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Theory and implementation

The MSD can be related to the incoherent intermediate scattering function via the cumulant expansion [49,50]
\begin{displaymath}
{\cal F}^{g}_{\mathrm{inc}}(\textbf{q},t) = \sum^{N_{species...
...a_{I,\mathrm{inc}} {\cal F}^{g}_{I,\mathrm{inc}}(\textbf{q},t)
\end{displaymath} (4.167)

where $N_{species}$ is the number of selected species, $n_I$ the number of atoms of species I, $\omega_{I,\mathrm{inc}}$ the weight for specie I (see Section 4.2.1 for more details) and
\begin{displaymath}
{\cal F}^{g}_{I,\mathrm{inc}}(\textbf{q},t) = \frac{1}{n_I}\...
...exp[-q^2\rho_{\alpha,1}(t) + q^4\rho_{\alpha,2}(t) \mp\ldots].
\end{displaymath} (4.168)

The cumulants $\rho_{\alpha,k}(t)$ are defined as
$\displaystyle \rho_{\alpha,1}(t)$ $\textstyle =$ $\displaystyle \frac{1}{2!}
\langle d^2_\alpha(t;\textbf{n}_q) \rangle$ (4.169)
$\displaystyle \rho_{\alpha,2}(t)$ $\textstyle =$ $\displaystyle \frac{1}{4!}\left[
\langle d_\alpha^4(t;\textbf{n}_q)\rangle - 3\langle d^2_\alpha(t;\textbf{n}_q)
\rangle^2\right]$ (4.170)
  $\textstyle \vdots$    

The vector $\textbf{n}_q$ is the unit vector in the direction of ${\bf q}$. In the Gaussian approximation the above expansion is truncated after the $q^2$-term. For certain model systems like the ideal gas, the harmonic oscillator, and a particle undergoing Einstein diffusion, this is exact. For these systems the incoherent intermediate scattering function is completely determined by the MSD.

nMOLDYN allows one to compute the total and partial incoherent intermediate scattering function in the Gaussian approximation by discretizing equation 4.167:

\begin{displaymath}
{\cal F}^{g}_{\mathrm{inc}}(q_m,k\cdot\Delta t) \doteq \sum^...
...t\Delta t),
\qquad k = 0\ldots N_t - 1,\; m = 0\ldots N_q - 1.
\end{displaymath} (4.171)

with for each specie the following expression for the intermediate scattering function:
$\displaystyle F^{g}_{I,\alpha ,\mathrm{inc}}(q_m,k\cdot\Delta t)$ $\textstyle =$ $\displaystyle \frac{1}{n_I}\sum^{n_I}_{\alpha} \exp\left[-\frac{(q_m)^2}{6}
\Delta^2_\alpha(k\cdot\Delta t)\right]
\qquad\mbox{{\rm isotropic system}},$ (4.172)
$\displaystyle F^{g}_{I,\alpha ,\mathrm{inc}}(q_m,k\cdot\Delta t)$ $\textstyle =$ $\displaystyle \frac{1}{n_I}\sum^{n_I}_{\alpha} \exp\left[-\frac{(q_m)^2}{2}
\De...
...2_\alpha(k\cdot\Delta t;{\bf n})\right]
\quad\mbox{{\rm non-isotropic system}}.$ (4.173)

$N_t$ is the total number of time steps in the coordinate time series and $N_q$ is a user-defined number of q-shells. The $(q,t)$-grid is the same as for the calculation of the intermediate incoherent scatering function (see Section 4.2.5.4). The quantities $\Delta^2_\alpha(t)$ and $\Delta^2_\alpha(t;{\bf n})$ are the mean-square displacements, defined in Equations (4.16) and (4.17), respectively. They are computed by using the algorithm described in Section 4.2.4.1. nMOLDYN corrects the atomic input trajectories for jumps due to periodic boundary conditions. It should be noted that the computation of the intermediate scattering function in the Gaussian approximation is much `cheaper' than the computation of the full intermediate scattering function, $\mathcal{F}_{\mathrm{inc}}(q,t)$, since no averaging over different q-vectors needs to be performed. It is sufficient to compute a single mean-square displacement per atom.


next up previous contents
Next: Parameters Up: Dynamic Incoherent Structure Factor Previous: Dynamic Incoherent Structure Factor   Contents
pellegrini eric 2009-10-06