06 | Visualizing higher symmetry point groups

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Multiples of Rotations and a non-Albelian group

Now lest consider a few other examples of symmetry groups. Below is the stereographic projection, Fig. 6, of \(3m\) the point group of the molecular ammonia. Like \(mm2\) this point group is also polar meaning there is not symmetry operation that transforms the relates principal rotation axis (\(z\)-axis) to its inverse (the \(-z\)-axis). Note that unlike \(mm2\) the point group \(3m\) is not Albelian. In the stereographic projection again the three mirror planes are represented as thick solid lines, and the three-fold rotation is pointing out of the plane and represented as 3.

Fig. 6 (Click image) Generation of the general positions of \(3m\) by application of a \(3\), \(3^2\) and \(m\) operation.

This point group also has an until now unexpected operation in the multiplication table, Table 3. In addition to \(1\), \(3\), and the three mirror planes \(m\), \(m^{\prime}\), and \(m^{\prime\prime}\) there the operation \(3^2\) is also listed. This operation is just the application of two 3-fold rotations in series and is equivalent to the multiplication of their matrix representations. Likewise in symmetry groups with four-fold rotations there is a \(4\), \(4^2=2\), and a \(4^3\) proper rotation operation.

Table 3 Symmetry group multiplication table for \(3m\)

\(3m\)	\(1\)	\(3\)	\(3^2\)	\(m\)	\(m^{\prime}\)	\(m^{\prime\prime}\)
\(1\)	\(1\)	\(3\)	\(3^2\)	\(m\)	\(m^{\prime}\)	\(m^{\prime\prime}\)
\(3\)	\(3\)	\(3^2\)	\(1\)	\(m^{\prime}\)	\(m^{\prime\prime}\)	\(m\)
\(3^2\)	\(3^2\)	\(1\)	\(3\)	\(m^{\prime\prime}\)	\(m\)	\(m^{\prime}\)
\(m\)	\(m\)	\(m^{\prime\prime}\)	\(m^{\prime}\)	\(1\)	\(3^2\)	\(3\)
\(m^{\prime}\)	\(m^{\prime}\)	\(m\)	\(m^{\prime\prime}\)	\(3\)	\(1\)	\(3^2\)
\(m^{\prime\prime}\)	\(m^{\prime\prime}\)	\(m^{\prime}\)	\(m\)	\(3^2\)	\(3\)	\(1\)

Perpendicular \(2\) rotations and mirror planes

The eclipsed conformation of the molecule ethane, \(C_6H_6\), can be assigned to the point group \(\bar{6}m2\). The principal rotation in this case is teh 6-fold rotary inversion and by convention point out of the plan in the stereographic projection. It is illustrated as the symbol $ in Fig. 7. Two other symmetry operations we have not seen in a projection are shown. A perpendicular \(2\) is draw at the edges of the circle and is illustrated as half of a fi in Fig. 7. In this point group it just so happens that the \(\perp 2\) axes are coincident with three of the four mirror planes draw as solid lines. There is a fourth mirror plan in \(\bar{6}m2\) which is perpendicular to the \(\bar{6}\) axis and is draw Fig. 7 as a solid circle that encompasses the entire diagram. Take careful note of how the \(\bar{6}\) rotations acts upon the general positions of the group.

There are 12 general positions in the point group \(\bar{6}m2\). However there are only 2 carbon atoms and only 6 hydrogen atoms in ethane. How must they align with the elements of symmetry in \(\bar{6}m2\) for this to be true?

Fig. 7 (Click image) Generation of the general positions in \(\bar{6}m2\) by application of a \(\perp m\), \(\perp 2\), \(\bar{6}\), and \(\bar{6}^2 \equiv 3\). For clarity a \(,\) is only only to denote when handedness is inverted. In this point group general positions are perfectly eclipsed when projected along the principal axis of rotation. The circle is bisected to show both the positive general position (above the plane) and the negative general position (below the plane). There are 12 general positions in \(\bar{6}m2\).

Inversion

Much like the identity operation that of inversion \(1\) is not always shown in stereographic projections provided all of the general positions in the group can be generated without it. When it is show, for example in the point group \(\bar{1}\), the symbol o is used to denote its presence.

Naming point group with generators

Thus far we have identified elements of symmetry and a system by which we can illustrate their symmetry operations in diagrams and overlays of molecules. The names of point groups have not been discussed. They are not arbitrary but are based on what are call the generators or the group. That is, the point group symbol contains all of the necessary symmetry operations required to generate all of the general positions of the group as well as all other symmetry operations. Their notation changes depending on the symmetry of the crystal system. The term crystal system is in large part is synonymous with coordinate system. The point group of water, \(mm2\), is in the orthorhombic crystal system and in this case the symbol encodes three symmetry operations referenced to the \(x\), \(y\), and \(z\), axes. Specifically a mirror plane perpendicular to \(x\), a mirror plane perpendicular to \(y\), and a \(2\) along \(z\).

In Palmer and Ladd Table 1.5 you will find a summary table describing how all point group symbols in Hermann-Mauguin notation are defined. This same information is available in Table 2.2.4.1 of the International Tables of Crystallography A (2005). Section 10.1 of International Tables of Crystallography A (2005) also contains the stereographic projections of all 32 crystallographic point groups in three-dimensions. Can you derive \(3m\) and \(\bar{6}m2\) from their descriptions in this table?

Point groups symbols have 1–3 characters. Often these characters just denote which axes a rotation lies along (all symmetry operations are rotations in the Hermann-Mauguin system). However sometimes a charecter can be a fraction as in the tetragonal point group \(\frac{4}{m}mm\). In this case the first character is \(\frac{4}{m}\) and can be translated to mean there is a 4-fold rotation (along \(z\)) and a mirror plane perpendicular to the 4-fold axis. Alternative this first character would also read as “there is a 4-fold rotation (along “z”) and a “\bar{2}” rotation along the same axis. The conventions for the ordering of charecters in point group names and their meanings are described in Table 4

Table 4 Crystal systems and their allowed point groups

Crystal System	Examples	Interpretation
Triclinic	\(1\); \(\bar{1}\)	Only two point groups. Describes symmetry along all basis vectors
Monoclinic	\(2\); \(2/m \equiv \frac{2}{m}\)	Only one symbol character. Describes symmetry with respect to \(y\)-axis
Orthorhombic	\(mm2\); \(mmm\) \(222\)	Three symbol characters. Each describes symmetry relative to the \(x\), \(y\), and \(z\) axes respectively.
Tetragonal	\(\frac{4}{m}mm\)	4-fold conventionally along \(z\)-axis. Symbols have up to three characters. First refers to symmetry relative to the principal rotation axis, second is relative to \(x\)-axis (\(x \equiv y in tetragonal system\)). Third character refers to symmetry operations along the \(xy\)-dihedrals.
Cubic	\(m3\); \(432\); \(m3m\)	All three axes \(x\), \(y\), and \(z\) are equivalent. 1st character is relative to \(x\). 2nd character is for symmetry relative to the line intersecting the corners of a cube. The 3rd character is for symmetry along the xy-dihedral which is equivalent to the xz and yz dihedrals.
Hexagonal	\(6/m\); \(6mm\); \(\bar{6}\)	1st charecter is along the principal rotation axis, \(z\) by convention. The \(x\) and \(y\) axes are equivalent. 2nd character is symmetry relative to \(x\) or \(y\) axis. 3rd is \(\perp x\) in the \(xy\)-plane
Trigonal	\(\bar{3}\); \(32\); \(\bar{3}m\)	1st charecter is along the principal rotation axis, \(z\) by convention. The \(x\) and \(y\) axes are equivalent. 2nd character is symmetry relative to \(x\) or \(y\) axis.

References

1. International Tables of Crystallography A 5th Edition 2005

2. Ladd, M. F. C.; Palmer, R. A. Structure determination by X-ray crystallography Chapt. 1.4: External Symmetry of Crystals. New York: Plenum Press, 1977, 35–41.