A set of
matrices that multiply together in a way that mimics the multiplication table of the elements of a group is called a
representation of the group. The simplest method of obtaining a representation of molecular group transformations is to trace the movements of atoms in a molecule when symmetry operations are applied. For example, a water molecule belonging to the
C2v point group might have an oxygen atom labelled 1 and two hydrogen atoms labelled 2 and 3 as shown in the right hand column vector below. If the hydrogen atoms are imagined to rotate by 180 degrees about an axis passing through the oxygen atom we have the familiar
C2 operation of this point group. The oxygen atom in position number 1 stays in position but the atoms in positions 2 and 3 are moved to positions 3 and 2 in the resulting column vector. The matrix connecting the two provides a 3 × 3 representation for this operation. :{ \begin{bmatrix} 1 \\ 3 \\ 2 \\ \end{bmatrix} }= { \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} } \times { \begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} }_{} This point group only contains four operations and matrices for the other three operations are obtained similarly, including the identity matrix which just contains 1's on the leading diagonal (top left to bottom right) and 0's elsewhere. Having obtained the representation matrices in this way it is not difficult to show that they multiply out in exactly the same way as the operations themselves. : \underbrace{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} }_{C_{2}} \times \underbrace{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} }_{\sigma_\text{v}} = \underbrace{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} }_{\sigma'_\text{v}} Although an infinite number of such representations exist, the
irreducible representations (or "irreps") of the group are all that are needed as all other representations of the group can be described as a
direct sum of the irreducible representations. The first step in finding the irreps making up a given representation is to sum up the values of the leading diagonals for each matrix so, taking the identity matrix first then the matrices in the order above, one obtains (3, 1, 3, 1). These values are the
traces or characters of the four matrices. Asymmetric point groups such as
C2v only have 1-dimensional irreps so the character of an irrep is exactly the same is the irrep itself and the following table can be interpreted as irreps or characters. Looking again at the characters obtained for the 3D representation above (3, 1, 3, 1), we only need simple arithmetic to break this down into irreps. Clearly, E = 3 means there are three irreps and a C2 representation sum of 1 means there must be two A and one B irreps so the only combination that adds up to the characters derived is 2A1+ B1
Robert Mulliken was the first to publish character tables in English and so the notation used to label irreps in the above table is called Mulliken notation. For asymmetric groups it consists of letters A and B with subscripts 1 and 2 as above and subscripts g and u as in the C2h example below. (Subscript 3 also appears in D2) The irreducible representations are those matrix representations in which the matrices are in their most diagonal form possible and for asymmetric groups this means totally diagonal. One further thing to note about the irrep/character table above is the appearance of polar and axial base vector symbols on the right hand side. This tells us that, for example, cartesian base vector × transforms as irrep B1 under the operations of this group. The same collection of product base vectors is used for all asymmetric groups but symmetric and spherical groups use different sets of product base vectors. Point group C2h has the operations {
E, C2, i, σh } and the 1,5-dibromonapthalene (C10H6Br2) shown in the figure belongs to this symmetry group. It is possible to construct four 18 × 18 matrices representing the transformations of atoms during its symmetry operations in the style of the water molecule example above and reduce it to 18 1D irreps. Notice however that carbon atom number 1 either stays in place or it is exchanged with carbon atom number 5 and these two atoms can be analysed separately from all the other atoms in the molecule. The transformation matrix for these two atoms alone during the molecular C2 rotation is :{ \begin{bmatrix} 5 \\ 1 \\ \end{bmatrix} }= { \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} } \times { \begin{bmatrix} 1 \\ 5 \\ \end{bmatrix} }_{} with character 0. When this computation is carried out for each of the operations above the characters obtained are (2,0,0.2) because two operations leave the atoms in place and two move them. The irrep table for this group is below. The first column tells us there are two1D irreps, the second column (C2) that there is one A and one B while columns 3 and 4 reveal that one irrep has subscript g the other has to have subscript u. This means that the irreps resulting from the two atoms are Ag + Bu. In fact, the 18 atoms in this molecule are paired off in exactly the same way as carbon atoms 1 and 5 so that, from a symmetry perspective, the atom consists of 9 pairs of equivalent atoms related through symmetry. It follows that each pair contributes the same irreps as the pair examined above to give a total 18 dimensional irrep result of 9(Ag + Bu).
Symmetric point group representations Symmetric point groups are divided into systems based on the increasing order of the main rotational axis from three to infinity. Systems are in turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one dimensional representations as shown in the table of irreps and the number of irreps is equal to the order of the group. The irreps shown use standard notation for the rotational group of a class but Mulliken sometimes gave different symbols to other members of the same class even though they belong to the same abstract group and therefore have the same irreps. Dihedral point groups contain a cyclic group of the same rotational order: so group Dn always contains group Cn as an
index-2 subgroup. It follows that dihedral irreps are superimposed on cyclic irreps because the cyclic group within a dihedral one does not cease to be a cyclic group. A dihedral group also contains a 2-fold rotational axis at right angles to the main cyclic axis and this has two consequences. Firstly, the A and B cyclic irreps are split into pairs of one dimensional irreps identified by subscripts 1 and 2. Secondly, pairs of 1D E−x and E+x cyclic irreps combine to form single Ex 2D irreps in the dihedral group because the 2-fold horizontal rotation makes pairs of rotations equivalent. For example, a 60 degree rotation about the main axis becomes equivalent to a (360 − 60) degree rotation because the 2-fold horizontal rotation makes them equivalent. Combinations of this kind are said to form a class. Infinite order dihedral group irreps sometimes use Greek symbol descriptions,
Σ,
Π,
Δ that follow from early linear molecule calculations. Taking benzene as a simple example, we have a molecule that belongs to the series of point groups C6, D6 and D6h with increasing orders 6, 12 and 24. The six carbon atoms may be represented by a 6 × 6 matrices which in group C6 have irreps A,B, E+1,E-1,E+2 and E-2 because n objects in an n-fold cyclic group always produce one of each irrep. If these cyclic irreps are promoted to group D6 we obtain A1, Bx, E1 and E2 when subscripts are added to the 1D A and B irreps and the others merge to form 2D irreps. A1 is there because the most symmetric irrep has to occur once and only once in the result leaving only the B irrep subscript to be deduced. The character of the 2-fold horizontal rotation operation is 2 because 2 carbon atoms stay in place during the rotation, Char(C2) = 2 telling us that there are two more 1 subscripts than 2 subscripts so the result is A1, B1, E1 and E2. Finally, promotion to D6h requires the addition of g and u subscripts. Since Char(
i) = 0 there are an equal number of g and u subscripts, and A1g has to be present as the most symmetric group so B1u is mandatory. Furthermore, odd and even 2D irreps take u and g subscripts so the final result for the carbon atoms is (A1g, B1u, E1u, E2g), but with the hydrogen atoms we get 2(A1g, B1u, E1u , E2g). Boric acid and boron trifluoride provide further hexagonal examples in spite of their slightly misleading Schoenflies symbols C3h and D3h. Taking boric acid first, we have three sets of equivalent atoms: 1 boron, 3 oxygen and 3 hydrogen. Obviously the oxygen and hydrogen atoms produce the same irreps so only one has to be deduced. Applying the 6-fold cyclic group to (say) the hydrogen atoms produces characters (3,0,0,3,0,0) yielding irreps A + E+2 +E-2. Doubling up and adding an irrep for the central boron atom produces A + 2(A +E+2 +E-2) in standard Laue class notation. Unfortunately, Mulliken used a different notation for C3h and D3h irreps to that used for other groups and a conversion table would be needed if that was important. Boron trifluoride has a central boron atom with 3 fluorine atoms and belongs to cyclic subgroup C3h and the larger dihedral group D3h. Following the above reasoning, the irreps in C3h are A +(A + E+2 +E-2) and when promoted to the dihedral group this becomes A1 + (A1 + E2). Again, conversion to Mulliken notation is required if that is important.
Spherical point group representations Spherical classes are defined by the tetrahedral, octahedral and icosahedral rotational groups T, O and I. The first two of these, T and O, are related in much the same way as cyclic and dihedral groups are related in symmetric groups. Both tetrahedral and octahedral molecules are often shown with their atoms inscribed in the apices or faces of cubes and might be considered as a single "cubic" system. The first Laue class of this system contains only the tetrahedral rotational group T of order 12 and the direct product of this group with space inversion Th of order 24. Every point group in the following octahedral class contains the tetrahedral rotational group as a subgroup. Irreps of tetrahedral and octahedral groups are also related similarly to cyclic and dihedral groups and the table below shows how tetrahedral irreps are incorporated in octahedral irreps Tetrahedral symmetry has 3 one dimensional irreps (A, E+ ,E-) and one 3 dimensional irrep T then the A and T irreps split into two irreps with subscripts 1 and 2 while the two 1D E irreps combine into a single 2D irrep. Notice that the T irrep is always 3 dimensional but the E irrep only becomes 2 dimensional in the higher order group. Methane (CH4) is often used as an example and, although often described as a tetrahedral molecule because of the very visible rotational symmetry, it really belongs to the octahedral symmetry class. Considering methane first as a tetrahedral molecule the 12 operations of group T are {E, 3 × c, 4 × b, 4 × b3} where c is a 180 degree rotation along x, y and z axes and b is a 120 degree rotation about the apices of a cube. It is not difficult to convert 5 × 5 symmetry operation transformation matrices to reducible matrices and thence to molecular irreps but this not necessary. Methane has two sets of equivalent atoms: a single carbon atom and 4 hydrogen atoms. The atoms of each set are transformed into each other during operations. A single atom can only ever be transformed into itself and therefore always contributes the most symmetrical irrep to the end total irrep count. Additionally, there is a rule of group theory that the most symmetrical irrep must occur once and only once in the irreps of any equivalent atom set so the five dimensions of irreps being sought contain 2A and three others. Although E1 and E2 are 1 dimensional they have to occur together in the irreps of any equivalent set of atoms. it follows that he only way of filling the remaining three dimensions is to adopt 3D irrep T so the irreps are 2A + T. (E irreps have to be taken in pairs in physical molecular applications).
methane sulfur hexafluoride Extending this treatment to the octahedral group Td requires six 4-fold roto-inversion operations (f) about the main axes and six 2-fold roto-inversions (a), appearing as mirror reflections through opposite edges of the imaginary cube in which methane is placed. So half the operations of this group are rotational and half non-rotational. Rotational group T exists within the non-rotational group Td = {E, 3 × c, 8 × b/b3, 6 × f, 6 × a} so the irreps in group T in the expansion to Td. Again we have 2 sets of equivalent atoms and each set must contribute one and only one of the most symmetrical irrep, in this case A1. Reasoning as above, we know that the irreps in Td must be 2A1 + Tx so the last step is to find the 3D subscript. A brief look at the 4 × 4 transformation matrix for the 4-fold rotation operation f shows character Ch(f) = 0 and the subscript × has to be 2 to balance the 1 on the A irrep. so the final result is 2A1 + T2 Sulfur hexafluoride (SF6) can also be treated first as a tetrahedral molecule T, then as octahedral O and finally as centred molecule Oi. There are two sets of equivalent atoms consisting of a single sulfur atom and six fluorine atoms. Transformations of the fluorine atoms generate a six dimensional representation that can only reduce into the direct sum of tetrahedral irreps A, E+1, E-1 and T because the direct sum must include the most symmetrical irrep once and only once, leaving five dimensions that can only be satisfied in the way shown - a direct sum of 5 can only be made up from a 2 and a 3 - no other combination is possible. These irreps are "promoted" to 2A1 + E1 + Tx in group O. To get the × subscript observe that the 4-fold rotation in SF6 has character Ch(f) = 2 because two atoms stay in position and a glance at this column of the table suggests A1 + E1 + T1. Finally the inversion operation (i) applied go the fluorine atoms has character Ch(i) = 0 indicating equal numbers of
g and
u subscripts (because none of the atoms remains in position). Since the most symmetrical irrep must occur once the only possible result is A1g + E1g + T1u. The single sulfur atom always has the most symmetric irrep to the final reduction of the seven dimensional matrices to a direct sum is 2A1g + E1g + T1u.
A summary of possible point group irreducible representations Cyclic groups have Schoenflies symbols Cn, Sn, Cnh, Cs and Ci positioned together in rows of the Laue class table above. These point groups are represented by the one dimensional symbols A, B, E+x and E-x. If the group has a centre of symmetry the number if irreps doubles and subscripts g and u have to be added to each of the simple cyclic symbols. For example, the eight possible irreps of C4h are Ag, Bg, E+1g, E-1g, Au, Bu, E+1u and E-1u. Cotton For example, the possible irreps of D4 (also C4v and D2d) are A1, A2, B1,B2 and E. (E1 is normally shown simply as E). A group with a centre of symmetry also has irreps with g and u subscripts. Once the irreps of a cyclic group Cn are known it is a simple task to deduce the irreps of the corresponding dihedral group Dn. In addition to the one and two dimensional irreps so far described, tetrahedral groups can a three dimensional degenerate irrep T that expands in octahedral groups to yield irreps T1 and T2. The following reference of character tables uses symbols Zx for abstract cyclic groups Cx with A4 and S4 (alternating and symmetric permutations of 4 objects) for T and O. Many authors just use C, T and O in two senses, making it clear which is intended. ==Historical background==