Hartree-Fock Theory

In 1930 Fock [13] pointed out that the Hartree wavefunction was invalid as it did not satisfy the Pauli Exclusion Principle -- that the wavefunction must be antisymmetric with respect to electron interchange [14]. Fock also showed that a Hartree product could be made antisymmetric by appropriately adding and subtracting all possible permutations of the Hartree product, thereby forming the Hartree-Fock (HF) wavefunction. Later, Slater showed that the resulting wavefunction is simply the determinant of a matrix, called a Slater determinant [15,16]
$$\begin{align}\Psi = \frac{1}{\sqrt{N!}} \left\vert \begin{array}{cccc} \chi_{1}... ...f x}_{N}) & \cdots & \chi_{N}({\bf x}_{N}) \end{array} \right\vert. \end{align}$$
The prefactor normalizes the HF wavefunction (remembering that the $\chi_{i}$ are orthonormal).