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One of the basic rules of photochemistry is that the electronic configuration of a conjugated polyene does not change when excited thermally, but changes when photochemically excited, i.e. an electron moves from HOMO to LUMO.

Can anyone please explain this to me in terms of quantum mechanics?

I have done some calculations, which I am enclosing here.

(P.S.: Please do not provide explanation like thermal energy is very less to excite than photochemical, since it would imply that at very high temp (~1,00,00,000 C) the reaction is possible, which is not really the case).

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But what you say in your last sentence is true, and not specific to polyenes; it is quite general. The energy of a visible photon at $500$ nm is $20000~ \pu{cm^{-1}}$ but at room temperature the average thermal energy is only $\approx 210 \pu{~cm^{-1}}$. The probability of a molecule with $20000~ \pu{cm^{-1}}$ of thermal energy is $\approx e^{-100} = 3.7\cdot 10^{-44}$ so small! Thus there in effectively zero chance that by thermal excitation an excited state at 500 nm will be produced. (By excited state I mean an electron is transferred from the HOMO orbital into the LUMO or another, higher, unoccupied orbital).

In your calculation you cannot use $C_V\Delta T$ as its not an operator but in a collision transferring energy there has to be an operator so you cannot just remove it from the integral. The properties of the operator determine the selection rules for transitions.

It is possible by collisions between molecules to move up and down in energy, but because thermal energy is small usually this normally involves vibrational and rotational levels. (The effect is immediately obvious by cooling or heating samples in the appearance of hot bands and in relative rotational line intensities.) If there are very low lying electronic states then I can see no reason why these also could not be excited thermally.

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