The electronic energy of a system of fermions can be obtained from the second-order reduced density matrix through the contracted Schrödinger equation or its anti-Hermitian counterpart. Both energy expressions depend on the third-order reduced density matrix (3-RDM) which is usually approximated from lower-order densities. The accuracy of these methods depends critically on the set of N-representability conditions enforced in the calculation and the quality of the approximate 3-RDM. There are no benchmark studies including most 3-RDM approximations and, thus far, no assessment of the deterioration of the approximations with correlation effects has been performed. In this paper we introduce a series of tests to assess the performance of 3-RDM approximations in a model system with varying electron correlation effects, the three-electron harmonium atom. The results of this work put forward several limitations of the currently most used 3-RDM approximations for systems with important electron correlation effects.
The electronic energy of a system of fermions can be obtained from the second-order reduced density matrix through the contracted Schrödinger equation or its anti-Hermitian counterpart. Both energy expressions depend on the third-order reduced density matrix (3-RDM) which is usually approximated from lower-order densities. The accuracy of these methods depends critically on the set of N-representability conditions enforced in the calculation and the quality of the approximate 3-RDM. There are no benchmark studies including most 3-RDM approximations and, thus far, no assessment of the deterioration of the approximations with correlation effects has been performed. In this paper we introduce a series of tests to assess the performance of 3-RDM approximations in a model system with varying electron correlation effects, the three-electron harmonium atom. The results of this work put forward several limitations of the currently most used 3-RDM approximations for systems with important electron correlation effects.
Two-electron reduced density matrices (2-RDMs) of many-electron molecules are directly determined without calculation of their wave functions by solving the anti-Hermitian contracted Schrödinger equation. Approximation of the 3-RDM in the anti-Hermitian contracted Schrödinger equation by a corrected cumulant expansion [Mazziotti, Phys. Rev. A 60, 3618 (1999)] permits the direct calculation of the energy and 2-RDM with many high-order correlation effects included. The method is illustrated for the molecules BeH2, H2O, NH3, CH4, and CO as well as the dissociation of BH. Correlation energies are obtained within 95%-100% of full-configuration interaction, and 2-RDMs very nearly satisfy known N-representability conditions.
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 nth 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, C2H2,C3H8, and C4H10, and the excited states of the Be atom and Li2 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.
Correlation energies and reduced density matrices (RDMs) of atoms and molecules are directly computed by solving the 1,3-contracted Schr\"odinger equation (1,3-CSE). The solution of the 1,3-CSE synthesizes two optimization strategies recently employed for the direct determination of the 2-RDM: (i) variational minimization of the energy with respect to a 2-RDM constrained by positivity conditions [D. A. Mazziotti, Phys. Rev. A 65, 062511 (2002)] and (ii) the contracted power method for solving the 2,4-CSE [D. A. Mazziotti, J. Chem. Phys. 116, 1239 (2002)]. While both the 3- and the 4-RDMs in the 2,4-CSE are reconstructed from the 2-RDM by cumulant expansions, similar techniques cannot be directly applied to the 1,3-CSE because constructing the 2-RDM from the 1-RDM with cumulant theory does not improve upon the mean-field approximation. We, however, establish a unique mapping from the 1-RDM to the 2-RDM by searching for the 2-RDM, constrained by contraction and N-representability conditions, which minimizes the energy. The 2-RDM constrained search is practically implemented through recent advances in positive semidefinite programming. With the variational reconstruction of the 2-RDM and a cumulant reconstruction of the 3-RDM, the 1,3-CSE may be solved via a contracted power method for the ground-state energy and RDMs. The initial RDMs, it is shown, need not be N representable for the contracted power method to converge; this allows us to choose the original RDMs from a variational calculation with approximate N-representability conditions on the 2-RDM. Application of the 1,3-CSE algorithm to atoms and molecules yields highly accurate correlation energies both near and far from equilibrium geometries.
The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.