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1. The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory

   https://doi.org/10.1146/annurev-physchem-062422-013259  

   Kaplan, Aaron D.; Levy, Mel; Perdew, John P. (April 2023, Annual Review of Physical Chemistry)

   Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that ( a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and ( b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development. 

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2. Efficient Embedded Cluster Density Approximation Calculations with an Orbital-Free Treatment of Environments

   https://doi.org/10.1021/acs.jctc.0c01133  

   Chi, Yu-Chieh; Tameh, Maliheh Shaban; Huang, Chen (April 2021, Journal of chemical theory and computation)

   Gagliardi, Laura (Ed.)

   The computational cost of the Kohn–Sham density functional theory (KS-DFT), employing advanced orbital-based exchange–correlation (XC) functionals, increases quickly for large systems. To tackle this problem, we recently developed a local correlation method in the framework of KS-DFT: the embedded cluster density approximation (ECDA). The aim of ECDA is to obtain accurate electronic structures in an entire system. With ECDA, for each atom in a system, we define a cluster to enclose that atom, with the rest atoms treated as the environment. The system’s electron density is then partitioned among the cluster and the environment. The cluster’s XC energy density is then calculated based on its electron density using an advanced orbital-based XC functional. The system’s XC energy is obtained by patching all clusters’ XC energy densities in an atom-by-atom manner. In our previous formulation of ECDA, environments were treated by KS-DFT, which makes the following two tasks computationally expensive for large systems. The first task is to partition the system’s electron density among a cluster and its environment. The second task is to solve the environments’ Sternheimer equations for calculating the system’s XC potential. In this work, we remove these two computational bottlenecks by treating the environments with the orbital-free (OF) DFT. The new method is called ECDA-envOF. The performance of ECDA-envOF is examined in two systems: ester and Cl-tetracene, for which the exact exchange (EXX) is used as the advanced XC functional. We show that ECDA-envOF gives results that are very close to the previous formulation in which the environments were treated by KS-DFT. Therefore, ECDA-envOF can be used for future large-scale simulations. Another focus of this work is to examine ECDA-envOF’s performance on systems having different bond types. With ECDA-envOF, we calculate the energy curves for stretching/compressing some covalent, metallic, and ionic systems. ECDA-envOF’s predictions agree well with the benchmarks by using reasonably large clusters. These examples demonstrate that ECDA-envOF is nearly a black-box local correlation method for investigating heterogeneous materials in which different bond types exist. 

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3. Capturing the electron–electron cusp with the coupling-constant averaged exchange–correlation hole: A case study for Hooke’s atoms

   https://doi.org/10.1063/5.0173370  

   Hou, Lin; Irons, Tom J.; Wang, Yanyong; Furness, James W.; Wibowo-Teale, Andrew M.; Sun, Jianwei (January 2024, The Journal of Chemical Physics)

   In density-functional theory, the exchange–correlation (XC) energy can be defined exactly through the coupling-constant (λ) averaged XC hole n̄xc(r,r′), representing the probability depletion of finding an electron at r′ due to an electron at r. Accurate knowledge of n̄xc(r,r′) has been crucial for developing XC energy density-functional approximations and understanding their performance for molecules and materials. However, there are very few systems for which accurate XC holes have been calculated since this requires evaluating the one- and two-particle reduced density matrices for a reference wave function over a range of λ while the electron density remains fixed at the physical (λ = 1) density. Although the coupled-cluster singles and doubles (CCSD) method can yield exact results for a two-electron system in the complete basis set limit, it cannot capture the electron–electron cusp using finite basis sets. Focusing on Hooke’s atom as a two-electron model system for which certain analytic solutions are known, we examine the effect of this cusp error on the XC hole calculated using CCSD. The Lieb functional is calculated at a range of coupling constants to determine the λ-integrated XC hole. Our results indicate that, for Hooke’s atoms, the error introduced by the description of the electron–electron cusp using Gaussian basis sets at the CCSD level is negligible compared to the basis set incompleteness error. The system-, angle-, and coupling-constant-averaged XC holes are also calculated and provide a benchmark against which the Perdew–Burke–Ernzerhof and local density approximation XC hole models are assessed. 

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4. Free-energy orbital-free density functional theory: recent developments, perspective, and outlook

   https://doi.org/10.1088/2516-1075/adadd4  

   Karasiev, Valentin V; Hilleke, Katerina P; Trickey, S B (February 2025, Electronic Structure)

   Abstract By summarizing the constraint-based development of orbital-free free-energy density functional approximations, we provide a perspective on progress over the last 15 years, the limitations of existing functionals, and the challenges awaiting resolution. We outline the chronology of the development of noninteracting and exchange-correlation free-energy orbital-free functionals and summarize the theoretical basis of existing local density approximation, second-order approximation, generalized gradient approximation (GGA), and meta-GGAs. We discuss limitations and challenges such as problems with thermodynamic derivatives, free-energy nonadditivity and the closely related issue of all-electron versus valence-only local pseudo-potential performance. 

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5. The Lieb-Oxford Lower Bounds on the Coulomb Energy, Their Importance to Electron Density Functional Theory, and a Conjectured Tight Bound on Exchange

   John P. Perdew; Jianwei Sun (July 2022, The Elliott Lieb Anniversary Volume)

   M. Lewin, Rupert L. (Ed.)

   Abstract: Lieb and Oxford (1981) derived rigorous lower bounds, in the form of local functionals of the electron density, on the indirect part of the Coulomb repulsion energy. The greatest lower bound for a given electron number N depends monotonically upon N, and the N→∞ limit is a bound for all N. These bounds have been shown to apply to the exact density functionals for the exchange- and exchange-correlation energies that must be approximated for an accurate and computationally efficient description of atoms, molecules, and solids. A tight bound on the exact exchange energy has been derived therefrom for two-electron ground states, and is conjectured to apply to all spin-unpolarized electronic ground states. Some of these and other exact constraints have been used to construct two generations of non-empirical density functionals beyond the local density approximation: the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the strongly constrained and appropriately normed (SCAN) meta-GGA. 

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