The Symmetry Groups may be systematically classified by considering how to build them up using increasingly more elaborate combinations of symmetry operations. The nomenclature to denote a unique symmetry group is known as Schönflies notation. Named after the German mathematician Alfred Moritz Schönflies (1853 - 1928).

Alfred Moritz Schönflies

The following outline is non-rigorous, but does give a practical scheme for use by most chemists.

**Step 1: Identify very easy to classify space groups by inspection**

a. Linear molecules

b. Very high symmetry molecules with multiple high order axes

a. Linear molecules: There are only two kinds of symmetry for linear molecules

i. Those with identical ends such as H_{2}, O_{2}, or HC≡CH. These molecules have an infinite-fold rotation axis (C_{∞}) which coincides with the molecular axis and an infinite number of vertical symmetry planes (σ_{v}). In addition, there is a horizontal plane of symmetry (σ_{h}) and an infinite number of C_{2} axes perpendicular to C_{∞}. **The point group is D _{∞h}**

ii. A linear molecule with different ends has only C_{∞} and the σ_{v}'s but no σ_{h}. Examples: CO, HCl. **The point group is C _{∞v} **

b. Very high symmetry molecules

In practice molecules with very high symmetry are also relatively easy to identify by inspection. The most common of these are molecules built on a central tetrahedron, octahedron, cubooctahedron, cube, pentagonal dodecahedron or icosahedron. Of these the tetrahedron (**T _{d}**) and octahedron (

**O**) are the most important since they are found in many kinds of molecules.

_{h}The Tetrahedron (T_{d})

Examples: CH_{4}, SiF_{4}, ClO_{4}^{-}, Ni(CO)_{4}, Ir_{4}(CO)_{12}

A view of some of the symmetry elements of a regular tetrahedron is shown below. The point group has a total of 24 symmetry operations. These include three S_{4} axes, four C_{3} axes, six reflection planes (σ_{d}). Note that despite the high symmetry, there is no inversion center (i) in T_{d}.

"Tetrahedron". Licensed under CC BY-SA 3.0 via Wikimedia Commons.

You can explore the various symmetry operations in more detail by going to the Otterbein University Website and clicking on any one of the T_{d} examples.

The Octahedron and the Cube

These two structures have the same symmetry elements and have a total of 48 symmetry operations. Point group symbol O_{}_{h}. There are three C_{4} axes, four C_{3} axes, six C_{2} axes bisecting opposite edges, three σ_{h} planes, and six σ_{d}. The C_{}_{4} axes are also S_{4} axes, and the C_{}_{3} axes are S_{6} axes. There is an inversion center (i). O_{h} is an important type of symmetry since octahedral complexes such as IrCl_{6}^{3-} and [Co(NH_{3})_{6}]^{3+} and molecules such as SF_{6} are very common.

The Pentagonal Dodecahedron and the Icosahedron

These structures are of even high symmetry and are related to each other in the same way as the octahedron and the cube - the vertices of one define the face centers of the other. They have a total of 120 symmetry operations. The icosahedral group has the symbol I_{h}. This is a key structural unit in boron chemistry (eg. B_{12}H_{12}^{2-}).

**Step 2: The next step is to look for proper or improper axes of rotation**

If none is found, look for a plane or center of symmetry. If a plane only is present, the point group is C_{s}. If a center only is present, the point group is C_{i}. If no symmetry element is present at all (only E), the group is the trivial one C_{1}.

**C**** _{s}**: E, σ - fairly common, examples include thionyl chloride (SOCl

_{2}) and secondary amines

**C**** _{i}**: E, i - rare

**C**** _{1}**: E - very common, lots of organic molecules

**Step 3: The S _{n} Groups**

In rare circumstances a molecule contains an S_{n} axis as its only symmetry element. It can be shown that two possibilities now exist: 1) S_{n} where n is odd (3, 5, 7, etc.), and 2) S_{n} where n is even (2, 4, 6, etc.). For n odd, it can be shown that the set of operations is actually the same as C_{nh}. It is only for n even, that new point groups exist (S_{4}, S_{6}, etc.) (Note S_{2} = i and that point group would be C_{i}). Also an S_{4} point group has a C_{2} axis and an S_{6} point group has a C_{3} axis. If additional operations are possible, we are then dealing with D_{nd} or D_{nh} point groups.

**Step 4: A proper or improper axis is present**

If a proper or improper axis is found which is not a consequence of the presence of S_{2n} (step 3) then we are dealing with C, or D type point groups. First, find the highest order proper axis C_{n}. Once identified, we look for a set of n C_{2} axes perpendicular to it. If this is the case then the point group is one of D_{n}, D_{nh}, or D_{nd}. If n C_{2} axes are not present, the molecule belongs to one of the C type point groups: C_{n}, C_{nv}, or C_{nh}.

**C _{n} **

If there are no symmetry elements other than the C_{n} axis the point group is C_{n}. Molecules with this type of axial symmetry, other than C_{2}, are rare. Examples are triphenyl phosphine (PPh_{3} point group **C**** _{3}**) and non-planar hydrogen peroxide (H

_{2}O

_{2}point group

**C**).

_{2}**C _{nv}**

This has a proper axis (order n) plus a set of n vertical planes and is fairly common. Examples: H_{2}O (**C**** _{2v}**), NH

_{3}(

**C**)

_{3v}**C _{nh}**

This also has a proper axis (order n) plus a horizontal plane of symmetry. These are relatively rare, examples: *trans*-planar H_{2}O_{2} (**C**** _{2h}**), boric acid (

**C**

**), and**

_{3h}*trans*- 1,2 - dichloroethylene.

**Step 5: D Point Groups**

If we have a vertical C_{n} axis, plus a set of n C_{2} axes, perpendicular to it the point group is D_{n}. This is rare. One set of examples is the metal tris chelates such as tris (oxalato) iron (III) (**D _{3}**).

**D _{nh} **

If there are additional symmetry operations possible when we get to D_{n} then D_{nh} or D_{nd} are possible. For example, if there is a horizontal plane of symmetry (σ_{h}) then the group is D_{nh}. Examples include benzene (**D**** _{6h}**), ferrocene (Cp

_{2}Fe eclipsed) (

**D**

_{5}**), PtCl**

_{h}_{4}

^{2-}(

**D**

**), BF**

_{4h}_{3}(

**D**

**). Additional symmetry operations are reflections in vertical planes which**

_{3h}*contain*the n C

_{2}axes.

**D _{nd }**

If we add a set of vertical planes of symmetry that *bisect *the angles between pairs of C_{2} axes, we have D_{nd}. The easy way to distinguish this from D_{nh} is the absence of a horizontal plane of symmetry. Examples are staggered ethane (**D**** _{3d}**), and staggered ferrocene (

**D**

**).**

_{5d}**Step 5: The S _{n} Groups**

In rare circumstances a molecule contains an S_{n} axis as its only symmetry element. It can be shown that two possibilities now exist: 1) S_{n} where n is odd (3, 5, 7, etc.), and 2) S_{n} where n is even (2, 4, 6, etc.). For n odd, it can be shown that the set of operations is actually the same as C_{nh}. It is only for n even that new point groups exist (S_{4}, S_{6}, etc.) (Note S_{2} = i and that point group would be C_{i}). Also an S_{4} point group has a C_{2} axis and an S_{6} point group has a C_{3} axis. If additional operations are possible, we are then dealing with D_{nd} or D_{nh} point groups.

The 5 Steps described above can be summarized as follows:

1. Special groups? Linear (C_{∞v}, D_{∞h}), or high symmetry (T_{d}, O_{h})

2. No proper or improper rotation axes (C_{1}, C_{s}, C_{i})

3. Only S_{n} (n even) axis present (S_{4}, S_{6}, S_{8}...)

4. C_{n} axis present not a consequence of S_{2n}.

No C_{2}'s perpendicular to C_{n} (C_{nh}, C_{nv}, C_{n})

5. C_{n} axis present not a consequence of S_{2n}.

Has C_{2}'s perpendicular to C_{n} (D_{nh}, D_{nd}, D_{n})

***At this point, it would be a good idea for you to test your knowledge by assigning point groups for various molecules. Numerous examples can be found in the homework assignments.***

Next: Character Tables

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