9 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 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 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 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 . The Green functions are given by The exchange and Hartree potentials are given by which is an approximation due to the truncation of the columns of U. Finally the GW self-energy can be deter- mined by 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 P ≶ qq ′ = ∑ ij,kl √ σ$_{q}$U ∗ $_{ij,q}$P ≶ $_{ij,kl}$U$_{kl,q}$ ′ √ σ$_{q}$ ′ . (A5) W $^{<}$( ε ) = W $^{>}$( ε ) − W $^{r}$( ε ) + W r $^{†}$( ε ) , (A9) W $^{>}$( ε ) = W $^{r}$( ε ) P $^{>}$( ε ) W r $^{†}$( ε ) , (A8) − W $^{r}$( ε ) = [ I V P $^{r}$( ε )] $^{1}$V, (A7) P $^{r}$( t ) = θ ( t ) ( P $^{>}$( t ) − P $^{<}$( t ) , (A6) W ≶ $_{ij,kl}$( t ) ≈ ∑ qq ′ U$_{ij,q}$ √ σ$_{q}$W ≶ qq ′ ( t ) √ σ$_{q}$ ′ U ∗ kl,q ′ , (A10) Σ ≶ GW, $_{ij}$( t ) = i ∑ kl G ≶ $_{kl}$( t ) W ≶ $_{ik,jl}$( t ) (A11) Σ r $_{GW}$( t ) = θ ( t ) ( Σ > $_{GW}$( t ) − Σ < $_{GW}$( t ) + δ ( t )Σ$_{x}$ . (A12) Σ$_{x,ij}$ = i ∑ kl V$_{ik,jl}$G < $_{kl}$( t = 0) (A13) Σ$_{H}$$_{,ij}$ = − 2 i ∑ kl V$_{ij,kl}$G < $_{kl}$( t = 0) (A14) G $^{r}$( ε ) =[( ε + iη ) S − H$_{KS}$ + v$_{xc}$ − Δ v$_{H}$ − Σ r $_{GW}$( ε )] − 1 (A15) G $^{<}$( ε ) = − f$_{FD}$ ( ε − µ )[ G $^{r}$( ε ) − G $^{r}$( ε ) $^{†}$] (A16) G $^{>}$( ε ) =(1 − f$_{FD}$ ( ε − µ ))[ G $^{r}$( ε ) − G $^{r}$( ε ) $^{†}$] (A17) P < $_{ij,kl}$( t ) = 2 iG < $_{ik}$( t ) G > ∗ jl ( t ) , (A3) P > $_{ij,kl}$( t ) = P < ∗ $_{ji,lk}$( t ) . (A4) n$_{q}$ ( r ) = ∑ ij n$_{ij}$ ( r ) U$_{ij,q}$/ √ σ$_{q}$, (A2) V$_{qq}$ ′ = 〈 n$_{q}$ | 1 | r − $^{$_{r}$′}$| | n$_{q}$ ′ 〉 , (A1) V. ACKNOWLEDGMENTS Appendix A: The GW self-energy where n$_{q}$ ( r ) are the optimized pair orbitals