5
$\begingroup$

I am learning quantum chemistry and I am trying to understand the statement in a .pdf I'm reading which states "The energy transition associated with the spectral transitions observed is: $$\Delta E = E_{LUMO} - E_{HOMO} = \frac{(N+1)h^2}{8mL^2}$$ where $L$ is the length of the "box" (it's talking about particle in a box), $m$ is the mass of the electron, and $N$ is the number of $\pi$-electrons in the system. I know due to the solution of the Schrodinger equation that $$E = \frac{n^2 h^2}{8mL}$$ So, $$\Delta E = E_{HOMO} - E_{LUMO} = E_{k+1} - E_{k} = \frac{(2k+1)h^2}{8mL^2}$$ So, I want to figure out why $2k = N$?

$\endgroup$
  • $\begingroup$ Welcome to Chemistry.SE. Is there any possibility that you could link to the pdf in question? $\endgroup$ – Ben Norris Sep 28 '13 at 1:15
3
$\begingroup$

Your calculation is indeed correct. However, because each electron in the $\pi$ system can have two spin states, there are two electrons per energy level of the “particle in a box” Hamiltonian. Hence, $N=2k$.

$\endgroup$
  • $\begingroup$ I've said that $N$ is the number of $\pi$-electrons. That's all my source says. $\endgroup$ – Samuel Reid Sep 28 '13 at 15:37
  • $\begingroup$ @SamuelReid sorry, I read too fast… answer amended $\endgroup$ – F'x Sep 28 '13 at 15:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.