Publications

Systematizing Approximate Density Functional Design

Correlation entropy of the H2 molecule

Correlation of a quantum many-body state makes the one-particle density matrix nonidempotent. Therefore, the Shannon entropy of the natural occupation numbers measures the correlation strength on the one-particle level. Here, it is shown how this general idea of a correlation entropy must be adapted for two-electron systems in view of conservation laws which mix Slater determinants even in the noninteracting limit. Results are presented for the correlation entropy s of H2 as a function of the nucleus-nucleus separation R. In the ground state, the entropy of the spatial factor of the wave function maximizes 1.7 bohr beyond the Coulson-Fischer separation. The role of the correlation entropy in density functional theory is also discussed. © 1997 John Wiley & Sons, Inc.

Self-Interaction Correction

Meta-generalized gradient approximations: What they can and cannot do for you

Why Density-Gradient Corrections Improve Atomization Energies and Barrier Heights

Formation energies of metallic voids, edges, and steps: Generalized liquid-drop model

The void formation energy is the work needed to create the curved surface of a void. For a spherical hole in a homogeneous metal (jellium or stabilized jellium), the void formation energy is calculated for large radii from the liquid-drop model (surface plus curvature terms), and for small radii from perturbation theory. A Pad\'e approximation is proposed to link these limits. For radii greater than or equal to that of a single atom or monovacancy, the liquid-drop model is found to be usefully accurate. Moreover, the predicted monovacancy formation energies for stabilized jellium agree reasonably well with those measured for simple metals. These results suggest a generalized liquid-drop model of possible high accuracy and explanatory value for the energetics of stable metal surfaces curved on the atomic scale (crystal faces, edges, corners, etc.). The bending energy per unit length for an edge at angle \ensuremath{\theta} is estimated to be \ensuremath{\gamma}(\ensuremath{\pi}-\ensuremath{\theta})/4, where \ensuremath{\gamma} is the intrinsic curvature energy. The step energy is estimated as (n-2+\ensuremath{\pi}/2)\ensuremath{\sigma}d, where \ensuremath{\sigma} is the intrinsic surface energy, n\ensuremath{\ge}1 is the number of atomic layers at the step, and d is the layer height.

Performance of scaled self-interaction correction to semilocal functionals

Why semilocal functionals work: Accuracy of the on-top pair density and importance of system averaging

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.

Surface chiral metal in the bulk half-integer quantum Hall effect of BaMnSb 2

Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy

We construct a Laplacian-level meta-generalized-gradient-approximation (meta-GGA) for the noninteracting (Kohn-Sham orbital) positive kinetic energy density $\ensuremath{\tau}$ of an electronic ground state of density $n$. This meta-GGA is designed to recover the fourth-order gradient expansion ${\ensuremath{\tau}}^{\mathit{GE}4}$ in the appropriate slowly varying limit and the von Weizs\"acker expression ${\ensuremath{\tau}}^{W}={\ensuremath{\mid}\ensuremath{\nabla}n\ensuremath{\mid}}^{2}∕(8n)$ in the rapidly varying limit. It is constrained to satisfy the rigorous lower bound ${\ensuremath{\tau}}^{W}(\mathbf{r})\ensuremath{\leqslant}\ensuremath{\tau}(\mathbf{r})$. Our meta-GGA is typically a strong improvement over the gradient expansion of $\ensuremath{\tau}$ for atoms, spherical jellium clusters, jellium surfaces, the Airy gas, Hooke's atom, one-electron Gaussian density, quasi-two-dimensional electron gas, and nonuniformly scaled hydrogen atom. We also construct a Laplacian-level meta-GGA for exchange and correlation by employing our approximate $\ensuremath{\tau}$ in the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA density functional. The Laplacian-level TPSS gives almost the same exchange-correlation enhancement factors and energies as the full TPSS, suggesting that $\ensuremath{\tau}$ and ${\ensuremath{\nabla}}^{2}n$ carry about the same information beyond that carried by $n$ and $\ensuremath{\nabla}n$. Our kinetic energy density integrates to an orbital-free kinetic energy functional that is about as accurate as the fourth-order gradient expansion for many real densities (with noticeable improvement in molecular atomization energies), but considerably more accurate for rapidly varying ones.

Erratum: Lattice constants from semilocal density functionals with zero-point phonon correction [Phys. Rev. B<b>85</b>, 014111 (2012)]

Received 24 February 2012DOI:https://doi.org/10.1103/PhysRevB.85.099903©2012 American Physical Society

The Constrained Search Formulation of Density Functional Theory

Discontinuity of the exchange-correlation potential: Support for assumptions used to find it

It was shown by J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz [Phys. Rev. Lett. 49, 1691 (1982)]; J. P. Perdew and M. Levy [Phys. Rev. Lett. 51, 1884 (1983)]; and L. J. Sham and M. Schl\"uter [Phys. Rev. Lett. 51, 1888 (1983)] that the exact Kohn-Sham exchange-correlation potential ${v}_{\text{xc}}(\stackrel{P\vec}{r})$ of an open system may jump discontinuously as the particle number crosses an integer, with important physical consequences. The original derivations of the size and nature of the discontinuity rely on an implicit assumption that, when $N\ensuremath{\rightarrow}J$ from above and below, the effective potential of the interacting system with particle number $N$ becomes identical (modulo a constant) to the effective potential at an integer particle number $J$ so that the Kohn-Sham orbitals change continuously as $N$ passes through $J$ at fixed external potential. We prove under mild assumptions that the noninteracting kinetic energy of the interacting system is a continuous function of the particle number (as is the density itself), and hence argue that the original assumption is likely correct; if so, then all energy components are continuous. A rigorous proof of the existence and size of the ${v}_{\text{xc}}$ discontinuity is presented in the special case where the particle number crosses 1, by showing that, even when the Levy-Lieb constrained search for a minimum kinetic energy expectation value is taken to go over ensembles instead of wave functions, the noninteracting kinetic energy is still given by the von Weizs\"acker functional for all densities with real particle number $N$ between 0 and 2 inclusive. We also prove that the von Weizs\"acker functional is a lower bound on the ensemble-search noninteracting kinetic energy for all real $N&gt;2$. To illustrate the ${v}_{\text{xc}}$ discontinuity and related results, we construct an analytic model density for the ${\text{H}}^{\ensuremath{-}}$ ion and plot the exchange-correlation potential as the particle number approaches 1 from above and below, and 2 from below. Finally, we discuss the behavior of the free-electron continuum as the particle number crosses an integer and the case where the particle number of a hydrogen or helium atom crosses $J=2$, the maximum number of electrons they can bind.

How accurate are the parametrized correlation energies of the uniform electron gas?

Density functional approximations to the exchange-correlation energy are designed to be exact for an electron gas of uniform density parameter ${r}_{s}$ and relative spin polarization $\ensuremath{\zeta}$, requiring a parametrization of the correlation energy per electron ${\ensuremath{\epsilon}}_{c}({r}_{s},\ensuremath{\zeta})$. We consider three widely used parametrizations [J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981) or PZ81, S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980) or VWN80, and J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992) or PW92] that interpolate the quantum Monte Carlo (QMC) correlation energies of Ceperley-Alder [Phys. Rev. Lett. 45, 566 (1980)], while extrapolating them to known high-$({r}_{s}\ensuremath{\rightarrow}0)$ and low- $({r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})$ density limits. For the physically important range $0.5\ensuremath{\le}{r}_{s}\ensuremath{\le}20$, they agree closely with one another, with differences of 0.01 eV (0.5%) or less between the latter two. The density parameter interpolation (DPI), designed to predict these energies by interpolation between the known high- and low-density limits, with almost no other input (and none for $\ensuremath{\zeta}=0$), is also reasonably close, both in its original version and with corrections for $\ensuremath{\zeta}\ensuremath{\ne}0$. Moreover, the DPI and PW92 at ${r}_{s}=0.5$ are very close to the high-density expansion. The larger discrepancies with the QMC of Spink et al. [Phys. Rev. B 88, 085121 (2013)], of order 0.1 eV (5%) at ${r}_{s}=0.5$, are thus surprising, suggesting that the constraint-based PW92 and VWN80 parametrizations are more accurate than the QMC for ${r}_{s}&lt;2$. For ${r}_{s}&gt;2$, however, the QMC of Spink et al. confirms the dependence upon relative spin polarization predicted by the parametrizations.

Electron Localization in Non-Compact Covalent Bonds Captured by the r2SCAN+V Approach

In density functional theory, the SCAN (Strongly Constrained and Appropriately Normed) and r2SCAN functionals significantly improve over generalized gradient approximation functionals such as PBE (Perdew-Burke-Ernzerhof) in predicting electronic, magnetic, and structural properties across various materials, including transition-metal compounds. However, there remain puzzling cases where SCAN and r2SCAN underperform, such as in calculating the band structure of graphene, the magnetic moment of Fe, the potential energy curve of the Cr2 molecule, and the bond length of VO2. This research identifies a common characteristic among these challenging materials: non-compact covalent bonding through s-s, p-p, or d-d electron hybridization. While SCAN and r2SCAN excel at capturing electron localization at local atomic sites, they struggle to accurately describe electron localization in non-compact covalent bonds, resulting in a biased improvement. To address this issue, we propose the r2SCAN+V approach as a practical modification that improves accuracy across all the tested materials. The parameter V is 4 eV for metallic Fe, but substantially lower for the other cases. Our findings provide valuable insights for the future development of advanced functionals.

Estimation, Computation, and Experimental Correction of Molecular Zero-Point Vibrational Energies

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).

Metal-cluster ionization energy: A profile-insensitive exact expression for the size effect

The ionization energy of a large spherical metal cluster of radius R is I(R)=W+(+c)/R, where W is the bulk work function and c\ensuremath{\approx}-0.1 is a material-dependent quantum correction to the electrostatic size effect. We present 'Koopmans' and 'displaced-profile change-in-self-consistent-field' expressions for W and c within the ordinary and stabilized-jellium models. These expressions are shown to be exact and equivalent when the exact density profile of a large neutral cluster is employed; these equivalences generalize the Budd-Vannimenus theorem. With an approximate profile obtained from a restricted variational calculation, the 'displaced-profile' expressions are the more accurate ones. This profile insensitivity is important, because it is not practical to extract c from solutions of the Kohn-Sham equations for small metal clusters.

Semilocal density functionals for exchange and correlation: Theory and applications