Most realistic calculations of moderately correlated materials begin with a ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT is used in about 40,000 scientific papers each year, the fundamental underpinnings are not widel
y appreciated. In this chapter, we analyze the inherent characteristics of DFT in their simplest form, using the asymmetric Hubbard dimer as an illustrative model. We begin by working through the core tenets of DFT, explaining what the exact ground-state density functional yields and does not yield. Given the relative simplicity of the system, almost all properties of the exact exchange-correlation functional are readily visualized and plotted. Key concepts include the Kohn-Sham scheme, the behavior of the XC potential as correlations become very strong, the derivative discontinuity and the difference between KS gaps and true charge gaps, and how to extract optical excitations using time-dependent DFT. By the end of this text and accompanying exercises, the reader will improve their ability to both explain and visualize the concepts of DFT, as well as better understand where others may go wrong.
We argue that the success of DFT can be understood in terms of a semiclassical expansion around a very specific limit. This limit was identified long ago by Lieb and Simon for the total electronic energy of a system. This is a universal limit of all
electronic structure: atoms, molecules, and solids. For the total energy, Thomas-Fermi theory becomes relatively exact in the limit. The limit can also be studied for much simpler model systems, including non-interacting fermions in a one-dimensional well, where the WKB approximation applies for individual eigenvalues and eigenfunctions. Summation techniques lead to energies and densities that are functionals of the potential. We consider several examples in one dimension (fermions in a box, in a harmonic well, in a linear half-well, and in the Poschl-Teller well. The effects of higher dimension are also illustrated with the three-dimensional harmonic well and the Bohr atom, non-interacting fermions in a Coulomb well. Modern density functional calculations use the Kohn-Sham scheme almost exclusively. The same semiclassical limit can be studied for the Kohn-Sham kinetic energy, for the exchange energy, and for the correlation energy. For all three, the local density approximation appears to become relatively exact in this limit. Recent work, both analytic and numerical, explores how this limit is approached, in an effort to deduce the leading corrections to the local approximation. A simple scheme, using the Euler-Maclaurin summation formula, is the result of many different attempts at this problem. In very simple cases, the correction formulas are much more accurate than standard density functionals. Several functionals are already in widespread use in both chemistry and materials that incorporate these limits, and prospects for the future are discussed.
Most torsional barriers are predicted to high accuracy (about 1kJ/mol) by standard semilocal functionals, but a small subset has been found to have much larger errors. We create a database of almost 300 carbon-carbon torsional barriers, including 12
poorly behaved barriers, all stemming from Y=C-X group, where X is O or S, and Y is a halide. Functionals with enhanced exchange mixing (about 50%) work well for all barriers. We find that poor actors have delocalization errors caused by hyperconjugation. These problematic calculations are density sensitive (i.e., DFT predictions change noticeably with the density), and using HF densities (HF-DFT) fixes these issues. For example, conventional B3LYP performs as accurately as exchange-enhanced functionals if the HF density is used. For long-chain conjugated molecules, HF-DFT can be much better than exchange-enhanced functionals. We suggest that HF-PBE0 has the best overall performance.
Classical turning surfaces of Kohn-Sham potentials, separating classically-allowed regions (CARs) from classically-forbidden regions (CFRs), provide a useful and rigorous approach to understanding many chemical properties of molecules. Here we calcul
ate such surfaces for several paradigmatic solids. Our study of perfect crystals at equilibrium geometries suggests that CFRs are absent in metals, rare in covalent semiconductors, but common in ionic and molecular crystals. A CFR can appear at a monovacancy in a metal. In all materials, CFRs appear or grow as the internuclear distances are uniformly expanded. Calculations with several approximate density functionals and codes confirm these behaviors. A classical picture of conduction suggests that CARs should be connected in metals, and disconnected in wide-gap insulators. This classical picture is confirmed in the limits of extreme uniform compression of the internuclear distances, where all materials become metals without CFRs, and extreme expansion, where all materials become insulators with disconnected and widely-separated CARs around the atoms.
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 Kohn-Sham ground state. This alternative scheme avoids either t
he 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.
Above the Kondo temperature, the Kohn-Sham zero-bias conductance of an Anderson junction has been shown to completely miss the Coulomb blockade. Within a standard model for the spectral function, we deduce a parameterization for both the onsite excha
nge-correlation potential and the bias drop as a function of the site occupation that applies for all correlation strengths. We use our results to sow doubt on the common interpretation of such corrections as arising from dynamical exchange-correlation contributions.
In the usual treatment of electronic structure, all matter has cusps in the electronic density at nuclei. Cusps can produce non-analytic behavior in time, even in response to perturbations that are time-analytic. We analyze these non-analyticities in
a simple case from many perspectives. We describe a method, the s-expansion, that can be used in several such cases, and illustrate it with a variety of examples. These include both the sudden appearance of electric fields and disappearance of nuclei, in both one and three dimensions. When successful, the s-expansion yields the dominant short-time behavior, no matter how strong the external electric field, but agrees with linear response theory in the weak limit. We discuss the relevance of these results to time-dependent density functional theory.
The exact ground-state exchange-correlation functional of Kohn-Sham density functional theory yields the exact transmission through an Anderson junction at zero bias and temperature. The exact impurity charge susceptibility is used to construct the e
xact exchange-correlation potential. We analyze the successes and limitations of various types of approximations, including smooth and discontinuous functionals of the occupation, as well as symmetry-broken approaches.
Transport through an Anderson junction (two macroscopic electrodes coupled to an Anderson impurity) is dominated by a Kondo peak in the spectral function at zero temperature. The exact single-particle Kohn-Sham potential of density functional theory
reproduces the linear transport exactly, despite the lack of a Kondo peak in its spectral function. Using Bethe ansatz techniques, we calculate this potential exactly for all coupling strengths, including the cross-over from mean-field behavior to charge quantization caused by the derivative discontinuity. A simple and accurate interpolation formula is also given.
Modern density functional theory (DFT) calculations employ the Kohn-Sham (KS) system of non-interacting electrons as a reference, with all complications buried in the exchange-correlation energy (Exc). The adiabatic connection formula gives an exact
expression for Exc. We consider DFT calculations that instead employ a reference of strictly-correlated electrons. We define a decorrelation energy that relates this reference to the real system, and derive the corresponding adiabatic connection formula. We illustrate this theory in three situations, namely the uniform electron gas, Hookes atom, and the stretched hydrogen molecule. The adiabatic connection for strictly-correlated electrons provides an alternative perspective for understanding density functional theory and constructing approximate functionals.
