Question

Does anyone know why the energy difference between HOMO and LUMO decreases as the extent of conjugation increases?

Answer 1

The degree of conjugation essentially determines the distance within which the movement of the electron is confined. Roughly speaking, the electron is a "particle in a box" where the length of the box is determined by the length of conjugation.

The gap between energy levels caused by confinement always decreases as the size of the confining container increases. (That's why when you confine an electron in macroscopic-sized boxes, like a piece of metal, you get infinitismally small gap between energy levels, essentially a continuous band of them.) So as the length of the conjugation "box" increases, all energy gaps, including the HOMO-LUMO gap, decrease in size.

As to why the gap decreases as the confinement length increases, a larger box means the longest possible de Broglie wavelength that fits in the box is longer, which means (recall lambda = h/p) a lower momentum and hence kinetic energy, hence total energy.

Answer 2

sorry for so late replying.