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Can we say that an electron with a specific set of quantum numbers of a fluorine atom is exactly the same as that of another electron with the same set of quantum numbers of another fluorine atom.

If no (which I don't think is the case), then on what basis will we differentiate them?

If yes, then how are we able to apply "Pauli's Exclusion Principle" to the Fluorine molecule. What happens when the two atoms combine? Do the atomic quantum numbers change into molecular quantum numbers? Even if so, is there any way to figure it out?

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    $\begingroup$ en.wikipedia.org/wiki/One-electron_universe may be of interest $\endgroup$
    – Ian Bush
    Commented Oct 2, 2023 at 11:06
  • $\begingroup$ You may want also review the related topic of Electron degeneracy pressure in context of white dwarfs and solid metals. $\endgroup$
    – Poutnik
    Commented Oct 2, 2023 at 11:10
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    $\begingroup$ Quantum numbers are for the whole system. In the first case the system is a totally isolated F atom, and you have two entirely disconnected systems, in different universes if you like. So You can distinguish the two electrons by being godlike and seeing which universe they are in. In the second case the quantum numbers are those from solving for the entire F2 molecule - those for the F atom are irrelevant as you can't simply factorize the solution into two separate contributions, partially because of the Pauli principle. Thus you apply the principle for the molecule, not two atoms. $\endgroup$
    – Ian Bush
    Commented Oct 2, 2023 at 12:23
  • $\begingroup$ @IanBush it's agreeable what you explained, but only before and after the formation of the molecule. What was happening during the formation of the molecule is not clear from the argument. What happened to the spin of the electrons and things like that? $\endgroup$
    – KeShAw
    Commented Oct 2, 2023 at 16:32
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    $\begingroup$ You (try to) solve the appropriate equations for the full system, get the quantum numbers and use the exclusion principle. The fundamental equations that describe Life, The Universe and Everything don't just apply at the beginning and end points, they are Universal. $\endgroup$
    – Ian Bush
    Commented Oct 2, 2023 at 16:55

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There are several forms of the 'Pauli exclusion principle', and you have described one that is a special case only applying to atoms.

More generally speaking, one should say that no two electrons can occupy the same state.*

In atoms, states happen to be defined by the four quantum numbers $(n, l, m_l, m_s)$, which means that no two electrons can have the same four quantum numbers.

In molecules, the states are labelled in a different way, no longer using the same quantum numbers as before. But the same principle holds: no two electrons can have the same state (i.e., be in the same orbital, with the same spin).

The opening section of the Wikipedia page goes into more detail about this: https://en.wikipedia.org/wiki/Pauli_exclusion_principle

Do the atomic quantum numbers change into molecular quantum numbers?

Well, the atomic states do slowly morph into molecular states. The atomic states are described using quantum numbers, but that is just a short way of writing the full states out (which are wavefunctions).


* Fermions, if you want to be really general.

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It will be useful to check the molecular orbital diagram of $\ce{F2}$, it is not difficult to understand the possible spatial states. Since the exclusion implies that two electrons should not have similar states, each molecular orbital is the state of two electrons with different spin up to the highest occupied molecular orbital. In the first quantization scheme one should not think about the electrons but about the states, the states are labeled with the quantum numbers for the atom, but when they overlap and form a molecule, the potential does not have the perfect spherical symmetry, it especially affects the angular momenta, while the first quantum number $n$ is specific to all discrete level systems. So to summarize, the orbitals overlap and form new molecular orbital states, what you now call electrons are simply indistinguishable and each has a specific molecular state.

There is no need to decipher the exclusion principle any further than the assumption that each spatial molecular orbital is the state of two electrons with different spins.

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    $\begingroup$ Glad to know what is indeed necessary and what not in MOT. $\endgroup$ Commented Oct 3, 2023 at 9:47

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