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In helium, the triplets ($S=1$) are lower in energy (more negative) than the singlets $S=0$.

One reason given by my lecturer is that in a triplet the spins of the two electrons are the same, then by the Pauli exclusion principle the electron wavefunctions must no overlap too strongly otherwise the there would be two electrons with the same spin state in the same location, pushing them apart so their Coulomb energy is lower (more tightly bound).

But if the electrons are further apart, shouldn't that imply that the electron in the excited state be further away from the nucleus and hence there would be less Coulomb attraction betwen the excited electron and nucleus, decreaseing the binding enegry? (i.e. the energy of the singlet should be more negative instead)

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  • $\begingroup$ can you give a link for these statements? I am confused by your notation. $\endgroup$
    – anna v
    Commented May 6, 2022 at 17:43
  • $\begingroup$ @anna v these are from my lecture note, sorry for any confusion. $\endgroup$
    – Chern-Simons
    Commented May 6, 2022 at 18:43
  • $\begingroup$ The electron-nuclear and electron-electron potentials are both on the same order of magnitude so you really can't rely on qualitative argumentsm, it just so happens that in this case the $e-e$ repulsion wins out and is more important. $\endgroup$
    – AfterShave
    Commented May 7, 2022 at 20:35

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The energy difference is caused by the exchange interaction. For the singlet the electron interaction is $J+K$ en for the triplet it is $J-K$, where $$ J = \int { \int { d^3 r_1 d^3 r_2 \frac{a^2(r_1) b^2(r_2)}{| r_1 - r_2 |}}} $$ and $$K= \int {\int {d^3 r_1 d^3 r_2 \frac{a(r_1) b(r_1) a(r_2) b(r_2)}{| r_1 - r_2 |}}}\,.$$

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  • $\begingroup$ I think I can understand it mathematically from the exchange integral. My question is more about the physical explanation for this. The triplets are more tightly bound and yet their electrons are pushed further apart from each other due to the exclusion principle, which is contradictory to me, I thought more tightly bound=electrons closer to each other? $\endgroup$
    – Chern-Simons
    Commented May 6, 2022 at 18:42
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    $\begingroup$ @Chern-Simons I am not aware of any intuitive physical explanation of exchange. Even Pauli himself struggled with it. nobelprize.org/prizes/physics/1945/pauli/lecture $\endgroup$
    – my2cts
    Commented May 6, 2022 at 18:54
  • $\begingroup$ @Chern-Simons The basic large energy levels are defined/calculable between negative electrons and positive nuclei. Seems to me you are discussing the fine structure induced on them by the more complex interactions that exist because of the coulomb potential between negative electrons . the fine structure is that has the contradictory behavior that puzzles you. $\endgroup$
    – anna v
    Commented May 7, 2022 at 4:18
  • $\begingroup$ seems to me that there can be what you deem as discrepancy in the partial energy level changes, because of the Pauli exclusion, $\endgroup$
    – anna v
    Commented May 7, 2022 at 4:38

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