  The correct description of nondynamic correlation by electronic structure
methods not belonging to the multireference family is a challenging issue. The
transition of $D_{2h}$ to $D_{4h}$ symmetry in H$_4$ molecule is among the most
simple archetypal examples to illustrate the consequences of missing nondynamic
correlation effects. The resurge of interest in density matrix functional
methods has brought several new methods including the family of Piris Natural
Orbital Functionals (PNOF). In this work we compare PNOF5 and PNOF6, which
include nondynamic electron correlation effects to some extent, with other
standard ab initio methods in the H$_4$ $D_{4h}/D_{2h}$ potential energy
surface. Thus far, the wrongful behavior of single-reference methods at the
$D_{2h}-D_{4h}$ transition of H$_4$ has been attributed to wrong account of
nondynamic correlation effects, whereas in geminal-based approaches it has been
assigned to a wrong coupling of spins and the localized nature of the orbitals.
We will show that actually $\textit{interpair}$ nondynamic correlation is the
key to a cusp-free qualitatively correct description of H$_4$ PES. By
introducing $\textit{interpair}$ nondynamic correlation, PNOF6 is shown to
avoid cusps and provide the correct smooth PES features at distances close to
equilibrium, total and local spin properties along with the correct electron
delocalization, as reflected by natural orbitals and multicenter delocalization
indices.
