Question

A conjugated molecule has the following structure: 1 B-Carotene The observed Amar for this molecule is 500 nm. Using the suitable quantum mechanics model, the predicted length of the conjugated portion of this molecule is

Answer 1

Linear conjugated system (diens or polyenes) can be treated quantum mechanically by free electron molecular orbital (FEMO) theory. According to this approach it can be assumed that pi-electrons in pi skeleton move freely in a constant potential along the molecular skeleton. This treatment is analogous to the motion of particle in a 1D-box of infinite height. And Energy for such a system can be given by

η?h2 ΔΕ = 8ml?

Where L is the length of linear conjugated system.

In case of β-carotene, λ_max = 500 nm and N=11 pi bonds.

There are 22e^- and they will occupy in 11 energy levels (n). So, n=11 will be HOMO and LUMO=n+1=12.

hc (6.626 x 10-34js)(3 x 108m/s) - 3.97 x 10-19 AE = = 500 x 10-9m

(122 - 112)h2 AE = (3.97 x 10-19]) = 8m2 23 x (6.626 X 10-34Js) 8x (9.1 x 10-31kg) x 22

L2 = 3x (9.1 x 23 x (6.626 x 10-54Js) 8x (9.1 x 10-31kg) x (3.97 x 10-197) = 3.49 x 10-18 m2

L = 3.49 X 10-18 m2 = 1.86 x 10-9m = 1.86nm

Hence, Predicted length of conjugated portion of β-carotene is 1.86nm.