ABINIT Discussion Forums

The calculation of the quasi-particle energies are calculated using an assumption that one can make a linear continuation of the quasi-particle energies from the Kohn-Sham energies along with the derivate of the self-energy. This approximation only works if the self-energy is linear in the energy, so that we can make the approximation as shown in the attachment below. I have two questions:

1) Where in the source code is this step (taking the derivate of the self-energy and linearizing with respect to the QP and KS energies)?

2) Is there some way to modify this step (hopefully with a change of some input variable) where the quasiparticle energies can be obtained better than this linear approximation?

Why not get the exact solution instead? It depends on what you want. As you know the central assumption in one-shot perturbative GW is that the ground-state (usually Kohn-Sham [KS]) orbitals are the same as the QuasiParticle (QP) orbitals. Thus the self energy $\Sigma$ is taken to be diagonal in the KS basis. Then we have the issue that the matrix elements $<i|\Sigma(E)|i>$ are energy dependent, and a simple linear Taylor expansion is made around the eigenvalue that we *do* know, i.e. the KS eigenvalue. So we need the derivative of the function $<i|\Sigma(E)|i>$ at the KS eigenvalue.

In the code this is estimated by default by calculating nine points around the KS eigenvalue and calculating a numerical derivative. This procedure is controlled by the variables "nomegasrd" and "omegasrdmax". However, you can plot the function in greater detail by specifying "nfreqsp", "freqspmax", "freqspmin" and even trying "gw_custom_freqsp". The points for the evaluation of the derivative are output as the _SGR file (actually it contains $Re[<i|\Sigma(E)|i>]$). With "nfreqsp" specified you get a _SIG file as well with both the real and imaginary parts of the self energy, and the spectral function. From the real part, depending on the density of points you've calculated, you can get any order of Taylor approximation you want. Or, you could spline the function from a sufficiently dense sampling and get very nearly the exact solution.

It is thus much easier to do what you propose as a post-processing step rather than modifying the code.

Note that in QuasiParticle Self-Consistent GW, the assumption is *not* made that the self energy is diagonal in the KS basis and $\Sigma$ is Hermitianised and diagonalised to find a new basis. For each step of the self-consistent cycle *both* the perturbative and the exact eigenvalues are reported, and when they are equal self-consistency is reached.

Hope this helps,
Martin Stankovski