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Molecular Symmetry 6 Molecular symmetry 6 Symmetry governs the bonding and hence the physical and spectroscopic properties of molecules. An introduction to symmetry In this chapter we explore some of the consequences of molecular symmetry and introduce the analysis systematic arguments of group theory. We shall see that symmetry considerations are essential for 6.1 Symmetry operations, elements constructing molecular orbitals and analysing molecular vibrations. They also enable us to extract and point groups information about molecular and electronic structure from spectroscopic data. 6.2 Character tables The systematic treatment of symmetry makes use of a branch of mathematics called group Applications of symmetry theory. Group theory is a rich and powerful subject, but we shall confine our use of it at 6.3 Polar molecules this stage to the classification of molecules in terms of their symmetry properties, the con- 6.4 Chiral molecules struction of molecular orbitals, and the analysis of molecular vibrations and the selection 6.5 Molecular vibrations rules that govern their excitation. We shall also see that it is possible to draw some general conclusions about the properties of molecules without doing any calculations at all. The symmetries of molecular orbitals 6.6 Symmetry-adapted linear combinations 6.7 The construction of molecular orbitals An introduction to symmetry analysis 6.8 The vibrational analogy That some molecules are ‘more symmetrical’ than others is intuitively obvious. Our aim Representations though, is to define the symmetries of individual molecules precisely, not just intuitively, 6.9 The reduction of a representation and to provide a scheme for specifying and reporting these symmetries. It will become 6.10 Projection operators clear in later chapters that symmetry analysis is one of the most pervasive techniques in inorganic chemistry. FURTHER READING EXERCISES PROBLEMS 6.1 Symmetry operations, elements and point groups Key points: Symmetry operations are actions that leave the molecule apparently unchanged; each sym- metry operation is associated with a symmetry element. The point group of a molecule is identified by noting its symmetry elements and comparing these elements with the elements that define each group. A fundamental concept of the chemical application of group theory is the symmetry opera- C tion, an action, such as rotation through a certain angle, that leaves the molecule apparent- 2 ly unchanged. An example is the rotation of an H2O molecule by 180º around the bisector * of the HOH angle (Fig. 6.1). Associated with each symmetry operation there is a symmetry element, a point, line, or plane with respect to which the symmetry operation is performed. Table 6.1 lists the most important symmetry operations and their corresponding elements. 180° All these operations leave at least one point unchanged (the centre of the molecule), and hence they are referred to as the operations of point-group symmetry. The identity operation, E, consists of doing nothing to the molecule. Every molecule has at least this operation and some have only this operation, so we need it if we are to classify all molecules according to their symmetry. * The rotation of an H2O molecule by 180º around a line bisecting the HOH angle (as in Fig. 6.1) is a symmetry operation, denoted C . In general, an n-fold rotation is a symmetry 2 Figure 6.1 An H2O molecule may be operation if the molecule appears unchanged after rotation by 360º/n. The corresponding rotated through any angle about the n C symmetry element is a line, an -fold rotation axis, n, about which the rotation is per- bisector of the HOH bond angle, but only a C rotation of 180Њ (the C operation) leaves it formed. There is only one rotation operation associated with a 2 axis (as in H2O) because 2 apparently unchanged. clockwise and anticlockwise rotations by 180º are identical. The trigonal-pyramidal NH3 180 6 Molecular symmetry * Table 6.1 Symmetry operations and symmetry elements C Symmetry operation Symmetry element Symbol 120° 3 Identity ‘whole of space’ E Њ nn C Rotation by 360 / -fold symmetry axis n Reflection mirror plane ␴ Inversion centre of inversion i Њ n n S Rotation by 360 / followed by -fold axis of improper rotation* n reflection in a plane perpendicular to the rotation axis S ␴ S i *Note the equivalences 1 = and 2 = . 120° * C3 C molecule has a threefold rotation axis, denoted 3, but there are now two operations associ- ated with this axis, one a clockwise rotation by 120º and the other an anticlockwise rotation C C 2 by 120º (Fig. 6.2). The two operations are denoted 3 and 3 (because two successive clock- 2 wise rotations by 120º are equivalent to an anticlockwise rotation by 120º), respectively. C3 C The square-planar molecule XeF4 has a fourfold 4 axis, but in addition it also has two C C Ј * pairs of twofold rotation axes that are perpendicular to the 4 axis: one pair ( 2 ) passes trans C Љ through each -FXeF unit and the other pair ( 2 ) passes through the bisectors of the FXeF angles (Fig. 6.3). By convention, the highest order rotational axis, which is called the Figure 6.2 A threefold rotation and the z corresponding C axis in NH . There are principal axis, defines the -axis (and is typically drawn vertically). 3 3 The reflection of an H O molecule in either of the two planes shown in Fig. 6.4 is a two rotations associated with this axis, 2 one through 120Њ (C ) and one through symmetry operation; the corresponding symmetry element, the plane of the mirror, is a 3 ␴ 240Њ (C 2). mirror plane, . The H2O molecule has two mirror planes that intersect at the bisector of 3 the HOH angle. Because the planes are ‘vertical’, in the sense of containing the rotational z ␴ ␴ Ј ( ) axis of the molecule, they are labelled with a subscript v, as in v and v . The XeF4 mol- ␴ ecule in Fig. 6.3 has a mirror plane h in the plane of the molecule. The subscript h signifies that the plane is ‘horizontal’ in the sense that the vertical principal rotational axis of the molecule is perpendicular to it. This molecule also has two more sets of two mirror planes v that intersect the fourfold axis. The symmetry elements (and the associated operations) are h ␴ ␴ denoted v for the planes that pass through the F atoms and d for the planes that bisect the angle between the F atoms. The d denotes ‘dihedral’ and signifies that the plane bisects d C Ј the angle between two 2 axes (the FXeF axes). To understand the inversion operation, i, we need to imagine that each atom is project- ed in a straight line through a single point located at the centre of the molecule and then C4 C2´ out to an equal distance on the other side (Fig. 6.5). In an octahedral molecule such as SF6, with the point at the centre of the molecule, diametrically opposite pairs of atoms at the corners of the octahedron are interchanged. The symmetry element, the point through C2” 1 Figure 6.3 Some of the symmetry elements of a square-planar molecule * 5 2 v such as XeF4. i 3 * 4 6 6 4 v' 3 * * 2 5 Figure 6.4 The two vertical mirror planes 1 ␴ ␴ Ј v and v in H2O and the corresponding operations. Both planes cut through the Figure 6.5 The inversion operation and the C centre of inversion i in SF . 2 axis. 6 An introduction to symmetry analysis 181 i which the projections are made, is called the centre of inversion, . For SF6, the centre of inversion lies at the nucleus of the S atom. Likewise, the molecule CO2 has an inversion centre at the C nucleus. However, there need not be an atom at the centre of inversion: an N2 molecule has a centre of inversion midway between the two nitrogen nuclei. An H2O molecule does not possess a centre of inversion. No tetrahedral molecule has a centre of inversion. Although an inversion and a twofold rotation may sometimes achieve the same (a) i effect, that is not the case in general and the two operations must be distinguished (Fig. 6.6). C An improper rotation consists of a rotation of the molecule through a certain an- 2 gle around an axis followed by a reflection in the plane perpendicular to that axis i (Fig. 6.7). The illustration shows a fourfold improper rotation of a CH4 molecule. In this case, the operation consists of a 90º (that is, 360°/4) rotation about an axis bisect- ing two HCH bond angles, followed by a reflection through a plane perpendicular to C the rotation axis. Neither the 90º ( 4) operation nor the reflection alone is a symmetry operation for CH4 but their overall effect is a symmetry operation. A fourfold improper S S S (b) rotation is denoted 4. The symmetry element, the improper-rotation axis, n ( 4 in the C2 example), is the corresponding combination of an n-fold rotational axis and a perpen- dicular mirror plane. S An 1 axis, a rotation through 360º followed by a reflection in the perpendicular plane, S ␴ ␴ is equivalent to a reflection alone, so 1 and h are the same; the symbol h is generally used S S rather than 1. Similarly, an 2 axis, a rotation through 180º followed by a reflection in the perpendicular plane, is equivalent to an inversion, i (Fig. 6.8); the symbol i is employed S Figure 6.6 Care must be taken not to rather than 2. confuse (a) an inversion operation with (b) a twofold rotation.
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