Group theory and its applications in inorganic chemistry

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 $ML_6$ (see diagram above), which belongs to the $O_h$ point group. Let’s assume that valence atomic orbitals (AO) of the transition metal $M$ participate in bonding with the valence orbitals of the ligands $L$ , specifically $p$ -orbitals with $\sigma$ symmetry. The wavefunctions of these ligand orbitals shall be denoted by $\sigma_i$ , where $i=1,\cdots,6$ . To generate a molecular orbital (MO) diagram for the complex, we need to perform the following steps:

1. Find the symmetry of the AOs of $M$ .
2. Determine the symmetry of the symmetry-adapted linear combinations (SALC) of $\sigma_i$ .
3. Work out which AOs of $M$ have non-zero vanishing integrals with valence orbitals of $L$ .

For step 1, the central atom $M$ lies on all the planes and axes of symmetry of the $O_h$ point group and is invariant to all symmetry operations $R_{\alpha}$ of the $O_h$ point group. Therefore, the valence AOs of $M$ must transform according to respective bases of the character table of the $O_h$ point group.

Valence $\boldsymbol{M}$ AOs	Irreducible representations of $\boldsymbol{O_h}$
$s$	$A_{1g}$
$p_x,p_y,p_z$	$T_{1u}$
$d_{z^2},d_{x^2-y^2}$	$E_g$
$d_{xy},d_{yz},d_{zx}$	$T_{2g}$

To accomplish step 2, we need to

1. Select a basis set to generate a reducible representation $\Gamma_{\sigma}$ of the $O_h$ point group.
2. Calculate the traces of the matrices of $\Gamma_{\sigma}$ .
3. Decompose $\Gamma_{\sigma}$ into irreducible representations of the $O_h$ point group.
4. Generate a set of orthogonal SALCs.

We shall employ the $\sigma$ -vectors in the $ML_6$ diagram as a basis set. The direction of each vector indicated in the diagram is regarded as the positive lobe of the ligand $p$ -orbital. Instead of carrying out the laborious task of letting $R_{\alpha}$ act on the basis set to produce matrices for the reducible representation, we can determine the traces by inspection. This is because each $\sigma$ -vector is transformed by $R_{\alpha}$ into another $\sigma$ -vector but not into a linear combination of $\sigma$ -vectors. Furthermore, a diagonal element of a matrix of $\Gamma_{\sigma}$ is equal to 1 when $R_{\alpha}$ leaves a $\sigma$ -vector invariant. For example, a $\hat{C}_4$ operation along the $z$ -axis leaves $L_5$ and $L_6$ invariant, giving a trace of 2. The result is

$\boldsymbol{O_h}$	$E$	$8C_3$	$6C_2$	$6C_4$	$3C'_2$	$i$	$6S_4$	$8S_6$	$3\sigma_h$	$6\sigma_d$
$\boldsymbol{T_{\sigma}}$	6	0	0	2	2	0	0	0	4	2

Using eq27a, $\Gamma_{\sigma}$ decomposes to $\Gamma_{\sigma}=A_{1g}\oplus E_{g}\oplus T_{1u}$ .

This implies that the matrices of $\Gamma_{\sigma}$ are block-diagonal matrices of the same form. Each block-diagonal matrix is composed of the direct sum of the three irreducible representations $A_{1g}$ , $E_g$ and $T_{1u}$ of the $O_h$ 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 $\Gamma_{\sigma}$ (one for $A_{1g}$ , two for $E_g$ , and three for $T_{1u}$ ).

Instead of generating the basis functions, known as SALCs, using the projection operator, we shall derive them by logic. The SALC $\phi_1$ that transforms according to $A_{1g}$ must totally symmetric. This is only possible if $\phi_1=\sigma_1+\sigma_2+\sigma_3+\sigma_4+\sigma_5+\sigma_6$ because the operation $R_{\alpha}$ simply permutates the order of $\sigma_i$ in $\phi_1$ , i.e. $R_{\alpha}\phi_1=\phi_1$ . There are three SALCs $\phi_2$ , $\phi_3$ and $\phi_4$ that transform according to $T_{1u}$ . Since non-zero vanishing integrals occur only when they overlap with AOs of $M$ belonging to $T_{1u}$ , these linear combinations must behave the same as the three $p$ -orbitals of $M$ under symmetry operations. By inspection, we find that $\phi_2=\sigma_1-\sigma_2$ transforms like $p_x$ of $M$ , $\phi_3=\sigma_3-\sigma_4$ transforms like $p_y$ of $M$ and $\phi_4=\sigma_5-\sigma_6$ transforms like $p_z$ of $M$ . It follows that the two SALCs $\phi_5$ and $\phi_6$ that transform according to $E_g$ have to behave the same as $d_{x^2-y^2}$ and $d_{z^2}$ of $M$ under symmetry operations (see diagram above). Therefore, $\phi_5=\sigma_1+\sigma_2-\sigma_3-\sigma_4$ because $d_{x^2-y^2}$ has positive lobes along the $x$ -axis and negative lobes along the $y$ -axis. The last SALC is $\phi_6=2\sigma_5+2\sigma_6-\sigma_1-\sigma_2-\sigma_3-\sigma_4$ and not $\phi_6=\sigma_5+\sigma_6-\sigma_1-\sigma_2-\sigma_3-\sigma_4$ because the latter is not orthogonal to the other SALCs. Therefore, the normalised set of SALCs are:

$\phi_1=\frac{1}{\sqrt{6}}(\sigma_1+\sigma_2+\sigma_3+\sigma_4+\sigma_5+\sigma_6)$

$\phi_1=\frac{1}{\sqrt{2}}(\sigma_1-\sigma_2)$

$\phi_1=\frac{1}{\sqrt{2}}(\sigma_3-\sigma_4)$

$\phi_1=\frac{1}{\sqrt{2}}(\sigma_5-\sigma_6)$

$\phi_5=\frac{1}{2}(\sigma_1+\sigma_2-\sigma_3-\sigma_4)$

$\phi_1=\frac{1}{2\sqrt{3}}(2\sigma_5+2\sigma_6-\sigma_1-\sigma_2-\sigma_3-\sigma_4)$

To construct the MO diagram, we refer to eq157 of the Hartree-Fock-Roothaan method. The total wavefunction is $\Psi_{\pi}=\hat{A}\Pi_{i=1}^{12}\psi_i$ , where $\hat{A}$ is the antisymmetriser and $\psi_i=\sum_{r=1}^{7}c_{ir}\varphi_r$ and $\varphi_r$ represents the AOs of $M$ and $\sigma_i$ in the SALCs. In general, a solution of eq157 for the $ML_6$ complex produces the following result:

The lowest six MOs of $a_{1g}$ , $t_{1u}$ and $e_{g}$ are bonding orbitals. They are occupied by 12 electrons, which are supplied by the six electron-donor ligands. The highest six MOs of $e^*_{g}$ , $a^*_{1g}$ and $t^*_{1u}$ are antibonding orbitals. This leaves three MOs of $t_{2g}$ symmetry as non-bonding MOs. The valence $d$ -electrons of $M$ 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 $p$ -orbitals (other than those of $\sigma$ symmetry) and the $s$ -orbitals of the ligands participate in bonding with the valence AOs of $M$ ?

Answer

Yes, they may participate in bonding. When selected as a basis set, the six $s$ -orbtials of the ligands generate a reducible representation with matrices that have the same traces as those of $\Gamma_{\sigma}$ . The SALCs also matches the six SALCs derived above. It follows that the MO diagram, when both $p$ -orbitals with $\sigma$ symmetry and $s$ -orbitals participate in bonding with $M$ , will be a superposition of two very similar MO diagrams.

The remaining $p$ -orbitals form $\pi$ -bonds with the $d_{xy},d_{yz},d_{zx}$ orbitals of $M$ . Such interactions will be discussed below.

Pi interactions

We shall utilise the remaining vectors in the $ML_6$ diagram as a basis set. The direction of each vector indicated in the diagram is regarded as the positive lobe of the ligand $p$ -orbital. Employing the same logic as we did for $\sigma$ interactions, we have

$\boldsymbol{O_h}$	$E$	$8C_3$	$6C_2$	$6C_4$	$3C'_2$	$i$	$6S_4$	$8S_6$	$3\sigma_h$	$6\sigma_d$
$\boldsymbol{T_{\pi}}$	12	0	0	0	-4	0	0	0	0	2

Using eq27a, $\Gamma_{\pi}$ is decomposes to $\Gamma_{\pi}=T_{1g}\oplus T_{2g}\oplus T_{1u}\oplus T_{2u}$ .

The orthonormal SALCs are

Irreducible representations	Orthonormal ligand SALCs	Metal atom AOs
$T_{1g}$	Non-bonding	Non-bonding
$T_{2g}$	$\phi_1=\frac{1}{2}(\pi_{1y}-\pi_{2y}+\pi_{3y}-\pi_{4x})$	$d_{xy}$
	$\phi_2=\frac{1}{2}(\pi_{3z}-\pi_{4z}+\pi_{5y}-\pi_{6y})$	$d_{yz}$
	$\phi_3=\frac{1}{2}(\pi_{1z}-\pi_{2z}+\pi_{5x}-\pi_{6x})$	$d_{zx}$
$T_{1u}$	$\phi_4=\frac{1}{2}(\pi_{3x}+\pi_{4x}+\pi_{5x}+\pi_{6x})$	$p_{x}$
	$\phi_5=\frac{1}{2}(\pi_{1y}+\pi_{2y}+\pi_{5y}+\pi_{6y})$	$p_{y}$
	$\phi_6=\frac{1}{2}(\pi_{1z}+\pi_{2z}+\pi_{3z}+\pi_{4z})$	$p_{z}$
$T_{2u}$	Non-bonding	Non-bonding

The MO diagram, which describes both sigma and pi interactions, has the following general form:

The $t_{1g}$ and $t_{2u}$ MOs are non-bonding, while the $t_{2g}$ 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 $\pi$ -acceptors or $\pi$ -donors. For instance, the energies of the $t^*_{2g}$ MOs are usually lower than those of the $e^*_{g}$ MOs for $\pi$ -donors ligands. Examples of ligands that can engage in both sigma and pi interactions include $CO$ and $CN^-$ , while examples of $\pi$ -acceptors and $\pi$ -donors ligands are $CO$ and $F^-$ , respectively. These MO diagrams provide the theoretical foundation for the ligand field theory.