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The local xc-potential, v$_{xc}$ , is subtracted to avoid dou- ble counting when adding the many-body self-energy, Σ$_{xc}$[ G ]. As indicated, the latter depends on the Green function and therefore Eq. (2) must in principle be solved self-consistently in conjuction with the self-energy. 

In the present study Σ$_{xc}$ is either the bare exchange potential or the GW self-energy. To be consistent with the code used for the calculations, we present the equa- tions for the GW self-energy on the so-called Keldysh contour. However, under the equilibrium conditions con- sidered here the Keldysh formalism is equivalent to the ordinary time-ordered formalism. The GW self-energy is defined by 

where τ and τ ′ are times on the Keldysh contour, C . The dynamically screened Coulomb interaction obeys the Dyson-like equation 

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and the polarization bubble is given by 

In the limit of vanishing polarization, P = 0, W reduces to the bare Coulomb interaction 

and the GW self-energy reduces to the exchange potential of HF theory. 

From the above equations for the contour-ordered quantities, the corresponding real time components, i.e. the retarded, advanced, lesser, and greater components, can be obtained from standard conversion rules $^{49,50}$. For completeness we give the expressions for the real time components of the GW equations in Appendix A. 

The time/energy dependence of the dynamical quan- tities G , W , P , and Σ, is represented on a uniform grid. We switch between time and energy domains using the Fast Fourier Transform in order to avoid time consuming convolutions. A typical energy grid used for the GW cal- culations in this work ranges from -150 to 150 eV with a grid spacing of 0.02 eV. The code is parallelized over basis functions and energy grid points. We use a Pulay mixing scheme for updating the Green function G r when iterating Eq. (2) to self-consistency as described in Ref. 47. 

We stress that no approximation apart from the fi- nite basis set is made in our implementation of the GW approximation. In particular the frequency dependence is treated exactly and analytic continuations from the imaginary axis are avoided since we work directly on the real frequency/time axis. The price we pay for this is the large size of the energy grid. 

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The single-particle excitation spectrum is contained in the spectral function 

For a molecule A ( ε ) shows peaks at the QP energies ε$_{n}$ = E$_{n}$ ( N + 1) − E$_{0}$ ( N ) and ε$_{n}$ = E$_{0}$ ( N ) − E$_{n}$ ( N − 1) corresponding to electron addition and removal energies, respectively. Here E$_{n}$ ( N ) denotes the energy of the n th excited state of the system with N electrons and N refers to the neutral state. 

When the Green function is evaluated in a non- orthogonal basis, like the augmented Wannier basis used here, the projected density of states for orbital φ$_{i}$ be- comes 

where matrix multiplication is implied. 48 Correspond- ingly, the total density of states, or quasiparticle spec- trum, is given by 

## D. Calculating Coulomb matrix elements 

The calculation of all of the Coulomb matrix elements, V$_{ij,kl}$ , is prohibitively costly for larger basis sets. For- tunately the matrix is to a large degree dominated by negligible elements. To systematically define the most significant Coulomb elements, we use the product basis technique of Aryasetiawan and Gunnarsson $^{51,52}$. In this approach, the pair orbital overlap matrix 

where n$_{ij}$ ( r ) = φ ∗ $_{i}$( r ) φ$_{j}$ ( r ) is used to screen for the sig- nificant elements of V . 

The eigenvectors of the overlap matrix Eq. (10) rep- resents a set of “optimized pair orbitals” and the eigen- values their norm. Optimized pair orbitals with insignif- icant norm must also yield a reduced contribution to the Coulomb matrix, and are omitted in the calculation of V . We limit the basis for V to optimized pair orbitals with a norm larger than 10 − $^{5}$a − 3 0 . This gives a signifi- cant reduction in the number of Coulomb elements that needs to be evaluated, and it reduces the matrix size of P ( ε ) and W ( ε ) correspondingly, see Appendix A. 

The evaluation of the double integral in Eq. (6) is efficiently performed in real space by solving a Poisson equation using multigrid techniques $^{45,53}$. 

All inputs to the GW/HF calculations, i.e. the selfcon- sistent Kohn-Sham Hamiltonian, H$_{KS}$ , the xc potential 

## C. Spectral function 

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## E. Valence-core exchange 

