each other. There is an 54 axis, as shown in Figure 4.8, but no a or i element, and therefore the molecule is not chiral. 

Examples are rare except for the S2 point group. This point group has only an S2 axis but, since S2 = i, it has only a centre of inversion, and the symbol generally used for this point group is C,. The isomer of the molecule ClFHC-CHFCl in which all pairs of identical FI, F or Cl atoms are trans to each other, shown in Figure 4.11(b), belongs to the C, point group. 

A C f, point group contains a C axis and a Of, plane, perpendicular to C . For n even the point group contains a centre of inversion i. It also contains other elements which may be generated from these. 

Molecules belonging to the 4 point group are very highly symmetrical, having 15 C2 axes, 10 C3 axes, 6 C5 axes, 15 n planes, 10 axes, 6 5io axes and a centre of inversion i. In addition to these symmetry elements are other elements which can be generated from them. 

The vibrations of acetylene provide an example of the so-called mutual exclusion mle. The mle states that, for a molecule with a centre of inversion, the fundamentals which are active in the Raman spectmm (g vibrations) are inactive in the infrared spectmm whereas those active in the infrared spectmm u vibrations) are inactive in the Raman spectmm that is, the two spectra are mutually exclusive. Flowever, there are some vibrations which are forbidden in both spectra, such as the torsional vibration of ethylene shown in Figure 6.23 in the >2 point group (Table A.32 in Appendix A) is the species of neither a translation nor a component of the polarizability. 

The first is the g or m symmetry property which indicates that ij/ is symmetric or antisymmetric respectively to inversion through the centre of the molecule (see Section 4.1.3). Since the molecule must have a centre of inversion for this property to apply, states are labelled g or m for homonuclear diatomics only. The property is indicated by a postsubscript, as in... 

In a molecule with a centre of inversion all hyper Raman active vibrations are u vibrations, antisymmetric to inversion. 

It should be noted that, whereas ferroelectrics are necessarily piezoelectrics, the converse need not apply. The necessary condition for a crystal to be piezoelectric is that it must lack a centre of inversion symmetry. Of the 32 point groups, 20 qualify for piezoelectricity on this criterion, but for ferroelectric behaviour a further criterion is required (the possession of a single non-equivalent direction) and only 10 space groups meet this additional requirement. An example of a crystal that is piezoelectric but not ferroelectric is quartz, and ind this is a particularly important example since the use of quartz for oscillator stabilization has permitted the development of extremely accurate clocks (I in 10 ) and has also made possible the whole of modern radio and television broadcasting including mobile radio communications with aircraft and ground vehicles. 

Note the absence of the g subscripts here. Although the d orbitals are still centro-symmetric, the tetrahedral environment lacks a centre of inversion. The d orbitals are therefore not classified with respect to a symmetry element which doesn t exist the absence of the g subscript does not imply the opposite - i.e. u (ungerade or odd). 

Again the left superscript indicates the spin-triplet nature of the arrangement. The letter A means that it is spatially (orbitally) one-fold degenerate and it is upper-case because we describe two-electron wavefunctions. The subscript is g because the product of d orbitals is even under the octahedral centre of inversion, and the right subscript 2 must remain a mystery for us once again. 

An S term, like an s orbital, is non-degenerate. Therefore, while the effect of a crystal field (of any symmetry) will be to shift its energy, there can be no question of its splitting. The ground term for the configuration is S. In an octahedral crystal field, this is relabelled Aig, in tetrahedral symmetry, lacking a centre of inversion, it is labelled M]. 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

The theoretical reason is as follows. Although the placing of the ligands in a tetrahedral molecule does not define a centre of symmetry, the d orbitals are nevertheless centrosymmetric and the light operator is still of odd parity and so d-d transitions remain parity and orbitally Al = 1) forbidden. It is the nuclear coordinates that fail to define a centre of inversion, while we are considering a... 

When discussing the origin of the large widths of d-d bands in Chapter 2, we noted that molecules are always vibrating. Some of these vibrations are such as to remove a centre of inversion. Consider just the one example in Fig. 4-3. This... 

Optical activity in metal complexes may also arise either if one of the ligands bound to the metal in the first co-ordination sphere is itself optically active or if the complex as a whole lacks a centre of inversion and a plane of symmetry. Thus all octahedral cts-complexes of the tris-or bis-chelate type have two isomeric forms related by a mirror plane, the d- and /-forms. These species have circular dichroism spectra of identical intensities but opposite in sign. The bands in the circular dichroism spectrum are, of course, modified if ligand exchange occurs but they are also exceedingly sensitive to the environment beyond the first co-ordination sphere. This effect has been used to obtain association constants for ion-pair formation. There also exists the possibility that, if such compounds display anti-tumour activity, only one of the mirror isomers will be effective. 

FIGURE 1.15 Symmetry in solids (a) two OF2 molecules related by a plane of symmetry, (b) three OF2 molecules related by a threefold axis of symmetry, and (c) two OF2 molecules related by a centre of inversion. 

Does the CF4 molecule in Figure 1.14 possess a centre of inversion What other rotation axis is coincident with the... 

Based on extensive studies of the symmetry in crystals, it is found that crystals possess one or more of the ten basic symmetry elements (five proper rotation axes 1,2,3, 4,6 and five inversion or improper axes, T = centre of inversion i, 2 = mirror plane m, I, and 5). A set of symmetry elements intersecting at a common point within a crystal is called the point group. The 10 basic symmetry elements along with their 22 possible combinations constitute the 32 crystal classes. There are two additional symmetry... 

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 ... 

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