Fourier tranform of infinite sum of Gaussians

Question

Suppose we have a 3d lattice of unit cells. In each unit cell we have a number of Gaussians $G(\boldsymbol{r}-\boldsymbol{R}_m-\boldsymbol{R}_n)$ sitting at a specific site $m$ in the unit cell with the lattice vector $ \boldsymbol{R}_n$. Now we want to calculate the Fourier transform of the sum of all these Gaussians. Suppose the sum goes to infinity, the sum should yield a 100% periodic function, so I thought this should not be a problem:

$\mathcal{F}\left(\sum\limits_{n=0}^\infty\underbrace{\sum\limits_{m=0}^{N_\mathrm{sites}} G(\boldsymbol{r}-\boldsymbol{R}_m-\boldsymbol{R}_n)}_{f(\boldsymbol{r}-\boldsymbol{R}_n)}\right)$

I thought about using the Poisson summation formula, but this gives me back only again a sum over Gaussians. I thought that maybe in reciprocal space we can get rid of the sum over unit cells by introducing a properly periodic function...

I add to this question that actually $\int_{\Omega} d\boldsymbol{r} f(\boldsymbol{r}) = 0$ for $\Omega$ being the unit cell volume. This is required to make the sum converge.

Answer 1

Just found the answer searching through the web (https://pavpanchekha.com/blog/gaussian-sine.html):

$\mathcal{F}\left[\sum\limits_{n=0}^{\infty} f(\boldsymbol{r}-\boldsymbol{R}_n\right] = \mathcal{F}\left[\int d\boldsymbol{r} \left[\sum\limits_{n=0}^\infty \delta(\boldsymbol{r}-\boldsymbol{R}_n)\right] f(\boldsymbol{r})\right] = \mathcal{F}[ f(\boldsymbol{r})*W(\boldsymbol{r})] = \mathcal{F}[f(\boldsymbol{r})]\mathcal{F}[W(\boldsymbol{r})]$

with $W$ being the Dirac comb. Both Fourier transforms give directly the same function back, so we have:

$\mathcal{F}[f(\boldsymbol{r})]\mathcal{F}[W(\boldsymbol{r})] = f(\boldsymbol{k}) W(\boldsymbol{k})$

which is a sum over delta-function scaled by the Fourier transform of function $f$.

$W(\boldsymbol{k}) = \sum\limits_{k=0}^\infty \delta(\boldsymbol{k}-\boldsymbol{k}_n)$

where $\boldsymbol{k}_n$ defines the periodicity of the lattice in reciprocal space. Putting it all together we can express the sum over unit cells by a Fourier sum using delta-functions:

$\sum\limits_{n=0}^\infty f(\boldsymbol{r}-\boldsymbol{R}_n) = \int d\boldsymbol{k} f(\boldsymbol{k}) W(\boldsymbol{k}) e^{i \boldsymbol{k}\boldsymbol{r}}$

The remaining question now is, can we truncate this sum in reciprocal space easier than in real space? In real space this converges badly which is why one uses the Ewald summation...