The two-body operator can also be expressed in terms of the anti-symmetrized matrix elements we discussed previously as $$ \begin{eqnarray} \hat{H}_I &=& \frac{1}{2} \sum_{\alpha\beta\gamma\delta} \langle \alpha \beta|\hat{v}|\gamma \delta\rangle a_\alpha^{\dagger} a_\beta^{\dagger} a_\delta a_\gamma \nonumber \\ &=& \frac{1}{4} \sum_{\alpha\beta\gamma\delta} \left[ \langle \alpha \beta|\hat{v}|\gamma \delta\rangle - \langle \alpha \beta|\hat{v}|\delta\gamma \rangle \right] a_\alpha^{\dagger} a_\beta^{\dagger} a_\delta a_\gamma \nonumber \\ &=& \frac{1}{4} \sum_{\alpha\beta\gamma\delta} \langle \alpha \beta|\hat{v}|\gamma \delta\rangle_{\mathrm{AS}} a_\alpha^{\dagger} a_\beta^{\dagger} a_\delta a_\gamma \tag{58} \end{eqnarray} $$