6
Molecular symmetry
Symmetry governs the bonding and hence the physical and spectroscopic properties of molecules.
In this chapter we explore some of the consequences of molecular symmetry and introduce the
systematic arguments of group theory. We shall see that symmetry considerations are essential for
constructing molecular orbitals and analysing molecular vibrations. They also enable us to extract
information about molecular and electronic structure from spectroscopic data.
An introduction to symmetry
analysis
The systematic treatment of symmetry makes use of a branch of mathematics called group
theory. Group theory is a rich and powerful subject, but we shall confine our use of it at
this stage to the classification of molecules in terms of their symmetry properties, the construction of molecular orbitals, and the analysis of molecular vibrations and the selection
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.
Applications of symmetry
6.1 Symmetry operations, elements
and point groups
6.2 Character tables
6.3 Polar molecules
6.4 Chiral molecules
6.5 Molecular vibrations
The symmetries of molecular orbitals
6.6 Symmetry-adapted linear
combinations
6.7 The construction of molecular orbitals
An introduction to symmetry analysis
That some molecules are ‘more symmetrical’ than others is intuitively obvious. Our aim
though, is to define the symmetries of individual molecules precisely, not just intuitively,
and to provide a scheme for specifying and reporting these symmetries. It will become
clear in later chapters that symmetry analysis is one of the most pervasive techniques in
inorganic chemistry.
6.1 Symmetry operations, elements and point groups
6.8 The vibrational analogy
Representations
6.9 The reduction of a representation
6.10 Projection operators
FURTHER READING
EXERCISES
PROBLEMS
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 operation, an action, such as rotation through a certain angle, that leaves the molecule apparently 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.
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 C2. In general, an n-fold rotation is a symmetry
operation if the molecule appears unchanged after rotation by 360º/n. The corresponding
symmetry element is a line, an n-fold rotation axis, Cn, about which the rotation is performed. There is only one rotation operation associated with a C2 axis (as in H2O) because
clockwise and anticlockwise rotations by 180º are identical. The trigonal-pyramidal NH3
C2
*
180°
*
Figure 6.1 An H2O molecule may be
rotated through any angle about the
bisector of the HOH bond angle, but only a
rotation of 180 (the C2 operation) leaves it
apparently unchanged.
180
6 Molecular symmetry
Table 6.1 Symmetry operations and symmetry elements
*
120°
C3
Symmetry operation
Symmetry element
Symbol
Identity
‘whole of space’
E
Rotation by 360/n
n-fold symmetry axis
Cn
Reflection
mirror plane
Inversion
centre of inversion
i
Rotation by 360/n followed by
reflection in a plane perpendicular
to the rotation axis
n-fold axis of improper rotation*
Sn
*Note the equivalences S1 = ␴ and S2 = i.
120°
*
C3
C32
*
Figure 6.2 A threefold rotation and the
corresponding C3 axis in NH3. There are
two rotations associated with this axis,
one through 120 (C3) and one through
240 (C32).
v
h
d
C4
C2´
C2”
molecule has a threefold rotation axis, denoted C3, but there are now two operations associated with this axis, one a clockwise rotation by 120º and the other an anticlockwise rotation
by 120º (Fig. 6.2). The two operations are denoted C3 and C32 (because two successive clockwise rotations by 120º are equivalent to an anticlockwise rotation by 120º), respectively.
The square-planar molecule XeF4 has a fourfold C4 axis, but in addition it also has two
pairs of twofold rotation axes that are perpendicular to the C4 axis: one pair (C2) passes
through each trans-FXeF unit and the other pair (C2) passes through the bisectors of the
FXeF angles (Fig. 6.3). By convention, the highest order rotational axis, which is called the
principal axis, defines the z-axis (and is typically drawn vertically).
The reflection of an H2O molecule in either of the two planes shown in Fig. 6.4 is a
symmetry operation; the corresponding symmetry element, the plane of the mirror, is a
mirror plane, ␴. The H2O molecule has two mirror planes that intersect at the bisector of
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 molecule 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
that intersect the fourfold axis. The symmetry elements (and the associated operations) are
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
the angle between two C2 axes (the FXeF axes).
To understand the inversion operation, i, we need to imagine that each atom is projected in a straight line through a single point located at the centre of the molecule and then
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
1
Figure 6.3 Some of the symmetry
elements of a square-planar molecule
such as XeF4.
*
2
5
v
i
*
4
3
6
6
4
'
v
*
3
*
5
2
Figure 6.4 The two vertical mirror planes
␴v and ␴v in H2O and the corresponding
operations. Both planes cut through the
C2 axis.
1
Figure 6.5 The inversion operation and the
centre of inversion i in SF6.
An introduction to symmetry analysis
which the projections are made, is called the centre of inversion, i. 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
effect, that is not the case in general and the two operations must be distinguished (Fig.
6.6).
An improper rotation consists of a rotation of the molecule through a certain angle around an axis followed by a reflection in the plane perpendicular to that axis
(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
the rotation axis. Neither the 90º (C4) operation nor the reflection alone is a symmetry
operation for CH4 but their overall effect is a symmetry operation. A fourfold improper
rotation is denoted S4. The symmetry element, the improper-rotation axis, Sn (S4 in the
example), is the corresponding combination of an n-fold rotational axis and a perpendicular mirror plane.
An S1 axis, a rotation through 360º followed by a reflection in the perpendicular plane,
is equivalent to a reflection alone, so S1 and ␴h are the same; the symbol ␴h is generally used
rather than S1. Similarly, an S2 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
rather than S2.
E X A M PL E 6 .1 Identifying symmetry elements
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,
we see that the eclipsed conformation of a CH3CH3 molecule (1) has the elements E, C3, C2, ␴h, ␴v, and S3.
The staggered conformation (2) has the elements E, C3, ␴d, i, and S6.
(a) i
C2
i
(b) C2
Figure 6.6 Care must be taken not to
confuse (a) an inversion operation with
(b) a twofold rotation. Although the two
operations may sometimes appear to have
the same effect, that is not the case in
general.
H
Self-test 6.1 Sketch the S4 axis of an NH ion. How many of these axes does the ion possess?
4
C
C3
(1) Rotate
1 A C3 axis
S1
C4
(2) Reflect
H
σ
(a)
C
(1) Rotate
h
S2
(b)
Figure 6.7 A fourfold axis of improper
rotation S4 in the CH4 molecule.
S6
(2) Reflect
i
Figure 6.8 (a) An S1 axis is equivalent to a
mirror plane and (b) an S2 axis is equivalent
to a centre of inversion.
181
2 An S6 axis
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
C1
E
SiHClBrF
C2
E, C2
H2O2
Cs
E, ␴
NHF2
C2v
E, C2, ␴v, ␴v
SO2Cl2, H2O
C3v
E, 2C3, 3␴v
NH3, PCl3, POCl3
C∞v
E, C2, 2C␸, ∞␴v
OCS, CO, HCl
D2h
E, 3C2, i, 3␴
N2O4, B2H6
D3h
E, 2C3, 3C2, ␴h, 2S3, 3␴v
BF3, PCl5
E, 2C4, C2, 2C 2′ , 2C 2′′ , i, 2S4, ␴h, 2␴v, 2␴d
XeF4,
trans-[MA4B2]
D∞h
E, ∞C2, 2C␸, i, ∞␴v, 2S␸
CO2, H2, C2H2
Td
E, 8C3, 3C2, 6S4, 6␴d
CH4, SiCl4
Oh
E, 8C3, 6C2, 6C4, 3C2, i, 6S4, 8S6, 3␴h, 6␴d
SF6
D4h
Shape
Examples
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
as C2v for a water molecule.
E X A M PL E 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
possesses the identity (E), a twofold rotation axis (C2), and two vertical mirror planes (␴v and ␴v). The set
An introduction to symmetry analysis
of elements (E,C2, ␴v, v) corresponds to the group C2v. (b) The symmetry elements of XeF4 are shown in
Fig. 6.3. XeF4 possesses the identity (E), a fourfold axis (C4), two pairs of twofold rotation axes that are
perpendicular to the principal C4 axis, a horizontal reflection plane ␴h in the plane of the paper, and two
sets of two vertical reflection planes, ␴v and ␴d. This set of elements identifies the point group as D4h.
183
C2
Self-test 6.2 Identify the point groups of (a) BF3, a trigonal-planar molecule, and (b) the tetrahedral
SO42– ion.
v
'
C2
v
Y C?N
n
u
c
le
o
M
Y
Y
Y
D∞h
i?
N
r?
a
e
in
L
N
Y σ ?N
h
Y Two or N
more
C n,
n > 2?
C∞v
Y
Linear groups
Y
C5?
i?
Y σ ?N
h
Select Cn with
N
highest n; then is
nC2 ⊥ Cn?
Y
Y nσ ? N
d
N
Y nσ ? N
v
N
v
Figure 6.10 The symmetry elements of
H2O. The diagram on the right is the view
from above and summarizes the diagram on
the left.
Y S ? N
2n
Dnh
N
Oh
i?
Yσ?N
h
Dnd
Dn
Cnh
Cnv
Cs
S2n
Ih
'
v
Ci
C1
O
C
Cn
Td
3 CO2 (D∞h)
Cubic groups
Figure 6.9 The decision tree for identifying a molecular point group. The symbols of each point refer to the
symmetry elements.
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 H2, CO2 (3), and HC⬅CH
belong to Dh. A molecule that is linear but has no centre of symmetry, such as HCl or
OCS (4) belongs to Cv. Tetrahedral (Td) and octahedral (Oh) molecules have more than
one principal axis of symmetry (Fig. 6.11): a tetrahedral CH4 molecule, for instance, has
four C3 axes, one along each CH bond. The Oh and Td point groups are known as cubic
groups because they are closely related to the symmetry of a cube. A closely related group,
the icosahedral group, Ih, 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
the most common groups for molecules are the low-symmetry groups C1 and Cs. There are
many examples of polar molecules in groups C2v (such as SO2) and C3v (such as NH3). There
are many linear molecules, which belong to the groups Cv (HCl, OCS) and Dh (Cl2 and
CO2), and a number of planar-trigonal molecules, D3h (such as BF3, 5), trigonal-bipyramidal
molecules (such as PCl5, 6), which are D3h, and square-planar molecules, D4h (7). So-called
‘octahedral’ molecules with two identical substituents opposite each other, as in (8), are
also D4h. 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
‘octahedral molecule’ belongs to the octahedral point group Oh only if all six groups and
the lengths of their bonds to the central atom are identical and all angles are 90º.
O
C
S
4 OCS (C∞v)
(a)
6.2 Character tables
Key point: The systematic analysis of the symmetry properties of molecules is carried out using
character tables.
We have seen how the symmetry properties of a molecule define its point group and how
that point group is labelled by its Schoenflies symbol. Associated with each point group is
(b)
Figure 6.11 Shapes having cubic symmetry.
(a) The tetrahedron, point group Td. (b) The
octahedron, point group Oh.
184
6 Molecular symmetry
C5
F
Cl
P
B
5 BF3 (D3h)
Figure 6.12 The regular icosahedron, point
group Ih, and its relation to a cube.
Cl
2–
Pt
6 PCl5 (D3h)
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
functions transform under the corresponding symmetry operations. A character table is
complete: every possible object or mathematical function relating to the molecule belonging to a particular point group must transform like one of the rows in the character table
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:
7 [PtCl4]2– (D4h)
Y
X
M
8 trans-[MX4Y2] (D4h)
Character
Significance
1
the orbital is unchanged
–1
the orbital changes sign
0
the orbital undergoes a more complicated change
For instance, the rotation of a pz orbital about the z axis leaves it apparently unchanged
(hence its character is 1); a reflection of a pz orbital in the xy 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
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 members of each class is shown in the heading of each column of the table, as in 2C3, 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
Table 6.3 The components of a character table
Name of
point
group*
Symmetry
operations R
arranged by
class (E, Cn, etc.)
Functions
Further
functions
Symmetry
species ()
Characters (␹)
Translations and
components of
dipole moments (x, y, z),
of relevance to IR
activity; rotations
Quadratic functions
such as z2, xy, etc.,
of relevance to Raman
activity
* Schoenflies symbol.
Order of
group, h
An introduction to symmetry analysis
185
by and π orbitals for linear molecules). The label in the first column is the symmetry
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
p orbitals (x,y,z) or rotations (Rx,Ry,Rz), 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.
E X A M PL E 6 . 3 Identifying the symmetry species of orbitals
Identify the symmetry species of the oxygen valence-shell atomic orbitals in an H2O molecule, which has
C2v symmetry.
Answer The symmetry elements of the H2O molecule are shown in Fig. 6.10 and the character table for C2v is
given in Table 6.4. We need to see how the orbitals behave under these symmetry operations. An s orbital on
the O atom is unchanged by all four operations, so its characters are (1,1,1,1) and thus it has symmetry species
A1. Likewise, the 2pz orbital on the O atom is unchanged by all operations of the point group and is thus
totally symmetric under C2v: it therefore has symmetry species A1. The character of the O2px orbital under
C2 is –1, which means simply that it changes sign under a twofold rotation. A px orbital also changes sign
(and therefore has character –1) when reflected in the yz-plane (v), but is unchanged (character 1) when
reflected in the xz-plane (␴v). It follows that the characters of an O2px orbital are (1,–1,1,–1) and therefore
that its symmetry species is B1. The character of the O2py orbital under C2 is –1, as it is when reflected in the
xz-plane (␴v). The O2py is unchanged (character 1) when reflected in the yz-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 Identify the symmetry species of all five d orbitals of the central Xe atom in XeF4 (D4h, Fig. 6.3).
The letter A used to label a symmetry species in the group C2v 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
group C3v (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 operations that can be carried out. For H2O, h = 4 and for NH3, h = 6. For highly symmetric
molecules, h is large; for example h = 48 for the point group Oh.
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
representation E. In other words, the N2px and N2py 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
E
2
T
3
Table 6.5 The C3v character table
C3v
E
2 C3
3 ␴v
h=6
A1
1
1
1
z
A2
1
1
–1
Rz
E
2
–1
0
(x, y) (Rx, Ry)
z2
(zx, yz) (x2 – y2, xy)
Table 6.4 The C2v character table
C2v
E
C2
v
v h = 4
A1
1
1
1
1
z
A2
1
1
–1
–1
Rz
B1
1
–1
1
–1
x, Ry
xy
B2
1
–1
–1
1
y, Rx
zx, yz
x2, y2, z2
186
6 Molecular symmetry
Figure 6.13 The nitrogen 2pz orbital in
ammonia is symmetric under all operations
of the C3v 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.
+
pz
–
px
–
+
py
+
–
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 example, the reflection through the vertical mirror plane containing the y-axis in NH3 results
in no change of the py orbital, but an inversion of the px orbital.
E X A M PL E 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
molecule is D3h. Reference to the character table for this group (Resource section 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 matching 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.
E X A M PL E 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
permanent electric dipole. Reference to Fig. 6.9 shows that a pentagonal prism belongs to the point group
D5h. Therefore, the molecule must be nonpolar.
9
Self-test 6.5 A conformation of the ferrocene molecule that lies 4 kJ mol1 above the lowest energy
configuration is a pentagonal antiprism (10). Is it polar?
Fe
10
6.4 Chiral molecules
H
Key point: A molecule cannot be chiral if it possesses an improper rotation axis (Sn).
A chiral molecule (from the Greek word for ‘hand’) is a molecule that cannot be superimposed 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
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
directions.
A molecule with an improper rotation axis, Sn, cannot be chiral. A mirror plane is an S1
axis of improper rotation and a centre of inversion is equivalent to an S2 axis; therefore,
molecules with either a mirror plane or a centre of inversion have axes of improper rotation and cannot be chiral. Groups in which Sn is present include Dnh, Dnd, and some of the
cubic groups (specifically, Td and Oh). Therefore, molecules such as CH4 and Ni(CO)4 that
belong to the group Td are not chiral. That a ‘tetrahedral’ carbon atom leads to optical
activity (as in CHClFBr) should serve as another reminder that group theory is stricter in
its terminology than casual conversation. Thus CHClFBr (11) belongs to the group C1, not
to the group Td; 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
mirror plane (and hence with no S1 or S2 axes) are usually chiral, but it is important to
verify that a higher-order improper-rotation axis is not also present. For instance, the quaternary ammonium ion (12) has neither a mirror plane (S1) nor an inversion centre (S2), but
it does have an S4 axis and so it is not chiral.
F
Br
Cl
11 CHClFBr (C1)
+
H
CH3
CH2
N
12 [N(CH2CH(CH3)CH(CH3)CH2)2]+
E X A M PL E 6 .6 Judging whether or not a molecule is chiral
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
group D3, which consists of the elements (E, C3, 3C2) and hence does not contain an Sn 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.
acac
Mn
13 [Mn(acac)3] (D3d)