The Inversion Center

If a molecule can be brought into an equivalent configuration by changing the coordinates (jr, y, z) of every atom, where the origin of coordinates lies at a point within the molecule, into ( jc, -y, -z), then the point at which the origin lies is said to be a center of symmetry or center of inversion. The symbol for the inversion center and for the operation of inversion is an italic /. Like a plane, the center is an element that generates only one operation. 

It may be noted that, when a center of inversion exists, restrictions are placed on the numbers of all atoms, or all but one atom, in the molecule. Since the center is a point, only one atom may be at the center. If there is an atom at the center, that atom is unique, since it is the only one in the molecule that is not shifted when the inversion is performed. All other atoms must occur in pairs, since each must have a twin with which it is exchanged when the inversion is performed. From this it follows that we need not bother to look for a center of symmetry in molecules that contain an odd number of more than one species of atom. 

The effect of carrying out the inversion operation n times may be expressed as It should be easily seen that / = when n is even, and / = i when n is odd. 

Some examples of molecules having inversion centers are octahedral AB6, planar AB4, planar and trans AB2C2, linear ABA, ethylene, and benzene. Two examples of otherwise fairly symmetrical molecules that do not have centers of inversion are C5H5 (plane pentagon) and tetrahedral AB4 (even though A is at the center and B s come in even numbers). 

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The two forms differ by the way they pack, a direct result being the different tilt angle of their molecular axis (24" and 30" for the low-temperature and high-temperature form, respectively). Another important difference is the fact that the inversion center of the molecule coincides with a center of symmetry of the unit cell in the HT form, whereas it does not in the LT form 84J. Direct consequences of this feature have not yet been identified. It will be of course of great interest to know what would be its influence on charge transport properties. 

The requirement for the existence of enantiomers is a chiral structure. Chirality is solely a symmetry property a rigid object is chiral if it is not superposable by pure rotation or translation on its image formed by inversion. Such an object contains no rotoinversion axis (or rotoreflection axis cf. Section 3.1). Since the reflection plane and the inversion center are special cases of rotoinversion axes (2 and 1), they are excluded. 

In organic stereochemistry the terms center of chirality or center of asymmetry are often used usually they refer to an asymmetrically substituted C atom. These terms should be avoided since they are contradictions in themselves a chiral object by definition has no center (the only kind of center existing in symmetry is the inversion center). 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

Carbon Monoxide. There are close similarities between carbon monoxide and nitrogen. The molecules are isoelectronic, and the bond lengths and dissociation energies are quite comparable. The phase diagrams of the two compounds show the same trends in the moderate pressure range with a variety of phase transitions between essentially alike crystal structures [333], when allowance is made for the lack of the inversion center and the presence of a weak electric dipole moment in carbon monoxide. However, the behavior and stability at higher... 

In biphenyls bridged at the 2 and 1 positions, the inversion center of the ideal planar biphenyl is removed and transitions can be allowed in both IPA and 2PA cases. As a consequence, the IPA and 2PA peaks are observed at approximately the same energy for fluorene (the lowest 2PA peak is located at 586 nm) [52]. Similar observations hold true for carbazole, dibenzofuran, and dibenzothiophene. However, the fine details of the spectra are hard to interpret. In a number of cases, 2PA peaks actually appear at the edge of the timing... 

The class of compounds that has been most extensively investigated from the point of view of two-photon absorption is that of so-called quadrupolar chromophores. hi essence, these molecifles are linear conjugated chains with electron donating or withdrawing substituents arranged symmetrically with respect to the center of the molecifle (Fig. 8, classes I-IV). With the inversion center being preserved, the lowest order moment supported by these molecules is the quadrupole moment. 

Where are the inversion centers in a triclinic lattice How many distinct ones are there ... 

Ostensibly, only allowed transitions should be observed experimentally. In many cases, however, transitions are observed which formally are forbidden. This is not as disastrous as it would appear. Usually it is our model of the molecular structure which is wrong we assume a static molecular skeleton and forget that vibrations can change this firm geometry and allow the molecule to have other structures. These other structures have different symmetry elements from those we worked with and give new and different selection rules. For example, we could destroy the inversion center and remove the parity restriction. 

The inversion operation is carried out by joining a point to the inversion center (or center of symmetry) and extending it an equal distance to arrive at an equivalent point. Molecules which possess an inversion center are termed centrosymmetric. Among the eight examples given so far, SF6 (Fig. 6.1.4), cyclohexane (Fig. 6.1.5), trans-N2F2 (Fig. 6.1.6), and BrFJ (Fig. 6.1.8) are centrosymmetric systems. Molecules lacking an inversion center are called non-centrosymmetric. 

As mentioned earlier, in a centrosymmetric complex, d-d transitions are Laporte forbidden. The fact that they are observed at all is due to a mechanism called vibronic interaction, which is a mixing of the vibrational and electronic wave-functions. Qualitatively, we may imagine that an electronic transition occurs at the very moment some vibrational modes of the complex distort the molecule in such a way that the center of symmetry is destroyed. When such a vibration takes place, the g character of the state is lost and the transition becomes (very slightly) allowed. Figure 8.10.1 shows two vibrations, with u symmetry, of an octahedral complex which remove the inversion center. 

Therefore, in compliance with the Law of Rational Indices, only n-axes with n = 1,2,3,4 and 6 are allowed in crystals. The occurrence of the inversion center means that the rotation-inversion axes I, 2(= m), 3, 4 and 6 are also possible. 

Fig. 6. Examples of electron micrographs of the inverse centered cubic structure38). Copolymer polystyrene-polyisoprene-polystyrene SIS. 1107 containing 10% polystyrene. White circles are polystyrene spheres.

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