Posted on February 18, 2017 at 1:10 PM |

For a system of fermions subject to one and two-particle forces the exact energy can be completely expressed in terms of the second-order reduced density matrix (2-RDM). Many authors have attempted to calculate the ground-state energy from the 2-RDM because it is a much simpler object than the electronic wavefunction. The use of the variational method to calculate the energy of a system involves the modification of the 2-RDM subject to the N-representability conditions. Among them, the contracted Schrödinger equation (CSE) and the antiHermitian counterpart (ACSE) have rekindled the interest in methods without wavefunctions. Both CSE and ACSE energy expressions depend on the third-order reduced density (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 two recent works (1 and 2) we had put forward two new approximations to the diagonal of the 3-RDM that were used to calculate 3c-indices in a series of molecules. Our approximations were compared against the Valdemoro, Nakatsuji and Mazziotti approximations, showing that one of our proposals was clearly superior to the others for the calculation of 3c-indices.

Now, in a paper recently published in *Phys. Chem. Chem. Phys.,* we introduce a series of tests (see the graphic below) 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 our work put forward several limitations of the currently most used 3-RDM approximations for systems with important electron correlation effects.

Our results show that the errors of the 3-RDM approximations increase as the inverse of the confinement strength (the parameter that regulates the electron correlation effects in harmonium). All approximations fail to satisfy several N-representability conditions and show significant deviations from the trace numbers upon inclusion of electron correlation. Surprisingly, Mazziotti's 3-RDM performs remarkably bad for the doublet state and very well for the quartet state. Valdemoro's approximation shows the most promising results but provides the largest termwise errors. In the paper we give a hint to improve the performance of 3-RDM approximations.

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