  In singlet two-electron systems the natural occupation numbers of the
one-particle reduced density matrix are given as squares of the natural
amplitudes which are defined as the expansion coefficients of the two-electron
wave function in a natural orbital basis. In this work we relate the sign of
the natural amplitudes to the nature of the two-body interaction. We show that
long-range Coulomb-type interactions are responsible for the appearance of
positive amplitudes and give both analytical and numerical examples that
illustrate how the long-distance structure of the wave function affects these
amplitudes. We further demonstrate that the amplitudes show an avoided crossing
behavior as function of a parameter in the Hamiltonian and use this feature to
show that these amplitudes never become zero, except for special interactions
in which infinitely many of them can become zero simultaneously when changing
the interaction strength. This mechanism of avoided crossings provides an
alternative argument for the non-vanishing of the natural occupation numbers in
Coulomb systems.
