### 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 G_{0}, 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.

| (13.1) |

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

| (13.2) |

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

| (13.3) |

here, Z_{n} is given by:

| (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 G_{0}W_{0} approximation. The Self–energy splits
in an energy independent exchange part Σ^{x} and a correlation part Σ^{c}(E) that does depend on
energy. Their matrix elements are given by:

and Where Z_{m} = Ω_{m} - iη are the excitation energies shifted infinitesimally into the complex plane.
The ρ_{m} are the corresponding excitation densities. More details, tests and benchmark calculations
are can be found in Ref. 157.