  We studied one-dimensional systems formed by $N$ identical particles confined
in a harmonic trap and subject to an inverse power law interaction potential
$\sim|x|^{-d}$. The correlation properties of a Wigner molecule with the lowest
energy are investigated in terms of their dependence on the number $N$ and the
power $d$, including the limit as $d\rightarrow 0$. There are $N$-particle
Wigner molecules with properties such that their correlations are mainly
manifested in the $N$ lowest natural orbitals. The values of the control
parameters of the system at which such states appear are identified. The
properties of Wigner molecules formed in the limit as $d$ approaches infinity
are also revealed.
