Challenging the Lieb–Oxford bound in a systematic way

Large-Z atoms in the strong-interaction limit of DFT: Implications for gradient expansions and for the Lieb-Oxford bound

We numerically study the strong-interaction limit of the exchange-correlation functional for neutral atoms and Bohr atoms as the number of electrons increases. Using a compact representation, we analyze the second-order gradient expansion, comparing it with the one for exchange (weak interaction limit). The two gradient expansions, at strong and weak interaction, turn out to be very similar in magnitude but with opposite signs. We find that the point-charge plus continuum model is surprisingly accurate for the gradient expansion coefficient at strong coupling, while generalized gradient approximations, such as Perdew-Burke-Ernzerhof (PBE) and PBEsol, severely underestimate it. We then use our results to analyze the Lieb-Oxford bound from the point of view of slowly varying densities, clarifying some aspects on the bound at a fixed number of electrons.

Semilocal Meta-GGA Exchange-Correlation Approximation from Adiabatic Connection Formalism: Extent and Limitations

The incorporation of a strong-interaction regime within the approximate semilocal exchange-correlation functionals still remains a very challenging task for density functional theory. One of the promising attempts in this direction is the recently proposed adiabatic connection semilocal correlation (ACSC) approach [Constantin, L. A.; Phys. Rev. B 2019, 99, 085117] allowing one to construct the correlation energy functionals by interpolation of the high and low-density limits for the given semilocal approximation. The current study extends the ACSC method to the meta-generalized gradient approximations (meta-GGA) level of theory, providing some new insights in this context. As an example, we construct the correlation energy functional on the basis of the high- and low-density limits of the Tao-Perdew-Staroverov-Scuseria (TPSS) functional. Arose in this way, the TPSS-ACSC functional is one-electron self-interaction free and accurate for the strictly correlated and quasi-two-dimensional regimes. Based on simple examples, we show the advantages and disadvantages of ACSC semilocal functionals and provide some new guidelines for future developments in this context.

Nonlocal Functionals Inspired by the Strongly Interacting Limit of DFT: Exact Constraints and Implementation

Capturing strong correlation effects remains a key challenge for the development of improved exchange-correlation (XC) functionals in density functional theory. The recently proposed multiple radii functional (MRF) [J. Phys. Chem. Lett. 2017, 8, 2799; J. Chem. Theory Comput. 2019, 15, 3580] was designed to capture strong correlation effects seamlessly, as its mathematical structure draws from that of the exact XC functional in the limit of infinite correlations. The MRF functional provides a framework for building approximations along the density-fixed adiabatic connection, delivers accurate XC energy densities in the standard DFT gauge (same as that of the exact exchange energy density), and is free of one-electron self-interaction errors. To facilitate the development of XC functionals based on the MRF, we examine the behavior of the MRF functional when applied to uniform and scaled densities and consider how it can be made exact for the uniform electron gas. These theoretical insights are then used to build improved forms for the fluctuation function, an object that defines XC energy densities within the MRF framework. We also show how the MRF fluctuation function for physical correlation can be easily readjusted to accurately capture the XC functional in the limit of infinite correlations, demonstrating the versatility of MRF for building approximations for different correlation regimes. We describe the implementation of MRF using densities expanded on Gaussian basis sets, which improves the efficiency of previous grid-based MRF implementations.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Approximate bounds and temperature dependence of adiabatic connection integrands for the uniform electron gas

Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. Methods that account for temperature dependence, such as Mermin–Kohn–Sham finite-temperature density functional theory and free energy density functional theory, are now employed with more regularity and available in many standard code packages. However, approximations from zero-temperature density functional theory are still often used in temperature-dependent simulations using thermally weighted electronic densities as an input to exchange–correlation functional approximations, a practice known to miss temperature-dependent effects in the exchange–correlation free energy of these systems. In this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange–correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb–Oxford inequality, which provides a lower bound for the ground-state exchange–correlation energy, bounds for the exchange–correlation at finite temperatures are approximated for various degrees of electronic correlation.

Gradient Expansions for the Large-Coupling Strength Limit of the Møller-Plesset Adiabatic Connection

The adiabatic connection that has, as weak-interaction expansion, the Møller-Plesset perturbation series has been recently shown to have a large coupling-strength expansion, in terms of functionals of the Hartree-Fock density with a clear physical meaning. In this work, we accurately evaluate these density functionals and we extract second-order gradient coefficients from the data for neutral atoms, following ideas similar to the ones used in the literature for exchange, with some modifications. These new gradient expansions will be the key ingredient for performing interpolations that have already been shown to reduce dramatically MP2 errors for large noncovalent complexes. As a byproduct, our investigation of neutral atoms with large number of electrons N indicates that the second-order gradient expansion for exchange grows as N log(N) rather than as N, as often reported in the literature.

Generalizing Double-Hybrid Density Functionals: Impact of Higher-Order Perturbation Terms

Connections between the Görling-Levy (GL) perturbation theory and the parameters of double-hybrid (DH) density functional are established via adiabatic connection formalism. Moreover, we present a more general DH density functional theory, where the higher-order perturbation terms beyond the second-order GL2 one, such as GL3 and GL4, also contribute. It is shown that a class of DH functionals including previously proposed ones can be formed using the present construction. Based on the proposed formalism, we assess the performance of higher-order DH and long-range corrected DH formed on the Perdew-Burke-Ernzerhof (PBE) semilocal functional and second-order GL2 correlation energy. The underlying construction of DH functionals based on the generalized many-body perturbation approaches is physically appealing in terms of the development of the non-local forms using more accurate and sophisticated semilocal functionals.

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.

van der Waals exchange-correlation functionals over bulk and surface properties of transition metals

The correct description of van der Waals (vdW) interaction forces is required for accurately describing dispersion bonded systems. Several approaches have been proposed to include London dispersion in density functional theory exchange-correlation functionals, where the family of so-called van der Waals (vdW-DF) exchange-correlation functionals have shown a better performance than local or semi local exchange-correlation functionals for describing molecular adsorption on metals. Despite the numerous benchmarks performed with these functionals, their performance in predicting bulk properties of transition metals has hitherto not been investigated in detail. We have therefore tested five vdW-DF exchange-correlation functionals, vdW-DF2, optPBE-vdW, BEEF-vdW, optB88-vdW and C09 _x -vdW to assess their performance in the prediction of lattice constants, bulk moduli, cohesive energies and surface energies of bulk Ni, Cu, Rh, Pd, Ag, Ir, Pt and Au (in fcc crystal structure). These transition metals are commonly used for benchmarking density functionals because they are important for applications in catalysis. The results are compared with experimental data and the PBE exchange-correlation functional. We found that both the optB88-vdW and the C09 _x -vdW exchange-correlation functionals estimate all properties with high accuracy, in better agreement with experimental data than PBE and other considered vdW functionals. The C09 _x -vdW functional clearly outperforms all other exchange-correlation functionals for surface energies for the (1 1 1) termination of different metals. We have also evaluated the interatomic electron density emerging from different functionals, and concluded that the observed differences are a result of the predicted lattice parameter, rather than a direct consequence of the functional form. Plane-wave and real-space grid-based expansions of the electron density are also compared, revealing good agreement between the two approaches for lattice parameters, cohesive energies, and surface energies, but more severe differences in bulk moduli. On the basis of our results, we recommend using the C09 _x -vdW for studying bulk properties and surface energies of transition metals.

Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy

From a simplified version of the mathematical structure of the strong coupling limit of the exact exchange-correlation functional, we construct an approximation for the electronic repulsion energy at physical coupling strength, which is fully nonlocal. This functional is self-interaction free and yields energy densities within the definition of the electrostatic potential of the exchange-correlation hole that are locally accurate and have the correct asymptotic behavior. The model is able to capture strong correlation effects that arise from chemical bond dissociation, without relying on error cancellation. These features, which are usually missed by standard density functional theory (DFT) functionals, are captured by the highly nonlocal structure, which goes beyond the "Jacob's ladder" framework for functional construction, by using integrals of the density as the key ingredient. Possible routes for obtaining the full exchange-correlation functional by recovering the missing kinetic component of the correlation energy are also implemented and discussed.

Testing Exact Upper Bounds to Exact Exchange

The exact exchange energy and its energy density are useful but computationally expensive ingredients in density functional approximations for Kohn-Sham density functional theory. We present detailed tests of some exact nonempirical upper bounds to exact exchange. These "Rung 3.5" upper bounds contract the Kohn-Sham one-particle density matrix with model density matrices used to construct semilocal model exchange holes and invoke the Cauchy-Schwarz inequality. The contraction automatically eliminates the computationally expensive long-range component of the exact exchange hole. Numerical tests show that the exchange upper bounds underestimate total exchange energies while predicting other properties with accuracy approaching standard hybrid approximations.