subset of the extended clutser constraints, at a fraction of the computational cost.

\[intro\]Theoretical Framework
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\[intro\_2DM\]Introduction to v2DM
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The Hamiltonian of a system interacting in up to two-body terms is: $$\label{two_body_ham}
\hat{H}_{\alpha\beta;\gamma\delta} = \sum_{\alpha\beta}t_{\alpha\beta}a^\dagger_\alpha a_\beta + \frac{1}{4}\sum_{\alpha\beta\gamma\delta}V_{\alpha\beta;\gamma\delta}a^\dagger_\alpha a^\dagger_\beta a_\delta a_\gamma~,$$ so the expectation value of the energy corresponding to an arbitrary $N$-particle state ${|\Psi^N\rangle}$ can be expressed using but the 2DM of that state: $$\label{E_2DM}
E(\Gamma) = \mathrm{Tr}~\Gamma H^{(2)} = \frac{1}{4}\sum_{\alpha\beta\gamma\delta}\Gamma_{\alpha\beta;\gamma\delta}H^{(2)}_{\alpha\beta;\gamma\delta}~,$$ with the 2DM defined as: $$\Gamma_{\alpha\beta;\gamma\delta} = {\langle \Psi^N|}a^\dagger_\alpha a^\dagger_\beta a_\delta a_\gamma {|\Psi^N\rangle}~,
\label{2DM}$$ and the reduced two-body Hamiltonian: $$\label{red_ham}
H^{(2)} = \frac{1}{N-1}\left[\delta_{\alpha\gamma}t_{\beta\delta}-\delta_{\alpha\delta}t_{\beta\gamma} -\delta_{\beta\gamma}t_{\alpha\delta}+\delta_{\beta\delta}t_{\alpha\gamma}\right] + V_{\alpha\beta;\gamma\delta}~.$$ The key idea underpinning v2DM is to use Eq. (\[E\_2DM\]) to determine the 2DM variationally. Once the approximate ground-state 2DM is found, we can extract all one- and two-body ground-state properties. This approach eliminates the need to reference the exponentially scaling wave function, and requires only the much more compact 2DM. A complication arises because the variation has to be performed over a limited set of 2DM’s, [*i.e.*]{} those that are derivable from an ensemble of $N$-particle wave functions. This is known as the $N$-representability problem [@coleman]. Using the variational principle and the fact that the set of $N$-representable 2DM’s is 