Active-Space $N$-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations

The ground-state energy of a system of fermions can be calculated by minimizing a linear functional of the two-particle reduced density matrix (2-RDM) if an accurate set of $N$-representability conditions is applied. In this Letter we introduce a class of linear $N$-representability conditions based on exact calculations on a reduced active space. Unlike wave-function-based approaches, the 2-RDM methodology allows us to combine information from calculations on different active spaces. By adding active-space constraints, we can iteratively improve our estimate for the ground-state energy. Applying our methodology to a 1D Hubbard model yields a significant improvement over traditional 2-positivity constraints with the same computational scaling.

Figure 1

The approximate energy versus the number of active-space constraints for a $L=6$ site Hubbard model with $U/t=20$. Initially, no additional constraints are added, yielding $E_{\mathrm{DGQ}}$. A significant fraction of the energy difference between the exact result and the DGQ answer can be recovered by imposing a small number of active-space constraints. The larger the active-space, the more accurate the final energy that is obtained.