Here we let \( \sum' \) indicate that the sums running over \( \alpha \) and \( \beta \) run over all single-particle states, while the summations \( \gamma \) and \( \delta \) run over all pairs of single-particle states. We wish to remove this restriction and since $$ \begin{equation} \langle \alpha\beta|\hat{v}|\gamma\delta\rangle = \langle \beta\alpha|\hat{v}|\delta\gamma\rangle \tag{49} \end{equation} $$ we get $$ \begin{eqnarray} \sum_{\alpha\beta} \langle \alpha\beta|\hat{v}|\gamma\delta\rangle a^{\dagger}_\alpha a^{\dagger}_\beta a_\delta a_\gamma &=& \sum_{\alpha\beta} \langle \beta\alpha|\hat{v}|\delta\gamma\rangle a^{\dagger}_\alpha a^{\dagger}_\beta a_\delta a_\gamma \tag{50} \\ &=& \sum_{\alpha\beta}\langle \beta\alpha|\hat{v}|\delta\gamma\rangle a^{\dagger}_\beta a^{\dagger}_\alpha a_\gamma a_\delta \tag{51} \end{eqnarray} $$ where we have used the anti-commutation rules.