(8) |

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

(9) |

Note a nice feature of this; if we try to put two electrons in the same orbital at the same time (i.e., set ), then . This is just a more sophisticated statement of the

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).