Chapter 3 An introduction to molecular symmetry TOPICS & Symmetry operators and symmetry elements & Point groups & An introduction to character tables & Infrared spectroscopy & Chiral molecules the resulting structure is indistinguishable from the first; 3.1 Introduction another 1208 rotation results in a third indistinguishable molecular orientation (Figure 3.1). This is not true if we Within chemistry, symmetry is important both at a molecu- carry out the same rotational operations on BF2H. lar level and within crystalline systems, and an understand- Group theory is the mathematical treatment of symmetry. ing of symmetry is essential in discussions of molecular In this chapter, we introduce the fundamental language of spectroscopy and calculations of molecular properties. A dis- group theory (symmetry operator, symmetry element, point cussion of crystal symmetry is not appropriate in this book, group and character table). The chapter does not set out to and we introduce only molecular symmetry. For qualitative give a comprehensive survey of molecular symmetry, but purposes, it is sufficient to refer to the shape of a molecule rather to introduce some common terminology and its using terms such as tetrahedral, octahedral or square meaning. We include in this chapter an introduction to the planar. However, the common use of these descriptors is vibrational spectra of simple inorganic molecules, with an not always precise, e.g. consider the structures of BF3, 3.1, emphasis on using this technique to distinguish between pos- and BF2H, 3.2, both of which are planar. A molecule of sible structures for XY2,XY3 and XY4 molecules. Complete BF3 is correctly described as being trigonal planar, since its normal coordinate analysis of such species is beyond the symmetry properties are fully consistent with this descrip- remit of this book. tion; all the FÿBÿF bond angles are 1208 and the BÿF bond distances are all identical (131 pm). It is correct to say that the boron centre in BF2H, 3.2,isinapseudo-trigonal planar environment but the molecular symmetry properties 3.2 Symmetry operations and symmetry are not the same as those of BF3. The FÿBÿF bond angle elements in BF2H is smaller than the two HÿBÿF angles, and the BÿH bond is shorter (119 pm) than the BÿF bonds In Figure 3.1, we applied 1208 rotations to BF and saw that (131 pm). 3 each rotation generated a representation of the molecule that F H was indistinguishable from the first. Each rotation is an example of a symmetry operation. 120˚ 120˚ 121˚ 121˚ B B A symmetry operation is an operation performed on an FF FF 120˚ 118˚ object which leaves it in a configuration that is (3.1) (3.2) indistinguishable from, and superimposable on, the original configuration. The descriptor symmetrical implies that a species possesses a number of indistinguishable configurations. When struc- The rotations described in Figure 3.1 were performed ture 3.1 is rotated in the plane of the paper through 1208, about an axis perpendicular to the plane of the paper and 80 Chapter 3 . An introduction to molecular symmetry F 120˚ rotationF 120˚ rotation F B B B FF F F F F Fig. 3.1 Rotation of the trigonal planar BF3 molecule through 1208 generates a representation of the structure that is indistinguishable from the first; one F atom is marked in red simply as a label. A second 1208 rotation gives another indistinguishable structural representation. passing through the boron atom; the axis is an example of a example, in square planar XeF4, the principal axis is a C4 symmetry element. axis but this also coincides with a C2 axis (see Figure 3.4). Where a molecule contains more than one type of Cn axis, A symmetry operation is carried out with respect to points, they are distinguished by using prime marks, e.g. C2, C2’ and lines or planes, the latter being the symmetry elements. C2’’. We return to this in the discussion of XeF4 (see Figure 3.4). Rotation about an n-fold axis of symmetry The symmetry operation of rotation about an n-fold axis Self-study exercises (the symmetry element) is denoted by the symbol C ,in n 1. Each of the following contains a 6-membered ring: benzene, 3608 which the angle of rotation is ; n is an integer, e.g. 2, 3 borazine (see Figure 12.19), pyridine and S6 (see Box 1.1). n Explain why only benzene contains a 6-fold principal rotation or 4. Applying this notation to the BF3 molecule in Figure axis. 3.1 gives a value of n ¼ 3 (equation 3.1), and therefore we 2. Among the following, why does only XeF4 contain a 4-fold say that the BF molecule contains a C rotation axis;in ÿ 3 3 principal rotation axis: CF ,SF, [BF ] and XeF ? this case, the axis lies perpendicular to the plane containing 4 4 4 4 ÿ C the molecule. 3. Draw the structure of [XeF5] . On the diagram, mark the 5 axis. The molecule contains five C axes. Where are these 3608 2 Angle of rotation ¼ 1208 ¼ ð3:1Þ axes? [Ans. for structure, see worked example 1.14] n 4. Look at the structure of B H in Figure 12.23b. Where is the In addition, BF also contains three 2-fold (C ) rotation 5 9 3 2 C axis in this molecule? axes, each coincident with a BÿF bond as shown in Figure 4 3.2. If a molecule possesses more than one type of n-axis, the axis of highest value of n is called the principal axis;itis Reflection through a plane of symmetry the axis of highest molecular symmetry. For example, in (mirror plane) BF , the C axis is the principal axis. 3 3 If reflection of all parts of a molecule through a plane In some molecules, rotation axes of lower orders than the produces an indistinguishable configuration, the plane is a principal axis may be coincident with the principal axis. For plane of symmetry; the symmetry operation is one of reflec- tion and the symmetry element is the mirror plane (denoted by ). For BF3, the plane containing the molecular frame- work (the yellow plane shown in Figure 3.2) is a mirror plane. In this case, the plane lies perpendicular to the vertical principal axis and is denoted by the symbol h. The framework of atoms in a linear, bent or planar molecule can always be drawn in a plane, but this plane can be labelled h only if the molecule possesses a Cn axis perpen- dicular to the plane. If the plane contains the principal axis, it is labelled v. Consider the H2O molecule. This possesses a C2 axis (Figure 3.3) but it also contains two mirror planes, one containing the H2O framework, and one perpendicular to it. Each plane contains the principal axis of rotation and so may be denoted as v but in order to distinguish between them, we use the notations v and v’.Thev label refers to the plane that bisects the HÿOÿH bond angle and the v’ label refers to the plane in which the molecule lies. Fig. 3.2 The 3-fold (C3) and three 2-fold (C2) axes of A special type of plane which contains the principal symmetry possessed by the trigonal planar BF3 molecule. rotation axis, but which bisects the angle between two Chapter 3 . Symmetry operations and symmetry elements 81 adjacent 2-fold axes, is labelled d. A square planar molecule such as XeF4 provides an example. Figure 3.4a shows that XeF4 contains a C4 axis (the principal axis) and perpen- dicular to this is the h plane in which the molecule lies. Coincident with the C4 axis is a C2 axis. Within the plane of the molecule, there are two sets of C2 axes. One type (the C2’ axis) coincides with F–Xe–F bonds, while the second type (the C2’’ axis) bisects the F–Xe–F 908 angle (Figure 3.4). We can now define two sets of mirror planes: one type (v) contains the principal axis and a C2’ axis (Figure 3.4b), while the second type (d) contains the principal axis and a C2’’ axis (Figure 3.4c). Each d plane bisects the angle between two C2’ axes. In the notation for planes of symmetry, , the subscripts h, v and d stand for horizontal, vertical and dihedral Fig. 3.3 The H2O molecule possesses one C2 axis and respectively. two mirror planes. (a) The C2 axis and the plane of symmetry that contains the H2O molecule. (b) The C2 axis and the plane of symmetry that is perpendicular to the plane Self-study exercises of the H2O molecule. (c) Planes of symmetry in a molecule are often shown together on one diagram; this representation for 1. N2O4 is planar (Figure 14.14). Show that it possesses three H2O combines diagrams (a) and (b). planes of symmetry. Fig. 3.4 The square planar molecule XeF4. (a) One C2 axis coincides with the principal (C4) axis; the molecule lies in a h plane which contains two C2’ and two C2’’ axes. (b) Each of the two v planes contains the C4 axis and one C2’ axis. (c) Each of the two d planes contains the C4 axis and one C2’’ axis. 82 Chapter 3 . An introduction to molecular symmetry 2. B2Br4 has the following staggered structure: (3.7) (3.8) Show that B2Br4 has one less plane of symmetry than B2F4 Self-study exercises which is planar. 1. Draw the structures of each of the following species and 3. Ga2H6 has the following structure in the gas phase: confirm that each possesses a centre of symmetry: CS2, ÿ ÿ [PF6] , XeF4,I2, [ICl2] .