Slater Determinants

For our two electron problem, we can satisfy the antisymmetry principle by a wavefunction like:

$$\Psi({\mathbf x}_1, {\mathbf x}_2) = \frac{1}{\sqrt{2}} \l... ...x}_2) - \chi_1({\mathbf x}_2) \chi_2({\mathbf x}_1) \right].$$ (8)

This is very nice because it satisfies the antisymmetry requirement for any choice of orbitals $\chi_1({\mathbf x})$ and $\chi_2({\mathbf x})$.

What if we have more than two electrons? We can generalize the above solution to $N$ electrons by using determinants. In the two electron case, we can rewrite the above functional form as

$$\Psi({\mathbf x}_1, {\mathbf x}_2) = \frac{1}{\sqrt{2}} \left\vert \begin{array}{cc} \chi_1({\bf x}_1)... ...i_2({\bf x}_1) \\ \chi_1({\bf x}_2) & \chi_2({\bf x}_2) \end{array}\right\vert$$ (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 $\chi_1 = \chi_2$), then $\Psi({\mathbf x}_1, {\mathbf x}_2) = 0$. This is just a more sophisticated statement of the Pauli exclusion principle, which is a consequence of the antisymmetry principle!

Now the generalization to $N$ electrons is then easy to see, it is just

$$\Psi = \frac{1}{\sqrt{N!}} \left\vert \begin{array}{cccc} ... ...f x}_N) & \cdots & \chi_N({\bf x}_N) \end{array} \right\vert.$$ (10)

A determinant of spin orbitals is called a Slater determinant after John Slater. An interesting consequence of this functional form is that the electrons are all indistinguishable, consistent with the strange results of quantum mechanics. Each electron is associated with every orbital! This point is very easily forgotten, especially because it is cumbersome to write out the whole determinant which would remind us of this indistinguishability. Speaking of which, it is time to introduce a more compact notation.

Since we can always construct a determinant (within a sign) if we just know the list of the occupied orbitals $\{ \chi_i({\bf x}), \chi_j({\bf x}), \cdots \chi_k({\bf x}) \}$, we can write it in shorthand in a ket symbol as $\vert \chi_i \chi_j \cdots \chi_k \rangle$ or even more simply as $\vert i j \cdots k \rangle$. 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).