The GW-Method for Quantum Chemistry Applications: Theory and Implementation M. J. van Setten,* ,† F. Weigend, †,‡ and F. Evers †,¶ † Institute of Nanotechnology, Karlsruhe Institute of Technology, P.O. Box 3640, D-76021 Karlsruhe, Germany ‡ Institute of Physical Chemistry, Karlsruhe Institute of Technology, P.O. Box 3640, D-76021 Karlsruhe, Germany ¶ Institute of Theoretical Condensed Matter, Karlsruhe Institute of Technology, P.O. Box 3640, D-76021 Karlsruhe, Germany * S Supporting Information ABSTRACT: The GW-technology corrects the Kohn−Sham (KS) single particle energies and single particle states for artifacts of the exchange-correlation (XC) functional of the underlying density functional theory (DFT) calculation. We present the formalism and implementation of GW adapted for standard quantum chemistry packages. Our implementation is tested using a typical set of molecules. We find that already after the first iteration of the self-consistency cycle, G 0 W 0 , the deviations of quasi-particle energies from experimental ionization potentials and electron affinities can be reduced by an order of magnitude against those of KS-DFT using GGA or hybrid functionals. Also, we confirm that even on this level of approximation there is a considerably diminished dependency of the G 0 W 0 -results on the XC-functional of the underlying DFT. 1. INTRODUCTION One of the most used approaches for the computational study of solids, nanoscale systems, and molecules is the density functional theory (DFT). 1 DFT has an ubiquitous appearance in computa- tional chemistry and materials sciences because it often offers the only possibility to obtain useful ab initio results for relevant system sizes. A well-known difficulty with DFT-calculations is related to approximations in the exchange-correlation (XC) functional, the most familiar one being the local density approximation (LDA). 2,3 The use of such local (or semilocal) approximations has several important consequences, in praxi. First, the neglect of the derivative-discontinuity in such approximate functionals implies uncertainties in the description of charge-transfer pro- cesses, because the alignment of Kohn−Sham (KS) levels of different subsystems is not properly accounted for. Second, neglecting nonlocal terms also implies that weak, van der Waals like, binding forces are being described badly or not at all. Moreover, (differences between) KS-single particle energies are often interpreted as physical excitation energies, because the KS- estimates of these energies tend to compare significantly better to experimental results than, for instance, those originating from Hartree−Fock (HF) theory. Still, a formal justification for this praxis (i.e., a KS-analog of the Koopmans’ theorem 4 ) does not exist in general. To identify situations when good quantitative estimates can be obtained, nevertheless, is the subject of ongoing research. 5,6 A method to systematically improve upon shortcomings of DFT-estimates of single particle excitation spectra, i.e., ionization potentials and electron affinities, is well established for electronic band structure calculations in solids: the GW-method. Its central object is the Green’s function G; its poles describe single particle excitation energies and lifetimes. The GW-approach is based on an exact representation of G in terms of a power series of the screened Coulomb interaction W, which is called the Hedin equations. 7,8 The GW-equations are obtained as an approx- imation to the Hedin-equations, in which the screened Coulomb interaction W is calculated neglecting so-called vertex correc- tions. 9−12 Effectively, one can say that the GW-approach, similar to other one electron Green’s function approaches, replaces in the DFT- calculation the problematic unknown XC-potential by a self- energy, Σ. In this process the KS-equations are transformed into a self-consistent set of quasi-particle equations. Similar to the XC- potentials of DFT, which are functionals of the electron density, Σ[G] is a functional of G and therefore typically needs to be updated in the iteration cycle that solves the quasi-particle equations. In contrast to XC-potentials, Σ is not Hermitian and depends on energy. Furthermore similar to the Fock-operator and unlike (semi) local XC-potentials, Σ is nonlocal in space. A key feature of Green’s functions is that their poles by construction define the single-particle (or quasi-particle) excitation energies. In particular, the GW-quasi-particle energies up to the highest occupied molecular orbital (HOMO) cor- respond to the primary vertical ionization energies. When using a basis that keeps track explicitly also of core states, we have access to the ionization energies relevant for core-level spectroscopies. The knowledge of the full quasi-particle spectrum and quasi- particle states also gives access to calculating other physical Received: July 25, 2012 Published: November 5, 2012 Article pubs.acs.org/JCTC © 2012 American Chemical Society 232 dx.doi.org/10.1021/ct300648t | J. Chem. Theory Comput. 2013, 9, 232−246