\sum_{abc}t^2_{abc}a^\dagger_a a^\dagger_b a_c + \sum_m s_m a^\dagger_m + \sum_{ab}x_{ab}a^\dagger_a a^\dagger_b h + \sum_{bc} y_{bc}p^\dagger a^\dagger_b a_c + \sum_{c}z_c g^\dagger a_c + \sum_{a}r_{a}a^\dagger_a d^\dagger ~,
\label{ecT2_red}$$ in which the different states outside of the cluster are defined as: $$\begin{aligned}
h = \sum_{\bar{c}}h_{\bar{c}}a_{\bar{c}}~,\qquad&\qquad p^\dagger = \sum_{\bar{a}}p_{\bar{a}}a^\dagger_{\bar{a}}~,\\
g^\dagger = \sum_{\bar{a}\bar{b}}g_{\bar{a}\bar{b}}a^\dagger_{\bar{a}}a^\dagger_{\bar{b}}~,\qquad&\qquad d^\dagger = \sum_{\bar{b}\bar{c}}d_{\bar{b}\bar{c}}a^\dagger_{\bar{b}}a_{\bar{c}}~.\end{aligned}$$ As before this leads to a new non-negativity constraint which includes the old $\mathcal{T}_2'$ on the cluster and extends it with four new terms. [$$\left(
\begin{matrix}
(\mathcal{T}_2)_{abc;dez} &  P_{abc;w} & X(h)_{abc;st} & Y(p)_{abc;pq} & Z(g)_{abc;v} & R(d)_{abc;l}\\
P^\dagger_{r;dez} & P^{(2)}_{r;w} &PX(h)_{r;st}& PY(p)_{r;pq}& PZ^\dagger(g)_{r;v}& PR^\dagger(d)_{r;l} \\ 
X^\dagger(h)_{mn;dez} & PX^\dagger(h)_{mn;w}& X^{(2)}(h)_{mn;st} & XY(h,p)_{mn;pq} & XZ(h,g)_{mn;v} & XR(h,d)_{mn;l} \\
Y^\dagger(p)_{xy;dez} &  PY^\dagger(p)_{xy;w}&XY^\dagger(h,p)_{xy;st} & Y^{(2)}(p)_{xy;pq} & YZ(p,g)_{xy;v}& YR(p,d)_{xy;l}\\
Z^\dagger(g)_{u;dez} & PZ^\dagger(g)_{u;w}  & XZ^\dagger(h,g)_{u;st}& YZ^\dagger(p,g)_{u;pq}& Z^{(2)}(g)_{u;v} & ZR(g,d)_{u;l}\\
R^\dagger(d)_{k;dez} &  PR^\dagger(d)_{k;w}& XR^\dagger(h,d)_{k;st}& YR^\dagger(p,d)_{k;pq}& ZR^\dagger(g,d)_{k;v}& R^{(2)}(d)_{k;l}
\end{matrix}
\right)
\succeq 0$$ ]{} in which the extra tensor terms are defined as: $$\begin{aligned}
X(h)_{abc;st} = \langle a^\dagger_a a^\dagger_b a_c h^\dagger a^\dagger_t a_s + h^\dagger a^\dagger_t a_s a^\dagger_a a^\dagger_b a_c \rangle~,&\quad Y(p)_{abc;pq} = \langle a^\dagger_a a^\dagger_b a_c a^\dagger_q a_p p + a^\dagger_q a_p p a^\dagger_a a^\dagger_b a_c  \rangle~,\\
Z(g)_{abc;v} = \langle a^\dagger_a a^\dagger_b a_c a^\dagger_v g + a^\dagger_v g a^\dagger_a a^\dagger_b a_c  \rangle~,&\quad R(d)_{abc;l} = \langle a^\dagger_a a^\dagger_b a_c 