Point factor group

Tabic 6-2. Correlation diagram of the C2/, point group of the isolated T6 molecule (left column) with the C2i, factor group for solid T(, (right column) via the site symmetry C group (center). L, M, and N indicate the principal molecular transition dipole moments, while a, b, and c arc the crystalline axes. 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

Table 2 Correlation of the molecular point group of Sg with the factor group of the orthorhombic crystal (E>4d—>C2— D2h) [88]...

Observed on the wing of the CS2 bending mode. Occurs in violation of the selection rules of the point group Dsd but is IR active under the Csi factor group of the crystal. Could also be a combination vibration or caused by the CS2 impurity which was present in the sample (see text)... 

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

Six comments are appropriate at this point. Firstly, it is the experience of the reviewer that chemically-similar compounds with very similar infrared and Raman spectra in factor-group split regions are isomorphous. This method is probably at least as reliable as X-ray powder methods when there is a significant change in scattering factors between the two compounds studied (e.g. bromo-derivatives of... 

Adams, D. M., Newton, D. C. Tables for factor group and point group analysis (Beckman-RIIC Limited)... 

Second, a multiplication table for the factor group is written down. The space group formed by the above symmetry elements is infinite, because of the translations. If we define the translations, which carry a point in one unit cell into the corresponding point in another unit cell, as equivalent to the identity operation, then the remaining symmetry elements form a group known as the factor, or unit cell, group. The factor... 

Both methods can easily be derived from a very simple model. Consider a unit cell which contains two XY molecules which are equivalent in the group-theoretical sense, i.e. they are transformed into one-another by the operations of the group of the unit cell (this group is the factor group of the space group 10 and is isomorphous with one of the 32 crystallographic point groups).8 ... 

In considering the vibronic side-bands to be expected in the optical spectra when we augment the static crystal field model by including the electron-phonon interaction, we must know the frequencies and symmetries of the lattice phonons at various critical points in the phonon density of states. We shall be particularly interested in those critical points which occur at the symmetry points T, A and at the A line in the Brillouin zone. Using the method of factor group for crystals we have ... 

It is necessary to define a factor group and to describe how it relates to a space group. In a crystal, one primitive cell or unit cell can be carried into another primitive cell or unit cell by a translation. The number of translations of unit cells then would seem to be infinite since a crystal is composed of many such units. If, however, one considers only one translation and consequently only two unit cells, and defines the translation that takes a point in one unit cell to an equivalent point in the other unit cell as the identity, one can define a finite group, which is called a factor group of the space group. 

The factor groups are isomorphic (one-to-one correspondence) with the 32 point groups and, consequently, the character table of the factor group can be obtained from the corresponding isomorphic point group. 

Point Group —> Site Group — Factor Group. 

We have illustrated the methods to obtain solid state selection rules. It should be mentioned that tables for factor group or point group analyses have been prepared by Adams and Newton (56,57) where one can read the number and type of species allowed directly from the table. Although useful, the approach neglects the procedures as how to obtain results in the tables. For further examples of the correlation method, see Refs. 58-61, and the Correlation Theory Bibliography. 

The factor group of any space group is isomorphic to one of the 32 point groups. The procedure for the classification of vibrations of free molecules can also be applied accordingly for the classification of the vibrations of molecular crystals, as explained by Table 2.7-1. 

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