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

$$ \hat{h}^{\mathrm{HF}}\psi_{\alpha} = \varepsilon_{\alpha}\psi_{\alpha} $$ with the Hartree-Fock Hamiltonian defined as $$ \hat{h}^{\mathrm{HF}}=\hat{t}+\hat{u}_{\mathrm{ext}}+\hat{u}^{\mathrm{HF}} $$

• 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

$$ \langle \hat{H} \rangle = E^{\mathrm{HF}}= \langle \Phi_0 | \hat{H}|\Phi_0 \rangle $$ that is to find a local minimum with a Slater determinant \( \Phi_0 \) being the ansatz for the ground state.

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