If we have two atoms with one electronic level that we put next to each other. If the electronic wavefunctions are called $|L>$ and $|R>$ respectively, what are all possible two electron wave functions?

If the energy between them is an interaction of the form $V(|\vec{r_{1}}-\vec{r_{2}}|)$, what are the first order correction energies for each wavefunction?

My attempt

My problem is that I really don't think I understand what they're asking, but taking into account spin, I said that the wavefunctions are:

$\sqrt{1/2}\,(|L>|R> -|R>|L>)|\uparrow\uparrow>$

$\sqrt{1/2}\,(|L>|R> -|R>|L>)|\downarrow\downarrow>$

$\sqrt{1/2}\,(|L>|R> - |R>|L>)1/\sqrt{2}(|\uparrow\downarrow>+|\downarrow \uparrow>)$

and $\sqrt{1/2}\,(|L>|R> + |R>|L>)1/\sqrt{2}(|\uparrow\downarrow>-|\downarrow \uparrow>)$

does anybody else think that this makes sense as per what I'm being asked?

and if so, the correction energies would just be $\langle\Psi|V|\Psi\rangle$ for each of the states which seems to simple too me? Any help would be appreciated here.

  • $\begingroup$ I think you are exactly on the right track. Your first three states belong to the spin-triplet, the last one is the spin-singlet. However, if the occupation of the $\left|\left. L\right>\right.$ or the $\left|\left. R\right>\right.$ orbital with two electrons is taken into consideration, then there are two additional states. $\endgroup$ – flaudemus Feb 24 at 19:05
  • $\begingroup$ what states could i be missing? $\endgroup$ – AWiltzer Feb 24 at 19:20
  • $\begingroup$ Ohhh you're saying |L>|L> or |R>|R>? $\endgroup$ – AWiltzer Feb 24 at 19:20

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