Kohn-Sham band gaps and potentials of solids from the optimised effective potential method within the random phase approximation

Going Beyond the GW Approximation Using the Time-Dependent Hartree-Fock Vertex

The time-dependent Hartree-Fock (TDHF) vertex of many-body perturbation theory (MBPT) makes it possible to extend TDHF theory to charged excitations. Here we assess its performance by applying it to spherical atoms in their neutral electronic configuration. On a theoretical level, we recast the TDHF vertex as a reducible vertex, highlighting the emergence of a self-energy expansion purely in orders of the bare Coulomb interaction; then, on a numerical level, we present results for polarizabilities, ionization energies (IEs), and photoemission satellites. We confirm the superiority of THDF over simpler methods such as the random phase approximation for the prediction of atomic polarizabilities. We then find that the TDHF vertex reliably provides better IEs than GW and low-order self-energies do in the light-atom, few-electron regime; its performance degrades in heavier, many-electron atoms instead, where an expansion in orders of an unscreened Coulomb interaction becomes less justified. New relevant features are introduced in the satellite spectrum by the TDHF vertex, but the experimental spectra are not fully reproduced due to a missing account of nonlinear effects connected to hole relaxation. We also explore various truncations of the self-energy given by the TDHF vertex, but do not find them to be more convenient than low-order approximations such as GW and second Born (2B), suggesting that vertex corrections should be carried out consistently both in the self-energy and in the polarizability.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Numerically Precise Benchmark of Many-Body Self-Energies on Spherical Atoms

We investigate the performance of beyond-GW approaches in many-body perturbation theory by addressing atoms described within the spherical approximation via a dedicated numerical treatment based on B-splines and spherical harmonics. We consider the GW, second Born (2B), and GW + second order screened exchange (GW+SOSEX) self-energies and use them to obtain ionization potentials from the quasi-particle equation (QPE) solved perturbatively on top of independent-particle calculations. We also solve the linearized Sham–Schlüter equation (LSSE) and compare the resulting xc potentials against exact data. We find that the LSSE provides consistent starting points for the QPE but does not present any practical advantage in the present context. Still, the features of the xc potentials obtained with it shed light on possible strategies for the inclusion of beyond-GW diagrams in the many-body self-energy. Our findings show that solving the QPE with the GW+SOSEX self-energy on top of a PBE or PBE0 solution is a viable scheme to go beyond GW in finite systems, even in the atomic limit. However, GW shows a comparable performance if one agrees to use a hybrid starting point. We also obtain promising results with the 2B self-energy on top of Hartree–Fock, suggesting that the full time-dependent Hartree–Fock vertex may be another viable beyond-GW scheme for finite systems.

Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues

Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Technically, σ-functionals are closely related to the well-known direct random phase approximation (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to reference sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calculations with the generalized gradient approximation (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The number of reference sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compounds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chemical accuracy for main group chemistry. For the 10 966 reactions from the highly accurate W4-11RE reference set, the B3LYP based σ-functional exhibits a mean average deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chemistry, accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calculation for typical systems.

Selfconsistent random phase approximation methods

This Perspective reviews recent efforts toward selfconsistent calculations of ground-state energies within the random phase approximation (RPA) in the (generalized) Kohn-Sham (KS) density functional theory context. Since the RPA correlation energy explicitly depends on the non-interacting KS potential, an additional condition to determine the energy as a functional of the density is necessary. This observation leads to the concept of functional selfconsistency (FSC), which requires that the KS density equals the interacting density defined as the functional derivative of the ground-state energy with respect to the external potential. While all existing selfconsistent RPA schemes violate FSC, the recent generalized KS semicanonical projected RPA (GKS-spRPA) method takes a step toward satisfying it. This leads to systematic improvements in densities, binding energy curves, reference state stability, and molecular properties compared to non-selfconsistent RPA as well as optimized effective potential RPA. GKS-spRPA orbital energies accurately approximate valence and core ionization potentials, and even electron affinities of non-valence bound anions. The computational cost and performance of GKS-spRPA are compared to those of related selfconsistent schemes, including GW and orbital optimization methods, and limitations are discussed. Large differences between KS and interacting densities observed in the absence of FSC and the well-rounded performance of GKS-spRPA suggest that the KS potential as a density functional should be defined via the FSC condition for explicitly potential-dependent density functionals.

Optimized effective potentials from the random-phase approximation: Accuracy of the quasiparticle approximation

The optimized effective potential (OEP) method presents an unambiguous way to construct the Kohn-Sham potential corresponding to a given diagrammatic approximation for the exchange-correlation functional. The OEP from the random-phase approximation (RPA) has played an important role ever since the conception of the OEP formalism. However, the solution of the OEP equation is computationally fairly expensive and has to be done in a self-consistent way. So far, large scale solid state applications have, therefore, been performed only using the quasiparticle approximation (QPA), neglecting certain dynamical screening effects. We obtain the exact RPA-OEP for 15 semiconductors and insulators by direct solution of the linearized Sham-Schlüter equation. We investigate the accuracy of the QPA on Kohn-Sham bandgaps and dielectric constants, and comment on the issue of self-consistency.

Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy

We introduce new functionals for the Kohn-Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct random phase approximation (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-constant and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of reference sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn-Sham calculations employing the exchange-correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chemical accuracy. For the 10 966 reactions of the W4-11RE reference set, the mean absolute deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calculation of the total energy requires less computational time than a density-functional calculation with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort.

Lieb-Oxford bound and pair correlation functions for density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem

Compliance with the Lieb-Oxford bound for the indirect Coulomb energy and for the exchange-correlation energy is investigated for a number of density-functional methods based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem to treat correlation. Furthermore, the correlation contribution to the pair density resulting from these methods is compared with highly accurate reference values for the helium atom and for the hydrogen molecule at several bond distances. For molecules, the Lieb-Oxford bound is obeyed by all considered methods. For the homogeneous electron gas, it is violated by all methods for low electron densities. The simplest considered ACFD method, the direct random phase approximation (dRPA), violates the Lieb-Oxford bound much earlier than more advanced ACFD methods that, in addition to the simple Hartree kernel, take into account the exchange kernel and an approximate correlation kernel in the calculation of the correlation energy. While the dRPA yields quite poor correlation contributions to the pair density, those from more advanced ACFD methods are physically reasonable but still leave room for improvements, particularly in the case of the stretched hydrogen molecule.

Screened hybrid meta-GGA exchange–correlation functionals for extended systems

Screened Hartree–Fock exchange integrated with semilocal exchange–correlation functionals often proficiently predict several solid-state properties.

Importance of the Kinetic Energy Density for Band Gap Calculations in Solids with Density Functional Theory

Recently, exchange-correlation potentials in density functional theory were developed with the goal of providing improved band gaps in solids. Among them, the semilocal potentials are particularly interesting for large systems since they lead to calculations that are much faster than with hybrid functionals or methods like GW. We present an exhaustive comparison of semilocal exchange-correlation potentials for band gap calculations on a large test set of solids, and particular attention is paid to the potential HLE16 proposed by Verma and Truhlar. It is shown that the most accurate potential is the modified Becke-Johnson potential, which, most noticeably, is much more accurate than all other semilocal potentials for strongly correlated systems. This can be attributed to its additional dependence on the kinetic energy density. It is also shown that the modified Becke-Johnson potential is at least as accurate as the hybrid functionals and more reliable for solids with large band gaps.

Comparing LDA-1/2, HSE03, HSE06 and G₀W₀ approaches for band gap calculations of alloys

It has long been known that the local density approximation and the generalized gradient approximation do not furnish reliable band gaps, and one needs to go beyond these approximations to reliably describe these properties. Among alternatives are the use of hybrid functionals (HSE03 and HSE06 being popular), the GW approximation or the recently proposed LDA-1/2 method. In this work, we compare rigorously the performance of these four methods in describing the band gaps of alloys, employing the generalized quasi-chemical approach to treat the disorder of the alloy and to obtain judiciously the band gap for the entire compositional range. Zincblende InGaAs and InGaN were chosen as prototypes due to their importance in optoelectronic applications. The comparison between these four approaches was guided both by the agreement between the predicted band gap and the experimental one, and by the demanded computational effort (time and memory). We observed that the HSE06 method provided the most accurate results (in comparison with experiments), whereas, surprisingly, the LDA-1/2 method gave the best compromise between accuracy and computational resources. Due to its low computational cost and good accuracy, we decided to double the supercell used to describe the alloys, and employing LDA-1/2 we observed that the bowing parameter changed remarkably, only agreeing with the measured one for the larger supercell, where LDA-1/2 plays an important role.

Singles correlation energy contributions in solids

The random phase approximation to the correlation energy often yields highly accurate results for condensed matter systems. However, ways how to improve its accuracy are being sought and here we explore the relevance of singles contributions for prototypical solid state systems. We set out with a derivation of the random phase approximation using the adiabatic connection and fluctuation dissipation theorem, but contrary to the most commonly used derivation, the density is allowed to vary along the coupling constant integral. This yields results closely paralleling standard perturbation theory. We re-derive the standard singles of Görling-Levy perturbation theory [A. Görling and M. Levy, Phys. Rev. A 50, 196 (1994)], highlight the analogy of our expression to the renormalized singles introduced by Ren and coworkers [Phys. Rev. Lett. 106, 153003 (2011)], and introduce a new approximation for the singles using the density matrix in the random phase approximation. We discuss the physical relevance and importance of singles alongside illustrative examples of simple weakly bonded systems, including rare gas solids (Ne, Ar, Xe), ice, adsorption of water on NaCl, and solid benzene. The effect of singles on covalently and metallically bonded systems is also discussed.

How close are the Slater and Becke-Roussel potentials in solids?

The Becke–Roussel (BR) potential [Phys. Rev. A 1989, 39, 3761] was proposed as an approximation to the Slater potential, which is the Coulomb potential generated by the exact exchange hole. In the present work, a detailed comparison between the Slater and BR potentials in solids is presented. It is shown that the two potentials usually lead to very similar results for the electronic structure; however, in a few cases, e.g., Si, Ge, or strongly correlated systems like NiO, the fundamental band gap or magnetic properties can differ markedly. Such differences should not be neglected when the computationally expensive Slater potential is replaced by the cheap semilocal BR potential in approximations to the exact-exchange Kohn–Sham potential, such as the one proposed by Becke and Johnson [J. Chem. Phys. 2006, 124, 221101].

Optimized effective potential method and application to static RPA correlation

The optimized effective potential (OEP) method is a promising technique for calculating the ground state properties of a system within the density functional theory. However, it is not widely used as its computational cost is rather high and, also, some ambiguity remains in the theoretical framework. In order to overcome these problems, we first introduced a method that accelerates the OEP scheme in a static RPA-level correlation functional. Second, the Krieger-Li-Iafrate (KLI) approximation is exploited to solve the OEP equation. Although seemingly too crude, this approximation did not reduce the accuracy of the description of the magnetic transition metals (Fe, Co, and Ni) examined here, the magnetic properties of which are rather sensitive to correlation effects. Finally, we reformulated the OEP method to render it applicable to the direct RPA correlation functional and other, more precise, functionals. Emphasis is placed on the following three points of the discussion: (i) level-crossing at the Fermi surface is taken into account; (ii) eigenvalue variations in a Kohn-Sham functional are correctly treated; and (iii) the resultant OEP equation is different from those reported to date.

Fock-exchange for periodic structures in the real-space formalism and the KLI approximation

The calculation of Fock-exchange interaction is an important task in the computation of molecule and solid properties.