Generalized Optimized Effective Potential for Orbital Functionals and Self-Consistent Calculation of Random Phase Approximations

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.

Reformulation of Thermally-Assisted-Occupation Density Functional Theory in the Kohn-Sham Framework

We reformulate the thermally-assisted-occupation density functional theory (TAO-DFT) into the Kohn-Sham single-determinant framework and construct two new post-self-consistent field (post-SCF) static correlation correction schemes, named rTAO and rTAO-1. In contrast to the original TAO-DFT with the density in an ensemble form, in which each orbital density is weighted with a fractional occupation number, the ground-state density is given by a single-determinant wavefunction, a regular Kohn-Sham density, and total ground state energy is expressed in the normal Kohn-Sham form with a static correlation energy formulated in terms of the Kohn-Sham orbitals. In post-SCF calculations with rTAO functionals, an efficient energy scanning to quantitatively determine $\theta$ is also proposed. The rTAOs provide a promising method to simulate systems with strong static correlation as original TAO, but simpler and more efficient.We show that both rTAO and rTAO-1 is capable of reproducing most results from TAO-DFT without the additional functional Eθ used in TAO-DFT. Furthermore, our numerical results support that, without the functional Eθ, both rTAO and rTAO-1 can capture correct static correlation profiles in various systems.

Multireference Density Functional Theory for Describing Ground and Excited States with Renormalized Singles

We applied renormalized singles (RS) in the multireference density functional theory (DFT) to calculate accurate energies of ground and excited states. The multireference DFT approach determines the total energy of the N-electron system as the sum of the (N - 2)-electron energy from a density functional approximation (DFA) and the two-electron addition energies from the particle-particle Tamm-Dancoff approximation (ppTDA), naturally including multireference description. The ppTDA@RS-DFA approach uses the RS Hamiltonian capturing all singles contributions in calculating two-electron addition energies, and its total energy is optimized with the optimized effective potential method. It significantly improves the original ppTDA@DFA. For ground states, ppTDA@RS-DFA properly describes dissociation curves tested and the double bond rotation of ethylene. For excited states, ppTDA@RS-DFA provides accurate excitation energies and largely eliminates the DFA dependence. ppTDA@RS-DFA thus provides an efficient multireference approach to systems with static correlation.

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.

Inverse Kohn-Sham Density Functional Theory: Progress and Challenges

Inverse Kohn-Sham (iKS) methods are needed to fully understand the one-to-one mapping between densities and potentials on which density functional theory is based. They can contribute to the construction of empirical exchange-correlation functionals and to the development of techniques for density-based embedding. Unlike the forward Kohn-Sham problems, numerical iKS problems are ill-posed and can be unstable. We discuss some of the fundamental and practical difficulties of iKS problems with constrained-optimization methods on finite basis sets. Various factors that affect the performance are systematically compared and discussed, both analytically and numerically, with a focus on two of the most practical methods: the Wu-Yang method (WY) and the partial differential equation constrained optimization (PDE-CO). Our analysis of the WY and PDE-CO highlights the limitation of finite basis sets. We introduce new ideas to make iKS problems more tractable, provide an overall strategy for performing numerical density-to-potential inversions, and discuss challenges and future directions.

Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition

In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-reference methods with qualitative concepts from ligand-field and molecular orbital theories. These insights provide the tools to reliably diagnose the multi-reference character, and our analysis reveals that while strong (i.e., static) correlation can be found in linear molecules (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for organic biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chemical bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs experiment for carbonyl-dissociation energies from double-hybrid density functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation.

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.

Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory

A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn-Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.

Preserving Symmetry and Degeneracy in the Localized Orbital Scaling Correction Approach

Symmetry is a fundamental concept that plays a critical role in many chemical and physical phenomena and processes, which highlights the importance of theoretical methods to correctly handle symmetry. The recently developed localized orbital scaling correction (LOSC1) shows great improvement on the description of band gaps, photoemission spectra, and dissociation limits of cationic species. However, issues remain with LOSC1 in dealing with the symmetry and degeneracy of electronic states, which are also relevant to other methods using localization. In this work, we utilize a new method that deals with the physical-space and the energy-space localization on an equal footing. The resulting localized orbitals, i.e., orbitalets, are able to maintain more symmetry and the desired state degeneracy, which is important in calculating the electronic structure of both molecules and periodic bulk systems. Furthermore, the curvature matrix is redefined to improve potential energy curves for systems with stretched bonds, while retaining the correct dissociation limits. This new approach, termed LOSC2, includes only two fitting parameters. It maintains accuracy similar to that of LOSC1 over many properties, while overcoming LOSC1's deficiencies in symmetry and degeneracy. Our tests have shown that LOSC2 orbitalets possess the full- or subgroup of molecular symmetry if allowed, which preserves the state degeneracy. Tests on differently sized planar annulenes, odd-numbered allenes, and triphenylene again verify that LOSC2 is able to maintain the state degeneracy, while LOSC1 cannot. All the tests demonstrate the advantage of LOSC2 in the calculation of molecular systems and its potential for application to periodic bulk systems.

Low-Scaling Self-Consistent Minimization of a Density Matrix Based Random Phase Approximation Method in the Atomic Orbital Space

An efficient minimization of the random phase approximation (RPA) energy with respect to the one-particle density matrix in the atomic orbital space is presented. The problem of imposing full self-consistency on functionals depending on the potential itself is bypassed by approximating the RPA Hamiltonian on the basis of the well-known Hartree-Fock Hamiltonian making our self-consistent RPA method completely parameter-free. It is shown that the new method not only outperforms post-Kohn-Sham RPA in describing noncovalent interactions but also gives accurate dipole moments demonstrating the high quality of the calculated densities. Furthermore, the main drawback of atomic orbital based methods, in increasing the prefactor as compared to their canonical counterparts, is overcome by introducing Cholesky decomposed projectors allowing the use of large basis sets. Exploiting the locality of atomic and/or Cholesky orbitals enables us to present a self-consistent RPA method which shows asymptotically quadratic scaling opening the door for calculations on large molecular systems.

Renormalized Singles Green's Function for Quasi-Particle Calculations beyond the G0 W0 Approximation

Quasi-particle energies and band gaps in particular are critical for investigating novel materials. Commonly used density functional approximations (DFAs) systematically underestimate band gaps, and GW approximation is the established method of choice for good accuracy and reliability. However, G₀ W₀ has some undesired dependence on the DFA, while self-consistent GW (sc GW) is expensive and not consistent in accuracy improvement. Here a simple and efficient G_RS W₀ approach has been developed: a subspace diagonalization of the Hartree-Fock (HF) Hamiltonian with the DFA density matrix provides the new reference Green's function G_RS that incorporates the effect of all single excitation contributions to the self-energy, thereby essentially eliminating the starting-point dependence. Calculations for molecules and large band gap solids demonstrate the significant improvement over G₀ W₀ and greatly reduced dependence on the initial DFA. G_RS W₀ approach also improve results for other bulks over G₀ W₀, but to a lesser extent, which could be due to the limitations in current implementation for bulks. The results demonstrate that to achieve good accuracy, it is not necessary to use hybrid DFA, which is expensive for bulks. This work should be greatly significant in making GW a more robust approach.

Wavefunction-like Correlation Model for Use in Hybrid Density Functionals

We present Unsöld-W12 (UW12), an approximation to the correlation energy of molecules that is an explicit functional of the single-particle reduced-density matrix. The approximation resembles one part of modern explicitly correlated second-order Møller-Plesset (MP2) theory and is intended as an alternative to MP2 in double-hybrid exchange-correlation functionals. Orbital optimization with UW12 is straightforward, and the UW12 energy is evaluated without a double summation over unoccupied orbitals, leading to a faster basis-set convergence than is seen in double-hybrid functionals. We suggest a one-parameter hybrid exchange-correlation functional XCH-BLYP-UW12. XCH-BLYP-UW12 is similar to double-hybrid functionals, but contains UW12 correlation instead of MP2 correlation. We find that XCH-BLYP-UW12 is more accurate than the existing double-hybrid functional B2-PLYP for small-molecule main-group reaction barrier heights and has roughly the same accuracy as the existing hybrid functional B3LYP for atomization energies.