KEY POINTS Symmetry operations are actions that leave the mol- ecule 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 180° around the bisector of the HOH angle (Fig. 3.1).
Associated with each symmetry operation there is a sym- metry element—a point, line, or plane with respect to which the symmetry operation is performed. Table 3.1 lists the most important symmetry operations and their correspond- ing elements. All these operations leave at least one point unchanged 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.
TABLE 3.1 Symmetry operations and symmetry elements Symmetry operation Symmetry element Symbol
Identity ‘whole of space’ E
Rotation by 360°/n n-fold symmetry axis Cn
Reflection mirror plane σ
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 = σ and S2 = i.
The rotation of an H2O molecule by 180° around a line bisecting the HOH angle (as in Fig. 3.1) is a symmetry operation, denoted C2. In general, an n-fold rotation is a symmetry operation if the molecule appears unchanged after rotation by 360°/n. The corresponding symmetry ele- ment is a line, an n-fold rotation axis, Cn, about which the rotation is performed. So for the H2O molecule a two- fold rotation leaves the molecule unchanged after rotation by 360°/2 or 180°. There is only one rotation operation associated with a C2 axis (as in H2O) because clockwise and anticlockwise rotations by 180° are identical. The trigonal-pyramidal NH3 molecule has a three-fold rotation axis, denoted C3, on rotation of the molecule through 360°/3 or 120°. There are now two operations associated with this axis, a clockwise rotation by 120° and an anti- clockwise rotation by 120° (Fig. 3.2). The two operations are denoted C3 and C32 (because two successive clockwise rotations by 120° are equivalent to an anticlockwise rota- tion by 120°), respectively.
The square-planar molecule XeF4 has a four-fold axis, C4, but in addition it also has two pairs of two-fold rota- tion 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. 3.3). By convention, the highest order rotational axis, which is called the principal axis, defines the z-axis (and is typically drawn vertically). For XeF4 the C4 axis is the principal axis. The C42 operation is equivalent to a C2 rotation, and this is normally listed separately from the C4 operation as ‘C2(=C42)’.
C2
180°
*
*
FIGURE 3.1 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.
C3
120°
120°
*
C3 *
*
C32
FIGURE 3.2 A three-fold 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.
The reflection of an H2O molecule in either of the two planes shown in Fig. 3.4 is a symmetry operation; the cor- responding symmetry element is a mirror plane, σ. The H2O molecule has two mirror planes that intersect at the bisector of the HOH angle. Because the planes are ‘verti- cal’, in the sense of containing the rotational (z) axis of the molecule, they are labelled with a subscript v, as in σv and σv′. The XeF4 molecule in Fig. 3.3 has a mirror plane σh 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 four-fold axis. The symmetry ele- ments (and the associated operations) are denoted σv for the planes that pass through the F atoms and σd for the planes that bisect the angle between the F atoms. The v denotes that the plane is ‘vertical’ and the d denotes ‘dihe- dral’ and signifies that the plane bisects the angle between two C2′ axes (the FXeF axes).
To understand the inversion operation, i, we need to imag- ine 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. 3.5).
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. In general, under inversion, an atom with coordinates (x, y, z) moves to (−x, −y, −z). The symmetry element, the point through 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 and the S42+ ion (1) has a centre of inversion in the middle of the square ion. An H2O molecule does not pos- sess a centre of inversion, and no tetrahedral molecule can have a centre of inversion. Although an inversion and a two-fold rotation may sometimes achieve the same effect, that is not the case in general and the two operations must be distinguished (Fig. 3.6).
S
2+
1 The S42+ cation
An improper rotation consists of a rotation of the mol- ecule through a certain angle around an axis followed by a reflection in the plane perpendicular to that axis (Fig. 3.7).
The illustration shows a four-fold improper rotation of a FIGURE 3.3 Some of the symmetry elements of a square-planar
molecule such as XeF4.
C′2
C′′2
C4
σh σv
σd
*
*
*
* σv
σ ′v
FIGURE 3.4 The two vertical mirror planes σv and σv′ in H2O and the corresponding operations. Both planes cut through the C2 axis.
1
2
3
4 5
6
1 6
3
5 4
2 i
FIGURE 3.5 The inversion operation and the centre of inversion i in SF6.
their overall effect is a symmetry operation. A four-fold improper rotation is denoted S4. The symmetry element, the improper-rotation axis, Sn (S4 in the example), is the cor- responding combination of an n-fold rotational axis and a perpendicular mirror plane.
An S1 axis, a rotation through 360° followed by a reflec- tion in the perpendicular plane, is equivalent to a reflection alone, so S1 and σh are the same; the symbol σh is used rather than S1. Similarly, an S2 axis, a rotation through 180° fol- lowed by a reflection in the perpendicular plane, is equiva- lent to an inversion, i (Fig. 3.8); the symbol i is employed rather than S2.
By identifying the symmetry elements of the molecule, and referring to Table 3.2 we can assign a molecule to its point group. 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. 3.9 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 nor- mally its Schoenflies symbol, such as C3v for an ammonia molecule.
EXAMPLE 3.1 Identifying symmetry elements Identify the symmetry elements in the eclipsed conformation 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 (2) has the elements E (do nothing), C3 (a three-fold rotation axis), 3C2 (three two-fold CH4 molecule. In this case, the operation consists of a 90°
(i.e. 360°/4) rotation about an axis bisecting two HCH bond angles, followed by a reflection through a plane perpendicu- lar to the rotation axis. Neither the 90° (C4) operation nor the reflection alone is a symmetry operation for CH4 but FIGURE 3.6 Care must be taken not to confuse (a) an inversion operation with (b) a two-fold rotation. Although the two operations may sometimes appear to have the same effect, that is not the case in general, as can be seen when the four terminal atoms of the same element are coloured differently.
i (a) i
C2
(b) C2
C4
σh
FIGURE 3.7 A four-fold axis of improper rotation S4 in the CH4 molecule. The four terminal atoms of the same element are coloured differently to help track their movement.
(1) Rotate
(2) Reflect
S1
(a)
(1) Rotate (2) Reflect
S2
(b)
i σ
FIGURE 3.8 (a) An S1 axis is equivalent to a mirror plane and (b) an S2 axis is equivalent to a centre of inversion.
TABLE 3.2 The composition of some common groups
Point group Symmetry elements Shape Examples
C1 E SiHClBrF
C2 E C2 H2O2
Cs E σ NHF2
C2v E C2 σv σ′v SO2Cl2, H2O
C3v E 2C3 3σv NH3, PCl3, POCl3
C∞v E 2C∞ ∞σv OCS, CO, HCl
D2h E 3C2 i 3σ N2O4, B2H6
D3h E 2C3 3C2 σh 2S3 3σv BF3, PCl5
D4h E 2C4 C2C2′2C2′′ i 2S4 σh 2σv 2σd XeF4, trans-[MA4B2]
D∞h E ∞C2′ 2C∞ i ∞σv 2S∞ CO2, H2, C2H2
Td E 8C3 3C2 6S4 6σd CH4, SiCl4
Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh σd SF6
rotation axes running through the C–C bond), σh (a horizontal mirror plane bisecting the C–C bond), 3σv (three separate vertical mirror planes running along each C–H bond), and S3 (an improper rotation on rotating around the three-fold axis of symmetry followed by reflection in the plane perpendicular to it). We can see that the staggered conformation (3) additionally
has the elements i (inversion) and S6 (an improper rotation around the six-fold axis of symmetry arising from the six staggered H atoms).
Self-test 3.1 Sketch the S4 axis of an NH4+ ion. How many of these axes does the ion possess?
Molecule
Linear?
i?
i?
Two or more
Cn n > 2?
C5?
Cn?
D∞h C∞v
Ih Oh
Dnh Cnv
S2n Cnh
Dnd Dn
Cn
Cs Ci C1
Td Y
Y
Y
Y Y Y
Y Y Y
Y Y Y
Y Y
N
N N
N
N N N
N
N
N N
N N N
Select Cn with highest n; then is
nC2⊥ Cn?
S2n?
i?
Linear groups
Cubic groups
h?
σ σh?
h? σ
n σd? n σv?
′
FIGURE 3.9 The decision tree for identifying a molecular point group. The symbols of each point refer to the symmetry elements.
C H
C3
C H
2 A C3 axis
C H
S6
C H
3 An S6 axis
EXAMPLE 3.2 Identifying the point group of a molecule
To what point groups do H2O and XeF4 belong?
Answer We need to either work through Table 3.2 or use Fig. 3.9. (a) The symmetry elements of H2O are shown in Fig. 3.10. H2O possesses the identity (E), a two-fold rotation axis (C2), and two vertical mirror planes (σv and σ′v). The set of elements (E, C2, σv, σ′v) corresponds to those of the group C2v listed in Table 3.2. Alternatively we can work through Fig. 3.9:
the molecule is not linear; does not possess two or more Cn with n > 2; does possess a Cn (a C2 axis); does not have 2C2 ⊥ to the C2; does not have σh; does not have 2σv; it is therefore C2v.
FIGURE 3.10 The symmetry elements of H2O. The diagram on the right is the view from above and summarizes the diagram on the left.
C2
C2
z
yz plane
xz plane σv
σv
σv
′
σ ′v
(a)
(b)
FIGURE 3.11 Shapes having cubic symmetry: (a) the tetrahedron, point group Td; (b) the octahedron, point group Oh.
C5
FIGURE 3.12 The regular icosahedron, point group Ih, and its relation to a cube.
(b) The symmetry elements of XeF4 are shown in Fig. 3.3. XeF4 possesses the identity (E), a four-fold axis (C4), two pairs of two-fold rotation axes that are perpendicular to the principal C4 axis, a horizontal reflection plane σh in the plane of the paper, and two sets of two vertical reflection planes, σv and σd. Using Table 3.2, we can see that this set of elements identifies the point group as D4h. Alternatively we can work through Fig.
3.9: the molecule is not linear; does not possess two or more Cn with n > 2; does possess a Cn (a C4 axis); does have 4C2 ⊥ to the C4; and does have σh; it is therefore D4h.
Self-test 3.2 Identify the point groups of (a) BF3, a trigonal- planar molecule, and (b) the tetrahedral SO42− ion.
It is very useful to be able to recognize immediately the point groups of some common molecules. Linear mol- ecules with a centre of symmetry, such as H2, CO2 (4), and HC≡CH belong to D∞h. A molecule that is linear but has no centre of symmetry, such as HCl or OCS (5) belongs to C∞v. Tetrahedral (Td) and octahedral (Oh) molecules have more than one principal axis of symmetry (Fig. 3.11): a tetrahe- dral 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 symme- try of a cube. A closely related group, the icosahedral group, Ih, characteristic of the icosahedron, has 12 five-fold axes (Fig. 3.12). The icosahedral group is important for boron
compounds (Section 13.11) and the C60 fullerene molecule (Section 14.6).
C O
4 CO2 C
O S
5 OCS
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 molecules in groups C2v (such as SO2) and C3v (such as NH3). There are many linear molecules, which belong to the groups C∞v (HCl, OCS) and D∞h (Cl2 and CO2), and a number of planar-trigonal molecules (such as BF3, 6), which are D3h; trigonal-bipyramidal molecules (such as PCl5, 7), which are also D3h; and square-planar mol- ecules, which are D4h (8). An ‘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 identi- cal and all angles are 90°. For instance, ‘octahedral’ mol- ecules with two identical substituents opposite each other, as in (9), are actually 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 but say little about symmetry.
B F
6 BF3, D3h
by its Schoenflies symbol. Associated with each point group is a character table. A character table displays all the symme- try elements of the point group together with a description of how various objects or mathematical functions transform under the corresponding symmetry operations. In simple terms, it summarizes how each of the symmetry elements transforms the molecule. A character table is complete: every possible object or mathematical function relating to the mol- ecule 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 3.3. The entries in the main part of the table are called characters, χ (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
−1 The orbital changes sign
0 The orbital undergoes a more complicated change, or is the sum of changes of degenerate orbitals
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 symme- try operations of the same geometrical type: the two (clock- wise and anticlockwise) three-fold 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, denot- ing that there are two members of the class of three-fold rotations. All operations of the same class have the same character. The order, h, of the group is the total number of symmetry operations that can be carried out.
Each row of characters corresponds to a particular irre- ducible representation of the group. An irreducible represen- tation has a technical meaning in group theory but, broadly speaking, it is a fundamental type of symmetry in the group.
The label in the first column is the symmetry species of that irreducible representation. The two columns on the right contain examples of functions that exhibit the characteristics
TABLE 3.3 The components of a character table Name of point group* Symmetry operations R
arranged by class (E, Cn, etc.)
Functions Further functions Order of
group, h Symmetry species (Γ) Characters (χ) 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.
P Cl
7 PCl5, D3h
Pt
Cl 2–
8 [PtCl4]2−, D4h
M X
Y
9 trans-[MX4Y2], D4h