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...

We construct exact Kohn-Sham potentials for the ensemble density-functional theory (EDFT) from the ground and excited states of helium. The exchange-correlation (XC) potential is compared with the quasi-localdensity approximation and both single-determinant and symmetry-eigenstate ghost-corrected exact exchange approximations. Symmetry-eigenstate Hartree exchange recovers distinctive features of the exact XC potential and is used to calculate the correlation potential. Unlike the exact case, excitation energies calculated from these approximations depend on ensemble weight, and it is shown that only the symmetry-eigenstate method produces an ensemble derivative discontinuity. Differences in asymptotic and near-ground-state behavior of exact and approximate XC potentials are discussed in the context of producing accurate optical gaps.

A very specific ensemble of ground and excited states is shown to yield an exact formula for any excitation energy as a simple correction to the energy difference between orbitals of the KohnSham ground state. This alternative scheme avoids either the need to calculate many unoccupied levels as in time-dependent density functional theory (TDDFT) or the need for many self-consistent ensemble calculations. The symmetry-eigenstate Hartree-exchange (SEHX) approximation yields results comparable to standard TDDFT for atoms. With this formalism, SEHX yields approximate double-excitations, which are missed by adiabatic TDDFT.The Hohenberg-Kohn (HK) theorem [1-4] of groundstate density-functional theory (DFT) [1,5] has several parts. The most-used in practice is the establishment of an exact density functional, F [n], whose minimum yields the exact ground-state density and energy of a given system. Almost all practical calculations use the KohnSham (KS) scheme [5] to minimize F with an approximation to the small exchange-correlation contribution, E XC [n]. In fact, many properties of interest in a modern chemical or materials calculation can be extracted from knowledge of the ground-state energy as a function of nuclear coordinates, or in response to a perturbing field.However, except under very special circumstances, most optical excitation frequencies cannot be deduced. Hence there has always been interest in extending ground-state DFT to include such excitations. Moreover, another part of the HK theorem guarantees that such frequencies (and all properties) are indeed functionals of the ground-state density. In recent years, linearresponse time-dependent DFT (TDDFT) [6][7][8][9][10] has become a popular route for extracting low-lying excitation energies of molecules, because of its unprecedented balance of accuracy with computational speed [11]. For significantly sized molecules, more CPU time will be expended on a geometry optimization than a single TDDFT calculation on the optimized geometry.However, while formally exact, TDDFT with standard approximations is far from perfect. If the unknown exchange-correlation (XC) kernel of TDDFT is approximated by its zero-frequency (and hence ground-state) limit, no multiple excitations survive [11]. While a useful work-around exists for cases where a double is close to a single excitation [12,13], there is as yet no simple and efficient general procedure for extracting double excitations within adiabatic TDDFT [14].Ensemble DFT (EDFT) [15,16] applies the principles of ground-state DFT to a convex ensemble of the lowest M levels of a system, for which a KS system can be defined [17]. EDFT is formally exact, but practical calculations require approximations, and initial attempts yielded disappointing results [18]. Accuracy is greatly improved when so-called "ghost interactions" between distinct states are removed from the approximations [19]. EDFT remains an active research area because, being variational, it should not suffer from some of the limitations of standard TDDFT. Rece...

A new method for extracting ensemble Kohn-Sham potentials from accurate excited state densities is applied to a variety of two-electron systems, exploring the behavior of exact ensemble density functional theory. The issue of separating the Hartree energy and the choice of degenerate eigenstates is explored. A new approximation, spin eigenstate Hartree-exchange, is derived. Exact conditions that are proven include the signs of the correlation energy components and the asymptotic behavior of the potential for small weights of the excited states. Many energy components are given as a function of the weights for two electrons in a one-dimensional flat box, in a box with a large barrier to create charge transfer excitations, in a three-dimensional harmonic well (Hooke's atom), and for the He atom singlet-triplet ensemble, singlet-triplet-singlet ensemble, and triplet bi-ensemble.