Computing phonon dispersion from molecular dynamics simulations?

Question

How would one obtain the phonon dispersion curve from a molecular dynamics simulation trajectory? What are the steps involved? Are there any packages that does this? If so, please mention them.

Answer 1

You are basically looking for finding phonon frequencies with respect to $\mathbf{q}$ the scattering vector in reciprocal space. From fluctuation-dissipation theory, the force constants of the system in the reciprocal space is given by:

$$\Phi_{k\alpha,k^{'}\beta}(\mathbf{q}) = k_{B}T \mathbf{G}^{-1}_{k\alpha,k^{'}\beta}(\mathbf{q})$$

$\mathbf{G}$ is the green function defined as:

$$\mathbf{G}_{k\alpha,k^{'}\beta}(\mathbf{q}) = \langle \mathbf{u}_{k\alpha}(\mathbf{q}) \cdot \mathbf{u}_{k^{'}\beta}^{*}(\mathbf{q}) \rangle$$

$\langle...\rangle$ is the ensemble average and $\mathbf{u}_{k\alpha}$ is $\alpha$ component of atomic displacement for $k$-th atom:

$$\mathbf{u}_{k\alpha}(\mathbf{q}) = \sum_{\ell}\mathbf{u}_{k\alpha}^{\ell}\exp{(i\mathbf{q}\cdot \mathbf{r}_{\ell})}$$

We could calculate Green function based on the instantaneous position of atoms ($\mathbf{R}$) and their ensemble average as:

$$\mathbf{G}_{k\alpha,k^{'}\beta}(\mathbf{q}) = \langle \mathbf{R}_{k\alpha}(\mathbf{q}) \cdot \mathbf{R}^{*}_{k^{'}\beta}(\mathbf{q})\rangle - \langle \mathbf{R}\rangle_{k\alpha} (\mathbf{q}) \cdot \langle \mathbf{R} \rangle^{*}_{k^{'}\beta}(\mathbf{q})$$

Now the dynamical matrix $\mathbf{D}$ is calculated as:

$$\mathbf{D}_{k\alpha,k^{'}\beta}(\mathbf{q}) = (m_{k}m_{k^{'}})^{-\frac{1}{2}} \Phi_{k\alpha,k^{'}\beta}(\mathbf{q})$$

The eigenvalues of this matrix are phonon frequencies at $\mathbf{q}$ and if you plot them versus $\mathbf{q}$ you would get the phonon dispersion curve. This is done in LAMMPS by using fix phonon command.

Answer 2

I think one could try to utilize a functionality of hiPhive or TDEP to fit the data from MD-simulations. hiPhive is probably an easier way since it is installed pretty easy using pip. In case of hiPhive once the force-field trained on the MD data is created, generation of the phonon spectrum is straightforward and discussed in the tutorial section.