Eq. (13) holds for case (1) since the lefthand side is zero due to the Pauli principle. We write Eq. (11) as $$ \begin{equation} \langle\alpha_1\alpha_2 \dots \alpha_n|a_\alpha|\alpha_1'\alpha_2' \dots \alpha_m'\rangle = \langle \alpha_1\alpha_2 \dots \alpha_n|\alpha\alpha_1'\alpha_2' \dots \alpha_m'\rangle \tag{14} \end{equation} $$ Here we must have \( m = n+1 \) if Eq. (14) has to be trivially different from zero.