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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. 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_P; 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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3. Machine-learning Kohn–Sham potential from dynamics in time-dependent Kohn–Sham systems

   https://doi.org/10.1088/2632-2153/ace8f0  

   Yang, Jun; Whitfield, James (August 2023, Machine Learning: Science and Technology)

   Abstract The construction of a better exchange-correlation potential in time-dependent density functional theory (TDDFT) can improve the accuracy of TDDFT calculations and provide more accurate predictions of the properties of many-electron systems. Here, we propose a machine learning method to develop the energy functional and the Kohn–Sham potential of a time-dependent Kohn–Sham (TDKS) system is proposed. The method is based on the dynamics of the Kohn–Sham system and does not require any data on the exact Kohn–Sham potential for training the model. We demonstrate the results of our method with a 1D harmonic oscillator example and a 1D two-electron example. We show that the machine-learned Kohn–Sham potential matches the exact Kohn–Sham potential in the absence of memory effect. Our method can still capture the dynamics of the Kohn–Sham system in the presence of memory effects. The machine learning method developed in this article provides insight into making better approximations of the energy functional and the Kohn–Sham potential in the TDKS system. 

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4. Hardness of molecules and bandgap of solids from a generalized gradient approximation exchange energy functional

   https://doi.org/10.1063/5.0096678  

   Carmona-Espíndola, Javier; Flores, Anaid; Gázquez, José L; Vela, Alberto; Trickey, S B (September 2022, The Journal of Chemical Physics)

   The deviations from linearity of the energy as a function of the number of electrons that arise with current approximations to the exchange–correlation (XC) energy functional have important consequences for the frontier eigenvalues of molecules and the corresponding valence-band maxima for solids. In this work, we present an analysis of the exact theory that allows one to infer the effects of such approximations on the highest occupied and lowest unoccupied molecular orbital eigenvalues. Then, we show the importance of the asymptotic behavior of the XC potential in the generalized gradient approximation (GGA) in the case of the NCAPR functional (nearly correct asymptotic potential revised) for determining the shift of the frontier orbital eigenvalues toward the exact values. Thereby we establish a procedure at the GGA level of refinement that allows one to make a single calculation to determine the ionization potential, the electron affinity, and the hardness of molecules (and its solid counterpart, the bandgap) with an accuracy equivalent to that obtained for those properties through energy differences, a procedure that requires three calculations. For solids, the accuracy achieved for the bandgap lies rather close to that which is obtained through hybrid XC energy functionals, but those also demand much greater computational effort than what is required with the simple NCAPR GGA calculation. 

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5. How the choice of exchange–correlation functional affects DFT-based simulations of the hydrated electron

   https://doi.org/10.1063/5.0253369  

   Borrelli, William R; Liu, Xiaoyan; Schwartz, Benjamin J (March 2025, The Journal of Chemical Physics)

   Hydrated electrons are anionic species that are formed when an excess electron is introduced into liquid water. Building an understanding of how hydrated electrons behave in solution has been a long-standing effort of simulation methods, of which density functional theory (DFT) has come to the fore in recent years. The ability of DFT to model the reactive chemistry of hydrated electrons is an attractive advantage over semi-classical methodologies; however, relatively few density functional approximations (DFAs) have been used for the hydrated electron simulations presented in the literature. Here, we simulate hydrated electron systems using a series of exchange–correlation (XC) functionals spanning Jacob’s ladder. We calculate a variety of experimental and other observables of the hydrated electron and compare the XC functional dependence for each quantity. We find that the formation of a stable localized hydrated electron is not necessarily limited to hybrid XC functionals and that some hybrid functionals produce delocalized hydrated electrons or electrons that react with the surrounding water at an unphysically fast rate. We further characterize how different DFAs impact the solvent structure and predicted spectroscopy of the hydrated electron, considering several methods for calculating the hydrated electron’s absorption spectrum for the best comparison between structures generated using different density functionals. None of the dozen or so DFAs that we investigated are able to correctly predict the hydrated electron’s spectroscopy, vertical detachment energy, or molar solvation volume. 

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