For the other terms on the rhs we obtain similar expressions and summing over all terms we obtain $$ \begin{eqnarray} H_I |\alpha_1\alpha_2\dots\alpha_n\rangle &=& \sum_{\alpha_1', \alpha_2'} \langle \alpha_1'\alpha_2'|\hat{v}|\alpha_1\alpha_2\rangle |\alpha_1'\alpha_2'\dots\alpha_n\rangle \nonumber \\ &+& \dots \nonumber \\ &+& \sum_{\alpha_1', \alpha_n'} \langle \alpha_1'\alpha_n'|\hat{v}|\alpha_1\alpha_n\rangle |\alpha_1'\alpha_2\dots\alpha_n'\rangle \nonumber \\ &+& \dots \nonumber \\ &+& \sum_{\alpha_2', \alpha_n'} \langle \alpha_2'\alpha_n'|\hat{v}|\alpha_2\alpha_n\rangle |\alpha_1\alpha_2'\dots\alpha_n'\rangle \nonumber \\ &+& \dots \tag{46} \end{eqnarray} $$