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and thus obtain a direct and unbiased assessment of the GW approximation itself. We found that the inclusion of valence-core exchange interactions, as facilitated by the PAW method, is important and affect the HF/GW HOMO levels by -1.2 eV on average. 

The position of the HOMO for a series of 33 molecules was calculated using fully selfconsistent GW, single-shot G$_{0}$W$_{0}$, Hartree-Fock, DFT-PBE0, and DFT-PBE. Both PBE and PBE0 eigenvalues grossly overestimate the HOMO energy with a mean absolute error (MAE) with respect to the experimental ionization potentials (IP) of 4.4 and 2.5 eV, respectively. Hartree-Fock underesti- mates the HOMO energy but improves the agreement with experiments yielding a MAE of 0.8 eV. GW and G$_{0}$W$_{0}$ overcorrects the Hartree-Fock levels slightly lead- ing to a small overestimation of the HOMO energy with a MAE relative to experiments of 0.4-0.5 eV. This shows that although screening is a weak effect in molecular sys- tems its inclusion at the GW level improves the electron removal energies by 30-50% relative to the unscreened Hartree-Fock. The best IPs were obtained from one-shot G$_{0}$W$_{0}$ calculations starting from the HF Green’s func- tion where the overscreening is least severe. Very similar conclusions were reached by comparing GW, G$_{0}$W$_{0}$ and HF to exact diagonalization for conjugated molecules de- scribed by the semi-empirical PPP model. 37 

We thank Mikkel Strange for useful discussions and assistance with the projected Wannier functions. We ac- knowledge support from the Danish Center for Scientific Computing and The Lundbeck Foundation’s Center for Atomic-scale Materials Design (CAMD). 

Let U denote the rotation matrix that diagonalizes the pair orbital overlap S$_{ij,kl}$ = 〈 n$_{ij}$ | n$_{kl}$ 〉 , i.e. U $^{†}$SU = σI . The columns of U are truncated to those which have corresponding eigenvalues σ$_{q}$ < 10 − $^{5}$a − 3 0 . We then only calculate the reduced number of Coulomb elements 

where n$_{q}$ ( r ) are the optimized pair orbitals 

which are mutually orthonormal, i.e. 〈 n$_{q}$ | n$_{q}$ ′ 〉 = δ$_{qq}$ ′ . Determining the GW self-energy proceeds by calculat- ing first the full polarization matrix in the time domain 

## V. ACKNOWLEDGMENTS 

## Appendix A: The GW self-energy 

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The factor 2 appears for spin-paired systems from sum- ming over spin indices. This is then downfolded to the reduced representation 

The screened interaction can be determined from the lesser and greater polarization matrices, and the static interaction V$_{qq}$ ′ , via the relations 

where all quantities are matrices in the optimized pair orbital basis and matrix multiplication is implied. We obtain the screened interaction in the original orbital ba- sis from 

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which is an approximation due to the truncation of the columns of U. Finally the GW self-energy can be deter- mined by 

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The exchange and Hartree potentials are given by 

The Green functions are given by 

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where f$_{FD}$ ( ε − µ ) is the Fermi-Dirac function and Δ v$_{H}$ = Σ$_{H}$[ G ] − Σ$_{H}$[ G$_{DFT}$ ] is the difference between the GW Hartree potential and the DFT Hartree potential. For self-consistent calculations, equation (A3)-(A17) are it- erated untill convergence in G . 

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