In recent years, the improvements in computer hardware and software have allowed the simulation of molecules with an increasing number of atoms. Unfortunately, the most accurate electronic structure methods based on N-particle wavefunctions (WFN) remain computationally too expensive to be applied to large systems. The most efficient method is undoubtedly the density functional theory (DFT). However, current implementations of DFT suffer from several problems, for instance, in the description of multi-reference systems. An alternative to both DFT and WFN methods lies in the development of a functional theory based on the one-particle reduced density matrix in its spectral expansion, known as natural orbital functional (NOF) theory. Several functionals of this type have been proposed, for which validation is necessary. A well-known tool for calibration, testing, and benchmarking of an approximate electronic structure method is the harmonium atom.
In the harmonium atom, the electron-nucleus potential is replaced by a harmonic confinement, but the electron-electron Coulomb interaction remains. By varying the strength of the harmonic potential, the correlation regime of this system can be tuned, making possible the transition from the weakly to the strongly correlated regime. Accordingly, the harmonium stands as an adequate system for studying the behavior of approximate NOFs, since it is possible to contrast them with their exact counterparts obtained from the analytic solution. In this presentation, the comparison between the quasi-exact and approximate electron-electron repulsion energy provided by eight known NOFs, in the singlet state of the four-electron harmonium atom with varying confinements, is analyzed in some detail. The present approach, which will appear soon in the Journal of Chemical Physics 143, not only reveals the failures of the functionals but also pinpoints the causes. In general, the functional PNOF6 shows the most consistent behavior, with decent accuracy, along all confinement regimes studied.