Spin Orbitals and the Pauli Principle

Example 12.3 for the helium atom assumed that both electrons have a principal quantum number of 1. If the hydrogen-like wavefunction analogy were taken further, we might say that both electrons are in the s subshell of the first shell—that they are in Is orbitals. Indeed, there is experimental evidence (mostly spectra) for this assumption. What about the next element, Li It has a third electron. Would this third electron also go into an approximate Is hydrogen-like orbital Experimental evidence (spectra) shows that it doesn t. Instead, it occupies what is approximately the s subshell of the second principal quantum shell It is considered a 2s electron. Why doesn t it occupy the Is shell  

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We begin with the assumption that the electrons in a multielectron atom can in fact be assigned to approximate hydrogen-like orbitals, and that the wavefunction of the complete atom is the product of the wavefunctions of each occupied orbital. These orbitals can be labeled with the quantum number labels Is, 2s, 2p, 3s, 3p, and so on. Each s,p,d,f. subshell can also be labeled by an quantum number, where ranges from — to T (2T + 1 possible values). But it can also be labeled with a spin quantum number m either -f or —The spin part of the wavefunction is labeled with either a or p, depending on the value of for each electron. Therefore, there are several simple possibilities for the approximate wavefunction for, say, the lowest-energy state (the ground state) of the helium atom  

Because spin is a vector and because vectors can add and subtract from each other, one can easily determine a total spin for each possible helium spin orbital. (It is actually a total z component of the spin.) For the first spin orbital equation above, both spins are a, so the total spin is (-l- ) + (+ ) = 1. Similarly, for the last spin orbital, the total spin is (— ) + (— ) = — 1. For the middle two spin orbitals, the total (ar-component) spin is exactly zero. To summarize  

Which wavefunction of the two is acceptable, or are they both One can suggest that both wavefunctions are acceptable and that the helium atom is doubly degenerate. This turns out to be an unacceptable statement because, in part, it implies that an experimenter can determine without doubt that electron 1 has a certain spin wave-function and that electron 2 has the other spin wavefunction. Unfortunately, we cannot tell one electron from another. They are indistinguishable. 

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