  The $N$-particle wavefunction has too many dimensions for a direct time
propagation of a many-body system according to the time-dependent Schr\"odinger
equation (TDSE). On the other hand, time-dependent density functional theory
(TDDFT) tells us that the single-particle density is, in principle, sufficient.
However, a practicable equation of motion (EOM) for the accurate time evolution
of the single-particle density is unknown. It is thus an obvious idea to
propagate a quantity which is not as reduced as the single-particle density but
less dimensional than the $N$-body wavefunction. Recently, we have introduced
time-dependent renormalized-natural-orbital theory (TDRNOT). TDRNOT is based on
the propagation of the eigenfunctions of the one-body reduced density matrix
(1-RDM), the so-called natural orbitals. In this paper we demonstrate how
TDRNOT is related to the multi-configurational time-dependent Hartree-Fock
(MCTDHF) approach. We also compare the performance of MCTDHF and TDRNOT vs the
TDSE for single-photon double ionization (SPDI) of a 1D helium model atom. SPDI
is one of the effects where TDDFT does not work in practice, especially if one
is interested in correlated photoelectron spectra, for which no explicit
density functional is known.

Computational Physics;Atomic Physics;Quantum Physics