The GW approximation has recently gained increasing attention as a viable method for the computation of deep core-level binding energies as measured by X-ray photoelectron spectroscopy. We present a comprehensive benchmark study of different GW methodologies (starting point optimized, partial and full eigenvalue-self-consistent, Hedin shift, and renormalized singles) for molecular inner-shell excitations. We demonstrate that all methods yield a unique solution and apply them to the CORE65 benchmark set and ethyl trifluoroacetate. Three GW schemes clearly outperform the other methods for absolute core-level energies with a mean absolute error of 0.3 eV with respect to experiment. These are partial eigenvalue self-consistency, in which the eigenvalues are only updated in the Green’s function, single-shot GW calculations based on an optimized hybrid functional starting point, and a Hedin shift in the Green’s function. While all methods reproduce the experimental relative binding energies well, the eigenvalue self-consistent schemes and the Hedin shift yield with mean absolute errors <0.2 eV the best results.

We combine the renormalized singles (RS) Green's function with the T-matrix approximation for the single-particle Green's function to compute quasiparticle energies for valence and core states of molecular systems. The G RS T 0 method uses the RS Green's function that incorporates singles contributions as the initial Green's function. The G RS T RS method further calculates the generalized effective interaction with the RS Green's function by using RS eigenvalues in the T-matrix calculation through the particle−particle random phase approximation. The G RS T RS method provides significant improvements over one-shot methods G 0 T 0 and G 0 W 0 as demonstrated in calculations for GW100 and CORE65 test sets. It also systematically eliminates the dependence of G 0 T 0 on the choice of density functional approximations. For valence states, the G RS T RS method provides excellent accuracy, which is better than that of G 0 T 0 and G 0 W 0 . For core states, the G RS T RS method identifies correct peaks in the spectral function and significantly outperforms G 0 T 0 on core-level binding energies (CLBEs) and relative CLBEs.

We applied localized orbital scaling correction (LOSC) in Bethe-Salpeter equation (BSE) to predict accurate excitation energies for molecules. LOSC systematically eliminates the delocalization error in the density functional approximation and is capable of approximating quasiparticle (QP) energies with accuracy similar or better than the GW Green's function approach and with much less computational cost. The QP energies from LOSC instead of commonly used G0 W0 and ev GW are directly used in BSE. We show that the BSE/LOSC approach greatly outperforms the commonly used BSE/ G0W0 approach for predicting excitations with different characters. For the calculations for Truhlar-Gagliardi test set containing valence, charge transfer (CT) and Rydberg excitations, BSE/LOSC with the Tamm-Dancoff approximation provides a comparable accuracy to time-dependent density functional theory (TDDFT) and BSE/ev GW. For the calculations of Stein CT test set and Rydberg excitations of atoms, BSE/LOSC considerably outperforms both BSE/ G0W0 and TDDFT approaches with a reduced starting point dependence. BSE/LOSC is thus a promising and efficient approach to calculate excitation energies for molecular systems.

We applied renormalized singles (RS) in the multireference density functional theory (DFT) to calculate accurate energies of ground and excited states. The multireference DFT approach determines the total energy of the N-electron system as the sum of the (N – 2)-electron energy from a density functional approximation (DFA) and the two-electron addition energies from the particle–particle Tamm–Dancoff approximation (ppTDA), naturally including multireference description. The ppTDA@RS-DFA approach uses the RS Hamiltonian capturing all singles contributions in calculating two-electron addition energies, and its total energy is optimized with the optimized effective potential method. It significantly improves the original ppTDA@DFA. For ground states, ppTDA@RS-DFA properly describes dissociation curves tested and the double bond rotation of ethylene. For excited states, ppTDA@RS-DFA provides accurate excitation energies and largely eliminates the DFA dependence. ppTDA@RS-DFA thus provides an efficient multireference approach to systems with static correlation.

We apply the renormalized singles with the correlation (RSc) Green function in the GW approximation for accurate quasiparticle (QP) energies and orbitals. The RSc Green function includes singles contributions from the associated density functional approximation (DFA) and considers correlation contributions perturbatively. G RSc W RSc uses the RSc Green function as the new starting point and in the formulation of the screened interaction. G RSc W 0 fixes the screened interaction at the DFA level. For the calculations of ionization potentials, G RSc W RSc and G RSc W 0 significantly reduce the starting point dependence and provide accurate results with errors around 0.2  eV. For the calculations of core-level binding energies, G RSc W RSc slightly overestimates the results because of underscreening, but G RSc W 0 with GGA functionals provides the optimal accuracy with errors of 0.40  eV. We also show that G RSc W RSc predicts accurate dipole moments. G RSc W RSc and G RSc W 0, are computationally favorable compared with any self-consistent GW methods. The RSc approach is promising for making GW and other Green function methods efficient and robust.