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.
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:
An approximation to the solution of this equation can be obtained by linearizing it:
here, Zn is given by:
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: