13.1 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)Ψ-†l,n(r′,z)----
G(r,r;z) =    z - εn(z)+ iηsgn(εn - μ).
           n
(13.1)

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:

εn = ϵn + ⟨n|Σ[GKS](εn) - Vxc|n⟩
(13.2)

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

ε  = ϵ + Z  ⟨n |Σ (ϵ )- V  |n⟩
 n    n   n      n    xc
(13.3)

here, Zn is given by:

     [            |       ]-1
             ∂Σ(E)||
Zn =  1 - ⟨n| ∂E  |E=ϵ |n⟩
                      n
(13.4)

reducing the computational effort to a single iteration.

The self–energy Σ appearing in Eqn. (13.2) 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:

                 ∑
⟨n|Σx|n′⟩  =  = -    (ni|in′),                   (13.5)
                  i
and
    c           ∑  ∑        |(nn|ρm )|2
⟨n|Σ (ϵn)|n⟩  =        ϵn-- ϵn---Zmsgn(ϵn--μ).          (13.6)
                 m  n      --         --
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. 157.