When using periodic boundary conditions, the discrete-momentum single-particle basis functions $$ \phi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot \mathbf{r}}/L^{d/2} $$ are associated with the single-particle energy $$ \begin{align} \varepsilon_{n_{x}, n_{y}} = \frac{\hbar^{2}}{2m} \left( \frac{2\pi }{L}\right)^{2}\left( n_{x}^{2} + n_{y}^{2}\right) \tag{5} \end{align} $$ for two-dimensional sytems and $$ \begin{align} \varepsilon_{n_{x}, n_{y}, n_{z}} = \frac{\hbar^{2}}{2m} \left( \frac{2\pi }{L}\right)^{2} \left( n_{x}^{2} + n_{y}^{2} + n_{z}^{2}\right) \tag{6} \end{align} $$ for three-dimensional systems.