Magnetism from Coulomb interaction

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.

• 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. – flaudemus Feb 24 at 19:05
• what states could i be missing? – AWiltzer Feb 24 at 19:20
• Ohhh you're saying |L>|L> or |R>|R>? – AWiltzer Feb 24 at 19:20