An interesting fundamental problem in density-functional theory of electronic structure of matter is to construct the exact Kohn-Sham (KS) potential for a given density. The exact potential can then be used to assess the accuracy of approximate functionals and the corresponding potentials. Besides its practical usefulness, such a construction by itself is a challenging inverse problem. Over the past three decades, many seemingly disjoint methods have been proposed to solve this problem. We show that these emanate from a single algorithm based on the Euler equation for the density. This provides a mathematical foundation for all different density-based methods that are used to construct the KS system from a given density and reveals their universal character.

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An accurate semianalytic wavefunction is proposed for the Hookium and two‐electron atoms for varying strength of αVee(0≤α≤1) where α is the strength parameter and Vee is coulomb interaction between two electrons. The wavefunction leads to energies that are as accurate as those from the Coupled cluster singles and doubles (CCSD) calculations. Using this wavefunction, we construct the external potential vextα such that the density of the system remains unchanged as α is varied. The work thus gives a unified picture of adiabatic connection for these systems based on an easy to use wavefunction and complements the past investigations done in this direction. Using the potential obtained, we explicitly calculate the energy of the corresponding positive ions and show that the chemical potential—calculated as the difference between the energies of the two‐electron system and its positive ion—is equal to the experimental ionization energy and remains unchanged as α is varied. Furthermore, using total energies E(α) of these systems as a function of α, we provide a new perspective into a variety of hybrid functionals.

Over the past few years it has been pointed out that direct inversion of accurate but approximate ground state densities leads to Kohn–Sham exchange–correlation (xc) potentials that can differ significantly from the exact xc potential of a given system. On the other hand, the corresponding wavefunction based construction of exchange-correlation potential as done by Baerends et al and Staroverov et al obviates such problems and leads to potentials that are very close to the true xc potential. In this paper, we provide an understanding of why the wavefunction based approach gives the exchange–correlation potential accurately. Our understanding is based on the work of Levy, Perdew and Sahni (LPS) who gave an equation for the square root of density (density amplitude) and the expression for the associated effective potential in the terms of the corresponding wavefunction. We show that even with the use of approximate wavefunctions the LPS expression gives accurate effective and exchange–correlation potentials. Based on this we also identify the source of difference between the potentials obtained from a wavefunction and those given by the inversion of the associated density. Finally, we suggest exploring the possibility of obtaining accurate ground-state density from an approximate wavefunction for a system by making use of the LPS effective potential.

A systematic way of improving exchange-correlation energy functionals of density functional theory has been to make them satisfy more and more exact relations. Starting from the initial GGA functionals, this has culminated into the recently proposed SCAN(Strongly constrained and appropriately normed) functional that satisfies several known constraints and is appropriately normed. The ultimate test for the functionals developed is the accuracy of energy calculated by employing them. In this paper, we test these exchange-correlation functionals −the GGA hybrid functionals B3LYP and PBE0, and the meta-GGA functional SCAN− from a different perspective. We study how accurately these functionals reproduce the exchange-correlation energy when electron-electron interaction is scaled as αV ee with α varying between 0 and 1. Our study reveals interesting comparison between these functionals and the associated difference T c between the interacting and the non-interacting kinetic energy for the same density.