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Using a black-box optimization software to determine the second-order reduced density matrices of atoms and molecules by N-representability conditions

Mituhiro Fukuda

Published Sep 1, 2008• 0 citations
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The ground-state fermion second-order reduced density matrix (2-RDM) is determined variationally using itself as a basic variable. As necessary conditions of the N-representability, we used the positive semidefiniteness conditions, P, Q, and G conditions that are described in terms of the 2-RDM. The variational calculations are performed by using recently developed semidefinite programming algorithm (SDPA). The calculated energies of various closed- and open-shell atoms and molecules are excellent, overshooting only slightly the full-CI energies. There was no case where convergence was not achieved. The calculated properties also reproduce well the full-CI results.

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Direct Determination of the Quantum-Mechanical Density Matrix Using the Density Equation

Hiroshi Nakatsuji, Koji Yasuda

With the use of the density equation [H. Nakatsuji, Phys. Rev. A 14, 41 (1976)], the second-order density matrices are directly determined without any use of the wave functions. The third- and fourth-order reduced density matrices (RDM's) are decoupled into lower-order ones using the Green's function technique. This method is applied to Be, Ne, ${\mathrm{H}}_{2}$O, ${H}_{3}{O}^{+}$, N${\mathrm{H}}_{3}$, C${\mathrm{H}}_{4}$, ${\mathrm{BH}}_{4}^{\phantom{\rule{0ex}{0ex}}\ensuremath{-}}$, ${\mathrm{NH}}_{4}^{\phantom{\rule{0ex}{0ex}}+}$, and C${\mathrm{H}}_{3}$F, and the results are successfully compared with the full configuration interaction results. The convergence is fairly good, and the calculated second-order RDM's almost satisfy the necessary conditions of the N representability, the P, Q, and G conditions, and the first-order RDM's are exactly N representable. These results show that the present method is very promising.

Physical Review Letters•Feb 12, 1996•268 citations

VARIATIONAL APPROACH TO ELECTRONIC STRUCTURE CALCULATIONS ON SECOND-ORDER REDUCED DENSITY MATRICES AND THE N-REPRESENTABILITY PROBLEM

Maho Nakata, Mituhiro Fukuda, Katsuki Fujisawa

Lecture notes series, Institute For Mathematical Sciences•Jun 19, 2013•2 citations
The ground-state fermion second-order reduced density matrix (2-RDM) is determined variationally using itself as a basic variable. As necessary conditions of the N-representability, we used the positive semidefiniteness conditions, P, Q, and G conditions that are described in terms of the 2-RDM. The variational calculations are performed by using recently developed semidefinite programming algorithm (SDPA). The calculated energies of various closed- and open-shell atoms and molecules are excellent, overshooting only slightly the full-CI energies. There was no case where convergence was not achieved. The calculated properties also reproduce well the full-CI results.
With the use of the density equation [H. Nakatsuji, Phys. Rev. A 14, 41 (1976)], the second-order density matrices are directly determined without any use of the wave functions. The third- and fourth-order reduced density matrices (RDM's) are decoupled into lower-order ones using the Green's function technique. This method is applied to Be, Ne, H2{\mathrm{H}}_{2}H2​O, H3O+{H}_{3}{O}^{+}H3​O+, NH3{\mathrm{H}}_{3}H3​, CH4{\mathrm{H}}_{4}H4​, ${\mathrm{BH}}_{4}^{\phantom{\rule{0ex}{0ex}}\ensuremath{-}}$, NH4+{\mathrm{NH}}_{4}^{\phantom{\rule{0ex}{0ex}}+}NH4+​, and CH3{\mathrm{H}}_{3}H3​F, and the results are successfully compared with the full configuration interaction results. The convergence is fairly good, and the calculated second-order RDM's almost satisfy the necessary conditions of the N representability, the P, Q, and G conditions, and the first-order RDM's are exactly N representable. These results show that the present method is very promising.