DFT: A Theory Full of Holes?

Abstract

This article is a rough, quirky overview of both the history and present state of the art of density functional theory. The field is so huge that no attempt to be comprehensive is made. We focus on the underlying exact theory, the origin of approximations, and the tension between empirical and non-empirical approaches. Many ideas are illustrated on the exchange energy and hole. Features unique to this article include how approximations can be systematically derived in a non-empirical fashion and a survey of warm dense matter. The number of DFT citations has exploded (as have ab initio methods). PBE is the number of citations of Ref.[28], and B3LYP of Ref. [24]. Dark indicates papers using either of these approximations without citing the original papers, while other is all other DFT papers. All numbers are estimates. Contrast with Fig. 1 of Ref. [7], which missed almost 2/3 of these. I. WHAT IS THIS ARTICLE ABOUT?The popularity of density functional theory (DFT) as an electronic structure method is unparalleled, with applications that stretch from biology[1] to exoplanets [2]. However, its quirks of logic and diverse modes of practical application have led to disagreements on many fronts and from many parties. Developers of DFT are guided by many different principles, while applied practitioners (a.k.a. users) are suspicious of DFT for reasons both practical (how can I pick a functional with so many choices? [3]) and cultural (with so many choices, why would I call this first-principles? ).A modern DFT calculation[4] begins with the purchase of a computer, which might be as small as a laptop, and a quantum chemical code. Next, a basis set is chosen, which assigns predetermined functions to describe the electrons on each atom of the molecule being studied. Finally, a DFT approximation to something called the exchange-correlation energy (XC) is chosen, and the code starts running. For each guess of the nuclear positions, the code calculates an approximate energy [4]. A geometry optimization should find the minimum energy configuration. With variations on this theme [5,6], one can read out all molecular geometries, dissociation energies, reaction barriers, vibrational frequencies, etc. A modern desktop may do a calculation for a 100-atom system within a day. A careful user will repeat the most important parts of the calculation with bigger basis sets, to check that answers don't change significantly. II. WHERE DOES DFT COME FROM?Although DFT's popularity has skyrocketed since applications to chemistry became useful and routine, its roots stretch back much further [7][8][9].Ye olde DFT: Developed without reference to the Schrödinger equation [10], Thomas-Fermi (TF) theory [11][12][13] was the first DFT. It is pure DFT, relying only on the electronic density, ρ(r), as input. The kinetic energy was approximated as that of a uniform electron gas, while the repulsion of the electrons was modeled with the classical electrostatic Coulomb repulsion, again depending only on the electronic density as an input.Mixing in orbit...

The Importance of Being Inconsistent

“…All practical calculations generalize the preceding formulas for arbitrary spin using spin-DFT [BH72]. The computational ease of DFT calculations relative to more accurate wavefunction methods usually allows much larger systems to be calculated 2 , leading to DFT's immense popularity today [PGB15]. However, all these approximations fail in the paradigm case of stretched H 2 , the simplest example of a strongly correlated system [B01].…”

“…Density functional theory(DFT) is used in more than 30,000 scientific papers per year [PGB15]. Most of these applications are routine, where the calculation yields sufficiently accurate results as to provide insight into some scientific or technological problem.…”

Does the gradient‐regulated connection improve the description of correlated metal bond properties?

“…The Kohn–Sham approach of density functional theory (KS‐DFT) is an unmissable player in computational chemistry . Throughout the years, its popularity has never stopped growing for many reasons.…”

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

“…The mainstream DFT (i.e., Kohn–Sham DFT (KS‐DFT)) is a slight alteration of the original work of Hohenberg and Kohn (HK‐DFT) where it was proved that the ground state of any many‐electron system is completely characterized by its density and the energy functional which permits the system to attain its minimum at the density corresponding to the ground state. However, representing the contribution of the kinetic energy as a density functional (KEDF) ( ) where is the kinetic energy density (KED) has proven to be challenging as the accuracy and applicability of the presently proposed KEDF are generally not sufficient for reliable calculations . So, Kohn and Sham (KS‐DFT) suggested an approximate approach where the “orbitals” are reintroduced such that the sum of the orbitals densities equals to the exact density of the real system and the kinetic energy is defined as the kinetic energy of the introduced “fictitious” system.…”

“…where t x ð Þ is the kinetic energy density (KED) has proven to be challenging as the accuracy and applicability of the presently proposed KEDF are generally not sufficient for reliable calculations. [3,[6][7][8][9][10] So, Kohn and Sham (KS-DFT) suggested an approximate approach where the "orbitals" are reintroduced such that the sum of the orbitals densities equals to the exact density of the real system and the kinetic energy is defined as the kinetic energy of the introduced "fictitious" system. Computation-wise, this results in a conversion of the problem from 3-dimensional (3D) to 3N-dimensional where N orbitals are determined by solving the governing N23D equations self-consistently and N is the number of electrons in the calculations.…”