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I have found a similar question on a different website upon searching on google but it didn't clear my confusion (which might be because I didn't understand the explanation). (PS: I am a 12th STD Student)

So, how exactly do they differ? Can they ever be equal?

HOMO- Highest Occupied Molecular Orbital LUMO- Lowest Unoccupied Molecular Orbital

Forbidden gap - Energy difference between the highest energy conduction band and lowest energy valence band.

So, why aren't these the same?

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The HOMO-LUMO gap is a computational number. It gives the energy of a band gap excitation without any relaxation and many electron correlation effects such an excitation may exhibit. These effects are very hard to calculate. I don't know of any calculation that correctly predicts band gaps to within say 20%. Maybe my literature knowledge needs an update, though.

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