The impossibility of expanding the square root of the electron density as a linear combination of elements of a complete set of basis functions

In orbital-free density functional theory (OFDFT), an equation exists for $\psi =\sqrt{n}$, the square root of the ground state electron density $n$. We show that $\psi$ cannot be expanded as a linear combination of elements of a complete set of basis functions except in the case of one or two electron systems. This is unlike the case for the ground state of a system of identical bosons in which the square root of the ground state bosonic density can have an expansion as a linear combination of elements of a complete set of basis functions.