Sampling phonon spectra from molecular-dynamics simulations

Sampling phonon DOS from molecular dynamics simulation

The phonon density of states can be obtained as the power spectrum from the normalized velocity auto correlation function. The normalized velocity auto correlation function for a $N$-particle system is given by \begin{equation} f(t)=\sum_{s=1}^{types}f_{s}(t)=\frac{\langle \sum_{s=1}^{types}\sum_{i=1}^{N_{s}}\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T+t) \rangle}{\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T)}. \end{equation} The brackets $\langle ,\rangle$ denotes a thermal average which has to be computed over different trajectories and and starting times $\Delta T$ within each trajectory. The sum over $i$ runs over the atoms within each species and the sum $s$ is over all atomic species contained in the simulated system. From this the phonon density of states is obtained by computing the power spectrum of $f_{s}(t)$: \begin{equation} g(\omega)=g_{s}(\omega)=\sum_{s=1}^{types}g_{s}(\omega)=\left| \int_{-\infty}^{\infty}\sum_{s=1}^{types}f_{s}(t)e^{-i\omega t}\right|^{2}. \end{equation} To properly sample the phonon density of states from molecular dynamics simulations the following the steps have to be accomplished:

\begin{tabular}{|c|l|}

   \hline
   \textbf{Step} & \textbf{Description} \\
   \hline
   1 & Generate start configurations \\
   \hline
   2 & Make NVE ensembles for every NVT start configuration \\
   \hline
   3 & Compute self correlation for every NVE trajectory \\
   \hline
   4 & Compute power spectrum of velocity autocorrelation for every start configuration \\
   \hline

\end{tabular}

code	publication
DSLEAP	Lahnsteiner et.al.
phq	Zhang et.al.
DynaPhoPy	Carreras et.al.

Sampling phonon DOS from molecular dynamics simulation

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