Variational calculation of second-order reduced density matrices by strong N-representability conditions and an accurate semidefinite programming solver

Abstract

The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition. Using these N-representability conditions, we can usually calculate correlation energies in percentage ranging from 100% to 101%, whose accuracy is similar to CCSD(T) and even better for high spin states or anion systems where CCSD(T) fails. Highly accurate calculations are carried out by handling equality constraints and/or developing multiple precision arithmetic in the semidefinite programming (SDP) solver. Results show that handling equality constraints correctly improves the accuracy from 0.1 to 0.6 mhartree. Additionally, improvements by replacing T2 condition with T2(') condition are typically of 0.1-0.5 mhartree. The newly developed multiple precision arithmetic version of SDP solver calculates extraordinary accurate energies for the one dimensional Hubbard model and Be atom. It gives at least 16 significant digits for energies, where double precision calculations gives only two to eight digits. It also provides physically meaningful results for the Hubbard model in the high correlation limit.

The second-order reduced density matrix method and the two-dimensional Hubbard model

“…This model is interesting not only because of its simplicity, but also its capability of describing strong electron correlation. The RDM method has been applied to the Hubbard model by Hammond et al [12], Nakata et al [9], and Verstichel et al [13]. Their results very accurately described total energies as well as other properties.…”

“…Fortunately, several necessary conditions are known. Using only the P , Q, G, T 1, and T 2 ′ (necessary) conditions have been shown to reliably obtain chemical accuracy [7,9].…”

“…They used the P , Q [2],and G conditions [6] as the N-representability constraints that resulted in 120% of correlation energies. These promising results led Zhao et al three years later to include the T 1, and the T 2-conditions in addition to the P , Q, and G conditions in the RDM method giving results with similar accuracy to coupled-cluster singles and doubles with perturbative triples (CCSD(T)) for atomic and molecular systems [7][8][9]. Since then, research along these lines has spread with enthusiasm and several papers have been published [10].…”

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

“…In fact, (23) is equivalent to (22). If we assume that strict inequality holds for some inequality constraint (w * ji ) 2 − v ji (z * ) ≥ 0 at the optimal solution (z * , w * ) of (23), it contradicts the fact that (z * , w * ) is the optimal solution.…”

“…The elapsed time for the computation is listed in "time" of the Tables 3 and 6. In the experiments whose results are summarized in Table 5, we used SDPA-GMP [22,30]. SDPA-GMP employs a multiple precision library and can carry out similar algorithms as SeDuMi does, but with arbitrary arithmetic precision.…”