Benzene symmetry operations

Let us eonsider the vibrational motions of benzene. To eonsider all of the vibrational modes of benzene we should attaeh a set of displaeement veetors in the x, y, and z direetions to eaeh atom in the moleeule (giving 36 veetors in all), and evaluate how these transform under the symmetry operations of D6h- For this problem, however, let s only inquire about the C-H stretehing vibrations. 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

The molecules shown in Fig. 1 are planar thus, the paper on which they are drawn is an element of symmetry and the reflection of all points through the plane yields an equivalent (congruent) structure. The process of carrying out the reflection is referred to as the symmetry operation a. However, as the atoms of these molecules are essentially point masses, the reflection operations are in each case simply the inversion of the coordinate perpendicular to the plane of symmetry. Following certain conventions, the reflection operation corresponds to z + z for BF3 and benzene, as it is the z axis that is chq ep perpendicular to die plane, while it is jc —> —x for water. It should be evident that the symmetry operation has an effect on the chosen coordinate systems, but not on the molecule itself. 

Although all molecules are in constant thermal motion, when all of their atoms are at their equilibrium positions, a specific geometrical structure can usually be assigned to a given molecule. In this sense these molecules are said to be rigid. The first step in the analysis of the structure of a molecule is the determination of the group of operations that characterizes its symmetry. Each symmetry operation (aside from the trivial one, E) is associated with an element of symmetry. Thus for example, certain molecules are said to be planar. Well known examples are water, boron trifluoride and benzene, whose structures can be drawn on paper in the forms shown in Fig. 1. 

Character tables for many point groups are listed in Section 9.12 at the end of this chapter. As an example, consider the character table for 6D6/, the point group of benzene. This group is of order 24. The symmetry operations are found to be divided into the 12 classes (1) E (2) C6, C ... 

Scheme 33. Generation of the Twin States by Mixing of the Kekule Structures for Benzene (a) and Cyclobutadiene (b) as Archetypal Aromatic and Antiaromatic Species, (c) Transformation Properties of Ki and K2 with Respect to Symmetry Operations of the Dgh and Dih Point Groups...

The three classical Kekule structures (already alluded to in section III.E) of naphthalene are shown in Scheme 36a. Two of them are designated as Ki and K2 and represent the annulenic resonance along the perimeter of the naphthalene, while the third one, Kc, has a double bond in the center and transforms as the totally symmetric irreducible representation, Ag of the Dzh group. The Ki and K2 structures are mutually interchangeable by the i, C2, and ov symmetry operations of the point group, much as in the case of benzene. An in-phase combination transforms, therefore, as Ag, whereas an out-of-phase one transforms as B2u. These symmetry adapted wave func-... 

The number of totally symmetric Heilbronner modes follows from counting of orbits. An orbit is a set of equivalent (structureless) objects, which are permuted amongst themselves by symmetry operations of the group every operation of the group either leaves a given member of the orbit in place and unchanged, or moves it to another location. Thus the six edges of benzene, the two face centres of pentalene and the pair of terminal vertices of an [w]-polyene chain, all form orbits. Each orbit has an associated permutation representation that contains the totally symmetric representation Jo exactly once. 

Figure 7.7a shows the four classical Kekule structures of anthracene (16,19,24). Two of the structures involve resonance in the central benzenic ring and are therefore labeled as KiB and K2B. The other two involve annulenic resonance along the molecule perimeter, and are labeled accordingly as K1A and K2a- The structures of the types A and B form two symmetry subsets, and within each subset, the two structures are mutually transformable by the D2h symmetry operations (i, C2, and ov). Therefore, as shown in Fig. 7.7b, within each subset there will be a positive combination that transforms as Ag and a negative one that transforms as B2u. 

A symmetry operation is the actual event of converting one item into another apparently identical item. This symmetry operation can take place about a point, a line, or a plane of symmetry. Thus, an object is symmetrical when a symmetry operation applied to it gives an object indistinguishable from the original. The point, line or plane about which the symmetry operation is performed is defined as a symmetry element. For example, an axis of symmetry is a symmetry element that describes a line about which all parts of an object are symmetrically disposed. We are all familiar with the vertical reflection plane (or near reflection plane) in the human face. We are also familiar with the sixfold axis of symmetry that lies perpendicular to the plane of a benzene molecule and passes through the center of it (Figure 4.2). 

FIGURE 11.6. (a) Intramolecular and (b) intermolecular distances in A in crystalline-benzene. In (b) the symmetry operation for each carbon atom is shown. 

FIGURE 11.6 (cont d). (c) View onto a benzene molecule in the crystal structure, showing the intermolecular interactions listed in (b). Carbon and hydrogen atoms drawn with filled circles are those in one asymmetric unit. Positions of all other atoms are generated from these by application of the symmetry operations of the space group. 

Crystals of tetracene and pentacene are triclinic, with two molecules per unit cell. The dimensions of the unit cell in the directions of the a and b axes are similar to those of naphthalene and anthracene the dimension in the direction of the c-axis increases proportionally to the number of benzene rings in the molecule. In all four crystal structures the long axes of both molecules in the unit cell are approximately parallel to the c crystal axis. However, in the triclinic structures of tetracene and pentacene, the only crystal symmetry operation is inversion. The results of calculation for tetracene crystals are shown in Table 3.6. Analogous results for pentacene can be found in (60), (61). The experimental data of the splitting cm-1 for the io-o transition in the tetracene crystal are in the interval 600-700 cm-1. For the /q o transition in a crystal of pentacene the... 

The final symmetry operation in this Book is inversion through a centre of symmetry. You met this operation when you were studying homonuclear diatomic molecules, but it is not confined to diatomic molecules. Structure 6.7, for example, has a centre of symmetry in the middle of the benzene ring ... 

As a first example, we shall discuss (Figure 3, Thble 1) the benzene molecule (planar regular hexagcm) which has 12 symmetry operations (expressed as cyclic permutations). 

The third point symmetry operation is reflection, which occurs when an equivalent configuration results if the object is reflected an equal distance through a mirror plane. The reflection operation is given the symbol a. The benzene molecule, for instance, has a rather obvious mirror plane that lies in the plane of the molecule itself and is perpendicular to the principal axis, as shown in Figure 8.2(a). Benzene also exhibits a set of mirror planes that contain the principal axis and pass through... 

In many applications the positions of the nuclei are fixed (clamped nuclei approximation, Chapter 6), often in a high-symmetry configuration (cf. Appendix C, p. 903). For example, the benzene molecule in its ground state (after minimizing the energy with respect to the positions of the nuclei) has the symmetry of a regular hexagon. In such cases the electronic Hamiltonian additionally exhibits invariance with respect to some symmetry operations and therefore the wave functions are... 

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