In Hartree-Fock theory the multi-dimensionality of the wave function \psi of stdQM is reduced by restricting the minimization of TE(\psi ) to anti-symmetric wave functions \psi (x_1,...,x_N) as linear combinations of products \psi_1(x_1)\times\psi_2(x_2)....\times \psi_N(x_N) of 3d wave functions \psi_j with global support named orbitals, in the form of Slater determinants:

  • \psi (x_1,x_2,...,x_N)=det (\psi_j(x_k)).

With proper choice of orbitals, typically as Hydrogen eigenfunctions, Hartree-Fock can give ground state energies in accordance (more or less) with observations, but ab initio computation is not feasible.

We compare with the wave function of realQM on summation form \psi (x)=\sum_j\psi_j(x) with the \psi_j(x) having disjoint local supports.