We now consider the possible combinations of symmetry elements. We cannot have arbitrary combinations of symmetry elements in a molecule. For example, suppose a molecule has one and only one $C{3}$ axis. Any symmetry operation must send this axis into itself. The molecule cannot, therefore, have a plane of symmetry at an arbitrary angle to the $C{3}$ axis; any plane of symmetry must either contain this axis or be perpendicular to it. (In $\mathrm{BF}{3}$ there are three $\sigma{v}$ planes and one $\sigma{h}$ plane.) The only possibility for a $C{n}$ axis noncoincident with the $C{3}$ axis is a $C{2}$ axis perpendicular to the $C{3}$ axis. The corresponding $\hat{C}{2}$ operation will send the $C{3}$ axis into itself. Since $\hat{C}{3}$ and $\hat{C}{3}^{2}$ are symmetry operations, if we have one $C{2}$ axis perpendicular to the $C{3}$ axis, we must have a total of three such axes (as in $\mathrm{BF}{3}$ ).
The set of all the symmetry operations of a molecule forms a mathematical group. A group is a set of entities (called the elements or members of the group) and a rule for combining any two members of the group to form the product of these members, such that certain requirements are met. Let $A, B, C, D, \ldots$ (assumed to be all different from one another) be the members of the group and let $B C$ denote the product of $B$ and $C$. The product $B C$ need not be the same as $C B$. The requirements that must be met to have a group are as follows: (1) The product of any two elements (including the product of an element with itself) must be a member of the group (the closure requirement). (2) There is a single element $I$ of the group, called the identity element, such that $K I=K$ and $I K=K$ for every element $K$ of the group. (3) Every element $K$ of the group has an inverse (symbolized by $K^{-1}$ ) that is a member of the group and that satisfies $K K^{-1}=I$ and $K^{-1} K=I$, where $I$ is the identity element. (4) Group multiplication is associative, meaning that $(B D) G=B (D * G)$ always holds for elements of the group.
The number of elements in a group is called the order of the group. A group for which $B C=C B$ for every pair of group elements is commutative or Abelian.
An example of a group is the set of all integers (positive, negative, and zero) with the rule of combination being ordinary addition. Closure is satisfied since the sum of two integers is an integer. The identity element is 0 . The inverse of the integer $n$ is the integer $-n$. Addition is associative. This group is of infinite order and is Abelian.
The set of all symmetry operations of a three-dimensional body, with the rule of combination of $\hat{R}$ and $\hat{S}$ being successive performance of $\hat{R}$ and $\hat{S}$, forms a group. Closure is satisfied because the product of any two symmetry operations must be a symmetry operation. The identity element of the group is the identity operation $\hat{E}$, which does nothing. Associativity is satisfied [Eq. (3.6)]. The inverse of a symmetry operation $\hat{R}$ is the symmetry operation that undoes the effect of $\hat{R}$. For example, the inverse of the inversion operation $\hat{i}$ is $\hat{i}$ itself, since $\hat{i} \hat{i}=\hat{E}$. The inverse of a $\hat{C}{3} 120^{\circ}$ counterclockwise rotation is a $120^{\circ}$ clockwise rotation, which is the same as a $240^{\circ}$ counterclockwise rotation: $\hat{C}{3} \hat{C}{3}^{2}=\hat{E}$ and $\hat{C}{3}^{-1}=\hat{C}_{3}^{2}$. Note that it is the symmetry operations of a molecule (and not the symmetry elements) that are the members (elements) of the group. We will make some use of group theory in Section 15.2, but a full development of group theory and its applications is omitted (see Cotton or Schonland).
For any symmetry operation of a molecule, the point that is the center of mass remains fixed. Hence the symmetry groups of isolated molecules are called point groups. For a crystal of infinite extent, we can have symmetry operations (for example, translations) that leave no point fixed, giving rise to space groups. Consideration of space groups is omitted.
Every molecule belongs to one of the symmetry point groups that we now list. For convenience the point groups have been classified into four divisions. Script letters denote point groups.
\(
\text { I. Groups with no } C{n} \text { axis : } \mathscr{C}{1}, \mathscr{C}{s}, \mathscr{C}{i}
\)
$\mathscr{C}{1}$ : If a molecule has no symmetry elements at all, it belongs to this group. The only symmetry operation is $\hat{E}$ (which is a $\hat{C}{1}$ rotation). CHFClBr belongs to point group $\mathscr{C}{1}$.
$\mathscr{C}{s}$ : A molecule whose only symmetry element is a plane of symmetry belongs to this group. The symmetry operations are $\hat{E}$ and $\hat{\sigma}$. An example is HOCl (Fig. 12.11).
$\mathscr{C}_{i}$ : A molecule whose only symmetry element is a center of symmetry belongs to this group. The symmetry operations are $\hat{i}$ and $\hat{E}$.
FIGURE 12.11 Molecules with no $C_{n}$ axis.
$\mathscr{C}{2}$
$\mathscr{C}{2 h}$
$\mathscr{C}_{3 h}$
FIGURE 12.12 Molecules with a single $C{n}$ axis.
$\mathrm{H}{2} \mathrm{O}{2}$; the $\mathrm{O}-\mathrm{O}$ bond is perpendicular to the plane of the paper
II. Groups with a single $C{n}$ axis : $\mathscr{C}{n}, \mathscr{C}{n h}, \mathscr{C}{n v}, \mathscr{S}{2 n}$
$\mathscr{C}{n}, n=2,3,4, \ldots$ : A molecule whose only symmetry element is a $C{n}$ axis belongs to this group. The symmetry operations are $\hat{C}{n}, \hat{C}{n}^{2}, \ldots, \hat{C}{n}^{n-1}, \hat{E}$. A molecule belonging to $\mathscr{C}{2}$ is shown in Fig. 12.12.
$\mathscr{C}{n h}, n=2,3,4, \ldots$ : If we add a plane of symmetry perpendicular to the $C{n}$ axis, we have a molecule belonging to this group. Since $\hat{\sigma}{h} \hat{C}{n}=\hat{S}{n}$, the $C{n}$ axis is also an $S{n}$ axis. If $n$ is even, the $C{n}$ axis is also a $C{2}$ axis, and the molecule has the symmetry operation $\hat{\sigma}{h} \hat{C}{2}=\hat{S}{2}=\hat{i}$. Thus, for $n$ even, a molecule belonging to $\mathscr{C}{n h}$ has a center of symmetry. (The group $\mathscr{C}{1 h}$ is the group $\mathscr{C}{s}$ discussed previously.) Examples of molecules belonging to groups $\mathscr{C}{2 h}$ and $\mathscr{C}{3 h}$ are shown in Fig. 12.12.
$\mathscr{C}{n v}, n=2,3,4, \ldots$ : A molecule in this group has a $C{n}$ axis and $n$ vertical symmetry planes passing through the $C{n}$ axis. (Group $\mathscr{C}{1 v}$ is the group $\mathscr{C}{s}$.) $\mathrm{H}{2} \mathrm{O}$ with a $C{2}$ axis and two vertical symmetry planes belongs to $\mathscr{C}{2 v}$. $\mathrm{NH}{3}$ belongs to $\mathscr{C}{3 v}$. (See Fig. 12.12.)
$\mathscr{S}{n}, n=4,6,8, \ldots: \mathscr{S}{n}$ is the group of symmetry operations associated with an $S{n}$ axis. First consider the case of odd $n$. We have $\hat{S}{n}=\hat{\sigma}{h} \hat{C}{n}$. The operation $\hat{C}{n}$ affects the $x$ and $y$ coordinates only, while the $\hat{\sigma}_{h}$ operation affects the $z$ coordinate only. Hence these operations commute, and we have
\(
\hat{S}{n}^{n}=\left(\hat{\sigma}{h} \hat{C}{n}\right)^{n}=\hat{\sigma}{h} \hat{C}{n} \hat{\sigma}{h} \hat{C}{n} \cdots \hat{\sigma}{h} \hat{C}{n}=\hat{\sigma}{h}^{n} \hat{C}_{n}^{n}
\)
Now $\hat{C}{n}^{n}=\hat{E}$, and, for odd $n, \hat{\sigma}{h}^{n}=\hat{\sigma}{h}$. Hence the symmetry operation $\hat{S}{n}^{n}$ equals $\hat{\sigma}{h}$ for odd $n$, and the group $\mathscr{S}{n}$ has a horizontal symmetry plane if $n$ is odd. Also,
\(
\hat{S}{n}^{n+1}=\hat{S}{n}^{n} \hat{S}{n}=\hat{\sigma}{h} \hat{S}{n}=\hat{\sigma}{h} \hat{\sigma}{h} \hat{C}{n}=\hat{C}_{n}, \quad n \text { odd }
\)
so the molecule has a $C{n}$ axis if $n$ is odd. We conclude that $\mathscr{S}{n}$ is identical to the group $\mathscr{C}{n h}$ if $n$ is odd. Now consider even values of $n$. Since $\hat{S}{2}=\hat{i}$, the group $\mathscr{C}{2}$ is identical to $\mathscr{C}{i}$. Thus it is only for $n=4,6,8, \ldots$ that we get new groups. The $S{2 n}$ axis is also a $C{n}$ axis: $\hat{S}{2 n}^{2}=\hat{\sigma}{h}^{2} \hat{C}{2 n}^{2}=\hat{E} \hat{C}{n}=\hat{C}{n}$.
III. Groups with one $C{n}$ axis and $n C{2}$ axes : $\mathscr{D}{n}, \mathscr{D}{n h}, \mathscr{D}{n d}$
$\mathscr{D}{n}, n=2,3,4, \ldots$ : A molecule with a $C{n}$ axis and $n C{2}$ axes perpendicular to the $C{n}$ axis (and no symmetry planes) belongs to $\mathscr{D}{n}$. The angle between adjacent $C{2}$ axes is $\pi / n$ radians. The group $\mathscr{D}{2}$ has three mutually perpendicular $C{2}$ axes, and the symmetry operations are $\hat{E}, \hat{C}{2}(x), \hat{C}{2}(y), \hat{C}_{2}(z)$.
FIGURE 12.13 Molecules with a $C{n}$ axis and $n C{2}$ axes.
$\mathscr{D}_{3 h}$
$\mathscr{D}{n h}, n=2,3,4, \ldots$ : This is the group of a molecule with a $C{n}$ axis, $n C{2}$ axes, and a $\sigma{h}$ symmetry plane perpendicular to the $C{n}$ axis. As in $\mathscr{C}{n h}$, the $C{n}$ axis is also an $S{n}$ axis. If $n$ is even, the $C{n}$ axis is a $C{2}$ and an $S{2}$ axis, and the molecule has a center of symmetry. Molecules in $\mathscr{D}{n h}$ also have $n$ vertical planes of symmetry, each such plane passing through the $C{n}$ axis and a $C{2}$ axis. (For the proof, see Prob. 12.29.) $\mathrm{BF}{3}$ belongs to $\mathscr{D}{3 h} ; \mathrm{PtCl}{4}^{2-}$ belongs to $\mathscr{D}{4 h}$; benzene belongs to $\mathscr{D}{6 h}$ (Fig. 12.13).
$\mathscr{D}{n d}, n=2,3,4, \ldots$ : A molecule with a $C{n}$ axis, $n C{2}$ axes, and $n$ vertical planes of symmetry, which pass through the $C{n}$ axis and bisect the angles between adjacent $C{2}$ axes, belongs to this group. The $n$ vertical planes are called diagonal planes and are symbolized by $\sigma{d}$. The $C{n}$ axis can be shown to be an $S{2 n}$ axis. The staggered conformation of ethane is an example of group $\mathscr{D}{3 d}$ (Fig. 12.13). [The symmetry of molecules with internal rotation (for example, ethane) actually requires special consideration; see H. C. Longuet-Higgins, Mol. Phys., 6, 445 (1963).]
\(
\text { IV. Groups with more than one } C{n} \text { axis, } n>2: \mathscr{T}{d}, \mathscr{T}, \mathscr{T}{h}, \mathscr{O}{h}, \mathcal{O}, \mathscr{I}{h}, \mathscr{I}^{\prime}, \mathscr{K}{h}
\)
These groups are related to the symmetries of the Platonic solids, solids bounded by congruent regular polygons and having congruent polyhedral angles. There are five such solids: The tetrahedron has four triangular faces, the cube has six square faces, the octahedron has eight triangular faces, the pentagonal dodecahedron has twelve pentagonal faces, and the icosahedron has twenty triangular faces.
$\mathscr{T}{d}$ : The symmetry operations of a regular tetrahedron constitute this group. The prime example is $\mathrm{CH}{4}$. The symmetry elements of methane are four $C{3}$ axes (each $\mathrm{C}-\mathrm{H}$ bond), three $S{4}$ axes, which are also $C{2}$ axes (Fig. 12.5), and six symmetry planes, each such plane containing two $\mathrm{C}-\mathrm{H}$ bonds. (The number of combinations of 4 things taken 2 at a time is $4!/ 2!2!=6$.)
$\mathbb{O}{h}$ : The symmetry operations of a cube or a regular octahedron constitute this group. The cube and octahedron are said to be dual to each other; if we connect the midpoints of adjacent faces of a cube, we get an octahedron, and vice versa. Hence the
$\Phi_{h}$
FIGURE 12.14 Molecules with more than one $C{n}$ axis, $n>2$. (For $\mathrm{B}{12} \mathrm{H}{12}^{2-}$, the hydrogen atoms have been omitted for clarity.)
cube and octahedron have the same symmetry elements and operations. A cube has six faces, eight vertices, and twelve edges. Its symmetry elements are as follows: a center of symmetry, three $C{4}$ axes passing through the centers of opposite faces of the cube (these are also $S{4}$ and $C{2}$ axes), four $C{3}$ axes passing through opposite corners of the cube (these are also $S{6}$ axes), six $C{2}$ axes connecting the midpoints of pairs of opposite edges, three planes of symmetry parallel to pairs of opposite faces, and six planes of symmetry passing through pairs of opposite edges. Octahedral molecules such as $\mathrm{SF}{6}$ belong to $\mathbb{O}{h}$.
$\Phi{h}$ : The symmetry operations of a regular pentagonal dodecahedron or icosahedron (which are dual to each other) constitute this group. The $\mathrm{B}{12} \mathrm{H}{12}^{2-}$ ion belongs to group $\mathscr{I}{h}$. The twelve boron atoms lie at the vertices of a regular icosahedron (Fig. 12.14). The soccer-ball-shaped molecule $\mathrm{C}{60}$ (buckminsterfullerene) belongs to $\mathscr{I}{h}$. Its shape is a truncated icosahedron formed by slicing off each of the 12 vertices of a regular icosahedron (Fig. 12.14), thereby generating a figure with 12 pentagonal faces ( 5 faces meet at each vertex of the original icosahedron), 20 hexagonal faces (formed from the 20 triangular faces of the original icosahedron), and $12 \times 5=60$ vertices ( 5 new vertices are formed when one of the original vertices is sliced off).
$\mathscr{K}{h}$ : This is the group of symmetry operations of a sphere. (Kugel is the German word for sphere.) An atom belongs to this group.
For completeness, we mention the remaining groups related to the Platonic solids; these groups are chemically unimportant. The groups $\mathscr{T}, \mathcal{O}$, and $\mathscr{I}$ are the groups of symmetry proper rotations of a tetrahedron, cube, and icosahedron, respectively. These groups do not have the symmetry reflections and improper rotations of these solids or the inversion operation of the cube and icosahedron. The group $\mathscr{T}_{h}$ contains the symmetry rotations of a tetrahedron, the inversion operation, and certain reflections and improper rotations.
What groups do linear molecules belong to? A rotation by any angle about the internuclear axis of a linear molecule is a symmetry operation. A regular polygon of $n$ sides has a $C{n}$ axis, and taking the limit as $n \rightarrow \infty$ we get a circle, which has a $C{\infty}$ axis. The internuclear axis of a linear molecule is a $C{\infty}$ axis. Any plane containing this axis is a symmetry plane. If the linear molecule does not have a center of symmetry (for example, $\mathrm{CO}, \mathrm{HCN}$ ), it belongs to the group $\mathscr{C}{\infty v}$. If the linear molecule has a center of symmetry (for example, $\mathrm{H}{2}, \mathrm{C}{2} \mathrm{H}{2}$ ), then it also has a $\sigma{h}$ symmetry plane and an infinite number of $C{2}$ axes perpendicular to the molecular axis. Hence it belongs to $\mathscr{D}{\infty h}$.
How do we find what point group a molecule belongs to? One way is to find all the symmetry elements and then compare with the above list of groups. A more systematic procedure is given in Fig. 12.15 [J. B. Calvert, Am. J. Phys., 31, 659 (1963)]. This procedure is based on the four divisions of point groups.
FIGURE 12.15 How to determine the point group of a molecule.
*If there are three mutually $\perp C{2}$ axes, check each axis for two $\sigma{v}$ planes.
We begin by checking whether or not the molecule is linear. Linear molecules are classified in $\mathscr{D}{\infty h}$ or $\mathscr{C}{\infty v}$ according to whether or not there is a center of symmetry. If the molecule is nonlinear, we look for two or more rotational axes of threefold or higher order. If these are present, the molecule is classified in one of the groups related to the symmetry of the regular polyhedra (division IV). If these axes are not present, we look for any $C{n}$ axis at all. If there is no $C{n}$ axis, the molecule belongs to one of the groups $\mathscr{C}{s}, \mathscr{C}{i}, \mathscr{C}{1}$ (division I). If there is at least one $C{n}$ axis, we pick the $C{n}$ axis of highest order as the main symmetry axis before proceeding to the next step. (If there are three mutually perpendicular $C{2}$ axes, we may pick any one of these axes as the main axis.) We next check for $n C{2}$ axes at right angles to the main $C{n}$ axis. If these are present, we have one of the division III groups. If these are absent, we have one of the division II groups. If we find the $n C_{2}$ axes,
(a)
(b)
we look for a symmetry plane perpendicular to the main $C{n}$ axis. If it is present, the group is $\mathscr{D}{n h}$. If it is absent, we check for $n$ planes of symmetry containing the main $C{n}$ axis (if the molecule has three mutually perpendicular $C{2}$ axes, we must try each axis as the main axis in looking for the two $\sigma{v}$ planes; the three $C{2}$ axes are equivalent in the groups $\mathscr{D}{n h}$ and $\mathscr{D}{n}$, but not in $\mathscr{D}{n d}$ ). If we find $n \sigma{v}$ planes, the group is $\mathscr{D}{n d}$; otherwise it is $\mathscr{D}{n}$. If the molecule does not have $n C{2}$ axes perpendicular to the main $C{n}$ axis, we classify it in one of the groups $\mathscr{C}{n h}, \mathscr{C}{n v}, \mathscr{S}{2 n}$, or $\mathscr{C}{n}$, by looking first for a $\sigma{h}$ plane, then for $n \sigma{v}$ planes, and, finally, if these are absent, checking whether or not the $C{n}$ axis is an $S{2 n}$ axis. The procedure of Fig. 12.15 does not locate all symmetry elements. After classifying a molecule, check that all the required symmetry elements are indeed present. Although the above procedure might seem involved, it is really quite simple and is easily memorized.
The most common error students make in classifying a molecule is to miss the $n$ $C{2}$ axes perpendicular to the $C{n}$ axis of a molecule belonging to $\mathscr{D}{n d}$. For example, it is easy to see that the $\mathrm{C}=\mathrm{C}=\mathrm{C}$ axis of allene is a $C{2}$ axis, but the other two $C{2}$ axes (Fig. 12.16) are often overlooked. Molecules with two equal halves "staggered" with respect to each other generally belong to $\mathscr{D}{n d}$. Models may be helpful for those with visualization difficulties.