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1. Density Functional Analysis: The Theory of Density-Corrected DFT

   https://doi.org/10.1021/acs.jctc.9b00826  

   Vuckovic, Stefan; Song, Suhwan; Kozlowski, John; Sim, Eunji; Burke, Kieron (October 2019, Journal of Chemical Theory and Computation)
2. Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory

   https://doi.org/10.1039/D0FD00102C  

   Jin, Ye; Su, Neil Qiang; Chen, Zehua; Yang, Weitao (December 2020, Faraday Discussions)

   null (Ed.)

   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. 

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3. Decomposition of the Electronic Energy in Terms of Density, Density Coherence, and the Connected Part of the Two-Body Reduced Density Matrix

   https://doi.org/10.1021/acs.jctc.1c00679  

   Zhang, Dayou; Truhlar, Donald G. (September 2021, Journal of Chemical Theory and Computation)
4. Emergence of the temperature–density relation in the low-density intergalactic medium

   https://doi.org/10.1093/mnras/stae401  

   Wells, Alexandra; Robinson, David; Avestruz, Camille; Gnedin, Nickolay Y (February 2024, Monthly Notices of the Royal Astronomical Society)

   We examine the evolution of the phase diagram of the low-density intergalactic medium during the Epoch of Reionization in simulation boxes with varying reionization histories from the Cosmic Reionization on Computers project. The probability density function (PDF) of gas temperature at fixed density exhibits two clear modes: a warm and a cold temperature mode, corresponding to the gas inside and outside of ionized bubbles. We find that the transition between the two modes is ‘universal’ in the sense that its timing is accurately parametrized by the value of the volume-weighted neutral fraction for any reionization history. This ‘universality’ is more complex than just a reflection of the fact that ionized gas is warm and neutral gas is cold: it holds for the transition at a fixed value of gas density, and gas at different densities transitions from the cold to the warm mode at different values of the neutral fraction, reflecting a non-trivial relationship between the ionization history and the evolving gas density PDF. Furthermore, the ‘emergence’ of the tight temperature–density relation in the warm mode is also approximately ‘universally’ controlled by the volume-weighted neutral fraction for any reionization history. In particular, the ‘emergence’ of the temperature–density relation (as quantified by the rapid decrease in its width) occurs when the neutral fraction is 10−4 ≲ XH i ≲ 10−3 for any reionization history. Our results indicate that the neutral fraction is a primary quantity controlling the various properties of the temperature–density relation, regardless of reionization history. 

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5. Comparison of a Neutral Density Model With the SET HASDM Density Database

   https://doi.org/10.1029/2021SW002888  

   Weimer, Daniel R.; Tobiska, W. Kent; Mehta, Piyush M.; Licata, R. J.; Drob, Douglas P.; Yoshii, Jean (December 2021, Space Weather)