Definitions and notations

We obtain $$ \begin{align} \int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \frac{1}{2}\sum_{\mu=1}^A\sum_{\nu=1}^A &\left[ \int \psi_{\mu}^*(x_i)\psi_{\nu}^*(x_j)\hat{v}(r_{ij})\psi_{\mu}(x_i)\psi_{\nu}(x_j) dx_idx_j \right. \tag{9}\\ &\left. - \int \psi_{\mu}^*(x_i)\psi_{\nu}^*(x_j) \hat{v}(r_{ij})\psi_{\nu}(x_i)\psi_{\mu}(x_j) dx_idx_j \right]. \tag{10} \end{align} $$ The first term is the so-called direct term. It gives rise to the Hartree term in Hartree-Fock theory, while the second is due to the Pauli principle and is called the exchange term and gives rise to the Fock term in the Hartree-Fock equations. The factor \( 1/2 \) is introduced because we now run over all pairs twice.