Representation of the molecular orbitals of benzene

Representation of the p molecular orbitals of benzene

The NLUMO is the b_2u orbital shown below.

The LUMO is a doubly degenerate set shown above with symmetry e_2u.

The HUMO is also a doubly degenerate set with symmetry e_1g.

For completeness we also include the lowest p orbital of a_2u symmetry.

The molecular orbital diagram is shown below.

The configuration of the ground state is A_1g. This is always the case for a closed shell molecule since the combination of the two spin paired electrons yields a direct product of the symmetry terms that is always totally symmetric. The symmetry of the first p-p* excited state is given by the direct product of the irreducible symmetry representations for those states. For benzene, the configurations are:

ground state (a_2u)²(e_1g)⁴

first excited state (a_2u)²(e_1g)³(e_2u)

We consider the direct product of (e_1g)(e_2u). In D_6h this transforms as:

E	2C₆	2C₃	C₂	3C₂'	3C₂''	i	2S₃	2S₆	s_h	3s_d	3 s_v
4	-1	1	-4	0	0	-4	1	-1	4	0	0

This can be determined by group theoretical methods

G_A1g = (1/24)[(1)(4)+2(1)(-1)+2(1)(1)+(1)(-4)+(1)(-4)+2(1)(1)+2(1)(-1)+(1)(4)] = 0

G_A2g = (1/24)[(1)(4)+2(1)(-1)+2(1)(1)+(1)(-4)+(1)(-4)+2(1)(1)+2(1)(-1)+(1)(4)] = 0

G_B1g = (1/24)[(1)(4)+2(-1)(-1)+2(1)(1)+(-1)(-4)+(1)(-4)+2(-1)(1)+2(1)(-1)+(-1)(4)] = 1

G_B2g = (1/24)[(1)(4)+2(-1)(-1)+2(1)(1)+(-1)(-4)+(1)(-4)+2(-1)(1)+2(1)(-1)+(-1)(4)] = 0

G_E1g = (1/24)[(2)(4)+2(1)(-1)+2(-1)(1)+(-2)(-4)+(2)(-4)+2(1)(1)+2(-1)(-1)+(-2)(4)] = 0

G_E2g = (1/24)[(2)(4)+2(-1)(-1)+2(-1)(1)+(2)(-4)+(2)(-4)+2(-1)(1)+2(-1)(-1)+(2)(4)] = 0

G_A1u = (1/24)[(1)(4)+2(1)(-1)+2(1)(1)+(1)(-4)+(-1)(-4)+2(-1)(1)+2(-1)(-1)+(-1)(4)] = 0

G_A2u = (1/24)[(1)(4)+2(1)(-1)+2(1)(1)+(1)(-4)+(-1)(-4)+2(-1)(1)+2(-1)(-1)+(-1)(4)] = 0

G_B1u = (1/24)[(1)(4)+2(-1)(-1)+2(1)(1)+(-1)(-4)+(-1)(-4)+2(1)(1)+2(-1)(-1)+(1)(4)] = 1

G_B2u = (1/24)[(1)(4)+2(-1)(-1)+2(1)(1)+(-1)(-4)+(-1)(-4)+2(1)(1)+2(-1)(-1)+(1)(4)] = 1

G_E1u = (1/24)[(2)(4)+2(1)(-1)+2(-1)(1)+(-2)(-4)+(-2)(-4)+2(-1)(1)+2(1)(-1)+(2)(4)] = 1

G_E2u = (1/24)[(2)(4)+2(-1)(-1)+2(-1)(1)+(2)(-4)+(-2)(-4)+2(1)(1)+2(1)(-1)+(-2)(4)] = 0

The excited singlet state configurations are ¹B_1u, ¹B_2u, and ¹E_1u.

By symmetry the ¹E_1u are allowed, and the others are forbidden. However, it turns out that the ¹B_2u state is much lower in energy. This is because of configuration interaction with the higher lying B_2u state (see the energy diagram above). This transition gives rise to observed 260 nm band in benzene. How can a "forbidden" state give rise to an absorption band? The answer lies in vibrational distortions of the molecule along coordinates that couple the forbidden state to a strongly allowed state. This mechanism is known as vibronic coupling. Vibronic coupling in absorption arises due to the coordinate dependence of the electronic transition moment.