arXiv:1001.1274v1 [cond-mat.mtrl-sci] 8 Jan 2010 Fully selfconsistent GW calculations for molecules C. Rostgaard, K. W. Jacobsen, and K. S. Thygesen Center for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK - 2800 Kgs. Lyngby, Denmark (Dated: October 25, 2018) We calculate single-particle excitation energies for a series of 33 molecules using fully selfcon- sistent GW, one-shot G$_{0}$W$_{0}$, Hartree-Fock (HF), and hybrid density functional theory (DFT). All calculations are performed within the projector augmented wave (PAW) method using a basis set of Wannier functions augmented by numerical atomic orbitals. The GW self-energy is calculated on the real frequency axis including its full frequency dependence and off-diagonal matrix elements. The mean absolute error of the ionization potential (IP) with respect to experiment is found to be 4.4, 2.6, 0.8, 0.4, and 0.5 eV for DFT-PBE, DFT-PBE0, HF, G$_{0}$W$_{0}$[HF], and selfconsistent GW, respectively. This shows that although electronic screening is weak in molecular systems its inclu- sion at the GW level reduces the error in the IP by up to 50% relative to unscreened HF. In general GW overscreens the HF energies leading to underestimation of the IPs. The best IPs are obtained from one-shot G$_{0}$W$_{0}$ calculations based on HF since this reduces the overscreening. Finally, we find that the inclusion of core-valence exchange is important and can affect the excitation energies by as much as 1 eV. PACS numbers: 31.15.A-,33.15.Ry,31.15.V- The hybrid 7–9 and screened hybrid 10 functionals, which admix around 25% of the (screened) Fock ex- change with the local DFT exchange, generally improve the description of band gaps in bulk semi-conductors and insulators $^{5,6}$. However, the orbital energies obtained for finite systems using such functionals still underes- timate the fundamental gap, I$_{p}$ − E$_{a}$ , (the difference between ionization potential and electron affinity) by up to several electron volts. In fact, for molecules the pure Hartree-Fock (HF) eigenvalues are usually closer to the true electron addition/removal energies than are the hybrid DFT eigenvalues. This is because HF is self-interaction free and because screening of the ex- change interaction is a relatively weak effect in molecular systems. $^{4,11,13}$. On the other hand, in extended systems the effect of self-interaction is less important and the long range Coulomb interaction becomes short ranged due to dynamical screening. As a consequence HF breaks down in extended systems leading to dramatically over- estimated band gaps and a qualitatively incorret descrip- tion of metals. 14–16 Most GW calculations to date rely on one or several approximations of more technical character. These in- clude the plasmon pole approximation, the linearized QP equation, neglect of off-diagonal matrix elements in the GW self-energy, analytic continuations from the imaginary to the real frequency axis, neglect of core states contributions to the self-energy, neglect of self- consistency. The range of validity of these approxima- tions has been explored for solid state systems by a num- ber of authors $^{38–43}$, however, much less is known about their applicability to molecular systems $^{24}$. Our imple- mentation of the GW method avoids all of these tech- nical approximations allowing for a direct and unbiased Density functional theory (DFT) 1 with the single- particle Kohn-Sham (KS) scheme 2 is today the most widely used approach to the electronic structure problem of real materials in both solid state physics and quantum chemistry. While properties derived from total energies (or rather total energy differences) are accurately pre- dicted by DFT, it is well known that DFT suffers from a band gap problem implying that the single-particle KS eigenvalues cannot in general be interpreted as real quasi- particle (QP) excitation energies. In particular, semilocal exchange-correlation functionals severely underestimate the fundamental gap of both insulators, semi-conductors, and molecules. 3–6 The many-body GW approximation of Hedin 17 has been widely and succesfully used to calculate QP band structures in metals, semi-conductors, and insulators. 3,18–20 The GW approximation can be viewed as HF with a dynamically screened Coulomb interac- tion. The fact that the screening is determined by the system itself instead of being fixed a priori as in the screened hybrid schemes, suggests that the GW method should be applicable to a broad class of systems rang- ing from metals with strong screening to molecules with weak screening. With the entry of nanoscience the use of GW has been extended to low-dimensional systems and nanostructures 21–31 and more recently even nonequilib- rium phenomena like quantum transport $^{32–36}$. In view of this trend it is important to establish the perfor- mance of the GW approximation for other systems than the crystalline solids. In this work we present first- principles benchmark GW calculations for a series of small molecules. In a closely related study we compared GW and Hartree-Fock to exact diagonalization results for semi-empirical PPP models of conjugated molecules $^{37}$. The main conclusions from the two studies regarding the qualities of the GW approximation in molecular systems, are very consistent. I. INTRODUCTION