Selection Rules for Electronic Spectra of Transition Metal Complexes

Selection Rules

There are various selection rules that govern the feasibility of a transition (i.e. transitions between electronic energy levels) for transition metal complexes. Some of the most important selection rules have been mentioned below-
1. ΔS = 0 The Spin Rule
2. Δl = ± 1 The Orbital Rule (or Laporte)

Spin Selection Rule

The Spin Rule says that allowed transitions must involve the promotion of electrons without a change in their spin.
i.e. ΔS = 0, Spin-allowed

Transitions between states of different spin multiplicities are spin forbidden.
i.e. ΔS ≠ 0, Spin-forbidden.
Therefore singlet → singlet state transitions and triplet → triplet state transitions are allowed while the transitions from singlet → triplet or triplet → singlet state are forbidden.

Example: for d² system in octahedral complexes, the spin multiplicity (2s + 1) is 3 (as S = 1). There are four states with the same spin multiplicity (3) in this system. One of which is a ground state ³T_1g and the other three are excited states [³T_2g, ³A_2g and ³A_1g(P)]. Thus, there are three transitions which are spin allowed and the transitions from ³T_1g to any other excited states are spin forbidden.
³T_2g ← ³T_1g
³A_2g ← ³T_1g
³T_1g(P) ← ³T_1g

Orbital or Laporte's Selection Rule

The Orbital Rule says that for a molecule having centre of symmetry, transitions within the same sub-shell are forbidden. As per this rule the p-p or d-d transitions are forbidden.
For any transition to take place, change in the value of total orbital angular momentum (ΔL) between the final and initial stage should be-
ΔL = ± 1 (ΔL = L_f – L_i)
Where ΔL is equal to L_f (total angular momentum of the final state) subtracted by L_i (total angular momentum of the initial state).
If ΔL = ± 1, transitions are Laporte allowed and absorbance may be high but if ΔL = 0,transitions are Laporte forbidden and absorbance is low.

The transitions that occurr between the states of opposite parity (g ↔ u or u ↔ g) are allowed. It means that s ↔ p, p ↔ d, d ↔ f transitions are allowed.
The transitions that occurr between the states of same parity (g ↔ g or u ↔ u) are disallowed. It means that p ↔ p, d ↔ d, f ↔ f transitions are not allowed or forbidden.
The d-orbitals which are gerade in terms of parity or symmetric with respect to the centre of inversion or wave function can not have transitions from one d-orbital to the other.
However, p-orbitals are ungerade that are asymmetric with respect to the centre of inversion or wave function and thus, transition can take place between d and p orbitals.
In very simple words, d-d transitions as well as p-p transitions are not allowed, whereas d to p (d → p) and p to d (p → d) are allowed.

Relaxation in Laporte's Selection Rule

The metal-ligands bonds in transition metal complexes are not rigid. When ultraviolet-visible light is incident on a complex, electronic transitions as well as vibrations occurs simultaneously. In octahedral complexes, ligand vibrate in such a way that the orbitals present can temporarily loose their center of symmetry and due to which a very small mixing of p-d orbitals occurs and in this case d-d transitions are not purely Laporte forbidden that means the transitions from d to d (d → d) orbitals are partially allowed. Although these transitions are observed but their molar extinction coefficient is extremely low (10 - 50 L mol^-1 cm^-1). Therefore, octahedral complexes show color of low intensity. These transitions are actually responsible for various colors of the transition metal complexes.

Tetrahedral complexes have no centre of symmetry so, p-d mixing is more pronounced because t₂ molecular orbitals are formed from atomic d(gerage) and p(ungerage) orbitals. Therefore, tetrahedral complexes absorb more strongly than octahedral complexes and gives more intense color.

Relaxation in Spin Selection Rule

Spin -orbit coupling can be relax the selection rule so the transition may be observe from ground state of spin multiplicity to excited state of different spin multiplicity. These spin forbidden transitions are generally much weaker than spin allowed transitions.The spin-orbit coupling depends on the charge on the nucleus and therefore, strength of spin-orbit coupling in lighter atom is lower than that of heavier atoms. The intensity of suchh bands increases with increase in atomic number.
Spin-orbit coupling in 4d and 5d transition metal complexes is quite strong and thus in these complexes, the spin multiplicity rule is relaxed to a good extent therefor, intensities of these complexes are greater than 3d metal complexes as spin-orbit coupling is very weak in 3d metal complexes.
A Laporte forbidden d-d transition is more intense than a spin forbidden transition.

The selection rules control the intensity of absorption bands. The molar absorption coefficient (ε) measures the absorption band intensities. The intensities of absorption bands are experimentally measured in terms of the molar absorption coefficient 'ε'. The 'ε' values for intensities of spectral bands for electronic transitions in 3d metal complexes are given below in the table-

Transition	ε M−1 cm−1
Spin-forbidden	< 1
Laporte-forbidden d-d	20-100
Laporte-allowed d-d	~250
Symmetry allowed	1,000-50,000

Laporte Rule	Spin Rule	Spectra	Molar Absorbance(dm3 cm−1 mol−1)	Examples
Spin-forbidden	Allowed	d-d	100-1000	[CoCl4]−2
Partially allowed	Allowed	d-d	8-10	[Ti(H2O)6]+3
Partially allowed	Forbidden	d-d	2-5	[MnCl4]−2
Forbidden	Forbidden	d-d	< 0.5	[Mn(H2O)6]+2

NOTE: This rule affects Octahedral and Square planar complexes as they have center of symmetry. Tetrahedral complexes do not have center of symmetry therefore, this rule does not apply.

Although d-d Transitions are Laporte Forbidden Transitions, they do occur in transition metal complexes, why?