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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}
\hat{T} \phi_\beta(\mathbf{r}_i ) \ ,
\end{displaymath} (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.


Peter D. Haynes 2002-10-31