Publications

On the Jacob’s Ladder of density functional approximations to the exchangecorrelation energy, each higher rung adds another local ingredient which can be used to satisfy further exact constraints. The rst rung is the local density approximation. The second rung or generalized gradient approximation (GGA) adds the gradient of the density. The third rung or meta-GGA further adds the Kohn-Sham orbital kinetic energy density. I will review the nonempirical construction of a recent meta-GGA [1] and its mostly-successful numerical tests for molecules [2,3] and solids [4].

\nFemtosecond transient absorption spectroscopy is performed to assess bridge effects on energy transfer and charge separation in molecular junctions. A short, conjugated bridge can facilitate charge separation from both donor and acceptor, whereas in longer bridges charge separation only occurs from the excited donor.\n

Density functional theory (DFT) is a widespread and effective tool in electronic structure calculations for ground-state electron systems. Its success has prompted exploration into the use of DFT for noncollective excited states. The delta self-consistent field (ΔSCF) method allows for the extension of DFT to excited-state energies by restricting the Kohn-Sham orbital occupations, producing an excited-state electron density, and then computing its energy. In this paper, we examine the performance of the LSDA, PBE generalized-gradient approximation (GGA), and SCAN/r²SCAN meta-GGA for the excitation energies of several important systems. We consider the energies of atoms with atomic numbers 1-18. For the hydrogen atom, where we use the exact electron density and have no multiplet splitting, we find significant improvement up the ladder from LSDA to PBE to SCAN. For the uniform gas, we find an effective mass different from the bare mass only with r²SCAN. We split the case of multielectron atoms into non-Aufbau excitations, where the highest-energy electron is excited to the lowest state in the next <i>nl</i> subshell (where accuracy is least limited by available basis sets), and spin-flip excitations, where the spin of an electron is flipped, leading to a higher energy state of the same <i>nl</i> configuration. We find reasonably accurate approximate excitation energies, except for the spin-flip cases where the auxiliary noninteracting wave function of a non-Hund's-rule spin state is not well described by a single determinant, but the single-determinant results are improved by spin purification.

The random phase approximation (RPA) stands on the top rung of the ladder of ground-state density functional approximations. The simple or direct RPA has been found to predict accurately many isoelectronic energy differences. A nonempirical local or semilocal correction to this direct RPA leaves isoelectronic energy differences almost unchanged, while improving total energies, ionization energies, etc., but fails to correct the RPA underestimation of molecular atomization energies. Direct RPA and its semilocal correction may miss part of the middle-range multicenter nonlocality of the correlation energy in a molecule. Here we propose a fully nonlocal, hybrid-functional-like addition to the semilocal correction. The added full nonlocality is important in molecules, but not in atoms. Under uniform-density scaling, this fully nonlocal correction scales like the second-order-exchange contribution to the correlation energy, an important part of the correction to direct RPA, and like the semilocal correction itself. For the atomization energies of ten molecules, and with the help of one fit parameter, it performs much better than the elaborate second-order screened exchange correction.

Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approximate density functional theory as symmetry-broken spin densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-density or charge-density waves at nonzero wavevector. In this sense, an approximate density functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not. The examples discussed here include the stretched H₂ molecule, antiferromagnetic solids, and the static charge-density wave/Wigner crystal phase of a low-density jellium. Time-dependent density functional theory is used to show quantitatively that the static charge-density wave is a soft plasmon. More precisely, the frequency of a related density fluctuation drops to zero, as found from the frequency moments of the spectral function, calculated from a recent constraint-based wavevector- and frequency-dependent jellium exchange-correlation kernel.

The density-gradient expansion of the fermion exchange hole is analyzed in real space. Unlike the local-density approximation, the second-order gradient-expansion approximation is found to violate two important properties of the exact hole: The exact hole is negative everywhere, and represents a deficit of one electron. Imposition of these exact constraints leads to an accurate new density functional for the exchange energy. Residual errors in the exchange energy for atoms are about 1% of this quantity. The new functional approximation may be generalized to include correlation.

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from metageneralized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations.

Unlike the local density approximation (LDA) and the generalized gradient approximation (GGA), calculations with meta-generalized gradient approximations (meta-GGA) are usually done according to the generalized Kohn-Sham (gKS) formalism. The exchange-correlation potential of the gKS equation is non-multiplicative, which prevents systematic comparison of meta-GGA bandstructures to those of the LDA and the GGA. We implement the optimized effective potential (OEP) of the meta-GGA for periodic systems, which allows us to carry out meta-GGA calculations in the same KS manner as for the LDA and the GGA. We apply the OEP to several meta-GGAs, including the new SCAN functional [Phys. Rev. Lett. 115, 036402 (2015)]. We find that the KS gaps and KS band structures of meta-GGAs are close to those of GGAs. They are smaller than the more realistic gKS gaps of meta-GGAs, but probably close to the gaps in the exact KS band structure. The well-known grid sensitivity of meta-GGAs is much more severe in OEP calculations.

The energy of a metallic crystal is expressed as a sum of volume, surface, and curvature terms. The fully self-consistent solution of a simplified problem shows that, in the absence of shell-structure effects, this expression can be accurate even for atomic-scale properties. Thus the liquid-drop model, originally developed for finite systems (nuclei), may actually be more appropriate for infinite ones (metals). First applications are made to the face dependence of the surface energy and to monovacancy-formation and cohesive energies. Predictions of the model may be tested by experiment or by fully self-consistent Kohn-Sham calculations.

The SCAN (strongly constrained and appropriately normed) meta-generalized gradient approximation (meta-GGA) for the exchange–correlation energy was constructed to satisfy 17 exact mathematical constraints and to fit non-bonded, normally correlated appropriate norms. It provides an excellent predictive description of the ground-state energies and electron spin densities of normally correlated systems in the absence of strong self-interaction error as in most sp atoms and covalent molecules at equilibrium geometries. A good self-interaction-corrected SCAN would be exact in all one-electron regions of space, without degrading SCAN’s accuracy in many-electron regions. In other words, it should be accurate for nearly all normally correlated systems. Is it possible that such a self-interaction corrected SCAN would also reliably describe the energetic effects of strong correlation through symmetry breaking, thus killing two birds with one stone? An extreme symmetry-broken limit is semi-classical, with a separated blob of one-electron density for each electron, and the approach to this limit can only be described correctly by a self-interaction-free density functional. This article discusses these hidden connections and speculates on the future possibility of a much more reliable and accurate Kohn–Sham density functional theory.