a^\dagger_c+\sum_{ab}x_{ab}a^\dagger_a a^\dagger_b p^\dagger+\sum_{c}z_c g^\dagger a^\dagger_c~.
\label{ecT1_red}$$ With this $B^\dagger$, a new cluster constraint can be constructed, containing the three-index $\mathcal{T}_1$ and extending it with two- and one-index terms: $$\left(
\begin{matrix}
(\mathcal{T}_1)_{abc;dez} & X(p)_{abc;st} & Z(g)_{abc;v}\\
X^\dagger(p)_{mn;dez} & X^{(2)}(p)_{mn;st}&(XZ)(p,g)_{mn;v}\\
Z^\dagger(g)_{u;dez} & (XZ)^\dagger(p,g)_{u;st}&Z^{(2)}(g)_{u;v}
\end{matrix}
\right)
\succeq 0~,$$ where the different tensors are defined as: $$\begin{aligned}
X(p)_{abc;st} =& \langle a^\dagger_a a^\dagger_b a^\dagger_c p a_t a_s + p a_t a_s a^\dagger_a a^\dagger_b a^\dagger_c\rangle~,\\
Z(g)_{abc;v} =& \langle a^\dagger_a a^\dagger_b a^\dagger_c a_v  g + a_v g a^\dagger_a a^\dagger_b a^\dagger_c\rangle~,\\
X^{(2)}(p)_{mn;st} =&  \langle a^\dagger_m a^\dagger_n p^\dagger p a_t a_s + p a_t a_s a^\dagger_n a^\dagger_m p^\dagger\rangle~,\\
XZ(p,g)_{mn;v} =&  \langle a^\dagger_m a^\dagger_n p^\dagger a_v g + a_v g a^\dagger_n a^\dagger_m p^\dagger\rangle~,\\
Z^{(2)}(g)_{u;v} =&  \langle g^\dagger a^\dagger_u a_v g + a_v g g^\dagger a^\dagger_u\rangle~.\end{aligned}$$ All these extra terms can be constructed using the full system 2DM and the knowledge of some predefined one- and two-particle states $p$ and $g$ outside the cluster.

\[eT2\]The extended $\mathcal{T}_2'$ constraint: $e\mathcal{T}_2'$
------------------------------------------------------------------

In an analogous way as for the $\mathcal{T}_1$ we can construct extensions which include cluster-system information to the $\mathcal{T}_2'$ condition. The two-particle-one-hole operator which constructs the full system $\mathcal{T}_2'$ (see Eq. (\[T2p\])) can be approximated on the cluster by: $$B^\dagger = 