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)=\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 the power spectrum as

\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}

Further information can be found on the following page. External tools as for example phonopy may also be considered. To compute the power spectra of the Fourier transformed projected velocity autocorrelations

[math]\displaystyle{ |G_{\nu}(\mathbf{q},\omega)|^{2}=\sum_{I,\alpha}\sum_{J,\beta} \int\left( \varepsilon_{I\nu}^{\beta}(\mathbf{q}) \sqrt{M_{I}}v_{I}^{\alpha}(t') \right )\left( \varepsilon_{J\nu}^{\beta}(\mathbf{q}) \sqrt{M_{J}}v_{J}^{\beta}(t'') \right )e^{i\mathbf{q} \cdot (\mathbf{R}_{I}(t')-\mathbf{R}_{J}(t''))}e^{-i\omega (t'-t'')}d(t'-t'') }[/math]

external tools are required. The following table summarizes a small list of codes which can compute projected velocity correlation functions from VASP output.

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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