Hartree-Fock (HF) theory is an algorithm for finding an approximative expression for the ground state of a given Hamiltonian. The basic ingredients are

- Define a single-particle basis \( \{\psi_{\alpha}\} \) so that

- The term \( \hat{u}^{\mathrm{HF}} \) is a single-particle potential to be determined by the HF algorithm.
- The HF algorithm means to choose \( \hat{u}^{\mathrm{HF}} \) in order to have

- The variational principle ensures that \( E^{\mathrm{HF}} \ge E_0 \), with \( E_0 \) the exact ground state energy.