Hartree product

The Hartree product is a system (many-particle) wavefunction, given as a combination of wavefunctions of the individual particle wavefunctions. It is inherently mean-field (assumes the particles are independent) and is the unsymmetrized version of the Slater determinant ansatz in the Hartree–Fock method.

For two particles,

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})=\chi _{1}(\mathbf {x} _{1})\chi _{2}(\mathbf {x} _{2}).$

This is not satisfactory for fermions, such as electrons, because the resulting wave function is not antisymmetric. An antisymmetric wave function can be mathematically described using the Slater determinant.