Mismatch between density- and energy-optimal one-electron reduced density matrices in finite basis sets

In principle, the exact ground-state one-electron reduced density matrix (1-RDM) gives both the exact electron density and ground-state energy. In a finite basis set, however, an approximate 1-RDM closely reproducing the exact density can give a surprisingly poor energy. We investigate one cause of this effect using the hydrogen atom, where functional and reference-density errors are absent. We find that density fitting over ensemble N-representable 1-RDMs produces weakly occupied natural orbitals with spurious oscillations that make anomalously large contributions to the total energy. Restricting the density optimization to pure states removes these high-energy contributions and yields total energies closer to the energy optimum. We also identify "density-energy consistent" basis sets for the H atom in which the ensemble density optimum is essentially pure, but these basis sets are neither transferable nor stable under augmentation. Although the pure-state restriction may not provide a general remedy for the density-energy mismatch, the energetic cost of weakly occupied, oscillatory natural orbitals could extend beyond one-electron systems.