The factors in front of the operator, either \( \frac{1}{4} \) or \( \frac{1}{2} \) tells whether we use antisymmetrized matrix elements or not.

We can now express the Hamiltonian operator for a many-fermion system in the occupation basis representation as $$ \begin{equation} H = \sum_{\alpha, \beta} \langle \alpha|\hat{t}+\hat{u}_{\mathrm{ext}}|\beta\rangle a_\alpha^{\dagger} a_\beta + \frac{1}{4} \sum_{\alpha\beta\gamma\delta} \langle \alpha \beta|\hat{v}|\gamma \delta\rangle a_\alpha^{\dagger} a_\beta^{\dagger} a_\delta a_\gamma. \tag{59} \end{equation} $$ This is the form we will use in the rest of these lectures, assuming that we work with anti-symmetrized two-body matrix elements.