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An introduction to molecular symmetry
Within chemistry, symmetry is important both at a molecular level and within crystalline systems, and an understanding of symmetry is essential in discussions of molecular spectroscopy and calculations of molecular properties. A discussion of crystal symmetry is not appropriate in this book, and we introduce only molecular symmetry. For qualitative purposes, it is suﬃcient to refer to the shape of a molecule using terms such as tetrahedral, octahedral or square planar. However, the common use of these descriptorsis not always precise, e.g. consider the structures of BF3, 4.1, and BF2H,
4.2, both of which are planar. A molecule of BF3 is correctly described as being trigonal planar, since its symmetry properties are fully consistent with this description; all the F B F bond angles are 1208 and the B F bond distances are all identical (131pm). It is correct to say that the boron centre in BF2H, 4.2,i s in a pseudo-trigonal planar environment but the molecular symmetry properties are not the same as those of BF3. The F B F bond angle in BF2Hi s smaller than the two H B F angles, and the B H bond is shorter (119pm) than the B F bonds (131pm).
The descriptor symmetrical implies that a species possesses a number of indistinguishable conﬁgurations. When structure 4.1 is rotated in the plane of the paper through 1208, the resulting structure is indistinguishable from the ﬁrst; another 1208 rotation results in a third indistinguishable molecular orientation (Figure 4.1). This is not true if we carry out the same rotational operations on BF2H. Group theory is the mathematical treatment of symmetry.
In this chapter, we introduce the fundamental language of group theory (symmetry operator, symmetry element, point group and character table). The chapter does not set out to givea comprehensivesurveyofmolecularsymmetry,butrather tointroducesome common terminologyand its meaning.We include in this chapteran introductionto the vibrationalspectra of simpleinorganicmolecules,for example, how to use this techniqueto distinguishbetween possible structuresfor XY2,
XY3 andXY4 molecules.Completenormalcoordinateanalysis of such speciesis beyond the remit of this book.
4.2 Symmetry operations and symmetry elements
In Figure 4.1, we applied 1208 rotations to BF3 and saw that each rotation generated a representation of the molecule that was indistinguishable from the ﬁrst. Each rotation is an example of a symmetry operation.
A symmetry operation is an operation performed on an object which leaves it in a conﬁguration that is indistinguishable from, and superimposable on, the original conﬁguration.
The rotations described in Figure 4.1 are performed about an axis perpendicular to the plane of the paper and passing through the boron atom; the axis is an example of a symmetry element.
A symmetry operation is carried out with respect to points, lines or planes, the latter being the symmetry elements.
& Symmetry operators and symmetry elements & Point groups & An introduction to character tables & Infrared spectroscopy & Chiral molecules
Rotation about an n-fold axis of symmetry
The symmetry operation of rotation about an n-fold axis (the symmetry element) is denoted by the symbol Cn,i n which the angle of rotation is 3608 or 4. Applying this notation to the BF3 molecule in Figure 4.1 gives a value of n ¼ 3 (equation 4.1), and therefore we say that the BF3 molecule contains a C3 rotation axis;i n this case, the axis lies perpendicular to the plane containing the molecule.
In addition, BF3 also contains three 2-fold (C2) rotation axes, each coincident with a B F bond as shown in Figure 4.2.
If a molecule possesses more than one type of n-axis, the axis of highest value of n is called the principal axis;i ti s the axis of highest molecular symmetry. For example, in
BF3, the C3 axis is the principal axis. In some molecules, rotation axes of lower orders than the principal axis may be coincident with the principal axis. For example, in square planar XeF4, the principal axis is a C4 axis but this also coincides with a C2 axis (see Figure 4.4).
Where a molecule contains more than one type of Cn axis with the same value of n, they are distinguished by using
1. Each of the following contains a 6-membered ring: benzene, borazine (see Figure 13.21), pyridine and S6 (see Box 1.1). Explain why only benzene contains a 6-fold principal rotation axis.
2. Among the following, why does only XeF4 contain a 4-fold principal rotation axis: CF4,S F4, [BF4] and XeF4?
3. Draw the structure of [XeF5] . On the diagram, mark the C5 axis. The molecule contains ﬁve C2 axes. Where are these axes? [Ans. for structure, see worked example 2.7]
4. Look at the structure of B5H9 in Figure 13.26a. Where is the C4 axis in this molecule?
Reflection through a plane of symmetry (mirror plane)
If reﬂection of all parts of a molecule through a plane produces an indistinguishable conﬁguration, the plane is a plane of symmetry; the symmetry operation is one of reﬂection and the symmetry element is the mirror plane (denoted by ). For BF3, the plane containing the molecular framework (the brown plane shown in Figure 4.2) 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 perpendicular to the plane. If the plane contains the principal axis, it is
labelled v. Consider the H2O molecule. This possesses a C2 axis (Figure 4.3) 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’.T he v label refers to the plane that bisects the H O H bond angle and the v’ label refers to the plane in which the molecule lies.
A special type of plane which contains the principal rotation axis, but which bisects the angle between two
Fig. 4.1 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.
Fig. 4.2 The 3-fold (C3) and three 2-fold (C2) axes of symmetry possessed by the trigonal planar BF3 molecule.
Chapter 4 . Symmetry operations and symmetry elements 89 adjacent 2-fold axes, is labelled d. A square planar molecule such as XeF4 provides an example. Figure 4.4a shows that XeF4 contains a C4 axis (the principal axis) and perpendicular 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 4.4). We can now deﬁne two sets of mirror planes:
one type ( v) contains the principal axis and a C2’ axis (Figure 4.4b), while the second type ( d) contains the principal axis and a C2’’ axis (Figure 4.4c). 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 respectively.
1. N2O4 is planar (Figure 15.15). Show that it possesses three planes of symmetry.
Fig. 4.3 The H2O molecule possesses one C2 axis and 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 of the H2O molecule. (c) Planes of symmetry in a molecule are often shown together on one diagram; this representation for
H2O combines diagrams (a) and (b).
Fig. 4.4 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.
90 Chapter 4 . An introduction to molecular symmetry
2. B2Br4 has the following staggered structure:
Show that B2Br4 has one less plane of symmetry than B2F4 which is planar.
3. Ga2H6 has the following structure in the gas phase:
Show that it possesses three planes of symmetry.
4. Show that the planes of symmetry in benzene are one h, three v and three d.
Reflection through a centre of symmetry (inversion centre)
If reﬂection of all parts of a molecule through the centre of the molecule produces an indistinguishable conﬁguration, the centre is a centre of symmetry, also called a centre of inversion (see also Box 2.1); it is designated by the symbol i.
1. Draw the structures of each of the following species and
2. [PtCl4]2 has a centre of symmetry, but [CoCl4]2 does not. One is square planar and the other is tetrahedral. Which is which?
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