TY - JOUR

T1 - Variational calculation of second-order reduced density matrices by strong N -representability conditions and an accurate semidefinite programming solver

AU - Nakata, Maho

AU - Braams, Bastiaan J.

AU - Fujisawa, Katsuki

AU - Fukuda, Mituhiro

AU - Percus, Jerome K.

AU - Yamashita, Makoto

AU - Zhao, Zhengji

N1 - Funding Information:
M.N. was supported by the Japan Science and Technology Agency (JST), and he and M.F. were also partially supported by the fellowships from the Japan Society for the Promotion of Science (JSPS). M.F. was supported by MEXT’s program “Promotion of Environmental Improvement for Independence of Young Researchers” under the Special Coordination Funds for Promoting Science and Technology, and in part by the JSPS Global COE program “Computationism as a Foundation for the Sciences.” M.N., M.F., and M.Y. were also partially supported by Grant-in-Aids for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology. B.J.B. was supported by U.S. Department of Energy under Grant No. DE-FG02-07ER5914 and Office of Naval Research under Grant No. N00014-05-1-0460. J.K.P. was supported by the Department of Energy (DOE) under Grant No. DE-FG02-02ER15292, and Z.Z. by DOE Grant No. DE-AC02-05CH11231. We are grateful to the Grid Technology Research Center (GTRC) in the National Institute of Advanced Industrial Science and Technology (AIST) for providing us the AIST Super Cluster for solving large-scale SDPs.

PY - 2008

Y1 - 2008

N2 - 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 T 2′ conditions. T 2′ 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 T 2′ 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.

AB - 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 T 2′ conditions. T 2′ 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 T 2′ 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.

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U2 - 10.1063/1.2911696

DO - 10.1063/1.2911696

M3 - Article

C2 - 18447427

AN - SCOPUS:42949093565

VL - 128

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 16

M1 - 164113

ER -