With the aim of constructing an electronic structure approach that systematically goes beyond the GW and random phase approximation (RPA) we introduce a vertex correction based on the exact-exchange (EXX) potential of time-dependent density functional theory. The EXX vertex function is constrained to be local but is expected to capture similar physics as the Hartree-Fock vertex. With the EXX vertex, we then unify different beyond-RPA approaches such as the various resummations of RPA with exchange and the second-order screened exchange approximation. The theoretical analysis is supported by numerical studies on the hydrogen dimer and the electron gas, and we discuss the role of including the vertex correction in both the screened interaction and the self-energy. Finally, we give details on our implementation within the plane-wave pseudo potential framework and demonstrate the excellent performance of the different RPA with exchange methods in describing the energetics of hydrogen and van der Waals bonds.
With the aim of constructing an electronic structure approach that systematically goes beyond the GW and random phase approximation (RPA) we introduce a vertex correction based on the exact-exchange (EXX) potential of time-dependent density functional theory. The EXX vertex function is constrained to be local but is expected to capture similar physics as the Hartree-Fock vertex. With the EXX vertex, we then unify different beyond-RPA approaches such as the various resummations of RPA with exchange and the second-order screened exchange approximation. The theoretical analysis is supported by numerical studies on the hydrogen dimer and the electron gas, and we discuss the role of including the vertex correction in both the screened interaction and the self-energy. Finally, we give details on our implementation within the plane-wave pseudo potential framework and demonstrate the excellent performance of the different RPA with exchange methods in describing the energetics of hydrogen and van der Waals bonds.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function L. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential W in the BSE for L leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
It is commonly accepted that the GW approximation for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approximation: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the atomic limit, in which, instead, correlation is directly responsible for the observed problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be corrected by a two-point vertex contribution to the self-energy derived from time-dependent density functional theory (TDDFT). We explain why the addition of a particle, instead, requires the use of a different approximate vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the atomic limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qualitatively. We discuss the effects of the self-screening error as well as the atomic limit using GW for the exactly solvable two-site Hubbard model.
We study the effects of local vertex corrections to the self energy of the electron gas. We find that a vertex derived from time-dependent density-functional theory can give accurate self energies without including the explicit time dependence of the exchange-correlation potential provided, however, that a proper decay at large momentum transfer (large q) is built into the vertex function. (The local-density approximation for the vertex fails badly.) Total energies are calculated from the Galitskii-Migdal formula and it is shown that a proper large-q behavior, results in a close consistency between the chemical potentials derived from these energies and those obtained directly from the self energy. We show that this internal consistency depends critically on including the same vertex correction in both the self-energy and the screening function. In addition the total energies become almost as accurate as those from elaborate quantum Monte-Carlo (QMC) calculations. We also study the accuracy and utility of the functional for the total energy proposed by Luttinger and Ward and a generalization by Almbladh, von Barth, and van Leeuwen. For the electron gas, even the simplest and readily evaluated approximations to these functionals yield total energies of similar quality as those of QMC calculations. The functionals depend on the one-electron Green's function and the screened Coulomb interaction and already rather crude approximations to these quantities produce accurate energies thus demonstrating the insensitivity of the functionals to their arguments. Different ways of incorporating vertex corrections beyond the GW level are studied in simple, exactly soluble polaron-like models. We study models of a structureless core electron coupled to valence electrons and a local polaron model by Cini, Hewson and Newns. Our model results indicate that the first vertex correction alone will in general not suffice to improve the spectrum away from the quasi-particle peak. By including a subsequence of Mahan's fractal vertex series, however, we obtain results with correct physical properties which agree better with exact model results.
The many-body theory of interacting electrons poses an intrinsically difficult problem that requires simplifying assumptions. For the determination of electronic screening properties of the Coulomb interaction, the random phase approximation (RPA) provides such a simplification. Here we explicitly show that this approximation is justified for band structures with sizable band gaps. This is when the electronic states responsible for the screening are energetically far away from the Fermi level, which is equivalent to a short electronic propagation length of these states. The RPA contains exactly those diagrams in which the classical Coulomb interaction covers all distances, whereas neglected vertex corrections involve quantum tunneling through the barrier formed by the band gap. Our analysis of electron-electron interactions provides a real-space analogy to Migdal's theorem on the smallness of vertex corrections in electron-phonon problems. An important application is the increasing use of constrained RPA calculations of effective interactions. We find that their usage of Kohn-Sham energies accounts for the leading local (excitonic) vertex correction in insulators.
We have calculated the correlation energy of the homogeneous electron gas (HEG) and the dissociation energy curves of molecules with covalent bonds from an efficient implementation of the adiabatic connection fluctuation dissipation expression including the exact-exchange (EXX) kernel. The EXX kernel is defined from first-order perturbation theory and used in the Dyson equation of time-dependent density-functional theory. Within this approximation (RPAx), the correlation energies of the HEG are significantly improved with respect to the random phase approximation (RPA) up to densities of the order of ${r}_{s}\ensuremath{\approx}10$. However, beyond this value, the RPAx response function exhibits an unphysical divergence and the approximation breaks down. Total energies of molecules at equilibrium are also highly accurate, but we find a similar instability at stretched geometries. Staying within an exact first-order approximation to the response function, we use an alternative resummation of the higher-order terms. This slight redefinition of RPAx fixes the instability in total energy calculations without compromising the overall accuracy of the approach.
The well-known and well-used Density Functional Theory (DFT) is a variational theory for the ground-state. There are no guarantees that it delivers good results for excited state properties such as quasiparticle energies or wave function. All that is guaranteed is that the total energy is exact, the chemical potential is correct, and that the eective electronic states (Kohn-Sham states) generate the right electron density at the variational minimum. Luttinger-Ward approaches to electronic structure are attractive as beyond-DFT schemes since, in principle, they oer a variational functional of the one-electron Green’s function that yields both the correct total energy and all one-particle properties at the variational extremum. However, compared to DFT, these methods face a number of additional challenges beyond the familiar problem of choosing an exchange-correlation functional: (a) they have a much greater computational cost compared to DFT, (b) one must choose some set of trial Green’s functions, and (c) one must locate an extremum (not necessarily a minimum). This talk aims both to be pedagogical and to provide a summary of our most recent work in this area. We focus mainly on the random-phase approximation (RPA) for the correlation functional which yields the GW self-energy for the Green’s function. We will describe ecient approaches for computing the RPA correlation energy; how various types of approximations to the correlation energy create a ladder of approximate self-energy operators and what type of physics is included in these approximations (exchange, screening, Hubbard U, etc.); and how the extremization is in fact highly problematic. We will end with some ideas for making practical progress with the extremization and their relation to the successful Quasiparticle Self-consistent GW (QSGW) scheme [1].