Are natural orbitals useful for generating an efficient expansion of the wave function?

We investigate whether the natural orbitals (NOs) minimize $\lVert\Psi - \Phi\rVert^2$, where $\Psi$ is a wave function and $\Phi$ is a full configuration interaction (CI) approximation to $\Psi$ in a truncated basis. We will show that the NOs rarely provide the optimal orbitals for $\Phi$, except when (1) there are only two particles or (2) only one basis function is removed in the case of fermions. Further, we will show that the CI expansion coefficients of $\Psi$ and $\Phi$ are identical up to a global scaling factor and demonstrate how the NOs can be used to generate the orbitals that minimize $\lVert\Psi - \Phi\rVert^2$.