The ground state of $N$ non-interacting fermions can be written using a Slater determinant as:

$$ \Phi\_{GS}(\textbf{r}\_{1}, ..., \textbf{r}\_{N}) = \frac{1}{\sqrt{N!}} 
\begin{vmatrix}
\phi\_{\mu\_{1}}(\textbf{r}\_{1}) & \cdots & \phi\_{\mu\_{N}}(\textbf{r}\_{1})\\ 
\vdots & \ddots & \vdots \\ 
\phi\_{\mu\_{1}}(\textbf{r}\_{N}) & \cdots & \phi\_{\mu\_{N}}(\textbf{r}\_{N})
\end{vmatrix}, $$

where $\phi\_{\mu} (\textbf{r}\_i)$ corresponds to single-particle wave functions, with $\mu$ orbital and $\textbf{r}\_{i}$ coordinate of the $i$th particle.

What (basis) functions can you choose to describe $\phi\_{k}$ ? Do they have to be orthogonal? For example, would $\phi\_{\mu} = e^{ik\_i r\_i}$ , $k\_i$ being the coordinate of the $i$th particle in momentum space, be a good choice?