07 | Assigning point groups to compounds#
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Much can be communicated about implicit properties of a molecule or materials by simply describing it’s composition, coordination numbers and symmetry. From the symmetry of a compound we can immediately describe its selection rules for spectroscopic transitions, systematic absences in X-ray diffraction, possible degeneracies within its electronic structure, and the topologies of a materials band structure. All of this can be known by simply identifying elements of symmetry in a material.
The procedure for naming a point group is straightforward. First it is important to identify what coordinate system we are in. For periodic materials (crystals, polymer…) these are called crystal systems and in three dimensions there are 7 crystal systems of note.
Identify the highest order symmetry of rotation (take care to check for rotary inversions)
If it is \(3\) the crystal system is Trigonal.
If it is \(6\) the crystal system is Hexagonal.
If it is \(4\) and there is also a \(3\), then it is Cubic.
If there is only a \(4\) then it is Tetragonal.
If it is either \(2\) or \(\bar{2}=m\) and there are sets of mutually orthogonal symmetry elements long \(x\) and \(y\) then it is Orthorhombic. If not then it is Monoclinic.
If is is \(1\) or \(\bar{1}\) then it is Triclinic and the point group’s name is the same as the highest symmetry element you found.
An easy means to identifying the crystal system is to know a few simple shapes that fit into each.
Triclinic |
No symmemtry, or a parallelepiped with only inversion symmetry |
Monoclinic |
A rectangular prism where the top and bottom faces are offset from one another by some angle \(\beta \neq 90°\). |
Orthorhombic |
A rectangular prism |
Tetragonal |
A square prism, an elongated octahedron |
Trigonal |
A triangular prism, a tetrahedral molecule with one elongated bond (iodomethane), an cube elongated or compressed along one of the 3-fold rotation axes. |
Hexagonal |
A hexagonal prism, or honeycomb structure |
Once the crystal system is know we have narrowed down number of allowed point groups considerably. There are only 32 points groups allowed in crystal structures. Certainly there structures in real crystals with non-crystallographic symmetries. The crystal structures of UF7 and ferrocene are known and both of these molecular units have 5 fold symmetry. However, we can not describe the symmetry of their crystal structures by including a 5 fold rotation. To put it simply, 5 fold rotations cannot tile the plane, nor can the periodically tile 3D space. So we are left with only 32 points groups that can describe a crystals and we’ll stick to those in this course.
Identify other symmetry elements relative to the basis vectors of your crystal system. Choose an orientation of your coordinate systems axes to make these assignments natural. You can do this by placing the principal rotation along z (conventionally \(y\) in monoclinic) then rotating the about this axis until the other two align well with the other symmetry elements of the compound. Identify the additional elements and write down their relationships to your basis vectors. Theses will typically be mirror planes perpendicular to or a rotation axis along a certain axis or dihedral.
Find the point group that contains all of the symmetry elements you identified. You usually wont need to identify all of the symmetry elements the generators of the group should be sufficient and once you know where to look (see table Table 4) this can be done quickly.
As with our identification of crystal systems the ability to visualize a shape of molecules fore each point group can be very helpful and is usually much faster to make an assignment based on an analogy to another shape of molecule and it is to identify all of the generators of the group. While less procedural, which a little practice it is more natural and efficient to make assignments using this method.
System |
Point Group |
Shapes |
Triclinic |
\(1\) |
No symmetry but the identity element \(1\). |
Triclinic |
\(\bar{1}\) |
No symmetry but inversion \(\bar{1}\), a parallelepiped |
Monoclinic |
\(2\) |
2-blade propeller, Peroxide (H-O-O-H), where the two O-H bonds are skew. |
Monoclinic |
\(m\) |
butterfly, thiazole |
Monoclinic |
\(\frac{2}{m}\) |
2-bladed 2D pinwheel, trans-butadiene |
Orthorhombic |
\(222\) |
twistane, biphenyl |
Orthorhombic |
\(mm2\) |
water |
Orthorhombic |
\(mmm\) |
rectangular prism |
Tetragonal |
\(4\) |
4-blade propeller, rare |
Tetragonal |
\(422\) |
rare |
Tetragonal |
\(\frac{4}{m}\) |
4-bladed 2D pinwheel, [Cu(N3)]2–) |
Tetragonal |
\(4mm\) |
square pyramid, bromine pentafluoride |
Tetragonal |
\(\frac{4}{m}mm\) |
square prism, square bipyrimid, CuCl42 |
Tetragonal |
\(\bar{4}\) |
rare |
Tetragonal |
\(\bar{4}m2\) |
tennis ball, allene |
Cubic |
\(23\) |
rare, chiral \(\bar{4}3m\) |
Cubic |
\(m3\) |
rare |
Cubic |
\(\bar{4}3m\) |
tetrahedron, methane |
Cubic |
\(432\) |
rare, chiral \(m3m\) |
Cubic |
\(m3m\) |
Cube, octahedron, SF6 |
Trigonal |
\(3\) |
3-blade propeller, triphenylmethane |
Trigonal |
\(32\) |
1,3,5-triphenylbenzene |
Trigonal |
\(3m\) |
triangular prism, ammonia |
Trigonal |
\(\bar{3}\) |
rare, 18-crown-6 |
Trigonal |
\(\bar{3}m\) |
Trigonal antiprism, staggard ethane |
Hexagonal |
\(6\) |
6-blade propeller, rare |
Hexagonal |
\(\frac{6}{m}\) |
rare, six bladed 2D pinwheel |
Hexagonal |
\(622\) |
rare, hexaphenylbenzene |
Hexagonal |
\(6mm\) |
rare, hexagonaly pyramid |
Hexagonal |
\(\frac{6}{m}mm\) |
Hexagonal prism |
Hexagonal |
\(\bar{6}\) |
three bladed 2D pinwheel, 1,3,5-trihydroxybenzene |
Hexagonal |
\(\bar{6}m2\) |
Triangular prism, trigonal bipyrimid, boron trifluoride, eclipsed ethane |