We combine the renormalized singles (RS) Greens function with the T-Matrix approximation for the single-particle Greens function to compute quasiparticle energies for valence and core states of molecular systems. The $G_{text{RS}}T_0$ method uses the RS Greens function that incorporates singles contributions as the initial Greens function. The $G_{text{RS}}T_{text{RS}}$ method further calculates the generalized effective interaction with the RS Greens function by using RS eigenvalues in the T-Matrix calculation through the particle-particle random phase approximation. The $G_{text{RS}}T_{text{RS}}$ method provides significant improvements over the one-shot T-Matrix method $G_0T_0$ 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 (DFAs). For valence states, the $G_{text{RS}}T_{text{RS}}$ method provides an excellent accuracy, which is better than $G_0T_0$ with Hartree-Fock (HF) or other DFAs. For core states, the $G_{text{RS}}T_{text{RS}}$ method correctly identifies desired peaks in the spectral function and significantly outperforms $G_0T_0$ on core level binding energies (CLBEs) and relative CLBEs, with any commonly used DFAs.
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