Molecular Symmetry 6

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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 symmetry 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 180 (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 nn C Rotation by 360 / -fold symmetry axis n Reflection mirror plane Inversion centre of inversion i 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 associated 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 * pairs of twofold rotation axes that are perpendicular to the 4 axis: one pair ( 2 ) passes trans C 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 120 (C ) and one through symmetry operation; the corresponding symmetry element, the plane of the mirror, is a 3 ␴ 240 (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 ␴ ␴ ( ) 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 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 ␴ ␴ 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 bisecting 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 perpendicular 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. Although the two operations may sometimes appear to have EXAMPLE 6.1 Identifying symmetry elements the same effect, that is not the case in general. Identify the symmetry elements in the eclipsed and staggered conformations of an ethane molecule. Answer We need to identify the rotations, reflections, and inversions that leave the molecule apparently unchanged. Don’t forget that the identity is a symmetry operation. By inspection of the molecular models, E C C ␴ ␴ S we see that the eclipsed conformation of a CH3CH3 molecule (1) has the elements , 3, 2, h, v, and 3. E C ␴ i S The staggered conformation (2) has the elements , 3, d, , and 6. H Self-test 6.1 Sketch the S axis of an NH ion. How many of these axes does the ion possess? 4 4 C

1 A C3 axis S (1) Rotate 1

C4 (2) Reflect H

σ C (a)

S2 S (1) Rotate 6

(2) Reflect h 2 An S axis i 6

(b) S Figure 6.8 (a) An 1 axis is equivalent to a Figure 6.7 A fourfold axis of improper S mirror plane and (b) an 2 axis is equivalent S rotation 4 in the CH4 molecule. to a centre of inversion. 182 6 Molecular symmetry

The assignment of a molecule to its point group consists of two steps: 1. Identify the symmetry elements of the molecule. 2. Refer to Table 6.2.

Table 6.2 The composition of some common groups

Point group Symmetry elements Shape Examples C E SiHClBrF 1

C E C H O 2 , 2 2 2

C E ␴ NHF s , 2

C E C ␴ ␴ SO Cl , H O 2v , 2, v, v 2 2 2

C E C ␴ NH , PCl , POCl 3v , 2 3, 3 v 3 3 3

C E C C ␴ OCS, CO, HCl ∞v , 2, 2 ␸, ∞ v D E C i ␴ N O , B H 2h , 3 2, , 3 2 4 2 6

D E C C ␴ ␴ BF , PCl 3h , 2 3, 3 2, h, 2S3, 3 v 3 5

D XeF , 4h E C C 2C ′ 2C ′′ i S ␴ ␴ ␴ 4 , 2 4, 2, 2 , 2 , , 2 4, h, 2 v, 2 d trans -[MA4B2]

D E C C i ␴ S CO , H , C H ∞h , ∞ 2 , 2 ␸, , ∞ v, 2 ␸ 2 2 2 2

T E C C S ␴ CH , SiCl d , 8 3, 3 2, 6 4, 6 d 4 4

O E C C C C i S S ␴ ␴ SF h , 8 3, 6 2, 6 4, 3 2, , 6 4, 8 6, 3 h, 6 d 6

In practice, the shapes in the table give a very good clue to the identity of the group to which the molecule belongs, at least in simple cases. The decision tree in Fig. 6.9 can also be used to assign most common point groups systematically by answering the questions at each decision point. The name of the point group is normally its Schoenflies symbol, such C as 2v for a water molecule.

EXAMPLE 6.2 Identifying the point group of a molecule

To what point groups do H2O and XeF4 belong? Answer We need to work through Fig. 6.9. (a) The symmetry elements of H2O are shown in Fig. 6.10. H2O E C ␴ ␴ possesses the identity ( ), a twofold rotation axis ( 2), and two vertical mirror planes ( v and v ). The set An introduction to symmetry analysis 183

E,C ␴ C of elements ( 2, v, v ) corresponds to the group 2v. (b) The symmetry elements of XeF4 are shown in E C Fig. 6.3. XeF4 possesses the identity ( ), a fourfold axis ( 4), two pairs of twofold rotation axes that are C ␴ perpendicular to the principal 4 axis, a horizontal reflection plane h in the plane of the paper, and two ␴ ␴ D sets of two vertical reflection planes, v and d. This set of elements identifies the point group as 4h. C2 Self-test 6.2 Identify the point groups of (a) BF3, a trigonal-planar molecule, and (b) the tetrahedral 2– SO4 ion. v'

v Y N Cn? Molecu Y σ N h? Select C with v Y n N v' highest n; then is Y N nC ⊥ C ? Y N Linear? 2 n i? Figure 6.10 The symmetry elements of H O. The diagram on the right is the view Y N Y N 2 Y N Y N σ ? σ ? i? Two or h h from above and summarizes the diagram on more Y σ N Y σ N the left. Cn, n ? n ? C d v D∞h ∞v n > 2? Y N Y N Linear groups i? S2n? D D D C C C C C Y N nh nd n nh nv s i 1 O C C5? S 2n Cn

Ih Oh Td 3 CO (D ) Cubic groups 2 ∞h Figure 6.9 The decision tree for identifying a molecular point group. The symbols of each point refer to the symmetry elements. O C S

It is very useful to be able to recognize immediately the point groups of some common molecules. Linear molecules with a centre of symmetry, such as H , CO (3), and HCϵCH 2 2 4 OCS (C ) D ∞v belong to h. A molecule that is linear but has no centre of symmetry, such as HCl or C T O OCS (4) belongs to v. Tetrahedral ( d) and octahedral ( h) molecules have more than

one principal axis of symmetry (Fig. 6.11): a tetrahedral CH4 molecule, for instance, has C O T four 3 axes, one along each CH bond. The h and d point groups are known as cubic groups because they are closely related to the symmetry of a cube. A closely related group, I the icosahedral group, h, characteristic of the icosahedron, has 12 fivefold axes (Fig. 6.12).

The icosahedral group is important for boron compounds (Section 13.11) and the C60 fullerene molecule (Section 14.6). The distribution of molecules among the various point groups is very uneven. Some of C C the most common groups for molecules are the low-symmetry groups 1 and s. There are C C many examples of polar molecules in groups 2v (such as SO2) and 3v (such as NH3). There C D are many linear molecules, which belong to the groups v (HCl, OCS) and h (Cl2 and D CO2), and a number of planar-trigonal molecules, 3h (such as BF3, 5), trigonal-bipyramidal D D (a) molecules (such as PCl5, 6), which are 3h, and square-planar molecules, 4h (7). So-called ‘octahedral’ molecules with two identical substituents opposite each other, as in (8), are D also 4h. The last example shows that the point-group classification of a molecule is more precise than the casual use of the terms ‘octahedral’ or ‘tetrahedral’ that indicate molecular geometry. For instance, a molecule may be called octahedral (that is, it has octahedral geometry) even if it has six different groups attached to the central atom. However, the O ‘octahedral molecule’ belongs to the octahedral point group h only if all six groups and the lengths of their bonds to the central atom are identical and all angles are 90º.

6.2 Character tables Key point: The systematic analysis of the symmetry properties of molecules is carried out using (b) character tables. Figure 6.11 Shapes having cubic symmetry. T We have seen how the symmetry properties of a molecule define its point group and how (a) The tetrahedron, point group d. (b) The O that point group is labelled by its Schoenflies symbol. Associated with each point group is octahedron, point group h. 184 6 Molecular symmetry

C5 F Cl

P B

5 BF3 (D3h) Figure 6.12 The regular icosahedron, point 6 PCl (D ) I 5 3h group h, and its relation to a cube.

a character table. A character table displays all the symmetry elements of the point group together with a description, as we explain below, of how various objects or mathematical Cl 2– functions transform under the corresponding symmetry operations. A character table is complete: every possible object or mathematical function relating to the molecule belong- ing to a particular point group must transform like one of the rows in the character table Pt of that point group. The structure of a typical character table is shown in Table 6.3. The entries in the main part of the table are called characters, ␹ (chi). Each character shows how an object or mathematical function, such as an atomic orbital, is affected by the corresponding symmetry operation of the group. Thus:

Character Significance 1 the orbital is unchanged

2– –1 the orbital changes sign 7 [PtCl4] (D4h) 0 the orbital undergoes a more complicated change

z Y For instance, the rotation of a pz orbital about the axis leaves it apparently unchanged xy (hence its character is 1); a reflection of a pz orbital in the plane changes its sign (character –1). In some character tables, numbers such as 2 and 3 appear as characters: this feature is explained later. The class of an operation is a specific grouping of symmetry operations of the same X M geometrical type: the two (clockwise and anticlockwise) threefold rotations about an axis form one class, reflections in a mirror plane form another, and so on. The number of mem- C bers of each class is shown in the heading of each column of the table, as in 2 3, denoting that there are two members of the class of threefold rotations. All operations of the same class have the same character. Each row of characters corresponds to a particular irreducible representation of the group. An irreducible representation has a technical meaning in group theory but, broadly speaking, it is a fundamental type of symmetry in the group (like the symmetries represented 8 trans-[MX4Y2] (D4h) Table 6.3 The components of a character table

Name of Symmetry Functions Further Order of point operations R functions group, h group* arranged by E C class ( , n, etc.) Symmetry Characters (␹) Translations and Quadratic functions species () components of such as z2, xy, etc., dipole moments (x, y, z), of relevance to Raman of relevance to IR activity activity; rotations

* Schoenflies symbol. An introduction to symmetry analysis 185 by and π orbitals for linear molecules). The label in the first column is thesymmetry species (essentially, a label, like and π) of that irreducible representation. The two columns on the right contain examples of functions that exhibit the characteristics of each symmetry species. One column contains functions defined by a single axis, such as translations or x,y,z R ,R ,R p orbitals ( ) or rotations ( x y z), and the other column contains quadratic functions such as d orbitals (xy, etc). Character tables for a selection of common point groups are given in Resource section 4.

Table 6.4 The C character table EXAMPLE 6.3 Identifying the symmetry species of orbitals 2v C EC h 2v 2 v v = 4 Identify the symmetry species of the oxygen valence-shell atomic orbitals in an H O molecule, which has 2 A 11 11 zx2, y2, z2 C symmetry. 1 2v R A2 1 1 –1 –1 z Answer The symmetry elements of the H O molecule are shown in Fig. 6.10 and the character table for C is 2 2v B 1–11–1x, R xy given in Table 6.4. We need to see how the orbitals behave under these symmetry operations. An s orbital on 1 y B 1–1–11 y, R zx, yz the O atom is unchanged by all four operations, so its characters are (1,1,1,1) and thus it has symmetry species 2 x

A1. Likewise, the 2pz orbital on the O atom is unchanged by all operations of the point group and is thus C totally symmetric under 2v: it therefore has symmetry species A1. The character of the O2px orbital under C 2 is –1, which means simply that it changes sign under a twofold rotation. A px orbital also changes sign yz (and therefore has character –1) when reflected in the -plane ( v ), but is unchanged (character 1) when xz ␴ reflected in the -plane ( v). It follows that the characters of an O2px orbital are (1,–1,1,–1) and therefore C that its symmetry species is B1. The character of the O2py orbital under 2 is –1, as it is when reflected in the xz ␴ yz -plane ( v). The O2py is unchanged (character 1) when reflected in the -plane ( v ). It follows that the

characters of an O2py orbital are (1,–1,–1,1) and therefore that its symmetry species is B2. Self-test 6.3 D Identify the symmetry species of all five d orbitals of the central Xe atom in XeF4 ( 4h, Fig. 6.3).

C The letter A used to label a symmetry species in the group 2v means that the function to which it refers is symmetric with respect to rotation about the twofold axis (that is, its character is 1). The label B indicates that the function changes sign under that rotation

(the character is –1). The subscript 1 on A1 means that the function to which it refers is also symmetric with respect to reflection in the principal vertical plane (for H2O this is the plane that contains all three atoms). A subscript 2 is used to denote that the function changes sign under this reflection.

Now consider the slightly more complex example of NH3, which belongs to the point C group 3v (Table 6.5). An NH3 molecule has higher symmetry than H2O. This higher symmetry is apparent by noting the order, h, of the group, the total number of symmetry op- h h erations that can be carried out. For H2O, = 4 and for NH3, = 6. For highly symmetric h h O molecules, is large; for example = 48 for the point group h.

Inspection of the NH3 molecule (Fig. 6.13) shows that whereas the N2pz orbital is unique (it has A1 symmetry), the N2px and N2py orbitals both belong to the symmetry 2 representation E. In other words, the N2px and N py orbitals have the same symmetry characteristics, are degenerate, and must be treated together. The characters in the column headed by the identity operation E give the degeneracy of the orbitals:

Symmetry label Degeneracy A, B 1 E2

C Table 6.5 The 3v character table C E C ␴ h 3v 2 3 3 v = 6 zz2 A1 11 1 R A2 11 –1 z x y R R zx yz x2 y2 xy E2–10 (, ) ( x, y)(, ) ( – , ) 186 6 Molecular symmetry

Figure 6.13 The nitrogen 2pz orbital in + pz px py ammonia is symmetric under all operations – – C + of the 3v point group and therefore has +

A1 symmetry. The 2px and 2py orbitals behave identically under all operations (they – cannot be distinguished) and are given the symmetry label E. A1 E

Be careful to distinguish the italic E for the operation and the roman E for the label: all operations are italic and all labels are roman. Degenerate irreducible representations also contain zero values for some operations because the character is the sum of the characters for the two or more orbitals of the set, and if one orbital changes sign but the other does not, then the total character is 0. For ex- y ample, the reflection through the vertical mirror plane containing the -axis in NH3 results in no change of the py orbital, but an inversion of the px orbital.

EXAMPLE 6.4 Determining degeneracy

Can there be triply degenerate orbitals in BF3? Answer To decide if there can be triply degenerate orbitals in BF3 we note that the point group of the D Resource section molecule is 3h. Reference to the character table for this group ( 4) shows that, because no character exceeds 2 in the column headed E, the maximum degeneracy is 2. Therefore, none of its orbitals can be triply degenerate. Self-test 6.4 The SF6 molecule is octahedral. What is the maximum possible degree of degeneracy of its orbitals?

Applications of symmetry Important applications of symmetry in inorganic chemistry include the construction and labelling of molecular orbitals and the interpretation of spectroscopic data to determine structure. However, there are several simpler applications, one being to use group theory to decide whether a molecule is polar or chiral. In many cases the answer may be obvious and we do not need to use group theory. However, that is not always the case and the following examples illustrate the approach that can be adopted when the result is not obvious. There are two aspects of symmetry. Some properties require a knowledge only of the point group to which a molecule belongs. These properties include its polarity and chirality. Other properties require us to know the detailed structure of the character table. These properties include the classification of molecular vibrations and the identification of their IR and Raman activity. We illustrate both types of application in this section.

6.3 Polar molecules Key point: A molecule cannot be polar if it belongs to any group that includes a centre of inversion, any of the groups D and their derivatives, the cubic groups (T, O), the icosahedral group (I), and their modifications.

A polar molecule is a molecule that has a permanent electric dipole moment. A molecule cannot be polar if it has a centre of inversion. Inversion implies that a molecule has match- ing charge distributions at all diametrically opposite points about a centre, which rules out a dipole moment. For the same reason, a dipole moment cannot lie perpendicular to any mirror plane or axis of rotation that the molecule may possess. For example, a mirror plane demands identical atoms on either side of the plane, so there can be no dipole moment across the plane. Similarly, a symmetry axis implies the presence of identical atoms at points related by the corresponding rotation, which rules out a dipole moment perpendicular to the axis. Applications of symmetry 187

In summary: 1. A molecule cannot be polar if it has a centre of inversion. 2. A molecule cannot have an electric dipole moment perpendicular to any mirror plane. 3. A molecule cannot have an electric dipole moment perpendicular to any axis of rotation.

EXAMPLE 6.5 Judging whether or not a molecule can be polar Ru

The ruthenocene molecule (9) is a pentagonal prism with the Ru atom sandwiched between two C5H5 rings. Can it be polar? Answer We should decide whether the point group is D or cubic because in neither case can it have a 9 permanent electric dipole. Reference to Fig. 6.9 shows that a pentagonal prism belongs to the point group D 5h. Therefore, the molecule must be nonpolar. Self-test 6.5 A conformation of the ferrocene molecule that lies 4 kJ mol1 above the lowest energy Fe configuration is a pentagonal antiprism 10( ). Is it polar?

10 6.4 Chiral molecules H S Key point: A molecule cannot be chiral if it possesses an improper rotation axis ( n).

A chiral molecule (from the Greek word for ‘hand’) is a molecule that cannot be superim- F Br posed on its own mirror image. An actual hand is chiral in the sense that the mirror image of a left hand is a right hand, and the two hands cannot be superimposed. A chiral molecule and its mirror image partner are called enantiomers (from the Greek word for ‘both Cl parts’). Chiral molecules that do not interconvert rapidly between enantiomeric forms are optically active in the sense that they can rotate the plane of polarized light. Enantiomeric pairs of molecules rotate the plane of polarization of light by equal amounts in opposite 11 CHClFBr (C1) directions. S S A molecule with an improper rotation axis, n, cannot be chiral. A mirror plane is an 1 S + axis of improper rotation and a centre of inversion is equivalent to an 2 axis; therefore, CH H molecules with either a mirror plane or a centre of inversion have axes of improper rota- 3 S D D tion and cannot be chiral. Groups in which n is present include nh, nd, and some of the T O cubic groups (specifically, d and h). Therefore, molecules such as CH4 and Ni(CO)4 that CH T 2 belong to the group d are not chiral. That a ‘tetrahedral’ carbon atom leads to optical N activity (as in CHClFBr) should serve as another reminder that group theory is stricter in C its terminology than casual conversation. Thus CHClFBr (11) belongs to the group 1, not T to the group d; it has tetrahedral geometry but not tetrahedral symmetry. When judging chirality, it is important to be alert for axes of improper rotation that might not be immediately apparent. Molecules with neither a centre of inversion nor a S S mirror plane (and hence with no 1 or 2 axes) are usually chiral, but it is important to verify that a higher-order improper-rotation axis is not also present. For instance, the qua- S S 12 [N(CH CH(CH )CH(CH )CH ) ]+ ternary ammonium ion (12) has neither a mirror plane ( 1) nor an inversion centre ( 2), but 2 3 3 2 2 S it does have an 4 axis and so it is not chiral.

EXAMPLE 6.6 Judging whether or not a molecule is chiral

acac The complex [Mn(acac)3], where acac denotes the acetylacetonato ligand (CH3COCHCOCH3 ), has the structure shown as (13). Is it chiral? Answer We begin by identifying the point group in order to judge whether it contains an improper-rotation axis either explicitly or in a disguised form. The chart in Fig. 6.9 shows that the ion belongs to the point Mn D E C C S group 3, which consists of the elements ( , 3, 3 2) and hence does not contain an n axis either explicitly or in a disguised form. The complex ion is chiral and hence, because it is long-lived, optically active. Self-test 6.6 Is the conformation of H2O2 shown in (14) chiral? The molecule can usually rotate freely about

the O–O bond: comment on the possibility of observing optically active H2O2. 13 [Mn(acac)3] (D3d)