2.2: Point Groups

The point group for a molecule is the complete set of symmetry operations such that their elements intersect at least as one fixed point. According to the multiplication rules for rotations, 3-d point groups consist of either (i) only proper rotations or (ii) equal numbers of proper and improper rotations. To verify the second conclusion, consider a group with n₁ proper rotations \(C^{(i)}\ (i = 1,\ldots,n_{1})\) and n₂ improper rotations \(S^{(j)}\ (j = 1,\ldots,n_{2})\). Choose one improper rotation \(S^{(k)}\), and multiply it with every member of the group. This procedure creates n₁ products \(S^{(k)}C^{(i)}\ (i = 1,\ldots,n_{1})\) that are distinct improper rotations, and n₂ products \(S^{(k)}S^{(j)}\ (j = 1,\ldots,n_{2})\) that are distinct proper rotations. By group multiplication rules and closure, n₁ = n₂. As a result, the subset of n₁ proper rotations is a subgroup of index 2 (one-half the order) of the original point group.

Figure 2.10: Location of poles for a 3-fold rotation axis on a sphere.

To systematically enumerate the 3-d point groups, we start by identifying the groups consisting of only proper rotations, a process that utilizes a geometrical argument involving poles of rotations. Every proper rotation \(C_{n}\) except \(C_{1}\) is described by an axis, which intersects a sphere in two diametrically opposite points called n-gonal poles, as shown here for a \(C_{3}\) operation.

Point Groups Containing just Proper Rotations

Consider a point group \(\mathcal{G}\) with order g that contains a n-fold rotation \(C_{n}\) with two n-gonal poles. The n operations \({C_{n}}^{m}\) (1 ≤ m ≤ n) leave these two poles fixed. All other rotations in \(\mathcal{G}\) transform each of these poles into other equivalent n-gonal poles. As a result, every set of equivalent n-gonal poles contains g/n poles. Now, each \(C_{n}\) axis has n−1 operations excluding the identity. Since there are 2 poles for each \(C_{n}\) axis, there are (n−1)/2 rotations for each n-gonal pole. Adding up the numbers of rotations for each set of equivalent poles gives g−1 operations because we have excluded just the identity operation in this counting exercise. This result can be expressed as follows:

\[\begin{align*} g-1 &= \sum_{\text{Pole Sets}\ i} (\#\ \text{poles in set}\ i)(\#\ \text{operations/pole}) \\[4pt] &= \sum_{\text{Pole Sets}\ i} \left( \frac{g}{n_{i}} \right) \left( \dfrac{n_{i} - 1}{2} \right) \\[4pt] &= \left( \dfrac{g}{2} \right) \sum_{\text{Pole Sets}\ i} \left( 1 - \dfrac{1}{n_{i}} \right) \\[4pt] &= \left( \dfrac{g}{2} \right)\sum_{\text{Pole Sets}\ i}\left( 1 - \dfrac{1}{n_{i}} \right) \end{align*} \nonumber \]

In these equations, n_i is the order of the rotation for pole set i, and i is an index starting with one. Rearranging this equation gives

\[2 - \frac{2}{g} = \sum_{\text{Pole Sets}\ i} \left( 1 - \dfrac{1}{n_{i}} \right). \nonumber \]

Solving this equation for possible values of n_i will yield all point groups composed of proper rotations. If g = 1, the point group contains just the identity \(\mathcal{C}_1 = 1\). Therefore, since g and each n_i must be positive integers greater than 1 with g ≥ n_i, the only possible solutions involve 2 or 3 sets of equivalent poles:

1. Two sets of inequivalent poles (i = 2):

\[2 - \frac{2}{g} = \left( 1 - \frac{1}{n_1} \right) + \left( 1 - \frac{1}{n_2} \right)\nonumber \] or \[\frac{2}{g} = \frac{1}{n_1} + \frac{1}{n_2}\ . \nonumber \]

There is only one solution, n₁ = n₂ = g. There is one g-fold axis with 2 inequivalent g-gonal poles diametrically opposite on the sphere. These point groups are the uniaxial cyclic groups \(\mathcal{C}_{n}\) or \(n\).

2. Three sets of inequivalent poles (i = 3):

\[2 - \frac{2}{g} = \left( 1 - \frac{1}{n_1} \right) + \left( 1 - \frac{1}{n_2} \right) + \left( 1 - \frac{1}{n_3} \right)\nonumber \] or \[1 + \frac{2}{g} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}\ . \nonumber \]

At least one of n₁, n₂, or n₃ must be 2. Therefore, set n₃ = 2, which gives:

\[\frac{1}{2} + \frac{2}{g} = \frac{1}{n_1} + \frac{1}{n_2}\nonumber \] or \[(n_1 - 2)( n_2 - 2 ) = 4 \left( 1 - \dfrac{n_1n_2}{g} \right). \nonumber \]

Since 0 ≤ (n₁ – 2)(n₂ – 2) < 4, there are four solutions:

• n₁ = g/2, n₂ = 2, n₃ = 2 (g = 2n₁): There is one g/2-fold axis with two equivalent poles, and two distinct sets of 2-gonal poles belonging to g 2-fold axes. If g/2 is even, there are two inequivalent sets of 2-fold axes. If g/2 is odd, all 2-fold axes are equivalent. These groups are the dihedral point groups \(\mathcal{D}_{n}\). In the International notation, they are designated \(n22\) (n even) or \(n2\) (n odd).
• n₁ = 3, n₂ = 3, n₃ = 2 (g = 12): There are four 3-fold axes, each with two inequivalent sets of 3-gonal poles, and three 2-fold axes with six equivalent 2-gonal poles. It is the tetrahedral group \(\mathcal{T} = 23\). In the International symbol, the 2-fold axes are parallel to the x-, y-, and z-axes, and the 3-fold axes are parallel to the body-diagonals of a cube (see below).
• n₁ = 4, n₂ = 3, n₃ = 2 (g = 24): There are three 4-fold axes, with six equivalent 4-gonal poles, four 3-fold axes with eight equivalent 3-gonal poles, and six 2-fold axes with twelve equivalent 2-gonal poles. It is the octahedral group \(\mathcal{O} = 432\): the 4-fold axes are parallel to the x-, y-, and z-axes; the 3-fold axes are parallel to the body-diagonals of a cube; the 2-fold axes are parallel to the face-diagonals of a cube (see below).
• n₁ = 5, n₂ = 3, n₃ = 2 (g = 60): There are six 5-fold axes, with twelve equivalent 5-gonal poles, ten 3-fold axes with twenty equivalent 3-gonal poles, and fifteen 2-fold axes with thirty equivalent 2-gonal poles. It is the icosahedral group \(\mathcal{I =}235\): three of the 2-fold axes are parallel to the x-, y-, and z-axes; four of the 3-fold axes are parallel to the body-diagonals of a cube (see below).

Figure 2.11: (left) \(\mathcal{T}\)T: C₂ axes; \(\mathcal{O}\)O: C₄ axes; \(\mathcal{I}\)I: C₂ axes. Figure 2.12: \(\mathcal{T}\)T, \(\mathcal{O}\)O, and \(\mathcal{I}\)I: C₃ axes. Figure 2.13: \(\mathcal{O}\)O: C₂ axes.

Point Groups Containing Improper Rotations

Since every improper rotation can be expressed as the product of inversion with a proper rotation, every point group \(\mathcal{G}\) containing improper rotations arises by using a group \(\mathcal{H}\) of proper rotations, \(\mathcal{H =}\left\{ C^{(j)},\ \ j = 1,\ldots,n \right\}\), and the inversion operation in one of two ways:

1. If \(\mathcal{G}\) contains the inversion, then \(\mathcal{G}=\left\{ C^{(j)},i \cdot C^{(j)},\ \ j = 1,\ldots,n \right\}=\mathcal{H}+ i\cdot\mathcal{H}\). This set includes the subset \(\left\{ E,i \right\}\), which is the point group \(\mathcal{C}_{i}\). Since the identity and inversion commute with every rotation, then \(\mathcal{G}=E\cdot\mathcal{H} +i\cdot\mathcal{H}=\mathcal{C}_{i}\cdot\mathcal{H}=\mathcal{H} \cdot\mathcal{C}_{i}\). The group \(\mathcal{H}\) of proper rotations includes \(\mathcal{C}_{n},\ \mathcal{D}_{n}\mathcal{,\ T,\ O,\ I}\) and the following groups are generated:

   \(\mathcal{C}_{n} \cdot \mathcal{C}_{i}\) (g = 2n):

   • \(\mathcal{C}_{nh} = n/m\) (n even) because \(C_{2}\) occurs in \(\mathcal{C}_{n}\) and \(i \cdot C_{2} = \sigma_{h}\).
   • \(\mathcal{S}_{2n} = \overline{2n}\) (n odd). For n = 1, \(\mathcal{S}_{2} \equiv \mathcal{C}_{i} = \overline{1}\).

   \(\mathcal{D}_{n} \cdot \mathcal{C}_{i}\) (g = 4n):

   • \(\mathcal{D}_{nh} = n/mmm\) (n even) because \(C_{2}\) occurs in \(\mathcal{D}_{n}\) and \(i \cdot C_{2} = \sigma_{h},\ \sigma_{v},\ \sigma_{d}\).
   • \(\mathcal{D}_{nd} = \overline{n}m\) (n odd).

   \(\mathcal{T} \cdot\mathcal{C}_{i}\) (g = 24):

   • \(\mathcal{T}_{h} = \overline{2}\overline{3} = m\overline{3}\) because \(i \cdot C_{2} = \sigma_{h}\) and \(\overline{2} = m\).

   \(\mathcal{O}\cdot\mathcal{C}_{i}\) (g = 48):

   • \(\mathcal{O}_{h} = \overline{4}\overline{3}\overline{2} = m\overline{3}m\) because \(i \cdot C_{2} = \sigma_{h}\), \({\overline{4}}^{2} = 2\), and \(\overline{2} = m\).

   \(\mathcal{I} \cdot \mathcal{C}_{i}\) (g = 120):

   • \(\mathcal{I}_{h} = \overline{2}\overline{3}\overline{5} = m\overline{3}\overline{5}\).

2. If \(\mathcal{G}\) does not contain the inversion, then \(\mathcal{G =}\left\{ C^{(j)},i \cdot {C'}^{(j)},\ \ j = 1,\ldots,n \right\}\), which is isomorphous¹ with the group of 2n proper rotations \(\mathcal{G}' = \left\{ C^{(j)},{C'}^{(j)},\ \ j = 1,\ldots,n \right\}\). Then, the set \(\mathcal{H =}\left\{ C^{(j)},\ \ j = 1,\ldots,n \right\}\) is a proper subgroup of \(\mathcal{G}'\) of index 2, and \(\mathcal{G}= \mathcal{H} + i \cdot \left( \mathcal{G}'\mathcal{- H} \right) \equiv \mathcal{G}':\mathcal{H}\). These groups are:

   \(\mathcal{C}_{2n}:\mathcal{C}_{n}\) (g = 2n):

   • \(\mathcal{S}_{2n} = \overline{2n}\) (n even).
   • \(\mathcal{C}_{nh} = n/m\) (n odd). \(C_{2}\) is not in \(\mathcal{C}_{n}\) but it is in \(\mathcal{C}_{2n}\) and \(i \cdot C_{2} = \sigma_{h}\). For n = 1, \(\mathcal{C}_{1h} \equiv \mathcal{C}_{s} = \left\{ E,\sigma \right\} = m\).

   \(\mathcal{D}_{n}:\mathcal{C}_{n}\) (g = 2n):

   • \(\mathcal{C}_{nv}\) because \(i \cdot C_{2}' = \sigma_{v}\) are parallel to the principal axis. The International symbol is \(nmm\) (n even) because there are two sets of equivalent \(\sigma_{v}\) planes; the symbol is \(nm\) (n odd) because all \(\sigma_{v}\) planes are equivalent.

   \(\mathcal{D}_{2n}:\mathcal{D}_{n}\) (g = 4n):

   • \(\mathcal{D}_{nd} = \overline{2n}2m\) (n even). \(C_{2}\) is in both \(\mathcal{D}_{2n}\) and \(\mathcal{D}_{n}\) and the set \(\left( \mathcal{D}_{2n} - \mathcal{D}_{n} \right)\) leaves one set of orthogonal \(C_{2}\) axes and vertical mirror planes \(\sigma_{d}\).
   • \(\mathcal{D}_{nh} = \overline{2n}m2\) (n odd). \(C_{2}\) is not in \(\mathcal{D}_{n}\) but it is in \(\mathcal{D}_{2n}\) and \(i \cdot C_{2} = \sigma_{h}\).

   \(\mathcal{O:T}\) (g = 24):

   • \(\mathcal{T}+ i \cdot \left( \mathcal{O}-\mathcal{T} \right) = \mathcal{T}_{d} = \overline{4}3\overline{2} = \overline{4}3m\). \(i \cdot \left( \mathcal{O}- \mathcal{T} \right)\) creates \(S_{4}\) and \(\sigma_{d}\).

This enumeration procedure reveals four distinct types of 3-d point groups as listed below:

Point Group	Notation			Order	Description
Type	Schönflies	International
Nonaxial	\(\mathcal{C}_{1}\)	1		1	No symmetry (Asymmetric)
	\(\mathcal{C}_{i}\)	\(\overline{1}\)		2	Inversion
Uniaxial	\(\mathcal{C}_{n}\)	\(n\)		\(n\)	One n-fold axis
	\(\mathcal{C}_{s}\)	\(m\)		2	One mirror plane
	\(\mathcal{S}_{2n}\)	\(\overline{2n}\)	(n even)	\(2n\)	One 2n-fold roto-reflection axis
		\(\overline{n}\)	(n odd)
	\(\mathcal{C}_{nh}\)	\(n/m\)	(n even)	\(2n\)	One n-fold axis + perpendicular mirror plane
		\overline{2n}\)	(n odd)
	\(\mathcal{C}_{nv}\)	nmm	(n even)	2n	One n-fold axis + n parallel mirror planes
		nm	(n odd)
Dihedral	\(\mathcal{D}_{n}\)	n22	(n even)	2n	One n-fold axis + n perpendicular 2-fold axes
		n2	(n odd)
	\(\mathcal{D}_{nh}\)	\(n/mmm\)	(n even)	4n	One n-fold axis + n perpendicular 2-fold axes + one perpendicular mirror plane
		\(\overline{2n}m2\)	(n odd)
	\(\mathcal{D}_{nd}\)	\(\overline{2n}2m\)	(n even)	4n	One n-fold axis + n perpendicular 2-fold axes + n parallel mirror planes
		\(\overline{n}m\)	(n odd)
Polyhedral	\(\mathcal{T}\)	\(23\)		12	Three 2-fold + four 3-fold axes
	\(\mathcal{T}_{h}\)	\(m\overline{3}\)		24	Three 2-fold + four 3-fold axes + inversion
	\(\mathcal{T}_{d}\)	\(\overline{4}3m\)		24	Three 4-fold roto-reflection axes + four 3-fold axes + six dihedral mirror planes
	\(\mathcal{O}\)	\(432\)		24	Three 4-fold + four 3-fold + six 2-fold axes
	\(\mathcal{O}_{h}\)	\(m\overline{3}m\)		48	Three 4-fold + four 3-fold axes + inversion
	\(\mathcal{I}\)	\(235\)		60	Fifteen 2-fold + ten 3-fold + six 5-fold axes
	\(\mathcal{I}_{h}\)	\(m\overline{35}\)		120	Fifteen 2-fold + ten 3-fold + six 5-fold axes + inversion

This table identifies a significant difference between the Schönflies and International notations. For the uniaxial and dihedral groups, the Schönflies notation prioritizes the highest-order n-fold proper rotation. The exceptions are the cyclic groups based upon a roto-reflection, i.e., \(\mathcal{S}_{2n}\) like \(\mathcal{S}_{4}\), \(\mathcal{S}_{6}\), etc. The International notation, on the other hand, emphasizes the highest-order rotation axis of either sort, as seen for \(\overline{6} = \mathcal{C}_{3h}\). Also, the International symbol depends on whether the principal axis is even-order or odd-order.

Although not listed in this table, the International symbols of certain point groups have a so-called long form: \(\mathcal{D}_{nh} = n/m\ 2/m\ 2/m\); \(\mathcal{T}_{h} = 2/m\ \overline{3}\); \(\mathcal{O}_{h} = 4/m\ \overline{3}\ 2/m\); and \(\mathcal{I}_{h} = 2/m\ \overline{3}\ \overline{5}\). All of these point groups include inversion, which is generated by the combination of a 2-fold proper rotation and a perpendicular reflection, i.e., by “\(2/m\)”. The short form is compact and identifies the operations that, when multiplied together, will generate all members of the point group.

Lastly, the polyhedral groups identify point symmetries for tetrahedral, octahedral, cubic, icosahedral, and dodecahedral structures. Each of these polyhedra can be inscribed in a cube that is oriented with its faces perpendicular to the Cartesian axes. As a result, the International symbols of these groups have two or three parts in the following order: (i) 2-, 4-, or \(\overline{4}\)-fold axes along the x, y, and z-directions; (ii) 3- or \(\overline{3}\)-fold axes along the body-diagonals of the cube; and (iii) 2-fold or 5-fold axes along or near the face-diagonals of the cube.

As an example of how the two notations compare, consider the point group \(\mathcal{D}_{2d}\). The Schönflies symbol identifies a principal \(C_{2}\) axis with perpendicular \(C_{2}\) axes and vertical dihedral mirror planes \(\sigma_{d}\), which bisect to perpendicular \(C_{2}\) axes. Multiplying these operations with each other generates a total of 8 operations: 4 proper rotations (in red) derived from the \(C_{2}\) rotations and 4 improper rotations (in green) including the reflections \(\sigma_{d}\) and an \(S_{4}\) roto-reflection. If the principal \(C_{2}\) axis lies along the z-direction, then the 8 members of this point group in both Schönflies and International notations are as follows:

\[\mathcal{D}_{\mathbf{2}\mathbf{d}}\mathbf{=}\overline{\mathbf{4}}\mathbf{2}\mathbf{m} \nonumber \]

Figure 2.13: Diagram of the 8 symmetry operation for point group D2d = 4-bar 2 m looking down z-axis. 4-bar axis is along z, denoted by square with inserted oval; 2-fold axes along x- and y-axes denoted by red arrows; vertical diagonal mirror planes denoted by green lines.

	Schönflies	International	Coordinates
(1)	\(E\)	\(1\)	x, y, z
(2)	\(C_{2z}=S_{4z}^2\)	\(2_z=\overline{4}_z^2\)	–x, –y, z
(3)	\(C_{2x}^\prime\)	\(2_x\)	x, –y, –z
(4)	\(C_{2y}^\prime\)	\(2_y\)	–x, y, –z
(5)	\(\sigma_{d1}\)	\(m_{x+y}\)	–y, –x, z
(6)	\(\sigma_{d2}\)	\(m_{x-y}\)	y, x, z
(7)	\(S_{4z}\)	\(\overline{4}_z^3\)	–y, x, –z
(8)	\(S_{4z}^3\)	\(\overline{4}_z\)	y, –x, –z

The International point group symbol is \(\overline{4}2m\), which also has three components: (1) the first part \(\overline{\mathbf{4}}2m\) identifies a 4-fold roto-inversion axis along the principal axis (the z-direction); (2) the second part \(\overline{4}\mathbf{2}m\) identifies the 2-fold rotation axes along the equivalent x- and y-directions; and (3) the final part \(\overline{4}2\mathbf{m}\) designates the equivalent reflections \(m_{x + y}\) and \(m_{x - y}\) that are perpendicular to the directions bisecting the x- and y-directions. Although the two notations emphasize features of the principal axis, the two sets of operations are identical.

There is another equivalent International symbol for \(\mathcal{D}_{2d}\), as noted in the following:

Figure 2.14: Diagram of the 8 symmetry operation for point group D2d = 4-bar m 2 looking down z-axis. 4-bar axis is along z, denoted by square with inserted oval; vertical mirror planes along x- and y-axes denoted by green lines; diagonal 2-fold rotation axes denoted by red arrows.

	Schönflies	International	Coordinates
(1)	\(E\)	\(1\)	x, y, z
(2)	\(C_{2z}=S_{4z}^2\)	\(2_z=\overline{4}_z^2\)	–x, –y, z
(3)	\(C_{2(x+y)}^\prime\)	\(2_{x+y}\)	y, x, –z
(4)	\(C_{2(x-y)}^\prime\)	\(2_{x-y}\)	–y, –x, –z
(5)	\(\sigma_{d1}\)	\(m_x\)	–x, y, z
(6)	\(\sigma_{d2}\)	\(m_y\)	x, –y, z
(7)	\(S_{4z}\)	\(\overline{4}_z^3\)	–y, x, –z
(8)	\(S_{4z}^3\)	\(\overline{4}_z\)	y, –x, –z

For this case, the dihedral reflections and 2-fold rotations perpendicular to the principal axis are rotated by 45º from the previous arrangement so that the International symbol is \(\overline{4}m2\). Therefore, the dihedral mirror planes are perpendicular to the equivalent x- and y-directions; the 2-fold axes lie along the equivalent x+y- and x–y-directions.

As this example highlights, the International notation constructs symbols that match the orientations of symmetry operations with specific directions in 3-dimensional space. For a molecule in solution or the gas phase, which is freely tumbling and moving, there are no fixed spatial directions except those local to the structure of the molecule. In this case, the Schönflies symbol is elegantly sufficient to identify all symmetry operations. On the other hand, for a molecule condensed in a crystal, there is a coordinate system derived from the unit cell. As a result, the International symbolism is better suited to describe the point group operations of the molecule in a crystal than the Schönflies notation, as shown for these two examples:

Figure 2.17: Li (gray atom) environment in LiFeCr₄O₈ is \(\bar{4}m2\)4m2 with 2-fold axes parallel to the a+b- and a–b-directions. Figure 2.16: Hg (black atoms) environment in Cu₂HgI₄ is \(\bar{4}2m\) with 2-fold axes parallel to the a- and b-axes