05 | Groups of point symmetries#
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Object’s very often posses more than one element of symmetry. The set of all symmetry elements that an objected possess is called a group. If all of these symmetry elements (points, axes, and planes) all intersect the same point then it is called a point group. The symmetry of a square from the previous examples defines one such point group with the name \(\frac{4}{m}mm\). We’ll see later that a group of symmetry elements that do not all intersect at the same point but rather intersect the same plane compose a plane groups, and if they all intersect the same 3D space they compose a space group.
In crystalline materials there are limits to which symmetry are allowed because the objects they form must tessellate to form a periodic lattice. Conventionally only 1, 2, 3, 4, and 6 fold rotational symmetries can tesselate limiting the overwhelming majority of crystals to only 32 “crystallographic” point groups. The symmetries of molecules (0D) as less restrictive since they do not need to also tesselate but still there are only ~50 observed point groups found in molecules.
Diagrams of point groups#
The symmetrey elements found in water include the operations: \(1\), \(2\), \(\bar{2}_{xz} \equiv m_{xz}\), \(\bar{2}_{yz} \equiv m_{yz}\) and no others, Fig. 3. These four symmetry operations compose the group \(mm2\).
Fig. 3 The symmetry elements and corresponding operations for water. The identity operation is not shown explicitly in these diagrams but it is a member of every symmetry group.#
For this relatively simple compound and corresponding point group this representation shows all of the symmetry operations relatively clearly, except that the 2-fold rotation axis and the \(m_{xz}\) reflection plane overlap. By adding a small break in the line these two elements can be better distinguished, but there as a better standard orientation for diagraming point symmetry.
Stereographic projections#
The stereographic projection, Fig. 4, of a point group describes the view of the molecule’s coodinate system as though we are looking directly along the axis of the principle (proper rotation).
Fig. 4 The stereographic projection of \(mm2\) overlaid on top of the molecule, water. The dotted line is only a reference representing a cross-section of a sphere. The thick solid lines represent the two mirror planes and fi represents the 2-fold rotation axis pointing out of the plane.#
The stereographic projection diagrams for all 32 crystallographic points groups, and all 17 plane groups can be found in Palmer and Ladd Ch. 1.4.1
General positions#
Instead of overlaying a stereographic projection of a symmetry group on top it is often useful to show the general positions within the group. Fig. 4. A general position is a point in space that does not intersect any elements of symmetry. Points in space that do intersect symmetry elements are called special positions. From a single general position all other symmetry equivalent general positions can be generated by applying the symmetry operations of the group to its coordinates recursively until no more unique points are generated. In the case of the water molecule in Fig. 4, the oxygen atom sits on one special position and the two hydrogen atoms sit on another symmetry equivalent special position. In the case of two hydrogen atoms they’re positions are said to be symmetry equivalent because after applying a symmetry operation the position of one hydrogen’s position can be generated from the other.Fig. 5.
Fig. 5 (Click image) Generation of the general positions of \(mm2\) by application of an \(m_yz\) operation followed by a \(2\) operation.#
In the case of water’s point group \(mm2\) there are 4 general positions. Any set of symmetry equivalent special positions within \(mm2\) will have less than 4 unique positions total. For example the special position that coincides with the oxygen atom in the structure has only one unique position.
In the diagram above general positions are represented as open circles and if needed a comma is added in the circle to denote relative handedness of the coordinate system. When transforming a set of points, either a coordinate system or object, by applying an operation of rotary inversion, the handedness of the coordinate systems will change from right to left handed. In the point group of water \(mm2\) the mirror planes change the handedness of the coordinate system and this is shown in yhe diagram in Fig. 5 by changing the direction that the comma curls.
Traditionally, to generate all the general position of a point group the first point is chosen arbitrarily to be near the top of the circle and slightly right of center by applying all of the symmetry operations of the groups recursively eventually of of the other points shown in Fig. 5 can be drawn and no others.
Properties of symmetry groups and operations#
Symmetry operations have many of the familiar properties or mathematical operations (addition, multiplication, matrix multiplication…). Ultimately, the symmetries of rotation and pseudo-rotation that compose the elements of a point group can be inter-related by symmetry operations that can be expressed by an \(n\times n\) transformation matrix that can be multiplied by a position vector to transform its location either the same location or another symmetry equivalent location. That is,
where W is an \(n\times n\) matrix representation of the symmetry operation is \(\mathbf{x}\) is the initial \(n\times 1\) position vector and \(\mathbf{\tilde x}\) is the transformed position.
Generally symmetry operations do not commute#
While some mathematical operations (e.g. addition, scalar multiplication) commute other do not (e.g. subtraction, division). Addition commutes such that \(4+2=2+4=6\) but subtraction does not as in \(4-2 != 2-4\). The same is true for symmetry operations where give the operations \(A,B\) then \(AB\neq BA\).
In the case of the point group \(mm2\) is just so happens that all of the symmetry operations do commute and because of this, it is referred to as an Albelian group. The point group of ammonia, \(3m\), we’ll see that all of the symmetry operations do not commute.
All groups contain the identity element, \(1\)#
The identity element, \(1\), is a member of all groups of symmmetry operations and commutes with all others. Given a symmetry operation \(S\), \(S1=1S=S\).
All symmetry operations are associative#
For a given three symmetry operations \(A,B,C\) it is true for any group of symmetry operations that \((AB)C = A(BC)\).
Every symmetry operation has a reciprocal#
In every symmetry group, for any symmetry operation \(S\) there is a reciprocal symmetry operation, \(R\), such that \(SR=RS=1\) where \(1\) is the identity operation.
Rearrangement Theorem of multiplication tables#
For a symmetry group, all of the symmetry elements can be written out as the column headers and row indices of a table. Each remaining cell contains the product of each pair of symmetry operations. Note here, that for a given position vector \(\mathbf{x}\), the operation in a row index is applied followed by the operation denoted in the column header. For example, the cell in the first row second column, could be represented as
In Table 1 the composition of each cell is shown only to illustration the order operations. Usually we would only show their products. This can be accomplished visually using Fig. 5 as a guide. By taking the initial general position in the top right we can apply the pair of symmetry operations \(11\). The first identity operation, \(1\) returns us to the same exact location we started at and then applying the second identity operation \(1\) also returns us to the exact same location such that \(11=1\). In Table 2 the multiplication table is written in its complete and proper form showing only the symmetry operation that is equivalent to each pairwise product.
\(\mathbf{mm2}\) |
\(\mathbf{1}\) |
\(\mathbf{2}\) |
\(\mathbf{m_{xz}}\) |
\(\mathbf{m_{yz}}\) |
\(\mathbf{1}\) |
\(11\) |
\(12\) |
\(1m_{xz}\) |
\(1m_{yz}\) |
\(\mathbf{2}\) |
\(21\) |
\(22\) |
\(2m_{xz}\) |
\(2m_{yz}\) |
\(\mathbf{m_{xz}}\) |
\(m_{xz}1\) |
\(m_{xz}2\) |
\(m_{xz}m_{xz}\) |
\(m_{xz}m_{yz}\) |
\(\mathbf{m_{yz}}\) |
\(m_{yz}1\) |
\(m_{yz}2\) |
\(m_{yz}m_{xz}\) |
\(m_{yz}m_{yz}\) |
\(\mathbf{mm2}\) |
\(\mathbf{1}\) |
\(\mathbf{2}\) |
\(\mathbf{m_{xz}}\) |
\(\mathbf{m_{yz}}\) |
\(\mathbf{1}\) |
\(1\) |
\(2\) |
\(m_{xz}\) |
\(m_{yz}\) |
\(\mathbf{2}\) |
\(2\) |
\(1\) |
\(m_{yz}\) |
\(m_{xz}\) |
\(\mathbf{m_{xz}}\) |
\(m_{xz}\) |
\(m_{yz}\) |
\(1\) |
\(2\) |
\(\mathbf{m_{yz}}\) |
\(m_{yz}\) |
\(m_{xz}\) |
\(2\) |
\(1\) |
While it is beyond the scope of this course to prove, we can appreciate that Rearrangement Theorem states that for each row and column in our multiplication table that each symmetry operation appears once and only once. This is equivalent in form to a latin square.
Exercises#
Draw an object or molecule with the point group \(mm2\) that contains symmetry equivalent points with a multiplicity of four. That is, points the occupy general positions in the point group.
For any general position shown in Fig. 5 (or on the shape you drew above) convince yourself that you can move to any of the other symmetry equivalent positions by applying a single symmetry operation.
References#
M. F. C. (Marcus Frederick Charles) Ladd. Structure determination by X-ray crystallography / M.F.C. Ladd and R.A. Palmer. Plenum Press, New York, 1977. ISBN 0306308444.
Notes#
Further reading#
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, 23–43.
Dresselhaus, Dresselhaus, and Jorio, Group Theory: Application to the Physics of Condensed Matter Berlin Heidelberg: Springer, 2010.
Resources that don’t use Hermann–Mauguin notation#
Grant Sanderson’s 3b1b video on group theory.
Miessler G. L.; Tarr, D. A. Inorganic Chemistry Chapter 4: Symmetry and Group Theory Harlow, Pearson, 2000.
Vincent, A. Molecular Symmetry and Group Theory. Chichester New York: Wiley, 2001.
Cotton, F. A. Chemical Applications of Group Theory
Miessler; G. L. Tarr, D. A. Inorganic Chemsitry
Drago Physical Methods in Chemistry