Direct determination of the quantum-mechanical density matrix: Parquet theory

The methods used to determine the reduced density matrix (RDM) of the ground and excited states, the finite-temperature systems, and the large systems without using the wave function by solving the density equation were discussed. We examined the foundations to reconstruct the higher-order RDMs of the ground and excited states and the finite-temperature systems in terms of the lower-order RDMs. We presented the equation to determine the RDMs of the finite-temperature systems directly and showed that only the exact RDMs satisfy the equation. Our previous approximation for third- and fourth-order RDMs of the ground state [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] was reformulated, and the accuracy of this approximation for the excited states was examined. The structure of the $n\mathrm{th}$ order energy density matrix $(n$-EDM) was analyzed, and the calculation method which sums up the Parquet diagram of the 2-EDM without explicitly constructing the third- and fourth-order RDMs was reported. This approximation is more accurate than the previous second-order approximation and also includes the infinite series of bubble and ladder Green’s function diagrams. Such a method is necessary to apply the density-equation method to large systems, such as polymers, metals, and semiconductors. The new approximation together with the density equation was applied to the ground states of some molecules including CO, ${\mathrm{C}}_2{\mathrm{H}}_2,$ ${\mathrm{C}}_3{\mathrm{H}}_8$, and ${\mathrm{C}}_4{\mathrm{H}}_{10},$ and the excited states of the Be atom and ${\mathrm{Li}}_2$ molecule. The calculated energies were as accurate as the exact or coupled-cluster single and double excitations with triples included noniteratively, and the energy errors of the second-order approximation were significantly reduced. The calculated 2-RDMs almost satisfied important representability conditions while the 1-RDMs were exactly ensemble representable. These results demonstrate that the density equation offers a new quantitative method for treating electron correlations. The relationship between the iterative procedure and the finite-temperature density-equation method was discussed.