the $\mathcal{T}_1$ condition is derived by demanding the non-negativity of the class of Hamiltonians: $$\hat{H} = \left\{B^\dagger,B\right\}~,\qquad\text{where}\qquad B^\dagger = \sum_{\alpha\beta\gamma}t^1_{\alpha\beta\gamma}a^\dagger_\alpha a^\dagger_\beta a^\dagger_\gamma~.$$ When we limit the $\mathcal{T}_1$ condition to a cluster we impose the non-negativity of a subclass of Hamiltonians generated by the three-particle operator: $$B^\dagger = \sum_{abc}t^1_{abc}a^\dagger_a a^\dagger_b a^\dagger_c~,$$ in which Latin indices $a$ are used to denote cluster states, overlined Latin indices $\bar{a}$ to denote the rest of the system and Greek indices $\alpha$ for general states. A better approximation to the full system $B^\dagger$ is obtained if one allows one or two creation operators to be outside the cluster: $$B^\dagger = \sum_{abc}t^1_{abc}a^\dagger_a a^\dagger_b a^\dagger_c+\sum_{ab\bar{c}}t^1_{ab\bar{c}}a^\dagger_a a^\dagger_b a^\dagger_{\bar{c}}+\sum_{\bar{a}\bar{b}c}t^1_{\bar{a}\bar{b}c}a^\dagger_{\bar{a}} a^\dagger_{\bar{b}} a^\dagger_c~.
\label{ecT1_full}$$ In this way it is clear that the hopping in and out of the cluster is included. However, we want to construct a condition which only depends on the cluster indices. This can be accomplished by factorizing the terms in Eq. (\[ecT1\_full\]) which contain both cluster and non-cluster indices: $$\begin{aligned}
t^1_{ab\bar{c}} =& x_{ab} p_{\bar{c}}~,\\
t^1_{\bar{a}\bar{b}c} =& g_{\bar{a}\bar{b}} z_{c}~.\end{aligned}$$ The new $B^\dagger$ operator only depends on cluster indices, a one-particle ($p^\dagger = \sum_{\bar{c}} p_{\bar{c}}a^\dagger_{\bar{c}}$) and a two-particle state ($g^\dagger = \sum_{\bar{a}\bar{b}}g_{\bar{a}\bar{b}}a^\dagger_{\bar{a}}a^\dagger_{\bar{b}}$) outside of the cluster: $$B^\dagger = \sum_{abc}t^1_{abc}a^\dagger_a a^\dagger_b 