by increasing the accuracy of the approximation by deriving new constraints [@zhao; @mazz_T_con; @dimi; @qsep; @shenvi]. A large variety of systems has been studied using v2DM, mostly atomic and molecular systems [@hammond; @nakata_last; @mazz_T_con; @Gido_T_con; @mazz_book; @braams_book], but also nuclear [@mihailovic; @rosina], and spin and lattice systems [@maz_hub; @barthel; @shenvi; @nakata_last; @hubbard_bv; @gutz_sdp]. Recently it was observed that this standard approach fails to describe the dissociation limit of certain diatomic molecules [@helen_1]. This failure was shown to originate from the lack of size-extensivity of the method [@qsep; @nakata_se], and several ways to cure this behaviour have been set forth [@qsep; @helen_2; @nakata_li].

This paper focuses on how the standard v2DM approach performs for the 2D Hubbard model, and proposes a computationally much cheaper way in which these results can be approximated. It should be noted that the proposed new conditions are approximations to the standard non-negativity constraints, and as such they do not intend to cure the non-size extensivity of the method.

The Hubbard model is a schematic model [@hubbard] developed to describe electron correlation in solids by means of the Hamiltonian: $$\hat{H} = -t\sum_{\langle ij \rangle}\left(a^\dagger_{i\sigma}a_{j\sigma} + a^\dagger_{j\sigma}a_{i\sigma}\right) + U\sum_in^\uparrow_in^\downarrow_i~,\qquad\text{with}\qquad n^\sigma_i = a^\dagger_{i\sigma}a_{i\sigma}~,
\label{hub_ham}$$ in which the sum $\langle 