A Finite-field Approach for GW Calculations Beyond the Random Phase Approximation He Ma,y,z Marco Govoni,y,{ Francois Gygi,x and Giulia Galli∗,y,z,{ yInstitute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States zDepartment of Chemistry, University of Chicago, Chicago, Illinois 60637, United States {Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, United States. xDepartment of Computer Science, University of California Davis, Davis, California 95616, United States. E-mail: [email protected] Abstract We describe a finite-field approach to compute density response functions, which allows for efficient G0W0 and G0W0Γ0 calculations beyond the random phase approx- imation. The method is easily applicable to density functional calculations performed with hybrid functionals. We present results for the electronic properties of molecules and solids and we discuss a general scheme to overcome slow convergence of quasipar- arXiv:1808.10001v2 [physics.chem-ph] 17 Dec 2018 ticle energies obtained from G0W0Γ0 calculations, as a function of the basis set used to represent the dielectric matrix. 1 1 Introduction Accurate, first principles predictions of the electronic structure of molecules and materi- als are important goals in chemistry, condensed matter physics and materials science.1 In the past three decades, density functional theory (DFT)2,3 has been successfully adopted to predict numerous properties of molecules and materials.4 In principle, any ground or excited state properties can be formulated as functionals of the ground state charge den- sity. In practical calculations, the ground state charge density is determined by solving the Kohn-Sham (KS) equations with approximate exchange-correlation functionals, and many important excited state properties are not directly accessible from the solution of the KS equations. The time-dependent formulation of DFT (TDDFT)5 in the frequency domain6 provides a computationally tractable method to compute excitation energies and absorption spectra. However, using the common adiabatic approximation to the exchange-correlation functional, TDDFT is often not sufficiently accurate to describe certain types of excited states such as Rydberg and charge transfer states,7 especially when semilocal functionals are used. A promising approach to predict excited state properties of molecules and materials is many-body perturbation theory (MBPT).8{10 Within MBPT, the GW approximation can be used to compute quasiparticle energies that correspond to photoemission and inverse photoemission measurements; furthermore, by solving the Bethe-Salpeter equation (BSE), one can obtain neutral excitation energies corresponding to optical spectra. For many years since the first applications of MBPT,9 its use has been hindered by its high computational cost. In the last decade, several advances have been proposed to improve the efficiency of MBPT calculations,11{13 which are now applicable to simulations of relatively large and com- plex systems, including nanostructures and heterogeneous interfaces.14{16 In particular, GW and BSE calculations can be performed using a low rank representation of density response functions,17{20 whose spectral decomposition is obtained through iterative diagonalization using density functional perturbation theory (DFPT).21,22 This method does not require the 2 explicit calculation of empty electronic states and avoids the inversion or storage of large dielectric matrices. The resulting implementation in the WEST code23 has been successfully applied to investigate numerous systems including defects in semiconductors,24,25 nanopar- ticles,26 aqueous solutions,15,27,28 and solid/liquid interfaces19,29 . In this work, we developed a finite-field (FF) approach to evaluate density response func- tions entering the definition of the screened Coulomb interaction W . The FF approach can be used as an alternative to DFPT, and presents the additional advantage of being applica- ble, in a straightforward manner, to both semilocal and hybrid functionals. In addition, FF calculations allow for the direct evaluation of density response functions beyond the random phase approximation (RPA). Here we first benchmark the accuracy of the FF approach for the calculation of several density response functions, from which one can obtain the exchange correlation kernel (fxc), defined as the functional derivative of the exchange-correlation potential with respect to the charge density. Then we discuss G0W0 calculations for various molecules and solids, carried out with either semilocal or hybrid functionals, and by adopting different approximations to include vertex corrections in the self-energy. In the last two decades a variety of meth- ods30{44 45 has been proposed to carry out vertex-corrected GW calculations, with different approximations to the vertex function Γ and including various levels of self-consistency be- tween G, W and Γ. Here we focus on two formulations that are computationally tractable fxc fxc also for relatively large systems, denoted as G0W0 and G0W0Γ0. In G0W0 , fxc is included in the evaluation of the screened Coulomb interaction W ; in G0W0Γ0, fxc is included in the calculation of both W and the self-energy Σ through the definition of a local vertex function. fxc Most previous G0W0 and G0W0Γ0 calculations were restricted to the use of the LDA func- 30,31,35,36 46 tional, for which an analytical expression of fxc is available. Paier et al. reported fxc 47 GW0 results for solids obtained with the HSE03 range-separated hybrid functional, and 34,48{50 the exact exchange part of fxc is defined using the nanoquanta kernel. In this work semilocal and hybrid functionals are treated on equal footing, and we present calculations us- 3 ing LDA,51 PBE52 and PBE053 functionals, as well as a dielectric-dependent hybrid (DDH) functional for solids.54 A recent study of Thygesen and co-workers55 reported basis set convergence issues when performing G0W0Γ0@LDA calculations, which could be overcome by applying a proper renor- 56{58 malization to the short-range component of fxc. In our work we generalized the renor- malization scheme of Thygesen et al. to functionals other than LDA, and we show that the convergence of G0W0Γ0 quasiparticle energies is significantly improved using the renormal- ized fxc. The rest of the paper is organized as follows. In Section 2 we describe the finite-field approach and benchmark its accuracy. In Section 3 we describe the formalism used to perform GW calculations beyond the RPA, including a renormalization scheme for fxc, and we compare the quasiparticle energies obtained from different GW approximations (RPA or vertex-corrected) for molecules in the GW100 test set59 and for several solids. Finally, we summarize our results in Section 4. 4 2 The finite-field approach We first describe the FF approach for iterative diagonalization of density response functions and we then discuss its robustness and accuracy. 2.1 Formalism Our G0W0 calculations are based on DFT single-particle energies and wavefunctions, ob- tained by solving the Kohn-Sham (KS) equations: HKS m(r) = "m m(r); (1) where the KS Hamiltonian HKS = T + VSCF = T + Vion + VH + Vxc. T is the kinetic energy operator; VSCF is the KS potential that includes the ionic Vion, the Hartree VH and the Pocc. 2 exchange-correlation potential Vxc. The charge density is given by n(r) = m j m(r)j . For simplicity we omitted the spin index. 0 We consider the density response function (polarizability) of the KS system χ0(r; r ) and 0 0 that of the physical system χ(r; r ); the latter is denoted as χRPA(r; r ) when the random phase approximation (RPA) is used. The variation of the charge density due to either a variation of the KS potential δVSCF or the external potential δVext is given by: Z δn(r) = K(r; r0)δV (r0)dr0; (2) 0 0 0 0 0 0 where K = χ0(r; r ) if δV (r ) = δVSCF(r ) and K = χ(r; r ) if δV (r ) = δVext(r ). The den- sity response functions of the KS and physical system are related by a Dyson-like equation: Z Z 0 0 00 000 00 00 000 00 000 000 0 χ(r; r ) = χ0(r; r ) + dr dr χ0(r; r )[vc(r ; r ) + fxc(r ; r )] χ(r ; r ) (3) 0 1 0 δVxc(r) where vc(r; r ) = jr−r0j is the Coulomb kernel and fxc(r; r ) = δn(r0) is the exchange- 5 correlation kernel. 0 Within the RPA, fxc is neglected and χ(r; r ) is approximated by: Z Z 0 0 00 000 00 00 000 000 0 χRPA(r; r ) = χ0(r; r ) + dr dr χ0(r; r )vc(r ; r )χ(r ; r ): (4) In the plane-wave representation (for simplicity we only focus on the Γ point of the 0 4πδ(G;G0) 4π 0 Brillouin zone), vc(G; G ) = jGj2 (abbreviated as vc(G) = jGj2 ). We use K(G; G ) to denote a general response function (K 2 fχ0; χRPA; χg), and define the dimensionless ~ 0 ~ 0 response function K(G; G )(K 2 fχ~0; χ~RPA; χ~g) by symmetrizing K(G; G ) with respect to vc: 1 1 0 2 0 2 0 K~ (G; G ) = vc (G)K(G; G )vc (G ): (5) The dimensionless response functionsχ ~RPA andχ ~0 (see eq 4) have the same eigenvectors, and their eigenvalues are related by: 0 RPA λi λi = 0 (6) 1 − λi RPA 0 where λi and λi are eigenvalues ofχ ~RPA andχ ~0, respectively. In general the eiegenvalues and eigenvectors ofχ ~RPA are different from those ofχ ~ due to the presence of fxc in eq 3. In our GW calculations we use a low rank decomposition of K~ : NPDEP ~ X K = λi jξii hξij (7) i where λ and jξi denote eigenvalue and eigenvectors of K~ , respectively. The set of ξ constitute a projective dielectric eigenpotential (PDEP) basis,17{19 and the accuracy of the low rank decomposition is controlled by NPDEP, the size of the basis.