I want to calculate the coherent intermediate scattering function, $S_{\text{coh}} (\mathbf{Q},t)$, from a molecular dynamics trajectory, based on its definition. The definition of the function is:

$$

S_{\text{coh}} (\mathbf{Q},t) = \frac{1}{N} \langle \sum_{j=1}^{N} \sum_{k=1}^{N} exp\left( i \mathbf{Q} \cdot \left[ \mathbf{r}_k (t) – \mathbf{r}_j (0) \right] \right) \rangle \nonumber = \\

= \frac{1}{N} \langle \sum_{j=1}^{N} \sum_{k=1}^{N}cos\left( \mathbf{Q} \cdot \left[ \mathbf{r}_k (t) – \mathbf{r}_j (0) \right] \right) + i \sum_{j=1}^{N} \sum_{k=1}^{N}sin\left( \mathbf{Q} \cdot \left[ \mathbf{r}_k (t) – \mathbf{r}_j (0) \right] \right)\rangle \nonumber \\

$$

where $N$ is the number of particles in the system, $\mathbf{Q}$, is a wave-vector, and

$\mathbf{r}_k (t)$ is the position of the $k^{th}$ particle at time $t$.

## When people plot $S_{\text{coh}} (\mathbf{Q},t)$ versus time, what is actually being plotted? Do they plot the complex modulus or just the real part?

$$

|S_{\text{coh}} (\mathbf{Q},t)| = \sqrt{Re^2 + Im^2}

$$

$Re$ and $Im$ are the real and imaginary parts of $S_{\text{coh}} (\mathbf{Q},t)$, respectively.

PS: The imagenary part of the coherent static structure factor $S_{\text{coh}} (\mathbf{Q},0)$ is zero by definition, but I have yet to find a reason that this should also apply for $S_{\text{coh}} (\mathbf{Q},t)$.

**Thank you…**

## Best Answer

The sum of sine terms is mathematically zero if you note that taking the ensemble average <...> requires random sampling of the directions of vector Q. Ususally true for liquids, this isotropic sampling over different solid angles causes each sine term to vanish.