Publications

In the standard interpretation of spin-density functional theory, a self-consistent Kohn-Sham calculation within the local spin density (LSD) or generalized gradient approximation (GGA) leads to a prediction of the total energy E, total electron density n(r)=${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$(r)+${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$(r), and spin magnetization density m(r)=${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$(r)-${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$(r). This interpretation encounters a serious ``symmetry dilemma'' for ${\mathrm{H}}_{2}$, ${\mathrm{Cr}}_{2}$, and many other molecules. Without changing LSD or GGA calculational methods and results, we escape this dilemma through an alternative interpretation in which the third physical prediction is not m(r) but the on-top electron pair density P(r,r), a quantity more directly related to the total energy in the absence of an external magnetic field. This alternative interpretation is also relevant to antiferromagnetic solids. We argue that the nonlocal exchange-correlation energy functional, which must be approximated, is most nearly local in the alternative spin-density functional theory presented here, less so in the standard theory, and far less so in total-density functional theory. Thus, in LSD or GGA, predictions of spin magnetization densities and moments are not so robust as predictions of total density and energy. The alternative theory helps to explain the surprising accuracy of LSD and GGA energies, and suggests that the correct solution of the Kohn-Sham equations in LSD or GGA is the fully self-consistent broken-symmetry single determinant of lowest total energy.

There is current interest in the random phase approximation (RPA), a "fifth-rung" density functional for the exchange-correlation energy. RPA has full exact exchange and constructs the correlation with the help of the unoccupied Kohn-Sham orbitals. In many cases (uniform electron gas, jellium surface, and free atom), the correction to RPA is a short-ranged effect that is captured by a local spin density approximation (LSDA) or a generalized gradient approximation (GGA). Nonempirical density functionals for the correction to RPA were constructed earlier at the LSDA and GGA levels (RPA+), but they are constructed here at the fully nonlocal level (RPA++), using the van der Waals density functional (vdW-DF) of Langreth, Lundqvist, and collaborators. While they make important and helpful corrections to RPA total and ionization energies of free atoms, they correct the RPA atomization energies of molecules by only about 1 kcal/mol. Thus, it is puzzling that RPA atomization energies are, on average, about 10 kcal/mol lower than those of accurate values from experiment. We find here that a hybrid of 50% Perdew-Burke-Ernzerhof GGA with 50% RPA+ yields atomization energies much more accurate than either one does alone. This suggests a solution to the puzzle: While the proper correction to RPA is short-ranged in some systems, its contribution to the correlation hole can spread out in a molecule with multiple atomic centers, canceling part of the spread of the exact exchange hole (more so than in RPA or RPA+), making the true exchange-correlation hole more localized than in RPA or RPA+. This effect is not captured even by the vdW-DF nonlocality, but it requires the different kind of full nonlocality present in a hybrid functional.

Global hybrids, which add a typically modest fraction of the exact exchange energy to a complement of semilocal exchange−correlation energy, are among the most widely used density functionals in chemistry and condensed matter physics. Here we briefly review the formal and practical advantages and disadvantages of global hybrids. We point out that empiricism seems unavoidable in the construction of global hybrids, as it is not for most other kinds of density functional. Then we use one to three parameters to hybridize many semilocal functionals (including recently developed nonempirical generalized gradient approximations or GGA's and meta-GGA's). We study the performance of these global hybrids for many properties of sp-bonded molecules composed from the lighter atoms of the periodic table: atomization energies, barrier heights, reaction energies, enthalpies of formation, total energies, ionization potentials, electron affinities, proton affinities, and equilibrium bond lengths. We find several new global hybrids that perform better in these tests than standard ones, and we correct some errors in literature assessments. We also discuss the representativity of small fitting sets and the adequacy of various Gaussian basis sets.

Semilocal density functionals for the exchange-correlation energy are needed for large electronic systems. The Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalized gradient approximation (meta-GGA) is semilocal and usefully accurate, but predicts too-long lattice constants. Recent "GGA's for solids" yield good lattice constants but poor atomization energies of molecules. We show that the construction principle for one of them (restoring the density gradient expansion for exchange over a wide range of densities) can be used to construct a "revised TPSS" meta-GGA with accurate lattice constants, surface energies, and atomization energies for ordinary matter.

We have computed the surface energies, work functions, and interlayer surface relaxations of clean (111), (110), and (100) surfaces of Al, Cu, Ru, Rh, Pd, Ag, Pt, and Au. Many of these metallic surfaces have technological or catalytic applications. We compare experimental reference values to those of the local density approximation (LDA), the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), the PBEsol (PBE for solids) GGA, the SCAN meta-GGA, and SCAN+rVV10 (SCAN with a long-range van der Waals or vdW correction). The closest agreement with uncertain experimental values is achieved by the simplest density functional (LDA) and by the most sophisticated general-purpose one (SCAN+rVV10). The long-range vdW interaction increases the surface energies by about 10%, and the work functions by about 1%. LDA works for metal surfaces through a stronger-than-usual error cancellation. PBE yields the most-underestimated and presumably least accurate surface energies and work functions. Surface energies within the random phase approximation (RPA) are also reported. Interlayer relaxations from different functionals are in reasonable agreement with one another, and usually with experiment.

The newly developed "strongly constrained and appropriately normed" (SCAN) meta-generalized-gradient approximation (meta-GGA) can generally improve over the non-empirical Perdew-Burke-Ernzerhof (PBE) GGA not only for strong chemical bonding, but also for the intermediate-range van der Waals (vdW) interaction. However, the long-range vdW interaction is still missing. To remedy this, we propose here pairing SCAN with the non-local correlation part from the rVV10 vdW density functional, with only two empirical parameters. The resulting SCAN+rVV10 yields excellent geometric and energetic results not only for molecular systems, but also for solids and layered-structure materials, as well as the adsorption of benzene on coinage metal surfaces. Especially, SCAN+rVV10 outperforms all current methods with comparable computational efficiencies, accurately reproducing the three most fundamental parameters---the inter-layer binding energies, inter-, and intra-layer lattice constants---for 28 layered-structure materials. Hence, we have achieved with SCAN+rVV10 a promising vdW density functional for general geometries, with minimal empiricism.

The properties of simple metals are fixed primarily by the equilibrium average valence-electron density parameter ${r}_{s},$ and secondarily by the valence z. The simplest level of theory that can account quantitatively for these trends invokes a ``universal'' local electron-ion pseudopotential, defined for each pair ${(r}_{s},z)$ and treated as a second-order perturbation. We construct this pseudopotential from two conditions: (1) The total energy should minimize at the equilibrium Wigner-Seitz radius ${z}^{1/3}{r}_{s}.$ (2) The bulk modulus should equal the realistic ${r}_{s}$-dependent prediction of the stabilized jellium model with effective valence ${z}^{*}=1.$ These conditions can be satisfied by an analytic local pseudopotential depending upon two parameters other than z; we show that the choice of the two-parameter form (evanescent core vs Heine-Abarenkov) is not important. Our universal local pseudopotential is applied to calculate realistic bulk binding energies, pressure derivatives of bulk moduli, Voigt shear moduli, and interstitial electron numbers, revealing their trends as functions of ${r}_{s}$ and z. Equilibrium crystal structures are mapped in the ${r}_{s}\ensuremath{-}z$ plane, where the Hume-Rothery rules for substitutional alloys are manifest. The effect of pressure on crystal structure is also examined.

Explicit functions are widely used to interpolate, extrapolate, and differentiate theoretical or experimental data on the equation of state (EOS) of a solid. We present two EOS functions which are theoretically motivated. The simplest realistic model for a simple metal, the stabilized jellium (SJ) or structureless pseudopotential model, is the paradigm for our SJEOS. A simple metal with exponentially overlapped ion cores is the paradigm for an augmented version (ASJEOS) of the SJEOS. For the three solids tested (Al, Li, Mo), the ASJEOS matches all-electron calculations better than prior equations of state. Like most of the prior EOS's, the ASJEOS predicts pressure P as a function of compressed volume $v$ from only a few equilibrium inputs: the volume ${v}_{0},$ the bulk modulus ${B}_{0},$ and its pressure derivative ${B}_{1}.$ Under expansion, the cohesive energy serves as another input. A further advantage of the new equation of state is that these equilibrium properties other than ${v}_{0}$ may be found by linear fitting methods. The SJEOS can be used to correct ${B}_{0}$ and the EOS found from an approximate density functional, if the corresponding error in ${v}_{0}$ is known. We also (a) estimate the typically small contribution of phonon zero-point vibration to the EOS, (b) find that the physical hardness $\mathrm{Bv}$ does not maximize at equilibrium, and (c) show that the ``ideal metal'' of Shore and Rose is the zero-valence limit of stabilized jellium.

As shown by Overhauser and others, the pair-distribution function $g(r)$ of a many-electron system may be found by solving a two-electron scattering problem with an effective screened electron-electron repulsion $V(r).$ We propose a simple physical picture in which this screened repulsion is the ``dressed-dressed'' interaction between two neutral objects, each an electron surrounded by its full-coupling exchange-correlation hole. For the effective interaction between two electrons of antiparallel spin in a high-density uniform electron gas of arbitrary spin polarization, we confirm that this picture is qualitatively correct. In contrast, the ``bare-dressed'' interaction is too repulsive, and does not have the expected symmetry ${V}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}{(r)=V}_{\ensuremath{\downarrow}\ensuremath{\uparrow}}(r).$ The simple original Overhauser model interaction, independent of the relative spin polarization $\ensuremath{\zeta},$ does not capture the $\ensuremath{\zeta}$ dependence of the correlation contribution to $g(r=0).$