Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary
N-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wave function. Here we introduce a route to enhancing necessary conditions for
N representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive
N-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wave functions, including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N
2 improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and nonequilibrium geometries. While more accurate, the rank-restricted
N-representability conditions are less expensive computationally than the full-rank conditions.
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an
N-particle wave function by two sets of
N-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by
N-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.
Variationally calculating the ground state of a many-electron quantum system using only the two-electron reduced-density-matrix (2-RDM) requires
N-representability conditions that constrain the 2-RDM to correspond to an
N-electron wave function. A systematic hierarchy of
N-representability conditions, known as
p-positivity conditions, has been developed [D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001)], and many-electron atoms and molecules in nonminimal basis sets have been solved with useful accuracy by a variational 2-RDM method with 2-positivity conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. This paper considers two forms of partial 3-positivity conditions, the lifting conditions and the
T1∕T2 conditions, to further enhance the accuracy of the 2-RDM methods without the computational cost of full 3-positivity conditions. Variational 2-RDM methods with different
N-representability constraints including 2-positivity conditions, the two types of partial 3-positivity conditions, as well as the complete 3-positivity conditions are applied to compute the ground state of the Lipkin spin model. The energies and 2-RDMs are compared to the results from full and truncated configuration interaction, many-body perturbation theory, and couple cluster theory with single and double excitations. Implications of using partial 3-positivity for variational 2-RDM calculations of many-electron atoms and molecules will be discussed.
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure
N-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure
N-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure
N-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble
N-representability conditions. Constraints on higher-order
p-RDMs where
p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.
Atomic and molecular ground-state energies are variationally determined by constraining the two-particle reduced density matrix (2-RDM) to satisfy positivity conditions. Because each positivity condition corresponds to correcting the ground-state energies for a class of Hamiltonians with two-particle interactions, these conditions collectively provide a new approach to many-body theory that, unlike perturbation theory, can capture significantly correlated phenomena including the multireference effects of potential-energy surfaces. The D, Q, and G conditions for the 2-RDM are extended through generalized lifting operators inspired from the formal solution of N-representability. These lifted conditions agree with the hierarchy of positivity conditions presented by Mazziotti and Erdahl [Phys. Rev. A 63, 042113 (2001)]. The connection between positivity and the formal solution explains how constraining higher RDMs to be positive semidefinite improves the N representability of the 2-RDM and suggests using pieces of higher positivity conditions that computationally scale like the D condition. With the D, Q, and G conditions as well as pieces of higher positivity the electronic energies for Be, LiH,
H2O, and BH are computed through a primal-dual interior-point algorithm for positive semidefinite programming. The variational method produces potential-energy surfaces that are highly accurate even far from the equilibrium geometry where single-reference perturbation-based methods often fail to produce realistic energies.
Ground-state energies and two-particle reduced density matrices (2-RDMs) corresponding to N-particle systems are computed variationally within the doubly occupied configuration interaction (DOCI) space by constraining the 2-RDM to satisfy a complete set of three-particle N-representability conditions known as three-positivity conditions. These conditions are derived and implemented in the variational calculation of the 2-RDM with standard semidefinite programming algorithms. Ground state energies and 2-RDMs are computed for N2, CO, CN−, and NO+ molecules at both equilibrium and nonequilibrium geometries as well as for pairing models at different repulsive interaction strengths. The results from the full three-positivity conditions are compared with those from the exact DOCI method and with approximated 2-RDM variational ones obtained within two-positivity and two-positivity plus a subset of three-positivity conditions, as recently reported [D. R. Alcoba et al., J. Chem. Phys. 148, 024105 (2018) and A. Rubio-García et al., J. Chem. Theory Comput. 14, 4183 (2018)]. The accuracy of these numerical determinations and their low computational cost demonstrate the usefulness of the three-particle variational constraints within the DOCI framework.