Determinantal wavefunction and the Pauli Exclusion Principle

In our treatment of helium, each electron occupies a ls-like orbital, and hence the wavefunctions given in eqs. (2.2.18), (2.2.19), and (2.2.6) may be written as 

However, we cannot handle atoms with more than two electrons in the same manner. For instance, for lithium, the wavefunction 

For atomic systems, it is often said that each electron is defined by four quantum numbers n, l, me, and ms. Actually, there is a fifth quantum number, s, which has the value of 1 /2 for all electrons. Quantum number ms can be either 1/2 or -1/2, corresponding to spin function a (spin up) and p (spin down), respectively. Spin functions a and p form an orthonormal set, 

However, function //p implies that electrons are distinguishable electron 1 has its spin up and electron 2 spin down. Similarly, we should also consider the following in the total wavefunction for helium  

There are two ways of combining //p and //q in order to make the two electrons indistinguishable  

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