Single-particle wavefunction in Slater determinant

Question

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?

Answer 1

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.

Answer 2

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.