Publications

SCAN+rVV10: A promising van der Waals density functional

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.

Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model

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.

New Meta-Generalized Gradient Approximation for Exchange and Correlation

Tests of a ladder of density functionals for bulk solids and surfaces

The local spin-density approximation (LSDA) and the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) are fully non-empirical realizations of the first two rungs of ``Jacob's ladder'' of exchange-correlation density functionals. The recently proposed non-empirical meta-GGA of Tao, Perdew, Staroverov, and Scuseria (TPSS), featuring the kinetic energy density as an additional local ingredient, completes the third rung. A hierarchy of these functionals, complemented by the meta-GGA of Perdew, Kurth, Zupan, and Blaha (PKZB), is tested in self-consistent Gaussian-type orbital calculations of equilibrium lattice constants, bulk moduli, and cohesive energies for 18 solids, and in studies of the jellium surface energy. The ascent of the ladder generally results in better performance, although most of the improvement for bulk solids occurs in the transition from LSDA to PBE. For the jellium surface energy, PBE is less accurate than LSDA, but PKZB and TPSS are more accurate. We support the idea that most of the error of these functionals for bulk solids arises in the description of core--valence interaction, by demonstrating that it can be removed through adjustment of the corresponding term in the equation of state. Overall, TPSS gives the best description of solids and surfaces, as it was found to do for molecules in earlier work.

The Lieb-Oxford Lower Bounds on the Coulomb Energy, Their Importance to Electron Density Functional Theory, and a Conjectured Tight Bound on Exchange

Lieb and Oxford (1981) derived rigorous lower bounds, in the form of local functionals of the electron density, on the indirect part of the Coulomb repulsion energy. The greatest lower bound for a given electron number N depends monotonically upon N, and the N-> infinity limit is a bound for all N. These bounds have been shown to apply to the exact density functionals for the exchange- and exchange-correlation energies that must be approximated for an accurate and computationally efficient description of atoms, molecules, and solids. A tight bound on the exact exchange energy has been derived therefrom for two-electron ground states, and is conjectured to apply to all spin-unpolarized electronic ground states. Some of these and other exact constraints have been used to construct two generations of non-empirical density functionals beyond the local density approximation: the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the strongly constrained and appropriately normed (SCAN) meta-GGA.

Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals

The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue'. (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.

Accurate van der Waals coefficients from density functional theory

The van der Waals interaction is a weak, long-range correlation, arising from quantum electronic charge fluctuations. This interaction affects many properties of materials. A simple and yet accurate estimate of this effect will facilitate computer simulation of complex molecular materials and drug design. Here we develop a fast approach for accurate evaluation of dynamic multipole polarizabilities and van der Waals (vdW) coefficients of all orders from the electron density and static multipole polarizabilities of each atom or other spherical object, without empirical fitting. Our dynamic polarizabilities (dipole, quadrupole, octupole, etc.) are exact in the zero- and high-frequency limits, and exact at all frequencies for a metallic sphere of uniform density. Our theory predicts dynamic multipole polarizabilities in excellent agreement with more expensive many-body methods, and yields therefrom vdW coefficients C(6), C(8), C(10) for atom pairs with a mean absolute relative error of only 3%.

Size-dependent ionization energy of a metallic cluster: Resolution of the classical image-potential paradox

The ionization energy for a metallic cluster of radius R is I(R)=W+\ensuremath{\alpha}${\mathit{e}}^{2}$/R+O(${\mathit{R}}^{\mathrm{\ensuremath{-}}2}$), where W is the work function. Two different classical values for \ensuremath{\alpha} have been derived: 1/2 from the spherical-capacitor approach, and 3/8 from the image-potential approach. We present a ``classical jellium'' or Wigner-crystal model in which \ensuremath{\alpha}=1/2 and W=9/10${\mathit{e}}^{2}$/${\mathit{r}}_{\mathit{s}}$. In the image-potential approach to the ``perfect-conductor'' model, we discover an additional work 1/8${\mathit{e}}^{2}$/R, resolving the contradiction in favor of \ensuremath{\alpha}=1/2. The experimental deviation from \ensuremath{\alpha}=1/2 is a quantum effect.

Orbital shifts: An easy way to calculate the exact Kohn-Sham exchange potential

Density Functionals for Non-relativistic Coulomb Systems in the New Century

Global Hybrid Functionals: A Look at the Engine under the Hood

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.

Phonon Frequencies from a Local Pseudopotential

Why Density Functionals Should Not Be Judged Primarily by Atomization Energies

While most molecules and solids are spin-unpolarized, most chemically-active atoms are partly spin-polarized. As a result, the errors of the spin-dependence of a density functional are much more troublesome for atomization energies than they are for typical reaction or formation energies. This observation explains why the atomization energy errors of approximate functionals do not correlate with their other errors, and why the errors of atomization energies for a given functional can be radically reduced by fitting the energies of the atoms. We present an illustrative example from the recent nonempirical construction of the SCAN meta-generalized gradient approximation.

Effect of strain on the band gap of monolayer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>MoS</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>

Monolayer molybdenum disulfide $({\mathrm{MoS}}_{2})$ under strain has many interesting properties and possible applications in technology. A recent experimental study examined the effect of strain on the band gap of monolayer ${\mathrm{MoS}}_{2}$ on a mildly curved graphite surface, reporting that under biaxial strain with a Poisson's ratio of 0.44, the band gap decreases at a rate of 400 meV/% strain. In this paper, we performed density functional theory calculations for a free-standing ${\mathrm{MoS}}_{2}$ monolayer, using the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE), the Heyd-Scuseria-Ernzerhof screened hybrid functional HSE06, and many-body perturbation theory with the $GW$ approximation using PBE wave functions $({G}_{0}{W}_{0}@\mathrm{PBE})$. For the unstrained monolayer, we found a standard level of agreement for the band gap between theory and experiment. For biaxial strain at the experimental Poisson's ratio, we found that the band gap decreases at rates of 63 meV/% strain (PBE), 73 meV/% strain (HSE06), and 43 meV/% strain $({G}_{0}{W}_{0}@\mathrm{PBE})$, which are significantly smaller than the experimental rate. We also found that PBE predicts a similarly smaller rate (90 meV/% strain) for a different Poisson's ratio of 0.25. Spin-orbit correction has little effect on the gap or its strain dependence. The strong disagreement between theory and experiment may reflect an unexpectedly strong effect of the substrate on the strain dependence of the gap. Additionally, we observed a transition from a direct to an indirect band gap under strain, and (under an equal biaxial strain of 10%) a semiconductor-to-metal transition, consistent with previous theoretical work.

A two-pronged approach to the cuprate pseudogap: A comparison of mode-coupling and first-principles (SCAN) results

Strongly Constrained and Appropriately Normed Semilocal Density Functional

The ground-state energy, electron density, and related properties of ordinary matter can be computed efficiently when the exchange-correlation energy as a functional of the density is approximated semilocally. We propose the first meta-generalized-gradient approximation (meta-GGA) that is fully constrained, obeying all 17 known exact constraints that a meta-GGA can. It is also exact or nearly exact for a set of "appropriate norms," including rare-gas atoms and nonbonded interactions. This strongly constrained and appropriately normed meta-GGA achieves remarkable accuracy for systems where the exact exchange-correlation hole is localized near its electron, and especially for lattice constants and weak interactions.

Escaping the symmetry dilemma through a pair-density interpretation of spin-density functional theory

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.

Accurate and Numerically Efficient r²SCAN Meta-Generalized Gradient Approximation

The recently proposed rSCAN functional [ <i>J. Chem. Phys.</i> 2019 150, 161101] is a regularized form of the SCAN functional [ <i>Phys. Rev. Lett.</i> 2015 115, 036402] that improves SCAN's numerical performance at the expense of breaking constraints known from the exact exchange-correlation functional. We construct a new meta-generalized gradient approximation by restoring exact constraint adherence to rSCAN. The resulting functional maintains rSCAN's numerical performance while restoring the transferable accuracy of SCAN.