Point Groups

Particularly we will consider the following point groups which molecules can belong to.

1. The groups $C_1$, $C_i$, and $C_s$.

A molecule belongs to the group $C_1$ if it has no elements other than identity $E$. Example: DNA. A molecule belongs to the group $C_i$, if it consist of two operations: the identity $E$ and the inversion $i$. Example: meso-tartaric acid. A molecule belongs to the group $C_s$, if it consists of two elements: identity $E$ and a mirror plane $\sigma$. Example: $OHD$.

2. The group $C_n$.

A molecule belongs to the group $C_n$ if it has a n-fold axis. Example: $H_2O_2$ molecule belongs to the $C_2$ group as it has the elements $E$ and $C_2$.

3. The group $C_{nv}$.

A molecule belongs to the group $C_{nv}$ if in addition to the identity $E$ and a $C_n$ axis, it has $n$ vertical mirror planes $\sigma_v$. Examples: $H_2O$ molecule belongs to the $C_{2v}$ group as it has the symmetry elements $E$, $C_2$, and two vertical mirror planes which are called $\sigma_v$ and $\sigma_v'$. The $NH_3$ molecule belongs to the $C_{3v}$ group as it has the symmetry elements $E$, $C_3$, and three $\sigma_v$ planes. All heteroatomic diatomic molecules and $OCS$ belong to the group $C_{\infty v}$ because all rotations around the internuclear axis and all reflections across the axis are symmetry operations.

4. The group $C_{nh}$.

A molecule belongs to the group $C_{nh}$ if in addition to the identity $E$ and a $C_n$ axis, it has a horizontal mirror plane $\sigma_h$. Example: butadiene $C_4H_6$, which belongs to the $C_{2h}$ group, while $B(OH)_3$ molecule belongs to the $C_{3h}$ group. Note, that presence of $C_2$ and $\sigma_h$ operations imply the presence of a center of inversion. Thus, the group $C_{2h}$ consists of a $C_2$ axis, a horizontal mirror plane $\sigma_h$, and the inversion $i$.

5. The group $D_n$.

A molecule belongs to the group $D_{n}$ if it has a n-fold principal axis $C_n$ and $n$ two-fold axes perpendicular to $C_n$. $D_1$ is of cause equivalent with $C_2$ and the molecules of this symmetry group are usually classified as $C_2$.

6. The group $D_{nd}$.

A molecule belongs to the group $D_{nh}$ if in addition to the $D_n$ operations it possess $n$ dihedral mirror planes $\sigma_d$. Example: The twisted, $90^o$ allene belongs to $D_{2d}$ group while the staggered confirmation of ethane belongs to $D_{3d}$ group.

7. The group $D_{nh}$.

A molecule belongs to the group $D_{nh}$ if in addition to the $D_n$ operations it possess a horizontal mirror plane $\sigma_h$. As a consequence, in the presence of these symmetry elements the molecule has also necessarily $n$ vertical planes of symmetry $\sigma_v$ at angles 360$^o$/2$n$ to one another. Examples: $BF_3$ has the elements $E$, $C_3$, $3C_2$, and $\sigma_h$ and thus belongs to the $D_{3h}$ group. $C_6H_6$ has the elements $E$, $C_6$, $3C_2$, $3C_2'$ and $\sigma_h$ and thus belongs to the $D_{6h}$ group. All homonuclear diatomic molecules, such as $O_2$, $N_2$, and others belong to the $D_{\infty h}$ group. Another examples are ethene $C_2H_4$ ($D_{2h}$), $CO_2$ ($D_{\infty h}$), $C_2H_2$ ($D_{\infty h}$).

8. The group $S_{n}$.

A molecule belongs to the group $S_{n}$ if it possess one $S_n$ axis. Example: tetraphenylmethane which belongs to the group $S_4$. Note, that the group $S_2$ is the same as $C_i$, so such molecules have been classified before as $C_i$.

9. The cubic groups.

There are many important molecules with more than one principal axes, for instance, $CH_4$ and $SF_6$. Most of them belong to the cubic groups, particularly to tetrahedral groups $T$, $T_d$, and $T_h$, or to the octahedral groups $O$ and $O_h$. If the object has the rotational symmetry of the tetrahedron, or octahedron, but has no their planes of reflection, then it belongs to the simpler groups $T$, or $O$. The group $T_h$ is based on $T$, but also has a center of inversion.

10. The full rotational group $R_3$.

This group consists of infinite number of rotational axes with all possible values of $n$. It is the symmetry of a sphere. All atoms belong to this symmetry group.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.