An overview of the GWR code

This page provides a quick introduction to the new GWR driver of ABINIT We discuss the technical details related to the implementation, the associated input variables. as well as the pros and cons with respect to the conventional GW implementation formulated in Fourier-space and frequency domain.

Why a new GW code?

The conventional GW algorithm has quartic scaling with the number of atoms whereas GWR scales cubically. The legacy GW code obtains the matrix elements of self-energy by performing a convolution in frequency domain, usually within the plasmon-pole approximation whereas GWR computes the self-energy elements in imaginary-time

Select the task to be performed when optdriver == 6 i.e. GWR code. while gwr_task defines the task

Requirements

• Scalapack
• Optmized FFT libraries (FFTW3 or MKL-DFTI)

Discuss single and double precision version. We recall that single-precision is the default

Formalism

The zero-temperature Green’s function in the imaginary-time domain is given by:

$$\begin{equation} G(\rr, \rr', i\tau) = \Theta(\tau) \Gove(\rr, \rr', i\tau) + \Theta(-\tau) \Gund (\rr, \rr', i\tau) \end{equation}$$

with $\Theta$ the Heaviside step-function and

$$\begin{equation} \Gove(\rr, \rr', i\tau) = -\sum_n^{\text{unocc}} \psi_n(\rr)\psi_n^*(\rr') e^{-\ee_n\tau} \qquad (\tau > 0) \end{equation}$$

$$\begin{equation} \Gund(\rr, \rr', i\tau) = \sum_n^{\text{occ}} \psi_n(\rr)\psi_n^*(\rr') e^{-\ee_n\tau} \qquad (\tau < 0) \end{equation}$$

For simplicity we have assumed a spin-unpolarized semiconductor with scalar wavefunctions and the Fermi level $\mu$ has been set to zero.

The GWR code constructs the Green’s function from the KS wavefunctions and eigenvalues stored in the WFK file specified via getwfk_filepath (getwfk in multi-dataset mode). This WFK file is usually produced by performing a NSCF calculation including empty states and the GWR driver provides a specialized option to perform a direct diagonalization of the KS Hamiltonian in parallel with Scalapack. (see section below) When computing $G^0$, the number of bands included in the sum over states is controlled by nband. Clearly, it does not make any sense to ask for more bands than the ones available in the WFK file.

The imaginary axis is sampled using a minimax mesh with gwr_ntau points. The other piece of information required for the selection of the minimax mesh is the ratio between the fundamental gap and the maximum transition energy i.e. the differerence between the highest KS eigenvalue for empty states that clearly depends on nband and the energy of the lowest occupied state.

Note that only $\Gamma$-centered $\kk$-meshes are supported in GWR, e,g:

• ngkpt 4 4 4
• nshiftk 1
• shiftk 0 0 0

The cutoff energy for the polarizability is given by ecuteps while ecutsigx defines the number of g-vectors for the exchange part of the self-energy.

Note that GWR also needs the GS density produced by a previous GS SCF run. This file can be read via getden_filepath.

$$\begin{equation} \chi(\rr,\RR', t) = G(\rr,\RR', i\tau) G^*(\rr,\RR', -i\tau) \end{equation}$$

As concerns the treatment of the long-wavelenght limit, we have the following input variables:

inclvkb, gw_qlwl, gwr_max_hwtene

GWR workflow for QP energies

A typical GWR workflow with arrows denoting dependencies between the different steps is schematically represented in the figure below:

In order to perform a standard one-shot GW calculation one has to:

1. Run a converged ground state calculation to obtain the density.

2. Perform a non self-consistent run to compute the KS eigenvalues and the eigenfunctions including several empty states. Note that, unlike standard band structure calculations, here the KS states must be computed on a regular grid of k-points.

3. Use optdriver = 3 to compute the independent-particle susceptibility $\chi^0$ on a regular grid of q-points, for at least two frequencies (usually, $\omega=0$ and a purely imaginary frequency - of the order of the plasmon frequency, a dozen of eV).

The Green’s function is constructed from the KS wavefunctions and eigenvalues stored in the WFK file hence the

The following physical properties can be computed:

• Imaginary part of ph-e self-energy in metals (eph_task 1) that gives access to:

  • Phonon linewidths induced by e-ph coupling

• Real and imaginary parts of the e-ph self-energy (eph_task 4) that gives access to:

  • Zero-point renormalization of the band gap

At the time of writing, the following features are not yet supported in GWR:

• PAW calculations
• Spinors (nspinor = 2)

Tricks to accelerate the computation and reduce the memory requirements