# 2.2: Point Groups

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## The Low Symmetry Point Groups

### C_{1 }Point Group

Overall, we divide point groups into three major categories: High symmetry point groups, low symmetry point groups, dihedral point groups, and rotational point groups. Let us begin with the low symmetry point groups. As the name says, these point groups only have few symmetry elements and operations. The point group C_{1} is the point group with the lowest symmetry. Molecules that belong to this point group only have the identity as symmetry element.

An example is the bromochlorofluromethane molecule (Fig. 2.2.1). It has no symmetry element, but the identity. The name C_{1} comes from the symmetry element C_{1}. A C_{1} operation is the same as the identity.

### C_{s} Point Group

The point group C_{s} has a mirror plane in a addition to the identity. An example is the 1,2-bromochloroethene molecule (Fig. 2.2.2).

This is a planar molecule and the mirror plane is within the plane of the molecule. This mirror plane does not move any atoms when the reflection operation is carried out, nonetheless it exists because any point of the molecule above the mirror plane will be found below the mirror plane after the execution of the operation. Vice versa, any point below the mirror plane will be above the mirror plane. This mirror plane does not have a vertical or horizontal mirror plane designation because no proper rotational axes exist.

### C_{i }Point Group

The point group C_{i} has the inversion as the only symmetry element besides the identity. The point group C_{i} is sometimes also called S_{2} because an S_{2} improper rotation-reflection is the same as an inversion. An example is the 1,2-dibromo 1,2-dichloro ethane (Fig. 2.2.3).

This molecule looks quite symmetric, but it has inversion center in the middle of the carbon-carbon bond as the only symmetry element. Upon execution of the inversion operation, the two carbons swap up their positions, and so do the two bromine, the two chorine, and the two hydrogen atoms.

## The High Symmetry Point Groups

The symmetry elements of the high symmetry point groups can be more easily understood when the properties of platonic solids are understood first. Platonic solids are polyhedra made of regular polygons. In a platonic solid all faces, edges, and vertices (corners) are symmetry-equivalent. We will see that this is a property that can be used to understand the symmetry elements in high symmetry point groups. There are only five possibilities to make platonic solids from regular polygons (Fig. 2.2.4).

The first possibility is to construct a tetrahedron from four regular triangles. The second platonic solid is the octahedron made of eight regular triangles. The third possibility is the icosahedron made of twenty triangles. In addition, six squares can be connected to form a cube, and twelve pentagons can be connected to form a dodecahedron. There are no possibilities to connect other regular polygons like hexagons to make a platonic solid.

The icosahedron is the most complex of all platonic solids. If you would like to see and study an icosahedron from the outside and inside, there is one for study on the playground of the Allentown Cedar Beach Park, in Allentown, Pennsylvania.

### The T_{d }Point Group

The tetrahedron, as well as tetrahedral molecules and anions such as CH_{4} and BF_{4}- belong to the high symmetry point group T_{d}. Let us find the symmetry elements and symmetry operations that belong to the point group T_{d}. First, we should not forget the identity operation, E. Next, it is useful to look for the principal axes.

The tetrahedron has four principal C_{3} axes (Fig. 2.2.6). It is a property of the high-symmetry point groups that they have more than one principal axis. The C_{3 }axes go through the vertices of the tetrahedron. Because each C_{3} axis goes through one vertex, there are four vertices, and we know that in a platonic solid all vertices are symmetry-equivalent, we can understand that there are four C_{3} axes. How many unique C_{3} operations are associated with these axes? After three rotations around 120° we reach the identity. Therefore C_{3}^{3}=E, and we only need to consider the C_{3}^{1} and the C_{3}^{2} rotation about 120 and 240° respectively. Because there are four C_{3 }axes, there are four C_{3}^{1} and four C_{3}^{2} operations and eight C_{3} operations overall. We can express this by writing the respective numbers as coefficients in front of the Schoenflies symbol for the operations (Fig. 2.2.7).

In addition to the C_{3} axes there are C_{2} axes (Fig. 2.2.8).

You can see that a C_{2} axis goes through two opposite edges in the tetrahedron. Because a tetrahedron has six edges, and each C_{2} axis go through two edges there are 6/2=3 C_{2} axes. There is only one C_{2} symmetry operation per C_{2} axis because we produce the identity already after two rotations. Therefore there are three C_{2}^{1} operations overall (Fig. 2.2.9).

In addition, the T_{d} point group has S_{4} improper rotation reflections. Like the C_{2} axes, they pass through the middle of two opposite edges. This also means that they are superimposing the C_{2} axes. Because there are six edges, and two S_{4} axes per edge there are 6/2=3 S_{4} axes (Fig. 2.2.10).

How many operations are associated with these S_{4} axes? The order of the axes are even, and therefore we need four S_{4} operations to produce the identity. The S_{4}^{2} operation is the same as a C_{2}^{1} operation because reflecting two times is equivalent to not reflecting at all, and rotating two times by 90° is the same as rotating about 180°. Therefore overall, only S_{4}^{1} and S_{4}^{3} operations are unique operations. S_{4}^{2} and S_{4}^{4 }can be expressed by the simpler operations C_{2}^{1} and E respectively. Because there are 3 S_{4} axes, there are three S_{4}^{1} and three S_{4}^{3} operations. Overall there are six S_{4} operations (Fig. 2.2.11).

There are also mirror planes (Fig. 2.2.11). The planes contain a single edge of the tetrahedron, thereby bisecting the tetrahedron. There a six edges in a tetrahedron, and therefore there are 6/1=6 mirror planes.

These planes are dihedral planes because each plane contains a C_{3} principal axis and is bisects the angle between two C_{2} axes. Overall, there are three C_{2} axes and three C_{2} operations. There is one reflection operation per mirror plane because reflecting two times produces the identity. Therefore, there are six σ_{d} reflection operations (Fig. 2.2.12, right and Fig. 2.2.13).

In sum (Fig.2.2.14 and 2.2.15) we can denote the overall symmetry of the T_{d} point group the following way: E, 8C_{3}, 3C_{2}, 6S_{4}, 6σ_{d}. In detail the unique symmetry operations are E, 4C_{3}^{1}, 4C_{3}^{2}, 3C_{2}^{1}, 3S_{4}^{1}, 3S_{4}^{3}, 6σ_{d}.

### The Rotational Subgroup T

The high symmetry point group T is the so-called rotational subgroup of the point group T_{d}. A rotational subgroup is a point group in which all symmetry operations but the identity and the proper rotations have been removed from a high-symmetry point group. For the point group T this leaves the 4C_{3}, the four C_{3}^{2} and the three C_{2} operations (Fig. 2.2.17). The S_{4} rotation-reflections and the mirror planes have been removed. The point group T is rare.

An example is the depicted Ga_{4}L_{6} cage (Fig. 2.2.16). The Ga atoms occupy the vertices of a tetrahedron, but the point group is not T_{d} but T because of the shape of the ligands that connect the four Ga atoms.

### The Octahedral Point Group O_{h}

Another high symmetry point group is the point group O_{h}. Both the octahedron as well as the cube belong to this point group despite their very different shape (Fig. 2.2.18). Because they belong to the same point group they must have the same symmetry elements and operations. There are many octahedrally shaped molecules, such as the SF_{6}.

Molecules with cubic shapes are far less common, because a cubic shape often leads to significant strain in the molecule. An example is cubane C_{8}H_{8}. Let us determine the symmetry elements and operations for the point group O_{h} using the example of the octahedron. If we used the cube, we would get exactly the same results.

There are three C_{4} principal axes in the octahedron. They go through two opposite vertices of the octahedron (Fig. 2.2.19). There are three C_{4} axes because an octahedron has six vertices which are all symmetry-equivalent because the octahedron is a platonic solid.

We can see that there are also C_{2} axes where the C_{4} axes run. This is because rotating two times around 90° is the same as rotating around 180°. What are the symmetry operations associated with these symmetry elements? Rotating four times around 90° using the C_{4} axes produces the identity. So we have to consider the operations C_{4}^{1}, C_{4}^{2}, C_{4}^{3} and C_{4}^{4}. How many of these are unique? C_{4}^{4} is the same as the identity, so it is not unique, In addition a C_{4}^{2} is identical to a C_{2}^{1}, and thus C_{4}^{2} is also not unique, and can be expressed by the simpler operation C_{2}^{1}. That leaves the C_{4}^{1} and the C_{4}^{3} as the only unique symmetry operations. Because we have three C_{4} axes, there are 2x3=6 C_{4} operations, in detail there are 3C_{4}^{1} and three C_{4}^{3} operations. In addition, there are the three C_{2}^{1} operations belonging the the three C_{2} axes (Fig. 2.2.20).

_{4 }and C

_{2 }symmetry operations in the octahedral point group O

_{h}

In addition, there are four C_{3} axes (Fig. 2.2.21). They are going through the center of two opposite triangular faces of the octahedron.

_{3 }axes in the octahedral point group O

_{h}(Attribution: symotter.org/gallery)

You see above a single C_{3} axis, and on the right hand side all four of these axes. How can we understand that there are four axes? An octahedron has overall eight triangular faces, and each C_{3} axis goes through two opposite faces, so there are 8/2=4 C_{3} axes. Each C_{3} axis has the C_{3}^{1} and the C_{3}^{2} as unique symmetry operations. The C_{3}^{3} is the same as the identity. So overall we have 4x2=8 operations, four of them are C_{3}^{1}, and four of them are C_{3}^{2 }(Fig. 2.2.22).

_{3 }in the octahedral point group O

_{h}

_{2}' axes in the octahedral point group O

_{h}(Attribution: symotter.org/gallery)

In addition to the C_{2} axes that superimpose the C_{4} axes, there are C_{2}’ axes which go though two opposite edges of the octahedron (Fig. 2.2.23). How many of them are there? An octahedron has twelve edges, and because each C_{2}’ passes through two edges, there must be 12/2=6 C_{2}’ axes. These axes have primes because they are not conjugate to the C_{2} axes that superimpose the C_{4} axes. For each C_{2}’ axis there is only the C_{2}’ ^{1} as the unique symmetry operation, and therefore there are overall 6 C_{2}’^{1} symmetry operations (Fig. 2.2.24).

_{2}' in the octahedral point group O

_{h}

Let us look at the mirror planes next (Fig. 2.2.25). There are horizontal mirror planes that stand perpendicular to the C_{4} principle axes. You can see a single one of them below on the left.

_{h}(Attribution: symotter.org/gallery)

Note that this mirror plane also contains two axes, in addition to the one to which it stands perpendicular. Because it contains two principal C_{4} axes, it has also properties of a vertical mirror plane. Nonetheless, we call it a horizontal mirror plane because it stands perpendicular to the third C_{4}. The horizontal properties trump the vertical ones, so to say. You can see that a single mirror plane contains four edges of the octahedron. Because there are twelve edges, there are 12/4=3 horizontal mirror planes. There is one mirror plane per principal C_{4} axis. There are three horizontal reflection operations because there is always only one reflection operation per mirror plane (Fig. 2.2.26).

_{h}

Next let us look for vertical mirror planes (Fig. 2.2.27). A vertical mirror plane is depicted below on the left.

_{h}(Attribution: symotter.org/gallery)

You can see that - contrast to the horizontal mirror planes - it does not contain any edges. Rather, it cuts through two opposite edges. You can see that this plane contains a C_{4} axis, but it does not stand perpendicular to the other two C_{4} axes. Therefore it has only the properties of a vertical mirror plane. You can see however, that the mirror plane bisects the angle between two C_{2}’ axes which also depicted. This makes the vertical mirror planes dihedral mirror planes, σ_{d}. How may of them do we have? As previously mentioned, each mirror plane cuts through two opposite edges. There are twelve edges in an octahedron, and thus there are 12/2=6 dihedral mirror planes. You can see all of them on the right side of Fig. 2.2.27. Each mirror plane is associated with one reflection operation, therefore there are six dihedral reflection operations (Fig. 2.2.28).

_{h}

Next we can ask if the point group O_{h} has an inversion center? Yes, there is one in the center of the octahedron (Fig. 2.2.29)!

Figure 2.2.29 The inversion center of the octahedral point group O_{h} (Attribution: symotter.org/gallery)

Each point in the octahedreon can be moved through the inversion center to the other side, and the produced octahedron will superimpose the original one. There is always one inversion operation associated with an inversion center (Fig. 2.2.30).

Next, let us look for rotation-reflections. You can see an S_{6} rotation-reflection operation below (Fig. 2.2.31, left).

_{6 }rotation-reflection element of the octahedral point group O

_{h}(Attribution: symotter.org/gallery)

The improper S_{6} axis passes though the centers of two opposite triangular faces. One can see that rotation about 60° alone does not make the octahedron superimpose. The reflection at a plane perpendicular to the improper axis is required to achieve superposition. Overall, the rotation-reflection swaps up the position of the two opposite triangular faces. How many S_{6} improper axes are there? Since each S_{6} passes through two faces, and an octahedron has 8 faces there must be 8/2=4 S_{4} axes. You can see all of them above (Fig. 2.2.31, right). Note that they are in the same position as the 4C_{3} axes we previously discussed. How many unique operations are associated with them? For an S_{6} axis we need to consider operations from S_{6}^{1} to S_{6}^{6}. S_{6}^{6} is the same as the identity so it is not unique. The S_{6}^{2} is the same as a C_{3}^{1} because rotating two times round 60° is the same as rotating around 120°, and reflecting twice is the same as not reflecting at all. Similarly, an S_{6}^{4} is the same as an C_{3}^{2}. Rotating four time by 60° is the same as rotating two times by 120° and reflecting four times is the same as not reflecting at all. Further, an S_{6}^{3} is the same as an inversion. After three 60° rotations we have rotated by 180°. If we reflect after that, then this is the same as an S_{2}^{1} operation which is the same as an inversion. Therefore, only the S_{6}^{1} and the S_{6}^{5} operations are unique, all other operations can be expressed by simpler operations (Fig. 2.2.32).

_{6 }of the octahedral point group O

_{h}

The octahedron also has S_{4} improper axes, and you can see one of them below (Fig. 2.2.33, right).

_{4 }improper axis of the octahedral point group O

_{h}(Attribution: symotter.org/gallery)

It goes through two opposite corners of the octahedron. The S_{4} improper axis seemingly does the same as the C_{4} axis that goes through the same two opposite vertices, but actually does not. While rotating around 90° already makes the octahedron superimpose with its original form, executing the reflection operation after the rotation swaps up the position of the two vertices, and generally all points of the octahedron above and below the plane, respectively. Overall the S_{4} moves the points within the object differently compared to the C_{4} which makes it an additional, unique symmetry element. There are overall three S_{4} improper axes because the octahedron has six vertices and one S_{4} passes through two vertices (Fig. 2.2.34).

_{4 }improper axis of the octahedral point group O

_{h}

_{h}(Attribution: symotter.org/gallery)

Here is an overview over all the symmetry elements and operations (Fig. 2.2.35 and 2.2.36). Overall, there are 48 different unique operations that one can perform!

_{h}

Like the point group T_{d} , also the point group O_{h} has a rotational subgroup, named O. It has the identity and the same proper rotations as the point group O_{h}, but no other symmetry operations (Fig. 2.2.38). An example is the polyoxometalate cluster core shown below (Fig. 2.2.37). Polyoxometalates are cluster anions of the group 5 and 6 elements.

Figure 2.2.37 Proper rotations in a V_{6}P_{8}O_{24} polyoxometalate cluster core. The C_{3} rotation is animated. (Attribution: symotter.org/gallery)

The point group O is generally rare.

_{h})

Another high symmetry point group is the point group T_{h}. It can also be derived from the point group O_{h}. In this case the S_{4}, the C_{4}, the C_{2}’, and the σ_{d} operations are removed from the octahedral symmetry. An example is the hexapyridyl iron (2+) cation (Fig. 2.2.39).

Figure 2.2.39 The hexapyridyl iron (2+) cation and its S_{6} and C_{2} symmetry elements. An S_{6} operation is animated.

You can see that the N-atoms of the pyridyl-ligands surround the Fe atoms octahedrally, but the symmetry is reduced from O_{h} to T_{h} because of the planar shape of the pyridyl-ligands. In particular the C_{4} symmetry is reduced to C_{2}. This reduction in symmetry leads to elimination of the S_{4}, the C_{2}’, and the σ_{d} symmetry elements (Fig. 2.2.40).

_{6}

^{2+}in the point group T

_{h}

### The I_{h }Point Group

The two remaining platonic solids, the icosahedron and the dodecahedron, belong both to the icosahedral point group I_{h}. This is despite they are made of different polygons (Fig. 2.2.41).

_{h }point group (Attribution: symotter.org/gallery)

Because they belong to the same point group, they have exactly the same symmetry operations. An example for a molecule with icosahedral shape is the molecular anion B_{12}H_{12}^{2-}. An example for a molecule with dodecahedral shape is the dodecahedrane C_{20}H_{20}.

_{12}H

_{12}

^{2- }and the dodecahedrane (Attribution: wikiwand.com https://upload.wikimedia.org/Wikipedia/commons/thumb/9/97/Dodecaborane-3D-balls.png/640px-Dodecaborane-3D-balls.png)

Let us determine the symmetry elements and symmetry operations for the example of the icosahedron. We could also use the dodecahedron, and the results would be the same. The principal axes of the icosahedron are the C_{5} axes. You can see one of them, going through the center of the pnetagon comprised of five triangular faces below (Fig. 2.2.43).

Figure 2.2.43 One of the C_{5 }axes of the icosahedron stands perpendicular to the paper plane going through the center of a pentagon of the icosahedron (Attribution: symotter.org/gallery)

You can understand that there is a C_{5} when considering that there are five triangular faces making a pentagon. The C_{5} axis sits in the center of the pentagon. We can see that when we rotate around this C_{5} axis, then the produced icosahedron superimposes the original one. The C_{5} axis goes through two opposite vertices of the icosahedron. Because an isosahedron has 12 vertices, there must be six C_{5} axes overall. You can see all of them below (Fig. 2.2.44).

_{5 }axes of the icosahedron (Attribution: symotter.org/gallery)

There are four unique symmetry operations associated with a single C_{5} axis, namely the C_{5}^{1}, the C_{5}^{2}, the C_{5}^{3}, and the C_{5}^{4}. The C_{5}^{5} is the same as the identity. Because there are six C_{5} axes, there are overall 6x4=24 C_{5} symmetry operations (Fig. 2.2.45).

_{5 }axis of icosahedral point group

In addition, there are C_{3} axes. One of them is shown below, and you can see that it passes through the centers of two opposite triangular faces (Fig. 2.2.46).

_{3 }axes of the icosahedral point group (Attribution: symotter.org/gallery)

As one rotates by 120° the atoms on the triangular faces change their position, and the resulting icosahedron superimposes the original one. As the name icosahedron says, there are twenty faces overall.

_{3 }axes of the icosahedral point group (Attribution: symotter.org/gallery)

Because one C_{3} passes through two opposite axes, there are 20/2=10 C_{3} axes overall (Fig. 2.2.47). Each C_{3} axis is associated with two symmetry operations, namely C_{3}^{1}, and C_{3}^{2}. Thus, there are overall 10x2=20 C_{3} symmetry operations.

_{3 }axes of the icosahedral point group

There are also C_{2} axes (Fig. 2.2.49). They pass through the centers of two opposite edges of the icosahedron. Rotating around the C_{2} axis shown makes the icosahedron superimpose.

_{2 }axes of the icosahedral point group (Attribution: symotter.org/gallery)

An isosahedron has overall 30 edges. Because one C_{2} axis passes through the centers of two opposite edges, we can understand that there are 30/2=15 C_{2} axes. There is one unique C_{2} operation per axis, and therefore there are 15 C_{2} operations (Fig. 2.2.50).

_{2 }axes of the icosahedral point group

We have now found all proper rotations. Let us look for mirror planes, next. You can see a mirror plane below (Fig. 2.2.51).

Figure 2.2.51 A mirror plane in the icosahedral point group (Attribution: symotter.org/gallery)

It contains two opposite edges. It also bisects two other edges. An icosahedron has overall 30 edges, therefore there are 30/2=15 mirror planes. You can see all of them below (Fig. 2.2.52 and Fig. 2.2.53)).

The icosahedron also has an inversion center in the center of the icosahedron (Fig. 2.2.54 and Fig. 2.2.55)).

Figure 2.2.54 The inversion center in the icosahedron (Attribution: symotter.org/gallery)

As we carry out the associated, one symmetry operation, all points in the isosahedron move through the inversion center to the other side.

Let us now look for improper rotations. The improper rotational axes with the highest order are S_{10} axes. They are located in the same position as the C_{5} axes, and go through two opposite corners (Fig. 2.2.56).

_{10 }improper rotational axes in the icosahedral point group (Attribution: symotter.org/gallery)

The S_{10} exists because in an icosahedron there are pairs of co-planar pentagons that are oriented staggered relative to each other. The rotation around 36° brings one pentagon in eclipsed position relative to the other, but superposition is only achieved after the reflection at the mirror plane perpendicular to the rotational axis. Because one S_{10} passes through two opposite vertices, and there are 12 vertices there are 6 S_{10} improper axes. For each axis there are four unique symmetry operations, the S_{10}^{1}, the S_{10}^{3}, the S_{10}^{7}, and the S_{10}^{9}. Therefore, there are overall 4x6=24 operations possible (Fig. 2.2.57).

_{10 }improper rotational axes of the icosahedral point group

Are the lower order improper rotational axes? Yes, there are S_{6} axes that pass through the centers of two opposite triangular faces (Fig. 2.2.58). This symmetry element exists because the two triangular faces are in staggered orientation to each other. Rotation alone brings one face in eclipsed orientation relative to the other, but reflection at a mirror plane perpendicular to the axis is required to achieve superposition. The S_{6} axes are in the same location as the C_{3} axes.

_{6 }rotation-reflections in the icosahedral point group (Attribution: symotter.org/gallery)

There are 10 S_{6} axes because there are twenty faces and one axis passes through two opposite faces. Only the S_{6}^{1} and the S_{6}^{5} operations are unique S_{6} operations, all others can be expressed by simpler operations. Therefore there are overall 10 S_{6}^{1}+10 S_{6}^{5} = 20 S_{6} operations (Fig. 2.2.59).

_{6 }rotation-reflections in the icosahedral point group

We have now found all symmetry operations for the I_{h} symmetry. There are overall 120 operations making the point group I_{h} the point group with the highest symmetry (Fig. 2.2.60 and Fig. 2.2.61).

Also the point group I_{h} has a rotational subgroup.

It is called I. An example of an object with this symmetry is the snub-dodecahedron (Fig. 2.2.62). It has the identity, and all the proper rotation operations of the point group O_{h}, but the inversion, the rotation-reflections, and the mirror planes are eliminated (Fig. 2.2.63).

_{h})

### Cyclic Point Groups

After having discussed high and low symmetry point groups, let us next look at cyclic point groups. They have the property that they have only a single proper n-fold rotational axis, but no other proper axes. In the most simple case they do not have any additional symmetry element such as mirror planes or rotation-reflections. These point groups are denoted C_{n} whereby n is the order of the proper axis. An example is the hydrogen peroxide molecule H_{2}O_{2} (Fig. 2.2.64).

Figure 2.2.64 The C_{2 }rotational axis of hydrogen peroxide

It has a so-called roof-structure due to its non-planarity. One hydrogen atom points toward us, and the other points away from us. This structure is due to the two electron-lone pairs at each sp^{3}-hybridized oxygen atom. These electron-lone pairs consume somewhat more space than the H atoms, and there is electrostatic repulsion between the electron lone pairs. Therefore, the electron lone pairs at the different oxygen atoms try to achieve the greatest distance from each other. This forces the H-atoms out of the plane, leading to the roof-structure of the hydrogen peroxide. Because the H_{2}O_{2} molecule is not planar, it only has a single C_{2} axis, but no other symmetry element besides the identity. The C_{2} axis passes through the center of the O-O bond. Execution of the C_{2} operation swaps up both the O and the H atoms.

Definition: Cyclic Groups C_{n}

Cyclic groups have one rotational axiPyramidal Groups

Another class of groups are the pyramidal groups, denoted C_{nv}. They have n vertical mirror planes containing the principal axis C_{n} in addition to the principal axis C_{n}. Generally molecules belonging to pyramidal groups are derived from an n-gonal pyramid. An n-gonal pyramid has an n-gonal polygon as the basis which is capped (Fig. 2.2.66).

For example a trigonal pyramid has a triangular basis which is capped, a tetragonal pyramid has a square which is capped, and so on. The proper axis associated with a specific pyramid has the order n and goes through the tip of the pyramid and the center of the polygon. An example of a molecule with a trigonal pyramidal shape is the NH_{3} (Fig. 2.2.67).

_{3 }axis and vertical mirror planes in NH

_{3 }

The three H atoms form the triangular basis of the pyramid, which is capped by the N atom. The NH_{3} molecule belongs to the point group C_{3v}. The C_{3} axis goes though the N atom which is the tip of the pyramid, and the center of the triangle defined by the H atoms. There are three vertical mirror planes that contain the C_{3} axis. Each of them goes through an N-H bond (Fig. 2.2.68).

_{3 }

Definition: Pyramidal Groups C_{nv}

Pyramidal groups have n vertical plane(s) in addition to the principal axis C_{n}.

### The Linear Group C_{∞v}

A special n-gonal polygon is the cone. A cone can be conceived as an n-gonal pyramid with an infinite number *n* of corners at the base (Fig. 2.2.69).

In this case the order of the rotational axis that passes through the tip of the cone and the center of the circular basis is infinite. This also means that there is an infinite number of vertical mirror planes that contain the C_{∞} axis (Fig. 2.2.71). The point group describing the symmetry of a cone is called the linear point group C_{∞v}. Polar, linear molecules such as CO, HF, N_{2}O, and HCN belong to this point group. You can see the HCN molecule with its C_{∞} axis and its infinite number of vertical mirror planes below (Fig. 2.2.70).

_{∞}axis of the HCN molecule

The infinite number of mirror planes, shown in blue are forming a cylinder that surround the molecule.

Definition: Linear group C_{oov}

The linear group C_{∞v }has an infinite number of vertical mirror planes containing a C_{∞ }axis

### Reflection Groups

If we add a horizontal mirror plane instead of n vertical mirror planes to a proper rotational axis C_{n} we arrive at a the reflection point group type C_{nh}. The presence of the horizontal mirror planes also generates an improper axis of the order n. This is because when one can rotate and reflect perpendicular to the rotational axes independently, then it must also be possible to do it in combination. An example of a molecule belong to a reflection group is the trans-difluorodiazene N_{2}F_{2 }(Fig. 2.2.72).

_{2 }axis and horizontal mirror plane in trans-N

_{2}F

_{2 }

It is a planar molecule with a C_{2} axis going through the middle of the N-N double bond, and standing perpendicular to the plane of the molecule. The horizontal mirror plane stands perpendicular to the C_{2} axis, and is within the plane of the molecule. There is an additional inversion center because an S_{2} must exist which is the same as an inversion center. The inversion center is in the middle of the N-N bonds. Overall, the molecule has the symmetry C_{2h}.

_{2}F

_{2 }

Definition: Reflection Group C_{nh}

A reflection group has a horizontal plane perpendicular to the principal axis C_{n}

### Dihedral Groups

Dihedral groups are point groups that have n additional C_{2} axes that stand perpendicular to the principal axis of the order n. If there are no other symmetry elements, then the point group is of the type D_{n}.

Figure 2.2.74 The tris-oxolato ferrate (3-) ion and its symmetry elements

For example in the point group D_{3} there is a C_{3} principal axis, and three additional C_{2} axes, but no other symmetry element (Fig. 2.2.75). The tris-oxolato ferrate (3-) ion belongs to this point group (Fig. 2.2.75). You can see that the C_{3} axis stands perpendicular to the paper plane, and there are three C_{2} axes in the paper plane.

_{3}

Definition: Dihedral Groups D_{n}

In a point group of the type D_{n} there is a principal axis of order n, n C_{2} axes, but no other symmetry elements.

If a horizontal mirror plane is added to the C_{n} axis and the n C_{2} axes we arrive at the prismatic point groups D_{nh} (Fig. 2.2.76). The addition of the horizontal mirror plane generates further symmetry elements namely an S_{n} and n vertical mirror planes.

_{nh}

Generally, molecules belonging to this point group derive from n-gonal prisms. The order of the principal axis is the same as the number of corners of the polygons the prism are made of.

Definition: Prismatic Groups D_{nh}

In prismatic point groups there is a horizontal mirror plane perpendicular to the principal axis C_{n}. There are also n C_{2} axes.

An example for a molecule belonging to a prismatic point group is PF_{5} (Fig. 2.2.77).

_{5 }molecule belonging to the point group D

_{3h }and its symmetry elements.

It has a trigonal bipyramidal shape. The C_{3} axis goes through the axial F atoms of the molecule, and the three C_{2} axes go through the three equatorial F atom. The horizontal mirror plane stands perpendicular to the principal C_{3} axis and is located within the equatorial plane of the molecule. In addition, there are the vertical mirror planes that contain the C_{3} axis, and go through the three equatorial P-F bonds. There is also an S_{3} axis which superimposes the C_{3} axis. In sum:

_{5}

A special case of a D_{nh} group is the linear group D_{∞h}. An object that has this symmetry is a cylinder. A cylinder can be conceived as a prism with an infinite number of vertices. Thus, the principal axis that passes through a cylinder has infinite order. Because of the infinite order of the principal axis, there is an infinite number of C_{2} axes that stand perpendicular to the principal axis. You can see one such C_{2} going though the cylinder (Fig. 2.2.79).

_{∞h}

There is now also an improper axis of infinite order, as well as an infinite number of vertical mirror planes. Non-polar linear molecules like H_{2}, CO_{2}, and acetylene C_{2}H_{2} belong to the point group D_{∞h}. You can see the C_{∞}** **axis passing through a CO_{2} molecule below (Fig. 2.2.80).

_{∞ }in a CO

_{2 }molecule

You can see the infinite number of vertical mirror planes as a blue cylinder. The infinite number of C_{2} axes is shown a yellow lines going around the molecule. In sum:

_{∞h}

Definition: Linear Group D_{ooh}

In the point group D_{∞h } there is an infinite number of n C_{2 }axes in addition to the principal axis of infinite order, an infinite number of vertical mirror planes, and one horizontal mirror plane.

If we add n vertical mirror planes to the principal axis and the n C_{2} axes, we arrive at the point group D_{nd}. The vertical mirror planes are dihedral mirror planes because they bisect the angle between the C_{2} axes. An example is the ethane molecule in staggered conformation which has the symmetry D_{3d }(Fig. 2.2.82).

_{nd}

The C_{3} axis goes along the C-C bond, and the 3C_{2} axes pass through the middle of the carbon-carbon bond, and bisect the angle between two hydrogens and one carbon atom. The three dihedral mirror planes pass through the C-H bonds. In addition, the ethane molecule has an S_{6} axis, and an inversion center. In sum:

Definition: D_{nd}

In this point group type there are n dihedral mirror planes that contain the C_{n }and bisect the angle between adjacent C_{2} axes

### Improper Rotational Point Groups

The last class of point groups to be discussed are the improper rotation point groups. The only have one proper rotational axis, and an improper rotational axis that has twice the order of the proper rotational axis (Fig. 2.2.85). There may be an inversion center present depending on the order of the proper and improper axes. An example the tetramethylcycloocta-tetraene molecule (Fig. 2.2.84).

Figure 2.2.84 The S_{4} and an C_{2 }axes of tetramethyl cycloocta-tetraene

It has an S_{4} and an C_{2} axis as the only symmetry elements besides the identity. Rotating by 90° alone does not superimpose the molecule because two C-C double bonds lie above the plane and two below the plane. In addition, two opposite methyl groups lie above and below the plane respectively. Therefore it needs the additional reflection to achieve superposition. There is also a C_{2} axis which is in the same locations as the S_{4} axis.

## Guide for the Determination of Point Groups

With the knowledge you have, you can unambiguously identify the point group of a molecule. The key to success is that you are able to see the symmetry elements in the molecule. This takes practice. With enough practice you can identify the point group of a molecule immediately. Until, you can use guides, that you can follow to identify a point group. Such a guide asks systematic questions about the presence or absence of a symmetry element. Depending on whether you answer the question with yes or no you can follow the guide in a particular direction. Eventually, after having answered enough questions the guide will lead you to the respective point group. You can see such a chart below (Fig. 2.2.86).

You can first ask if there is at least one C_{n} present. If not, then the molecule must be in a low symmetry point group. If there is an inversion center, the point group is C_{i}. If not we can next ask, if there is a mirror plane. If yes it is C_{s}, and if not the point group is C_{1}. If a low symmetry point group can be ruled out, then we can ask next, if there is a high symmetry point group. This is the case when there are either 4C_{3}, 3C_{4}, or 6C_{5} rotational axes present, standing for tetrahedral, octahedral, and icosahedral symmetry, respectively. If there is an inversion center in case a C_{5} is present, the point group is I_{h}. If not, it is I. Similarly, if there are 3C_{4} axes, and an inversion center, the point group must be O_{h}. If there is no inversion center the point group is O. If there are 4C_{3} axes, the point group must be T type. If there is a horizontal mirror plane in addition, then the point group must be T_{h}. If not, we can ask next is there are dihedral mirror planes. If yes, the point group is T_{d}, otherwise it is T. Now we have checked for all high symmetry point groups.

If a high-symmetry point groups can be ruled out, we ask if there are n C_{2} axes in addition to the C_{n} principal axis. If this is so, then we must have a dihedral group of the D type. Next we ask, if there is a horizontal mirror plane. If yes, the point group must be D_{nh}. If not, we ask if there are dihedral mirror planes. If yes, the point group is D_{nd}. If there are no mirror planes at all, then the point group is D_{n}. If there are no n C_{2} axes in addition to the C_{n}, the the group must be either a rotational group or an improper rotation group. We next ask, if there is a horizontal mirror plane. If yes, then the point group is of the type C_{nh}. If not, we ask if there are vertical mirror planes. If yes, then the point group is C_{nv}. If that is not the case, we ask if there is an S_{2n} in addition to the C_{n}. If yes, the point group is S_{2n}. If not, it is C_{n}.

### Example: Dibromonaphtalene

Let us practice the point group guide by one example. Let us look at the dibromonaphtalene molecule (Fig. 2.2.87).

The first question we would ask is: Can you see at least one proper rotational axis? The answer is yes. There is a C_{2} proper rotational axis that stands perpendicular to the plane of the molecule and goes through the center of the C-C bond that is shared by the two aromatic rings. Next we ask: Are the 6C_{5}, 3C_{4} or 4C_{3} axes? That is clearly not the case, and thus we do not have a high symmetry point group. Next we can think about if the there are 2C_{2} axes in addition to the C_{2} axes we already found. The answer is no, so the point group cannot be a dihedral group. Next, we would ask: Is there a horizontal mirror plane. This is indeed the case. There is a horizontal mirror plane in the plane of the molecule. It does not move any atoms around, but as we discussed before, a mirror does not need to do this to exist. This identifies the point group as C_{2h}.

## Chiral Point Groups

A chiral point group is a point group that only has proper rotation operations in addition to the identity. This is equivalent to the statement that no improper rotations must exist in a chiral point group. This includes mirror planes and inversion centers because a mirror plane is the same as an S_{1}, and an inversion center is the same as an S_{2}. If a molecule belongs to a chiral point group, then it has a mirror image that cannot be superimposed with the original molecule. The two mirror images are called enantiomers. An example is the bromochlorofluoromethane. You can see two enantiomers separated by a dotted line (Fig. 2.2.88).

The dotted line represents a mirror plane. Note that this mirror plane is not a mirror plane in the meaning of a symmetry element. You can see that the two molecules to the left and the right of the mirror plane are mirror images respective to each other. The molecules cannot be superimposed meaning that they are enantiomers. Note that the bromochlorofluoromethane molecule on the far right is not an enantiomer to the other two. It the same molecule as the enantiomer on the far left. We only need to rotate clockwise around the C-Cl axis to make the two molecules superimpose meaning that they are the same.

It should be pointed out that the fact that a molecule has a carbon with four different substituents is not sufficient to make it a chiral molecule. The dibromodichloroethane molecule shown above (Fig. 2.2.89) has four different substituents around the carbon atom, but it is not chiral because there is an inversion center in the middle of the C-C bond.

### Chiral High Symmetry Point Groups

Chiral groups not necessarily need to have low symmetry, in fact, the high symmetry rotational subgroups I, O, and T are chiral groups because they only have proper axes in addition to the identity (Fig. 2.2.90).

Dr. Kai Landskron (Lehigh University). If you like this textbook, please consider to make a donation to support the author's research at Lehigh University: Click Here to Donate.