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From the most fundamental to the most practical side of density functional theory (DFT), Kohn–Sham inversions (iKS) can contribute to the development of functional approximations and shed light on their performance and limitations. On the one hand, iKS allows for the direct exploration of the Hohenberg–Kohn and Runge–Gross density‐to‐potential mappings that provide the foundations for DFT and time‐dependent DFT. On the other hand, iKS can guide the analysis and development of approximate exchange–correlation and noninteracting kinetic energy functionals, and diagnose their errors. iKS can also play a similar role in the development of nonadditive functionals for modern density‐based embedding methods. Various strategies to perform iKS calculations have been explored since the inception of DFT. We introduce n2v, a density‐to‐potential inversion Python module that is capable of performing the most useful and state‐of‐the‐art inversion calculations. Currently based on NumPy, n2v was developed to be easy to learn by newcomers to the field. Its structure allows for other inversion methods to be easily added. The code offers a general interface that gives the freedom to use different software packages in the computational molecular sciences (CMS) community, and the current release supports the Psi4 and PySCF packages. Six inversion methods have been implemented into n2v and are reviewed here along with detailed numerical illustrations on molecules with numbers of electrons ranging from ~10 to ~100. This article is categorized under:Software > Quantum ChemistryElectronic Structure Theory > Density Functional Theory [ABSTRACT FROM AUTHOR]

The accuracy of charge‐transfer excitation energies, solvatochromic shifts, and other environmental effects calculated via various density‐embedding techniques depend critically on the approximations employed for the nonadditive noninteracting kinetic energy functional, Tsnadn. Approximating this functional remains an important challenge in electronic‐structure theory. To assist in the development and testing of approximations for Tsnadn, we derive two virial relations for fragments in molecules. These establish separate connections between the nonadditive kinetic energies of the noninteracting and interacting systems of electrons, and quantities such as the electron‐nuclear attraction forces, the partition (or embedding) energy and potential, and the Kohn‐Sham potentials of the system and its parts. We numerically verify both relations on diatomic molecules. [ABSTRACT FROM AUTHOR]

The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems. [ABSTRACT FROM AUTHOR]

Abstract. We construct an internally-consistent density-functional theory valid for noninteger electron numbers N by precise definition of a density functional that is continuous in N. In this theory, charge transfer between the atoms of a heteronuclear diatomic molecule, which have been separated adiabatically to infinity, is avoided because the hardness for fractional occupation of a single HOMO spin-orbital is negative. This N-continuous density functional makes possible a variational theory of "atoms" in "molecules" that exactly decomposes the molecular electron density into a sum of contributions from its parts. The parts are treated as though isolated. That theory, in turn, gives a deep foundation to the chemical reactivity theory provided that the hardness of entities with vanishing spin density is positive, as argued to be the case here. This transition from negative to positive hardness closely parallels the transition from the Heitler-London to the Hund-Mulliken picture of molecular bonding. [ABSTRACT FROM AUTHOR]

The ground‐state energy and density of 4 low‐energy conformations of the formic acid dimer were calculated via partition density functional theory (PDFT). The differences between isolated and PDFT monomer densities display similar deformation patterns for primary and secondary hydrogen bonds (HBs) among all 4 dimers. In contrast, the partition potential shows no transferable features in the bonding regions. These observations highlight the global character of the partition potential and the cooperative effect that occurs when a dimer is bound via more than 1 HB. We also provide numerical confirmation of the intuitive (but unproven) observation that fragment deformation energies are larger for systems with larger binding energies. Partition potentials for the 4 lowest‐energy conformations of the formic acid dimer. Guess which one corresponds to the global minimum.. [ABSTRACT FROM AUTHOR]