Gradient-corrected density functionals provide a common tool for electronic structure calculations in quantum chemistry and condensed matter physics. This article explains why local and semilocal approximations work for the exchange-correlation energy. We demonstrate the high accuracy of the local spin-density (LSD) approximation for the on-top pair density, which provides the missing link between real atoms and molecules and the uniform electron gas. Special attention is devoted to the leading correction to exchange in the high-density (or weakly correlated) limit. We give an improved analytic expression for the on-top pair density in the uniform electron gas, calculating its spin-polarization dependence exactly in the high-density limit. We find the exact form of the gradient expansion for the on-top pair density, using Levy’s scaling of the interacting wave function. We also discuss the importance of system averaging, which unweights spatial regions where the density varies most rapidly. We show how the depth of the on-top hole correlates with the degree of locality of the exchange-correlation energy. Finally, we discuss how well fully nonlocal approximations (weighted-density, self-interaction correction, and hybrid-exchange) reproduce the on-top hole.
For accurate thermochemical tests of electronic structure theory, accurate true anharmonic zero-point vibrational energies ZPVE(true) are needed. We discuss several possibilities to extract this information for molecules from density functional or wave function calculations and/or available experimental data: (1) Empirical universal scaling of density-functional-calculated harmonic ZPVE(harm)s, where we find that polyatomics require smaller scaling factors than diatomics. (2) Direct density-functional calculation by anharmonic second-order perturbation theory PT2. (3) Weighted averages of harmonic ZPVE(harm) and fundamental ZPVE(fund) (from fundamental vibrational transition frequencies), with weights (3/4, 1/4) for diatomics and (5/8,3/8) for polyatomics. (4) Experimental correction of the PT2 harmonic contribution, i.e., the estimate ZPVE(true)PT2 + (ZPVE(fund)expt - ZPVE(fund)PT2) for ZPVE(true). The (5/8,3/8) average of method 3 and the additive correction of method 4 have been proposed here. For our database of experimental ZPVE(true), consisting of 27 diatomics and 8 polyatomics, we find that methods 1 and 2, applied to the popular B3LYP and the nonempirical PBE and TPSS functionals and their one-parameter hybrids, yield polyatomic errors on the order of 0.1 kcal/mol. Larger errors are expected for molecules larger than those in our database. Method 3 yields errors on the order of 0.02 kcal/mol, but requires very accurate (e.g., experimental, coupled cluster, or best-performing density functional) input harmonic ZPVE(harm). Method 4 is the best-founded one that meets the requirements of high accuracy and practicality, requiring as experimental input only the highly accurate and widely available ZPVE(fund)expt and producing errors on the order of 0.05 kcal/mol that are relatively independent of functional and basis set. As a part of our study, we also test the ability of the density functionals to predict accurate equilibrium bond lengths and angles for a data set of 21 mostly polyatomic molecules (since all calculated ZPVEs are evaluated at the correspondingly calculated molecular geometries).
To evaluate the possible role of cytokines in human immunodeficiency virus (HIV)-associated muscular disorders, we performed immunocytochemistry for interleukin-1 alpha, -1 beta, and -6 and tumor necrosis factor-alpha on frozen muscle biopsy specimens from HIV-infected patients with various myopathies (HIV polymyositis in 5, HIV-wasting syndrome in 5, zidovudine myopathy in 10) and from seronegative individuals (normal muscle in 2, mitochondrial cytopathies in 10). The HIV-infected patients showed positive reactivities in vessels (interleukin-1) and in inflammatory cells (mainly interleukin-1 and tumor necrosis factor-alpha), including perivascular hemosiderin-laden macrophages in 5 patients. In zidovudine myopathy, a majority of AZT fibers (i.e., ragged-red fibers with marked myofibrillar changes) showed mild to marked expression of interleukin-1. Expression of interleukin-1 in the other mitochondrial myopathies was much weaker. Interleukin-1 beta messenger RNA was demonstrated in muscle fibers by in situ hybridization, implying that interleukin-1 was produced in muscle cells. Immunoelectron microscopy showed that interleukin-1 alpha was mainly bound to mitochondrial membranes in AZT fibers. Proinflammatory and destructive effects of the studied cytokines might be responsible for several myopathological changes observed in HIV-infected patients, including inflammation and hemosiderin deposits in muscle tissue, and prominent myofibrillar breakdown in AZT fibers.
From a global perspective, the density of an atom is strongly inhomogeneous and not at all like the density of a uniform or nearly-uniform electron gas. But, from the semi-local or myopic perspective of standard density functional approximations to the exchange-correlation energy,it is not so easy to tell an atom from an electron gas. We address the following problem: Given the ground-state electron density n and orbital kinetic energy density in the neighborhood of a point r, can we construct an "inhomogeneity index" w(r) which approaches zero for weakly-inhomogeneous densities and unity for strongly-inhomogeneous ones? The solution requires not only the usual local ingredients of a meta-generalized gradient approximation (n,rn,r2n, ),but also r and r2 . The inhomogeneity index is displayed for atoms, and for model densities of metal surfaces and bulk metals. Scaling behavior and a possible application to functional interpolation are discussed.
We discuss six questions related to the recent “strongly constrained and appropriately normed” (SCAN) meta‐generalized gradient approximation (meta‐GGA): (1) When and why should a semilocal approximation to the density functional for the exchange‐correlation energy be accurate? (2) What is the right dimensionless ingredient for a meta‐GGA, and why? (3) In the construction of density functional approximations, should we satisfy more or fewer exact constraints? (4) Is there a tight lower bound on the exchange energy for all spin‐unpolarized densities? (5) Should a semilocal approx‐ imation yield any intermediate‐range van der Waals interaction? (6) Do semilocal functionals make consistent predictions for the energy differences between different molecules (and thus presumably for reaction and formation energies)? © 2016 Wiley Periodicals, Inc.
We review some recent results concerning the probability that two electrons will be found close together in any interacting electronic system, and why this probability is usually well approximated by local (LSD) and semilocal spin density functional theories. The success of these approximations for the energy in "normal" systems is explained by the usual sum rule arguments on the system- and spherically-averaged exchange-correlation hole density <n xc (u)>, coupled with the nearly correct, but not exact, behavior of these approximations as the interelectronic separation u → 0. We argue that the accuracy of the LSD on-top hole density in "normal" systems is due to its accuracy in the noninteracting, weakly-interacting, and strongly-interacting limits.
Nanostructures can be bound together at equilibrium by the van der Waals (vdW) effect, a small but ubiquitous many-body attraction that presents challenges to density functional theory. How does the binding energy depend upon the size or number of atoms in one of a pair of identical nanostructures? To answer this question, we treat each nanostructure as a whole object, not as a collection of atoms. Our calculations start from an accurate static dipole polarizability for each considered nanostructure, and an accurate equilibrium center-to-center distance for the pair (the latter from experiment or from the vdW-DF-cx functional). We consider the competition in each term -C<sub>2k</sub>/d<sup>2k</sup> (k = 3, 4, 5) of the long-range vdW series for the interaction energy, between the size dependence of the vdW coefficient C<sub>2k</sub> and that of the 2kth power of the center-to-center distance d. The damping of these vdW terms can be negligible, but in any case, it does not affect the size dependence for a given term in the absence of non-vdW binding. To our surprise, the vdW energy can be size-independent for quasi-spherical nanoclusters bound to one another by vdW interaction, even with strong nonadditivity of the vdW coefficient, as demonstrated for fullerenes. We also show that, for low-dimensional systems, the vdW interaction yields the strongest size-dependence, in stark contrast to that of fullerenes. We illustrate this with parallel planar polycyclic aromatic hydrocarbons. The size dependences of other morphologies or bonding types lie between, as shown by sodium clusters.
The physics of the inward or outward relaxation of the first plane of ions at a metal surface is explored by means of semi-self-consistent calculations for Al, Mg, and Na. The adiabatic screening response of the electron density to a shift of the first lattice plane drastically reduces the curvature of the potential in which this plane sits, and thus cannot be ignored in any qualitatively correct theory of the equilibrium lattice configuration at the surface. Because the electronic screening is nearly perfect, the calculated face-dependent surface energies and work functions are nearly independent of small displacements of the first lattice plane. Just as in the calculation of the bulk-longitudinal-phonon frequencies, there is a delicate cancellation between Madelung forces which are treated exactly and electronic forces which are approximated. Indeed a quantitatively correct theory of the surface interplanar spacing must also be detailed enough to predict the correct bulk-phonon frequencies.
