Electron-phonon potential from supercells

The computation of the electron-phonon potential, [math]\displaystyle{ \partial_{\nu \mathbf{q}} V(\mathbf{r}) }[/math], is a prerequisite for the calculation of the electron-phonon matrix element:

[math]\displaystyle{ g_{mn \mathbf{k}, \nu \mathbf{q}} \equiv \langle \psi_{m \mathbf{k} + \mathbf{q}} | \partial_{\nu \mathbf{q}} V | \psi_{n \mathbf{k}} \rangle. }[/math]

In the direct interpolation approach, [math]\displaystyle{ \partial_{\nu \mathbf{q}} V(\mathbf{r}) }[/math] is computed from a supercell calculation by means of Fourier interpolation while the Bloch orbitals, [math]\displaystyle{ \psi_{n \mathbf{k}}(\mathbf{r}) }[/math], are computed directly in the primitive cell. Naturally, this process involves multiple VASP calculations in different cells, which can introduce additional complexities compared to just a single execution of VASP. This page tries to give a high-level overview of the general workflow associated with electron-phonon calculations using the direct interpolation approach.

Finite displacements in the supercell

The electron-phonon potential, [math]\displaystyle{ \partial_{\nu \mathbf{q}} V(\mathbf{r}) }[/math], is computed from finite atomic displacements in a sufficiently large supercell. In this case, sufficient means that the effects of an atomic displacement become negligible at about half the supercell size. For materials with long-range electrostatic interactions that go beyond reasonable supercell dimensions, a correction scheme exists.