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

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.