Expectation value constraints for the $N$-representability problem

It has long been known that reduced density matrices (RDM) can be used to obtain the exact ground state energy of many-particle Hamiltonians by performing a constrained minimization over the set of all $N$-representable reduced density matrices. In this paper, we present a set of $N$-representability constraints for both bosons and fermions based on expectation values. These expectation value constraints are derived from a simple yet nontrivial recursive formula, and can easily be included in RDM-based variational schemes. The expectation value constraints are shown to provide additional $N$-representability information when compared to existing $N$-representability constraints.

Figure 1

A schematic representation of the $N$-representability problem. For $N>2$, the space of all positive semidefinite matrices is larger than the $N$-representable subspace ${}^2\mathcal{D}_N$. Unconstrained minimization of the quantity $\mathrm{Tr}\left[{}^{2}K{}^{2}D\right]$ over the space of all $2$-RDMs leads to energies well below the ground state energy. As additional constraints are added to the minimization, the energy obtained under these constraints may approach the true ground state energy provided the constraints are well chosen: the more accurate the constraints, the more accurate the estimation of the ground state.

Figure 2

(Color online) Comparison of the bosonic bounds with the calculated maximal values of $⟨k\mid {}^{2}D\mid k⟩$ for $L=2$ and three different geminals $\{\mid k⟩\}$. Crosses show the real maxima ${\lambda }_N(\mid k⟩)$ obtained through numerical maximization of Eq. (13), circles show the numerical upper bound ${\Lambda }_N(\mid k⟩)$ obtained from the full recurrence relation Eq. (17), and the solid line shows the analytic bound in Eqs. (47, 48). Clearly, both the recurrence bound and the analytic bound are nearly tight.

Figure 3

(Color online) Comparison of the bosonic bounds with the calculated maximal values of $⟨k\mid {}^{2}D\mid k⟩$ for $L=3$ and three different geminals $\{\mid k⟩\}$. Crosses show the real maxima ${\lambda }_N(\mid k⟩)$ obtained through numerical maximization of Eq. (13), circles show the numerical upper bound ${\Lambda }_N(\mid k⟩)$ obtained from the full recurrence relation Eq. (17), and the solid line shows the analytic bound in Eq. (47) and Eq. (48). Comparison with Fig. 2 shows that the accuracy of our bounds has minimal dependence on $L$.

Figure 4

(Color online) Comparison of the fermionic bounds with the calculated maximal values of $⟨k\mid {}^{2}D\mid k⟩$ for $L=8$ and three different geminals $\{\mid {\kappa }_1⟩,\mid {\kappa }_2⟩,\mid {\kappa }_3⟩\}$. Crosses show the real maxima ${\lambda }_N^{\prime }(\mid k⟩)$ obtained through numerical maximization of Eq. (14). Circles show the upper bound obtained numerically from the full recurrence relation Eq. (54). The solid line shows the analytic bound from corollary 2.1, Eq. (56) and Eq. (57). This grows roughly as $N$ and therefore does not provide a useful bound for any of the geminals studied. On the other hand, the recursive bound is fairly accurate for small $N$.

Figure 5

(Color online) A comparison of the relative error of our recursive bounds (theorem 1) with the $G$-constraint, Eq. (11), for bosons with $L=3$. Our recursive bound gives a more accurate bound on ${\lambda }_N(\mid k⟩)$ than the $G$ constraint, demonstrating that it provides additional $N$-representability information. The results shown here for a randomly selected geminal $\mid k⟩$ are typical; the recursive bound consistently outperforms the $G$ constraint, although the difference between the two grows smaller as $N$ increases.

Figure 6

(Color online) A comparison of the relative error of our recursive bounds (theorem 2) with the $G,Q$ constraints, Eqs. (11, 12), for fermions with $L=8$. Neither of the methods preserves the particle-hole symmetry which requires that ${\lambda }_N^{\prime }(\mid k⟩)={\lambda }_{L-N+2}^{\prime }(\mid k⟩)$ (see the Appendix ).