d a_l + d a_l a^\dagger_a a^\dagger_b a_c  \rangle~,\\
X^{(2)}(h)_{mn;st} =\langle a^\dagger_m a^\dagger_n h h^\dagger a_t a_s  + h^\dagger a_t a_s a^\dagger_m a^\dagger_n h  \rangle~,&\quad XY(h,p)_{mn;pq} =\langle a^\dagger_m a^\dagger_n h a^\dagger_q a_p p + a^\dagger_q a_p p a^\dagger_m a^\dagger_n h \rangle~,\\
XZ(h,g)_{mn;v} =\langle a^\dagger_m a^\dagger_n h a^\dagger_v g + a^\dagger_v g a^\dagger_m a^\dagger_n h \rangle~,&\quad XR(h,d)_{mn;l} =\langle a^\dagger_m a^\dagger_n h  d a_l + d a_l a^\dagger_m a^\dagger_n h \rangle~,\\
XP(h)_{mn;w} =\langle a^\dagger_m a^\dagger_n h  a_w \rangle~,&\quad Y^{(2)}(p)_{xy;pq} =\langle p^\dagger a^\dagger_x a_y a^\dagger_q a_p p +   a^\dagger_q a_p p p^\dagger a^\dagger_x a_y \rangle\\
YZ(p,g)_{xy;v} =\langle p^\dagger a^\dagger_x a_y a^\dagger_v g +   a^\dagger_v g p^\dagger a^\dagger_x a_y \rangle~,&\quad YR(p,d)_{xy;v} =\langle p^\dagger a^\dagger_x a_y d a_l +   d a_l p^\dagger a^\dagger_x a_y \rangle\\
YP(p)_{xy;w} =\langle p^\dagger a^\dagger_x a_y a_w\rangle~,&\quad Z^{(2)}(g)_{u;v} = \langle g^\dagger a_u a_v g + a_v g g^\dagger a_u \rangle\\
ZR(g,d)_{u;l} = \langle g^\dagger a_u d a_l + d a_l g^\dagger a_u \rangle~,&\quad ZP(g)_{u;w} = \langle g^\dagger a_u a_w \rangle~,\\
R^{(2)}(d)_{k;l} = \langle a^\dagger_k d^\dagger d a_l  + d a_l a^\dagger_k d^\dagger \rangle~,&\quad RP(d)_{k;w} = \langle a^\dagger_k d^\dagger a_w \rangle~.\end{aligned}$$ which can all be constructed using the full system 2DM and the different states ($h,p,g$ and $d$) outside of the cluster.

