8.1 Theoretical background.

A method to systematically improve upon DFT-estimates of single particle excitation spectra, i.e., ionization potentials and electron affinities is the GW-method. Its central object is the single particle Green’s function G; its poles describe single particle excitation energies and lifetimes. In particular, the poles up to the Fermi-level correspond to the primary vertical ionization energies. The GW-approach is based on an exact representation of G in terms of a power series of the screened Coulomb interaction W, which is called the Hedin equations. The GW-equations are obtained as an approximation to the Hedin-equations, in which the screened Coulomb interaction W is calculated neglecting so called vertex corrections. In this approximation the self–energy Σ, which connects the fully interacting Green’s function G to a reference non-interacting Green’s function G0, is given by Σ = GW.

This approach can be used to perturbatively calculate corrections to the Kohn-Sham spectrum. To this end, the Green’s function is expressed in a spectral representation as a sum of quasi particle states.

                            †   ′
      ′     ∑  ---Ψr,n(r,z)Ψl,n(r-,z)---
G (r,r;z) =    z - εn(z)+ iηsgn(εn - μ ).
             n
(8.1)

Under the approximation that the KS-states are already a good approximation to these quasi–particle states Ψl,n the leading order correction can be calculated by solving the zeroth order quasi–particle equation:

ε = ϵ  + ⟨n|Σ[G   ](ε )- V  |n⟩
 n   n         KS   n     xc
(8.2)

An approximation to the solution of this equation can be obtained by linearizing it:

εn = ϵn + Zn⟨n|Σ (ϵn)- Vxc|n⟩
(8.3)

here, Zn is given by:

     [              |       ]-1
              ∂Σ(E-)||
Zn =   1- ⟨n|  ∂E   |    |n⟩
                     E=ϵn
(8.4)

reducing the computational effort to a single iteration.

The self–energy Σ appearing in Eqn. (8.2) is calculated in the GW approximation from the KS Green’s function and screening. This is the so-called G0W0 approximation. The Self–energy splits in an energy independent exchange part Σx and a correlation part Σc(E) that does depend on energy. Their matrix elements are given by:

    x|| ′⟩         ∑   (    ′)
⟨n |Σ  n    =  = -      ni|in  ,                  (8.5)
                    i
and
                 ∑   ∑                 2
⟨n|Σc(ϵn)|n⟩  =         -------|(nn-|ρm-)|-------.         (8.6)
                  m   n ϵn - ϵn - Zmsgn (ϵn - μ)
Where Zm = Ωm - are the excitation energies shifted infinitesimally into the complex plane. The ρm are the corresponding excitation densities. More details, tests and benchmark calculations are can be found in Ref. 105.