The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not. We present two related methods for the self-interaction correction (SIC) of any density functional for the energy; correction of the self-consistent one-electron potenial follows naturally from the variational principle. Both methods are sanctioned by the Hohenberg-Kohn theorem. Although the first method introduces an orbital-dependent single-particle potential, the second involves a local potential as in the Kohn-Sham scheme. We apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole, while substantially improving the description of its shape. We apply this method to a number of physical problems, where the uncorrected LSD approach produces systematic errors. We find systematic improvements, qualitative as well as quantitative, from this simple correction. Benefits of SIC in atomic calculations include (i) improved values for the total energy and for the separate exchange and correlation pieces of it, (ii) accurate binding energies of negative ions, which are wrongly unstable in LSD, (iii) more accurate electron densities, (iv) orbital eigenvalues that closely approximate physical removal energies, including relaxation, and (v) correct longrange behavior of the potential and density. It appears that SIC can also remedy the LSD underestimate of the band gaps in insulators (as shown by numerical calculations for the rare-gas solids and CuCl), and the LSD overestimate of the cohesive energies of transition metals. The LSD spin splitting in atomic Ni and $s\ensuremath{-}d$ interconfigurational energies of transition elements are almost unchanged by SIC. We also discuss the admissibility of fractional occupation numbers, and present a parametrization of the electron-gas correlation energy at any density, based on the recent results of Ceperley and Alder.

Self-interaction correction to density-functional approximations for many-electron systems

Current studies in density functional theory and density matrix functional theory are reviewed, with special attention to the possible applications within chemistry. Topics discussed include the concept of electronegativity, the concept of an atom in a molecule, calculation of electronegativities from the Xα method, the concept of pressure, Gibbs-Duhem equation, Maxwell relations, stability conditions, and local density functional theory.

Density-functional theory of atoms and molecules

Previous attempts to combine Hartree–Fock theory with local density‐functional theory have been unsuccessful in applications to molecular bonding. We derive a new coupling of these two theories that maintains their simplicity and computational efficiency, and yet greatly improves their predictive power. Very encouraging results of tests on atomization energies, ionization potentials, and proton affinities are reported, and the potential for future development is discussed.

A New Mixing of Hartree-Fock and Local Density-Functional Theories

We present an analysis of the performances of a parameter free density functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amount of exact exchange. The results obtained for structural, thermodynamic, kinetic and spectroscopic (magnetic, infrared and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chemistry and condensed matter physics.

Toward reliable density functional methods without adjustable parameters: The PBE0 model

/pdf/scalable-molecular-dynamics-with-namd-3j7xyc70g4.pdf

Scalable Molecular Dynamics with NAMD

Precision is given to the concept of electronegativity. It is the negative of the chemical potential (the Lagrange multiplier for the normalization constraint) in the Hohenberg–Kohn density functional theory of the ground state: χ=−μ=−(∂E/∂N)v. Electronegativity is constant throughout an atom or molecule, and constant from orbital to orbital within an atom or molecule. Definitions are given of the concepts of an atom in a molecule and of a valence state of an atom in a molecule, and it is shown how valence‐state electronegativity differences drive charge transfers on molecule formation. An equation of Gibbs–Duhem type is given for the change of electronegativity from one situation to another, and some discussion is given of certain relations among energy components discovered by Fraga.

Electronegativity: The density functional viewpoint

The Hohenberg-Kohn theorem is extended to fractional electron number $N$, for an isolated open system described by a statistical mixture. The curve of lowest average energy ${E}_{N}$ versus $N$ is found to be a series of straight line segments with slope discontinuities at integral $N$. As $N$ increases through an integer $M$, the chemical potential and the highest occupied Kohn-Sham orbital energy both jump from ${E}_{M}\ensuremath{-}{E}_{M\ensuremath{-}1}$ to ${E}_{M+1}\ensuremath{-}{E}_{M}$. The exchange-correlation potential $\frac{\ensuremath{\delta}{E}_{\mathrm{xc}}}{\ensuremath{\delta}n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}$ jumps by the same constant, and $\frac{{\mathrm{lim}}_{r\ensuremath{\rightarrow}\ensuremath{\infty}}\ensuremath{\delta}{E}_{\mathrm{xc}}}{\ensuremath{\delta}n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}g~0$.

Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy

The local-density approximation for the exchange-correlation potential understimates the fundamental band gaps of semiconductors and insulators by about 40%. It is argued here that underestimation of the gap width is also to be expected from the unknown exact potential of Kohn-Sham density-functional theory, because of derivative discontinuities of the exchange-correlation energy. The need for an energy-dependent potential in band theory is emphasized. The center of the gap, however, is predicted exactly by the Kohn-Sham band structure.

Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities

Universal variational functionals of densities, first-order density matrices, and natural spin-orbitals are explicitly displayed for variational calculations of ground states of interacting electrons in atoms, molecules, and solids. In all cases, the functionals search for constrained minima. In particular, following Percus [Formula: see text] is identified as the universal functional of Hohenberg and Kohn for the sum of the kinetic and electron—electron repulsion energies of an N-representable trial electron density ρ. Q[ρ] searches all antisymmetric wavefunctions Ψρ which yield the fixed. ρ. Q[ρ] then delivers that expectation value which is a minimum. Similarly, [Formula: see text] is shown to be the universal functional for the electron—electron repulsion energy of an N-representable trial first-order density matrix γ, where the actual external potential may be nonlocal as well as local. These universal functions do not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus, the v-representability problem, which is especially severe for trial first-order density matrices, has been solved. Universal variational functionals in Hartree—Fock and other restricted wavefunction theories are also presented. Finally, natural spin-orbital functional theory is compared with traditional orbital formulations in density functional theory.

https://www.pnas.org/content/pnas/76/12/6062.full.pdf

Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem

As an alternative to the standard Kohn-Sham procedure, other exact realizations of density-functional theory ~generalized Kohn-Sham methods! are presented. The corresponding generalized Kohn-Sham eigenvalue gaps are shown to incorporate part of the discontinuity D xc of the exchange-correlation potential of standard KohnSham theory. As an example, a generalized Kohn-Sham procedure splitting the exchange contribution to the total energy into a screened, nonlocal and a local density component is considered. This method leads to band gaps far better than those of local-density approximation and to good structural properties for the materials Si, Ge, GaAs, InP, and InSb.

Mel Levy

Papers

Electronegativity: The density functional viewpoint

Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy

Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities

Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem

Generalized Kohn-Sham schemes and the band-gap problem