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1. Reformulation of thermally assisted-occupation density functional theory in the Kohn–Sham framework

   https://doi.org/10.1063/5.0087012  

   Yeh, Shu-Hao; Yang, Weitao; Hsu, Chao-Ping (May 2022, The Journal of Chemical Physics)

   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 (KS) density, and total ground state energy is expressed in the normal KS form with a static correlation energy formulated in terms of the KS orbitals. In post-SCF calculations with rTAO functionals, an efficient energy scanning to quantitatively determine θ 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. 

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2. Correlation energy of the uniform electron gas determined by ground-state conditional probability density functional theory

   https://doi.org/10.1103/PhysRevB.105.165143  

   Perchak, Dennis; McCarty, Ryan J.; Burke, Kieron (April 2022, Physical review)

   Conditional-probability density functional theory (CP-DFT) is a formally exact method for finding correlation energies from Kohn-Sham DFT without evaluating an explicit energy functional. We present details on how to generate accurate exchange-correlation energies for the ground-state uniform gas. We also use the exchange hole in a CP antiparallel spin calculation to extract the high-density limit. We give a highly accurate analytic solution to the Thomas-Fermi model for this problem, showing its performance relative to Kohn-Sham and it may be useful at high temperatures. We explore several approximations to the CP potential. Results are compared to accurate parametrizations for both exchange-correlation energies and holes. 

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3. A deep learning framework to emulate density functional theory

   https://doi.org/10.1038/s41524-023-01115-3  

   del Rio, Beatriz G.; Phan, Brandon; Ramprasad, Rampi (August 2023, npj Computational Materials)

   Abstract Density functional theory (DFT) has been a critical component of computational materials research and discovery for decades. However, the computational cost of solving the central Kohn–Sham equation remains a major obstacle for dynamical studies of complex phenomena at-scale. Here, we propose an end-to-end machine learning (ML) model that emulates the essence of DFT by mapping the atomic structure of the system to its electronic charge density, followed by the prediction of other properties such as density of states, potential energy, atomic forces, and stress tensor, by using the atomic structure and charge density as input. Our deep learning model successfully bypasses the explicit solution of the Kohn-Sham equation with orders of magnitude speedup (linear scaling with system size with a small prefactor), while maintaining chemical accuracy. We demonstrate the capability of this ML-DFT concept for an extensive database of organic molecules, polymer chains, and polymer crystals. 

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4. Towards a density functional theory of molecular fragments. What is the shape of atoms in molecules?

   https://doi.org/http://dx.doi.org/10.18257/raccefyn.960  

   Chavez, Victor; Wasserman, Adam (March 2020, Revista de la Academia Colombiana de Ciencias Exactas Físicas y Naturales)

   In some sense, quantum mechanics solves all the problems in chemistry: The only thing one has to do is solve the Schrödinger equation for the molecules of interest. Unfortunately, the computational cost of solving this equation grows exponentially with the number of electrons and for more than ~100 electrons, it is impossible to solve it with chemical accuracy (~ 2 kcal/mol). The Kohn-Sham (KS) equations of density functional theory (DFT) allow us to reformulate the Schrödinger equation using the electronic probability density as the central variable without having to calculate the Schrödinger wave functions. The cost of solving the Kohn-Sham equations grows only as N3, where N is the number of electrons, which has led to the immense popularity of DFT in chemistry. Despite this popularity, even the most sophisticated approximations in KS-DFT result in errors that limit the use of methods based exclusively on the electronic density. By using fragment densities (as opposed to total densities) as the main variables, we discuss here how new methods can be developed that scale linearly with N while providing an appealing answer to the subtitle of the article: What is the shape of atoms in molecules 

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5. 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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