| \( n_{x}^{2}+n_{y}^{2}+n_{z}^{2} \) | \( n_{x} \) | \( n_{y} \) | \( n_{z} \) | \( N_{\uparrow \downarrow } \) |
| 0 | 0 | 0 | 0 | 2 |
| 1 | -1 | 0 | 0 | |
| 1 | 1 | 0 | 0 | |
| 1 | 0 | -1 | 0 | |
| 1 | 0 | 1 | 0 | |
| 1 | 0 | 0 | -1 | |
| 1 | 0 | 0 | 1 | 14 |
| 2 | -1 | -1 | 0 | |
| 2 | -1 | 1 | 0 | |
| 2 | 1 | -1 | 0 | |
| 2 | 1 | 1 | 0 | |
| 2 | -1 | 0 | -1 | |
| 2 | -1 | 0 | 1 | |
| 2 | 1 | 0 | -1 | |
| 2 | 1 | 0 | 1 | |
| 2 | 0 | -1 | -1 | |
| 2 | 0 | -1 | 1 | |
| 2 | 0 | 1 | -1 | |
| 2 | 0 | 1 | 1 | 38 |
| 3 | -1 | -1 | -1 | |
| 3 | -1 | -1 | 1 | |
| 3 | -1 | 1 | -1 | |
| 3 | -1 | 1 | 1 | |
| 3 | 1 | -1 | -1 | |
| 3 | 1 | -1 | 1 | |
| 3 | 1 | 1 | -1 | |
| 3 | 1 | 1 | 1 | 54 |
Continuing in this way we get for \( n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=4 \) a total of 12 additional states, resulting in \( ? \) as a new magic number. We can continue like this by adding more shells.
When performing calculations based on many-body perturbation theory, Coupled cluster theory or other many-body methods, we need then to add states above the Fermi level in order to sum over single-particle states which are not occupied.