A method to systematically improve upon DFT estimates of single-particle excitation
spectra, that is, 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:

G_{0}W_{0} is implemented in TURBOMOLE in the escf module supporting the following features.

- LDA, GGA and Hybrid functionals can be used for the underlying DFT calculation.
- RI approximation.
- In G
_{0}W_{0}, the linearized, Eqn. (13.3), and solved, Eqn. (13.2), quasi-particle equation. - Both RPA and TDDFT response functions can be used to screen the coulomb interaction in constructing W.
- Closed shell systems with rpas excitations and open shell systems with urpa excitations. Additionally, Kramers-restricted closed shell systems within the two-component relativistic framework (inclusion of spin-orbit coupling) can be treated using soghf excitations, see Chapter 8 along with ref. [177].

The general recipe for a G_{0}W_{0} calculation is as follows:

- 1.
- define session
- 2.
- Provide additional GW control flags
- 3.
- dscf or ridft calculation
- 4.
- escf calculation

Ad 1) The def2-TZVPP basis seems to be the most useful, it comes for all tested systems within
0.1 eV of the def2-QZVP result with about half the number of basis functions. In the
define session the calculation of the response function needs to be defined. In the final
define menu select the ex menu and select the calculation of RPA singlet excitations, or
urpa in case of open shell. Choose soghf to run a two-component calculation. Select
"all all" to get all excitations. For systems and basis set having less than 4000 rpas
excitations just set for all excitations. For large systems start to run G_{0}W_{0} with ≈4000
rpas excitations. In subsequent runs add more excitations until a converged result is
reached. escf will keep the converged roots, so not much time is lost using this restart
approach.

Ad 2) If $gw is set in the control file the quasi particle energies will be evaluated according to equation 13.3. Additional options are described in the keywords section, Section 20.2.15.

Ad 3) Symmetry up to D2h is available for both closed shell (rpas) and open shell system (urpa)
for GW. Moreover, the simplified methods xGW and sGW, which neglect contributions from
excitation vectors are available for all point groups implemented in Turbomole. Two-component
calculations can only be performed in C_{1}.

Ad 4) In the escf run the response function is calculated which is needed to determine the screened coulomb interaction. At the end of this run the actual GW calculation is performed.

Possible source of errors: When dscf or ridft is repeated after escf the sing_a file may not
be correct anymore, this may happen when degenerate levels are present. escf will
however not recognize this and continue using the previously converged data in sing_a
leading to nonsense values for Σ_{c}. Before running escf the old sing_a file has to be
removed.