  Electron correlations in molecules can be divided in short range dynamical
correlations, long range Van der Waals type interactions and near degeneracy
static correlations. In this work we analyze for a one-dimensional model of a
two-electron system how these three types of correlations can be incorporated
in a simple wave function of restricted functional form consisting of an
orbital product multiplied by a single correlation function $f(r_{12})$
depending on the interelectronic distance $r_{12}$. Since the three types of
correlations mentioned lead to different signatures in terms of the natural
orbital (NO) amplitudes in two-electron systems we make an analysis of the wave
function in terms of the NO amplitudes for a model system of a diatomic
molecule. In our numerical implementation we fully optimize the orbitals and
the correlation function on a spatial grid without restrictions on their
functional form. Due to this particular form of the wave function, we can prove
that none of the amplitudes vanishes and moreover that it displays a distinct
sign pattern and a series of avoided crossings as a function of the bond
distance in agreement with the exact solution. This shows that the wave
function Ansatz correctly incorporates the long range Van der Waals
interactions. We further show that the approximate wave function gives an
excellent binding curve and is able to describe static correlations. We show
that in order to do this the correlation function $f(r_{12})$ needs to diverge
for large $r_{12}$ at large internuclear distances while for shorter bond
distances it increases as a function of $r_{12}$ to a maximum value after which
it decays exponentially. We further give a physical interpretation of this
behavior.
