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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?

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  • $\begingroup$ It is not necessarily (or even typically) true that the ground state is always a Slater determinant. There are other antisymmetric configurations that may sometimes have lower energy expectations. $\endgroup$
    – Buzz
    Commented May 20, 2022 at 0:58

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The basis functions are usually assumed to be orthogonal to avoid unnecessary complications. The set can be any set but in practice the physics is in picking a “good” set, where the determinant will involve only a few functions.

In the case of a ground state, for instance, you would want to pick as part of your set the lowest $N/2$ energy states of some relevant non-interacting Hamiltonian, and maybe a few states above the $N/2$th, especially if some states are closely spaced near this energy. If you choose a poor set, you will need lots of determinants to get reasonable results.

Unfortunately the choice above is not always a “convenient” choice to compute matrix elements of the residual interaction: not all functions integrate “easily” and always accurately on a computer. Thus some often prefer to work with - say - a set of Gaussian states since the overall computational cost of integration over Gaussian states makes up for the larger set of functions required.

Thus it’s a bit of an art and several different “quasi-canonical” sets have been developed (especially for application to DFT). Some are better than others at predicting this or that aspect of molecules, for instance.

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  • $\begingroup$ Thank you for your answer. But would then $\phi_{\mu} = e^{ik_i r_i}$ be also a valid choice for single particle wave-function? $\endgroup$
    – relaxon
    Commented Feb 20, 2022 at 16:07
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    $\begingroup$ It's a terrible choice because it is an eigenstate of a free particle (not a bound state), it cannot be normalized in the standard way and the state $\phi_\mu$ does not satisfy any reasonable boundary conditions for your system. I'm not an expert at using "bad" choices: maybe it can be done but it would certainly be an overly laborious choice in practice. $\endgroup$ Commented Feb 20, 2022 at 16:12
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Actually you want to choose plane waves as single particle orbitals when you want to describe uniform bulk matter. The various sums in the Hartre Fock energy then becomes integrals over momentum weighted by the Fermi numebr distribution, which allows you to include both temperature and chemical potential in the calculation. Off course for finite systems like atoms etc. you want to choose normalizable functions.

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