Group theory plays a pivotal role in understanding molecular symmetry and electronic properties in inorganic chemistry, particularly when applied to transition metal compounds.
Sigma interactions
Consider an octahedral molecule (see diagram above), which belongs to the point group. Let’s assume that valence atomic orbitals (AO) of the transition metal participate in bonding with the valence orbitals of the ligands , specifically orbitals with symmetry. The wavefunctions of these ligand orbitals shall be denoted by , where . To generate a molecular orbital (MO) diagram for the complex, we need to perform the following steps:

 Find the symmetry of the AOs of .
 Determine the symmetry of the symmetryadapted linear combinations (SALC) of .
 Work out which AOs of have nonzero vanishing integrals with valence orbitals of .
For step 1, the central atom lies on all the planes and axes of symmetry of the point group and is invariant to all symmetry operations of the point group. Therefore, the valence AOs of must transform according to respective bases of the character table of the point group.
Valence AOs  Irreducible representations of 
To accomplish step 2, we need to

 Select a basis set to generate a reducible representation of the point group.
 Calculate the traces of the matrices of .
 Decompose into irreducible representations of the point group.
 Generate a set of orthogonal SALCs.
We shall employ the vectors in the diagram as a basis set. The direction of each vector indicated in the diagram is regarded as the positive lobe of the ligand orbital. Instead of carrying out the laborious task of letting act on the basis set to produce matrices for the reducible representation, we can determine the traces by inspection. This is because each vector is transformed by into another vector but not into a linear combination of vectors. Furthermore, a diagonal element of a matrix of is equal to 1 when leaves a vector invariant. For example, a operation along the axis leaves and invariant, giving a trace of 2. The result is
6  0  0  2  2  0  0  0  4  2 
Using eq27a, decomposes to .
This implies that the matrices of are blockdiagonal matrices of the same form. Each blockdiagonal matrix is composed of the direct sum of the three irreducible representations , and of the point group. Since the number of basis functions of an irreducible representation corresponds to the dimension of the representation, there are a total of six such functions for (one for , two for , and three for ).
Instead of generating the basis functions, known as SALCs, using the projection operator, we shall derive them by logic. The SALC that transforms according to must totally symmetric. This is only possible if because the operation simply permutates the order of in , i.e. . There are three SALCs , and that transform according to . Since nonzero vanishing integrals occur only when they overlap with AOs of belonging to , these linear combinations must behave the same as the three orbitals of under symmetry operations. By inspection, we find that transforms like of , transforms like of and transforms like of . It follows that the two SALCs and that transform according to have to behave the same as and of under symmetry operations (see diagram above). Therefore, because has positive lobes along the axis and negative lobes along the axis. The last SALC is and not because the latter is not orthogonal to the other SALCs. Therefore, the normalised set of SALCs are:
To construct the MO diagram, we refer to eq157 of the HartreeFockRoothaan method. The total wavefunction is , where is the antisymmetriser and and represents the AOs of and in the SALCs. In general, a solution of eq157 for the complex produces the following result:
The lowest six MOs of , and are bonding orbitals. They are occupied by 12 electrons, which are supplied by the six electrondonor ligands. The highest six MOs of , and are antibonding orbitals. This leaves three MOs of symmetry as nonbonding MOs. The valence electrons of occupied the five MOs in green. The relative order of some of the MOs may vary depending on the types of metal and ligand.
Question
Do the remaining orbitals (other than those of symmetry) and the orbitals of the ligands participate in bonding with the valence AOs of ?
Answer
Yes, they may participate in bonding. When selected as a basis set, the six orbtials of the ligands generate a reducible representation with matrices that have the same traces as those of . The SALCs also matches the six SALCs derived above. It follows that the MO diagram, when both orbitals with symmetry and orbitals participate in bonding with , will be a superposition of two very similar MO diagrams.
The remaining orbitals form bonds with the orbitals of . Such interactions will be discussed below.
Pi interactions
We shall utilise the remaining vectors in the diagram as a basis set. The direction of each vector indicated in the diagram is regarded as the positive lobe of the ligand orbital. Employing the same logic as we did for interactions, we have
12  0  0  0  4  0  0  0  0  2 
Using eq27a, is decomposes to .
The orthonormal SALCs are
Irreducible representations 
Orthonormal ligand SALCs  Metal atom AOs 
Nonbonding  Nonbonding  
Nonbonding  Nonbonding 
The MO diagram, which describes both sigma and pi interactions, has the following general form:
The and MOs are nonbonding, while the MOs are now bonding. The relative order of some of the MOs may vary depending on the type of complex and whether the ligands are acceptors or donors. For instance, the energies of the MOs are usually lower than those of the MOs for donors ligands. Examples of ligands that can engage in both sigma and pi interactions include and , while examples of acceptors and donors ligands are and , respectively. These MO diagrams provide the theoretical foundation for the ligand field theory.