Molecular symmetry

In this chapter we sharpen the concept of shape into a precise denition of symmetry,and show that symmetry may be discussed systematically. We see how to classify anymolecule according to its symmetry and how to use this classication to discuss molecularproperties. After describing the symmetry properties of molecules themselves, we turn to a consideration of the effect of symmetry transformations on orbitals and see that theirtransformation properties can be used to set up a labelling scheme. These symmetry labelsare used to identify integrals that necessarily vanish. One important integral is the overlap integral between two orbitals. By knowing which atomic orbitals may have nonzero overlap,we can decide which ones can contribute to molecular orbitals. We also see how to selectlinear combinations of atomic orbitals that match the symmetry of the nuclear framework.Finally, by considering the symmetry properties of integrals, we see that it is possible to derive the selection rules that govern spectroscopic transitions.

The systematic discussion of symmetry is called group theory. Much of group theoryis a summary of common sense about the symmetries of objects. However, becausegroup theory is systematic, its rules can be applied in a straightforward, mechanicalway. In most cases the theory gives a simple, direct method for arriving at useful con-clusions with the minimum of calculation, and this is the aspect we stress here. Insome cases, though, it leads to unexpected results.

The symmetry elements of objects

Some objects are more symmetrical than others. A sphere is more symmetrical thana cube because it looks the same after it has been rotated through any angle about anydiameter. A cube looks the same only if it is rotated through certain angles aboutspecic axes, such as 90, 180, or 270 about an axis passing through the centres ofany of its opposite faces (Fig. 11.1), or by 120 or 240 about an axis passing throughany of its opposite corners. Similarly, an NH3 molecule is more symmetrical than anH2O molecule because NH3 looks the same after rotations of 120 or 240 about theaxis shown in Fig. 11.2, whereas H2O looks the same only after a rotation of 180.

An action that leaves an object looking the same after it has been carried out iscalled a symmetry operation. Typical symmetry operations include rotations, reec-tions, and inversions. There is a corresponding symmetry element for each symmetryoperation, which is the point, line, or plane with respect to which the symmetry opera-tion is performed. For instance, a rotation (a symmetry operation) is carried outaround an axis (the corresponding symmetry element). We shall see that we can classify molecules by identifying all their symmetry elements, and grouping together

11The symmetry elements ofobjects

11.1 Operations and symmetryelements

11.2 The symmetry classication ofmolecules

11.3 Some immediateconsequences of symmetry

Applications to molecularorbital theory and spectroscopy

11.4 Character tables andsymmetry labels

11.5 Vanishing integrals and orbitaloverlap

11.6 Vanishing integrals andselection rules

Checklist of key equations

Discussion questions

Exercises

Problems

418 11 MOLECULAR SYMMETRY

molecules that possess the same set of symmetry elements. This procedure, for example, puts the trigonal pyramidal species NH3 and SO 3

2 into one group and theangular species H2O and SO2 into another group.

11.1 Operations and symmetry elements

Key points (a) Group theory is concerned with symmetry operations and the symmetry elementswith which they are associated; point groups are composed of symmetry operations that preserve

a single point. (b) A set of operations form a group if they satisfy certain criteria.

The classication of objects according to symmetry elements corresponding to opera-tions that leave at least one common point unchanged gives rise to the point groups.There are ve kinds of symmetry operation (and ve kinds of symmetry element) ofthis kind. When we consider crystals (Chapter 19), we shall meet symmetries arisingfrom translation through space. These more extensive groups are called space groups.

(a) Notation

The identity, E, consists of doing nothing; the corresponding symmetry element is theentire object. Because every molecule is indistinguishable from itself if nothing is doneto it, every object possesses at least the identity element. One reason for including theidentity is that some molecules have only this symmetry element (1); another reasonis technical and connected with the detailed formulation of group theory.

An n-fold rotation (the operation) about an n-fold axis of symmetry, Cn (the cor-responding element) is a rotation through 360/n. The operation C1 is a rotationthrough 360, and is equivalent to the identity operation E. An H2O molecule has onetwofold axis, C2. There is only one twofold rotation associated with a C2 axis becauseclockwise and counterclockwise 180 rotations have an identical outcome. An NH3molecule has one threefold axis, C3, with which is associated two symmetry opera-tions, one being 120 rotation in a clockwise sense and the other 120 rotation in acounterclockwise sense. A pentagon has a C5 axis, with two (clockwise and counter-clockwise) rotations through 72 associated with it. It also has an axis denoted C5

2, cor-responding to two successive C5 rotations; there are two such operations, one through144 in a clockwise sense and the other through 144 in a counterclockwise sense. Acube has three C4 axes, four C3 axes, and six C2 axes. However, even this high sym-metry is exceeded by a sphere, which possesses an innite number of symmetry axes(along any diameter) of all possible integral values of n. If a molecule possesses severalrotation axes, then the one (or more) with the greatest value of n is called the principalaxis. The principal axis of a benzene molecule is the sixfold axis perpendicular to thehexagonal ring (2).

A reection (the operation) in a mirror plane, (the element), may contain theprincipal axis of a molecule or be perpendicular to it. If the plane is parallel to theprincipal axis, it is called vertical and denoted v. An H2O molecule has two vertical

C2C3

C4

Fig. 11.1 Some of the symmetry elements of a cube. The twofold, threefold, andfourfold axes are labelled with theconventional symbols.

I

F

C

Br

Cl

1 CBrClFI

C6

2 Benzene, C6H6

(a)

(b)

C3

C2

Fig. 11.2 (a) An NH3 molecule has athreefold (C3) axis and (b) an H2Omolecule has a twofold (C2) axis. Both haveother symmetry elements too.

11.1 OPERATIONS AND SYMMETRY ELEMENTS 419

planes of symmetry (Fig. 11.3) and an NH3 molecule has three. A vertical mirror planethat bisects the angle between two C2 axes is called a dihedral plane and is denoted d(Fig. 11.4). When the plane of symmetry is perpendicular to the principal axis it iscalled horizontal and denoted h. A C6 H6 molecule has a C6 principal axis and a horizontal mirror plane (as well as several other symmetry elements).

In an inversion (the operation) through a centre of symmetry, i (the element), weimagine taking each point in a molecule, moving it to the centre of the molecule, andthen moving it out the same distance on the other side; that is, the point (x, y, z) istaken into the point (x, y, z). Neither an H2O molecule nor an NH3 molecule hasa centre of inversion, but a sphere and a cube do have one. A C6H6 molecule does havea centre of inversion, as does a regular octahedron (Fig. 11.5); a regular tetrahedronand a CH4 molecule do not.

An n-fold improper rotation (the operation) about an n-fold axis of improper rota-tion or an n-fold improper rotation axis, Sn (the symmetry element), is composed oftwo successive transformations, neither of which alone is necessarily a symmetry operation. The rst component is a rotation through 360/n, and the second is areection through a plane perpendicular to the axis of that rotation; neither operationalone needs to be a symmetry operation. A CH4 molecule has three S4 axes (Fig. 11.6).

(b) The criteria for being a group

In mathematics, a group has a special meaning and is the basis of the name grouptheory for the quantitative description of symmetry. A set of operations constitute agroup if they satisfy the following criteria:

The identity operation is a member of the set.

The inverse of each operation is a member of the set.

If R and S are members of the set, then the operation RS is also a member.

These criteria are satised by a large number of objects, but our concern is with sym-metry operations, and we conne our remarks to them.

It is quite easy to see that the symmetry operations of a molecule full the criteriathat let them qualify as a group. First, we have seen that every molecule possesses theidentity operation E. To judge whether the inverse of a symmetry operation is alwayspresent we need to note whether for each operation we can nd another operation (orthe same operation) that brings the molecule back to its original state. A reection applied twice in succession (which we denote ) is one example. A clockwise n-foldrotation followed by a counterclockwise n-fold rotation (denoted C n

C +n) is another

v

v

Fig. 11.3 An H2O molecule has two mirrorplanes. They are both vertical (i.e. containthe principal axis), so are denoted v and v.

d d d

Fig. 11.4 Dihedral mirror planes (d) bisectthe C2 axes perpendicular to the principalaxis.

Centre ofinversion, i

Fig. 11.5 A regular octahedron has a centreof inversion (i).

S4

h

h

C4

C6

S6

(a)

(b)

Fig. 11.6 (a) A CH4 molecule has a fourfoldimproper rotation axis (S4): the molecule is indistinguishable after a 90 rotationfollowed by a reection across the horizontalplane, but neither operation alone is asymmetry operation. (b) The staggeredform of ethane has an S6 axis composed of a 60 rotation followed by a reection.

420 11 MOLECULAR SYMMETRY

example. To every symmetry operation of a molecule there corresponds an inverseand, provided we include both, criterion 2 is satised.

The third criterion is very special, and is called the group property. It states that, if two symmetry operations are carried out in succession, then the outcome is equi-valent to a single symmetry operation. For example, two clockwise threefold rotationsapplied in succession, giving an overall rotation of 240, is equivalent to a single counterclockwise rotation, so we can write C 3

+C 3+

= C 3 and in this case two operations

applied in succession are equivalent to a single operation. A twofold rotation through180 followed by a reection in a horizontal plane is equivalent to an inversion, so wecan write hC2 = i. Once again, we see that successive operations are equivalent to asingle operation, as criterion 3 requires.

All the symmetry operations of molecules satisfy the three criteria for them con-stituting a group, so we are justied in calling the theory of symmetry group theoryand using the powerful apparatus that mathematicians have assembled.

11.2 The symmetry classication of molecules

Key point Molecules are classied according to the symmetry elements they possess.

To classify molecules according to their symmetries, we list their symmetry elementsand collect together molecules with the same list of elements. This procedure putsCH4 and CC14, which both possess the same symmetry elements as a regular tetrahe-dron, into the same group, and H2O into another group.

The name of the group to which a molecule belongs is determined by the sym-metry elements it possesses. There are two systems of notation (Table 11.1). TheSchoenies system (in which a name looks like C4v) is more common for the discus-sion of individual molecules, and the HermannMauguin system, or Internationalsystem (in which a name looks like 4mm), is used almost exclusively in the discussionof crystal symmetry. The identication of a molecules point group according to the Schoenies system, which we outline below, is simplied by referring to the owdiagram in Fig. 11.7 and the shapes shown in Fig. 11.8.

Table 11.1 The notation for point groups*

Ci

Cs m

C1 1 C2 2 C3 3 C4 4 C6 6

C2v 2mm C3v 3m C4v 4mm C6v 6mm

C2h 2m C3h C4h 4/m C6h 6/m

D2 222 D3 32 D4 422 D6 622

D2h mmm D3h 2m D4h 4/mmm D6h 6/mmm

D2d 2m D3d m S4 /m S6

T 23 Td 3m Th m3

O 432 Oh m3m

* In the International system (or HermannMauguin system) for point groups, a number n denotes thepresence of an n-fold axis and m denotes a mirror plane. A slash (/) indicates that the mirror plane isperpendicular to the symmetry axis. It is important to distinguish symmetry elements of the same type but ofdifferent classes, as in 4/mmm, in which there are three classes of mirror plane. A bar over a number indicatesthat the element is combined with an inversion. The only groups listed here are the so-calledcrystallographic point groups (Section 19.1).

11.2 THE SYMMETRY CLASSIFICATION OF MOLECULES 421

Molecule

Linear?Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

N

N

NN

N

N N

N

N

N

N

N

N

Ni?

i?

Two or moreCn, n > 2?

C5?

Cn?

Select Cn with the highest n; then, are there nC2 perpendicular to Cn?

h?

h?

n d?

n v?

S2nS2n

Dh

Cv

Ih Oh Td

Dnh

Dnd Dn

Cs

C1Ci

?

i?

Cnh

Cnv

Cn

Fig. 11.7 A ow diagram for determiningthe point group of a molecule. Start at thetop and answer the question posed in eachdiamond (Y = yes, N = no).

S2n

Dnh

Dnd

Dn

Cnh

Cnv

Cn

n = 2 3 4 5 6

ConePyramid

Plane or bipyramid

Fig. 11.8 A summary of the shapescorresponding to different point groups.The group to which a molecule belongs can often be identied from this diagramwithout going through the formalprocedure in Fig. 11.7.

422 11 MOLECULAR SYMMETRY

(a) The groups C1, Ci, and Cs

A molecule belongs to the group C1 if it has no element other than the identity, as in(1). It belongs to Ci if it has the identity and the inversion alone (3), and to Cs if it hasthe identity and a mirror plane alone (4).

(b) The groups Cn, Cnv, and Cnh

A molecule belongs to the group Cn if it possesses an n-fold axis. Note that the symbolCn is now playing a triple role: as the label of a symmetry element, a symmetry opera-tion, and the name of a group. For example, an H2O2 molecule has the elements E andC2 (5), so it belongs to the group C2.

If in addition to the identity and a Cn axis a molecule has n vertical mirror planes v,then it belongs to the group Cnv. An H2O molecule, for example, has the symmetry elements E, C2, and 2v, so it belongs to the group C2v. An NH3 molecule has the elements E, C3, and 3v, so it belongs to the group C3v. A heteronuclear diatomicmolecule such as HCl belongs to the group C

v because all rotations around the axisand reections across the axis are symmetry operations. Other members of the groupC

v include the linear OCS molecule and a cone.Objects that in addition to the identity and an n-fold principal axis also have a hor-

izontal mirror plane h belong to the groups Cnh. An example is trans-CHCl=CHCl(6), which has the elements E, C2, and h, so belongs to the group C2h; the moleculeB(OH)3 in the conformation shown in (7) belongs to the group C3h. The presence ofcertain symmetry elements may be implied by the presence of others: thus, in C2h theoperations C2 and h jointly imply the presence of a centre of inversion (Fig. 11.9).

OH

OH

H

H

COOH

COOH

Centre ofinversion

3 Meso-tartaric acid, HOOCCH(OH)CH(OH)COOH

N

4 Quinoline, C9H7N

O

H

C2

5 Hydrogen peroxide, H2O2

C2

Cl

Cl

h

6 trans-CHCl=CHCl

C3B

OHh

7 B(OH)3

h

i

C2

Fig. 11.9 The presence of a twofold axis anda horizontal mirror plane jointly imply thepresence of a centre of inversion in themolecule.

(c) The groups Dn, Dnh, and Dnd

We see from Fig. 11.7 that a molecule that has an n-fold principal axis and n twofoldaxes perpendicular to Cn belongs to the group Dn. A molecule belongs to Dnh if it alsopossesses a horizontal mirror plane. The planar trigonal BF3 molecule has the ele-ments E, C3, 3C2, and h (with one C2 axis along each BF bond), so belongs to D3h(8). The C6H6 molecule has the elements E, C6, 3C2, 3C 2 , and h together with someothers that these elements imply, so it belongs to D6h. The prime on 3C 2 indicates thatthese three twofold axes are different from the other three twofold axes. In benzene,three of the C2 axes bisect CC bonds and the other three pass through vertices of thehexagon formed by the carbon framework of the molecule. All homonuclear diatomicmolecules, such as N2, belong to the group Dh because all rotations around the axisare symmetry operations, as are end-to-end rotation and end-to-end reection; D

h

is also the group of the linear OCO and HCCH molecules and of a uniform cylinder.Other examples of Dnh molecules are shown in (9), (10), and (11).

A molecule belongs to the group Dnd if in addition to the elements of Dn it possessesn dihedral mirror planes d. The twisted, 90 allene (12) belongs to D2d, and the stag-gered conformation of ethane (13) belongs to D3d.

B

F

8 Boron trifluoride, BF3

C2

C2 h

9 Ethene, CH2=CH2 (D2h)

11.2 THE SYMMETRY CLASSIFICATION OF MOLECULES 423

A brief illustration

Host molecules, such as the bowl-shaped cryptophans, that encapsulate smaller guest

molecules have become a focus of interest for a wide variety of applications. Hostguest

complexes are an important means of constructing nanoscale devices, selectively separ-

ating mixtures of small molecules on the basis of chemical and physical properties,

delivering biologically active molecules to target cells, and providing unique environments

to catalyse reactions. The shape of the host can inuence both the encapsulation of guest

molecules and the potential application of the complex. The anti and syn cryptophan

isomers (14) and (15), for instance, belong to the groups D3 and C3h, respectively.

Another brief illustration

Cucurbiturils are pumpkin-shaped water-soluble compounds composed of six, seven,

or eight glycouril (16) units with a hydrophilic exterior and a hydrophobic interior

cavity. With six glycouril units, for example, the host (17) belongs to the group D6h.

P

Cl

C3

C2C2

C2h

10 Phosphorus pentachloride, PCl5 (D3h)

Cl

Au

C4C2

C2C2

h

11 Tetrachloroaurate(III) ion, [AuCl4]

, (D4h)

C2, S4

C2C2

12 Allene, C3H4 (D2d)

C3,S6

C2

d

13 Ethane, C2H6 (D3d)

OMeO MeO O

O

OMe

OO

O

OMeOMeMeO

14

OMeO OMeO OMe

MeOOMe

OMe

O

OOO

15

HN NH

HN NH

O

O

16 Glycouril

O

OO

O

OOO

17

N

O

424 11 MOLECULAR SYMMETRY

(d) The groups Sn

Molecules that have not been classied into one of the groups mentioned so far, butthat possess one Sn axis, belong to the group Sn. An example is tetraphenylmethane,which belongs to the point group S4 (18). Molecules belonging to Sn with n > 4 arerare. Note that the group S2 is the same as Ci, so such a molecule will already have beenclassied as Ci.

(e) The cubic groups

A number of very important molecules (e.g. CH4 and SF6) possess more than oneprincipal axis. Most belong to the cubic groups, and in particular to the tetrahedralgroups T, Td, and Th (Fig. 11.10a) or to the octahedral groups O and Oh (Fig. 11.10b).A few icosahedral (20-faced) molecules belonging to the icosahedral group, I(Fig. 11.10c), are also known: they include some of the boranes and buckminster-fullerene, C60 (19). The groups Td and Oh are the groups of the regular tetrahedron (forinstance, CH4) and the regular octahedron (for instance, SF6), respectively. If the objectpossesses the rotational symmetry of the tetrahedron or the octahedron, but none oftheir planes of reection, then it belongs to the simpler groups T or O (Fig. 11.11). Thegroup Th is based on T but also contains a centre of inversion (Fig. 11.12).

S4

Ph

Ph

PhPh

18 Tetraphenylmethane, C(C6H5)4 (S4)

19 Buckminsterfullerene, C60 (I)

(a) (b) (c)

Fig. 11.10 (a) Tetrahedral, (b) octahedral, and (c) icosahedral molecules are drawn in a way that shows their relation to a cube: they belong tothe cubic groups Td, Oh, and Ih, respectively.

(a) (b)

Fig. 11.11 Shapes corresponding to thepoint groups (a) T and (b) O. The presenceof the decorated slabs reduces thesymmetry of the object from Td and Oh,respectively.

11.3 SOME IMMEDIATE CONSEQUENCES OF SYMMETRY 425

Fig. 11.12 The shape of an object belongingto the group Th.

Ag

Ni

SC(CH3)2CH(NH2)CO2

5

20

Ru

Cp = C5H5

21 Ruthenocene, Ru(Cp)2

Cp = C5H5

Fe

22 Ferrocene, Fe(Cp)2

A brief illustration

The ion [Ag8Ni6{SC(Me)2CH(NH2)CO2}12Cl]5 (20) is a tetrahedral host belonging to

the group Th.

(f) The full rotation group

The full rotation group, R3 (the 3 refers to rotation in three dimensions), consists ofan innite number of rotation axes with all possible values of n. A sphere and an atombelong to R3, but no molecule does. Exploring the consequences of R3 is a very important way of applying symmetry arguments to atoms, and is an alternative approach to the theory of orbital angular momentum.

Example 11.1 Identifying a point group of a molecule

Identify the point group to which a ruthenocene molecule (21) belongs.

Method Use the ow diagram in Fig. 11.7.

Answer The path to trace through the ow diagram in Fig. 11.7 is shown by a greenline; it ends at Dnh. Because the molecule has a vefold axis, it belongs to the groupD5h. If the rings were staggered, as they are in an excited state of ferrocene that lies4 kJ mol1 above the ground state (22), the horizontal reection plane would be absent, but dihedral planes would be present.

Self-test 11.1 Classify the pentagonal antiprismatic excited state of ferrocene (22).[D5d]

11.3 Some immediate consequences of symmetry

Key points (a) Only molecules belonging to the groups Cn, Cnv, and Cs may have a permanentelectric dipole moment. (b) A molecule may be chiral, and therefore optically active, only if it does

not possess an axis of improper rotation, Sn.

Some statements about the properties of a molecule can be made as soon as its pointgroup has been identied.

426 11 MOLECULAR SYMMETRY

(a) Polarity

A polar molecule is one with a permanent electric dipole moment (HCl, O3, and NH3are examples). If the molecule belongs to the group Cn with n > 1, it cannot possess a charge distribution with a dipole moment perpendicular to the symmetry axis because the symmetry of the molecule implies that any dipole that exists in one direc-tion perpendicular to the axis is cancelled by an opposing dipole (Fig. 11.13a). For example, the perpendicular component of the dipole associated with one OH bondin H2O is cancelled by an equal but opposite component of the dipole of the secondOH bond, so any dipole that the molecule has must be parallel to the twofold sym-metry axis. However, as the group makes no reference to operations relating the twoends of the molecule, a charge distribution may exist that results in a dipole along theaxis (Fig. 11.13b), and H2O has a dipole moment parallel to its twofold symmetry axis.The same remarks apply generally to the group Cnv, so molecules belonging to any ofthe Cnv groups may be polar. In all the other groups, such as C3h, D, etc., there are sym-metry operations that take one end of the molecule into the other. Therefore, as wellas having no dipole perpendicular to the axis, such molecules can have none along theaxis, for otherwise these additional operations would not be symmetry operations.We can conclude that

Only molecules belonging to the groups Cn, Cnv, and Cs mayhave a permanent electric dipole moment.

For Cn and Cnv, that dipole moment must lie along the symmetry axis. Thus ozone,O3, which is angular and belongs to the group C2v, may be polar (and is), but carbondioxide, CO2, which is linear and belongs to the group Dh, is not.

(b) Chirality

A chiral molecule (from the Greek word for hand) is a molecule that cannot be superimposed on its mirror image. An achiral molecule is a molecule that can be superimposed on its mirror image. Chiral molecules are optically active in the sensethat they rotate the plane of polarized light. A chiral molecule and its mirror-imagepartner constitute an enantiomeric pair of optical isomers and rotate the plane of polarization in equal but opposite directions.

A molecule may be chiral, and therefore optically active, only if it does not possess an axis of improper rotation, Sn.

However, we need to be aware that such an axis may be present under a differentname, and be implied by other symmetry elements that are present. For example,molecules belonging to the groups Cnh possess an Sn axis implicitly because they possess both Cn and h, which are the two components of an improper rotation axis.Any molecule containing a centre of inversion, i, also possesses an S2 axis, because iis equivalent to C2 in conjunction with h, and that combination of elements is S2(Fig. 11.14). It follows that all molecules with centres of inversion are achiral andhence optically inactive. Similarly, because S1 = , it follows that any molecule with amirror plane is achiral.

A molecule may be chiral if it does not have a centre of inversion or a mirror plane,which is the case with the amino acid alanine (23), but not with glycine (24).However, a molecule may be achiral even though it does not have a centre of inver-sion. For example, the S4 species (25) is achiral and optically inactive: though it lacksi (that is, S2) it does have an S4 axis.

Criterion forbeing chiral

Criterion forbeing polar

(a) (b)

Fig. 11.13 (a) A molecule with a Cn axis cannothave a dipole perpendicular to the axis, but(b) it may have one parallel to the axis. Thearrows represent local contributions to theoverall electric dipole, such as may arisefrom bonds between pairs of neighbouringatoms with different electronegativities.

i

S2

Fig. 11.14 Some symmetry elements areimplied by the other symmetry elements in a group. Any molecule containing an inversion also possesses at least an S2element because i and S2 are equivalent.

COOH

CH3

H

NH2

23 L-Alanine, NH2CH(CH3)COOH

24 Glycine, NH2CH2COOH

COOH

H

H NH2