Permittivity

This page contains a guide on how to perform permittivity (dielectric constant) calculations using Quantum Espresso (tested using version 7.1).

The method outlined here involves employing multiple calculations using the PW executable of Quantum Espresso at different electric field strengths in order to compare the electronic and ionic dipoles that form in the system. This should align with the Modern Theory of Polarisation method (CITE).

We assume here that the user is looking for both the electronic and ionic components to the permittivity. However, to obtain just the electronic, no ionic relaxation can be turned off whilst sampling the electric fields (NOTE: only ionic relaxation can be performed using electric fields, variable cell relaxation (vc-relax) is not currently permitted in Quantum Espresso, as of version 7.1).

Whilst there are other forms of permittivity calculations (i.e. dielectric response, which varies with frequency), the approach here relates to obtaining the static (zero frequency or dielectric constant) and electronic (infinite frequency) dielectric response values. Other pages will be set up to detail how to perform the frequency-dependent calculations at a later date, but these involve using the EPW executable of Quantum Espresso.

There are two main steps to this approach.

1. Start with a fully-relaxed atomic structure. Apply a zero-strength electric field to the system and run the PW calculation. The necessary tags are:

 &control
   calculation = ‘relax’,
   forc_conv_thr = 1.d-4,
   lelfield = .true.,
   nberrycyc = 1
 /
 &electrons
   efield_cart(1) = 0.d0,
   efield_cart(2) = 0.d0,
   efield_cart(3) = 0.d0
 /

This will output the electronic and ionic dipoles inherent in the system. You can query this using the following in-terminal command (where <OUTPUT> is the name of the file that the Quantum Espresso output has been piped to):

 grep ‘Dipole’ <OUTPUT>|tail -n6

2. Now perform the same calculation as before, but altering the electric field values along the three cartesian coordinate axes (x=1, y=2, z=3). We recommend a small value, such as 0.001d0.

Once this has been performed, compare the zero-field and applied field electronic and ionic dipole values. Use the following equation to determine the static response:

$\varepsilon _{0}=1+4\pi {\frac {D(f)-D(0)}{f\Omega }}$,

where $f$ is the field strength (suggested 0.001) in atomic units (a.u.), $D$ is the dielectric field strength, and $\Omega$ is the volume of the cell in atomic units.

For the infinite frequency response, use the following equation:

.