Theory

For functions represented as values on a regular grid, integrals like the one of equation (3) can be calculated, or rather approximated to increasing accuracy, by a sum over grid points, as long as the value of the integrand is known at every grid point:

$$T_{\alpha \beta} \simeq -\frac{1}{2} w \sum_{\mathbf{r}_i} \phi^*_\alpha(\mathbf{r}_i) \hat{T} \phi_\beta(\mathbf{r}_i ) \ ,$$ (4)

where $\hat{T}$ is the Laplacian operator in the discrete representation, $w$ is the volume per grid point, and the sum formally goes over all the grid points in the simulation cell.