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Why do electrons in an $H_2$ molecule have opposite spins, while protons do not necessarily exhibit this behavior? Considering that both are fermions, shouldn't they both adhere to the Pauli exclusion principle?

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  • $\begingroup$ Who cares about the Pauli principle when the states and their respective wavefunctions cannot overlap. $\endgroup$
    – M06-2x
    Dec 25, 2023 at 12:31

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The Fermi exclusion principle does apply to the proton as well. The ground state of the hydrogen molecule has the proton spins in opposite directions. This is known as parahydrogen. To get the proton spins parallel you need to add energy to put the molecule into an excited state known as orthohydrogen.

The Pauli exclusion principle doesn't forbid the electrons having parallel spins. It just forbids then when both electrons are in the same molecular orbital. Since the ground state has both electrons in the same $1\sigma$ orbital the spins have to be opposite. However if we excite the molecule to the $1\sigma^1 1\sigma*^1$ state, where the electrons are in different orbitals, then the electron spins can be parallel instead of opposite.

This is basically the difference between para and orthohydrogen. As a rough approximation we can think of parahydrogen as having both protons in the same state so their spins have to be opposite. Then orthohydrogen has the protons in different states so their spins can be parallel.

For more on this see the Wikipedia article on the spin isomers of hydrogen.

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  • $\begingroup$ So in case of parahydrogen the protons basically have overlapping wave functions, but in orthohydrogen they don't? Is there even any significant overlapping in general, since the distance between the two nuclei is large in comparison to the size of the nuclei? $\endgroup$ Dec 25, 2023 at 14:03
  • $\begingroup$ The reason I said as a rough approximation is because you cannot treat the two protons separately from the electrons. The protons and electrons are all entangled and are described by a four particle wavefunction. The protons certainly do interact, after all their separation is about the same as the average separation between the electrons, but it would be stretching things to talk about proton orbitals as separate from the electron orbitals. $\endgroup$ Dec 25, 2023 at 14:20

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