What if we have more than two electrons? We can generalize the above
solution to electrons by using determinants. In the two electron
case, we can rewrite the above functional form as
Now the generalization to electrons is then easy to see, it is just
Since we can always construct a determinant (within a sign) if we just know the list of the occupied orbitals , we can write it in shorthand in a ket symbol as or even more simply as . Note that we have dropped the normalization factor. It's still there, but now it's just implied!
It is not at all obvious at this point, but it turns out that the assumption that the electrons can be described by an antisymmetrized product (Slater determinant) is equivalent to the assumption that each electron moves independently of all the others except that it feels the Coulomb repulsion due to the average positions of all electrons (and it also experiences a strange ``exchange'' interaction due to antisymmetrization). Hence, Hatree-Fock theory is also referred to as an independent particle model or a mean field theory. (Many of these descriptions also apply to Kohn-Sham density functional theory, which bears a striking resemblance to Hartree-Fock theory; one difference, however, is that the role of the Hamiltonian different in DFT).