Plane of symmetry, and

We next turn to monoclinic lattices, of which there are two types. A monoclinic lattice is one in which we require one vector to be perpendicular to the plane of the other two. The lattice then has twofold rotational symmetry about this unique vector and planes of symmetry perpendicular to it. A monoclinic lattice (so-called because there is only one nonorthogonal pair of... 

Chromophores which are asymmetric by nature are characterized by the absence of a center and plane of symmetry in the group of atoms participating in the optical transition. The rotational strength of these are usually larger when compared with chromophores that become optically active due to substitution. This is demonstrated in Mason and Schnepp s8 study of trans-cyclooctene, a-pinene and /1-pi none. They pointed out that the g (anisotropy factor, g = Ae/e) value of the major bands in trans -cyclooctene is relatively high as expected for an intrinsically asymmetric chromophore when compared with the other two olefins. 

The structure of BF3 is a trigonal plane as shown here with the rotation axes and planes of symmetry as indicated ... 

In this theory we abstract from the molecule its system of axes and planes of symmetry with their corresponding symmetry operations. The structure and properties of the symmetry group of the molecule depend only on the relations between its elements, the symmetry operations and these relations are completely determined by the spatial relations between the axes and planes of symmetry. Any two molecule no matter how different in form or complexity, which have the same system of axes and planes of symmetry will have the same symmetry group and those of their properties which depend on symmetry will be the same. 

This is one of the reasons for the power and generality of group theoretical methods in discussing the properties of molecules for although the number of different imaginable molecules is unbounded, this is not true of their possible systems of axes and planes of symmetry. These are severely restricted by geometrical considerations and it is possible to write down a list of all the molecular symmetry groups that can exist and to discuss... 

Before going on to discuss the molecular symmetry groups in more detail we note one feature that they all possess. A symmetry operation which rotates or reflects a molecule into itself must leave the centre of mass (centre of gravity) of the molecule unmoved if the molecule has a plane or axis of symmetry, the centre of mass must lie on this plane or axis. It follows that all the axes and planes of symmetry of a molecule must intersect in at least one common point and that at least one point remains fixed under all the symmetry operations of the molecule. For this reason, the symmetry group of molecule is generally referred to as its point group and we shall use this name, which is taken over from crystallography, from now on. 

Group D. Tf the horizontal plane containing the twofold axes of Dn is also a plane of symmetry, then n vertical planes containing the main axis of symmetry and a twofold axis are necessarily also planes of symmetry. This system of axes and planes of symmetry gives rise to the group Dnh which, in addition to the elements of... 

Cl (asymmetric) C (dissymmetric) D (dissymmetric) Cj (plane of symmetry) C( (center of symmetry) D (, (plane of symmetry) D i (plane of symmetry) S (improper axis) Tj (plane of symmetry) Oi, (center and plane of symmetry) //, (center and plane of symmetry) C (plane of symmetry)... 

Attention should perhaps be drawn to the characteristic symmetry of the cubic system which is not, as might be supposed, the 4-fold (or 2-fold) axes of symmetry or planes of symmetry but four 3-fold axes parallel to the body-diagonals of the cubic unit cell. This combination of inclined 3-fold axes introduces either three 2-fold or three 4-fold axes which are mutually perpendicular and parallel to the cubic axes. Further axes and planes of symmetry may be present but are not essential to cubic symmetry and do not occur in all the cubic point groups or space groups. 

We saw in Chapter 2 that the characteristic (minimum) symmetry of a cubic crystal is the set of 3-fold axes parallel to the body-diagonals of the cubic cell. These 3-fold axes also imply 2-fold axes parallel to the cube edges. The NaCl structure has the highest class of cubic symmetry, with 4-fold axes and planes of symmetry. An octahedral ion such as (TlCl ) also has full cubic symmetry (m3m) can occupy the Cl positions in the normal NaCl structure. Groups such as S2, (F—H-F) , and CN have lower symmetry and can form the fully symmetrical NaCl structure only if they are rotating or are randomly oriented with their centres... 

Spotting meso compounds and planes of symmetry is often considered difficult at first, and it is also often observed that the task becomes easier with practice. 

Table 3.3 shows the character table for the Cgv point group. The NH3 molecule possesses C3V symmetry, and worked example 3.2 illustrated the principal axis of rotation and planes of symmetry in NH3. In the character table, the presence of three cr planes in NH3 is represented by the notation 3ctv in the top line of the table. The notation 2C3 summarizes the two operations C3 and C3. The operation C3 is equivalent to the identity operator, E, and so is not specihed again. 

Figure 3.4 showed the proper axes of rotation and planes of symmetry in the square planar molecule XeF4. This has 7)41, symmetry. The 7)4], character table is given in Appendix 3, and the top row of the character table that summarizes the symmetry operations for this point group is as follows ... 

Another commonly used criterion for identifying a chiral species is the lack of an inversion centre, i, and plane of symmetry, a. However, both of these properties are compatible with the criterion given above, since we can rewrite the symmetry operations i and a in terms of the improper rotations S2 and Si respectively. (See problem 3.25 at the end of the chapter.) However, a word of caution there are a few species that are non-chiral (achiral) despite lacking an inversion centre, i, and plane of symmetry, a. 

It must be remembered, however, that while some crystals may possess a centre and several different axes and planes of symmetry, others may have no element of symmetry at all. 

Table 4.1 Characteristic S5mmetry elements of some important classes of point groups. The characteristic symmetry elements of the I d, 0[, and 4 are omitted because the point groups are readily identified (see Figures 4.8 and 4.9). No distinction is made in this table between and planes of symmetry. For complete lists of symmetry elements, eharaeter tables (Appendix 3) should be consulted.

As soon as we put together an Sc with chiral molecules, the centre of inversion and plane of symmetry are removed, leaving only the axis C2. Imagining a macroscopic electric polarisation in this direction, no symmetry operation can cancel it out. In these conditions, Curie s principle tells us that this polarisation must exist in a generic manner. This observation, made by R.B. Meyer in... 

In order to predict the number of expected signals in the H NMR spectrum of a compound, it is not necessary to compare all of the protons and drive yourself crazy looking for axes and planes of symmetry. In general, it is possible to determine the number of expected signals for a compound using a few simple rules ... 

Problem 1.1 In Section 1.2.3, Figure 1.19 shows the structure of the square planar complex [PtCL,], And and label all the proper rotation axes and planes of symmetry for this structure. Remember to consider the full set of operations for high-order axes. 

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