## Why Hartree-Fock?

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.