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

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

A method for accurate and efficient local density functional calculations (LDF) on molecules is described and presented with results The method, Dmol for short, uses fast convergent three‐dimensional numerical integrations to calculate the matrix elements occurring in the Ritz variation method The flexibility of the integration technique opens the way to use the most efficient variational basis sets A practical choice of numerical basis sets is shown with a built‐in capability to reach the LDF dissociation limit exactly Dmol includes also an efficient, exact approach for calculating the electrostatic potential Results on small molecules illustrate present accuracy and error properties of the method Computational effort for this method grows to leading order with the cube of the molecule size Except for the solution of an algebraic eigenvalue problem the method can be refined to quadratic growth for large molecules

An all‐electron numerical method for solving the local density functional for polyatomic molecules

https://www.lct.jussieu.fr/pagesperso/savin/papers/lyp/MieSavStoPre-89.pdf

Results obtained with the correlation energy density functionals of becke and Lee, Yang and Parr

Density functional approximations for the exchange‐correlation energy EDFAxc of an electronic system are often improved by admixing some exact exchange Ex: Exc≊EDFAxc+(1/n)(Ex−EDFAx). This procedure is justified when the error in EDFAxc arises from the λ=0 or exchange end of the coupling‐constant integral ∫10 dλ EDFAxc,λ. We argue that the optimum integer n is approximately the lowest order of Gorling–Levy perturbation theory which provides a realistic description of the coupling‐constant dependence Exc,λ in the range 0≤λ≤1, whence n≊4 for atomization energies of typical molecules. We also propose a continuous generalization of n as an index of correlation strength, and a possible mixing of second‐order perturbation theory with the generalized gradient approximation.

/pdf/rationale-for-mixing-exact-exchange-with-density-functional-19evr7wlkm.pdf

Rationale for mixing exact exchange with density functional approximations

The aim of this paper is to advocate the usefulness of the spin-density-functional (SDF) formalism. The generalization of the Hohenberg-Kohn-Sham scheme to and SDF formalism is presented in its thermodynamic version. The ground-state formalism is extended to more general Hamiltonians and to the lowest excited state of each symmetry. A relation between the exchange-correlation functional and the pair correlation function is derived. It is used for the interpretation of approximate versions of the theory, in particular the local-spin-density (LSD) approximation, which is formally valid only in the limit of slow and weak spatial variation in the density. It is shown, however, to give good account for the exchange-correlation energy also in rather inhomogeneous situations, because only the spherical average of the exchange-correlation hole influences this energy, and because it fulfills the sum rule stating that this hole should contain only one charge unit. A further advantage of the LSD approximation is that it can be systematically improved. Calculations on the homogeneous spin-polarized electron liquid are reported on. These calculations provide data in the form of interpolation formulas for the exchange-correlation energy and potentials, to be used in the LSD approximation. The ground-state properties are obtained from the Galitskii-Migdal formula, which relates the total energy to the one-electron spectrum, obtained with a dynamical self-energy. The self-energy is calculated in an electron-plasmon model where the electron is assumed to couple to one single mode. The potential for excited states is obtained by identifying the quasiparticle peak in the spectrum. Correlation is found to significantly weaken the spin dependence of the potentials, compared with the result in the Hartree-Fock approximation. Charge and spin response functions are calculated in the long-wavelength limit. Correlation is found to be very important for properties which involve a change in the spinpolarization. For atoms, molecules, and solids the usefulness of the SDF formalism is discussed. In order to explore the range of applicability, a few applications of the LSD approximation are made on systems for which accurate solutions exist. The calculated ionization potentials, affinities, and excitation energies for atoms propose that the valence electrons are fairly well described, a typical error in the ionization energy being 1/2 eV. The exchange-correlation holes of two-electron ions are discussed. An application to the hydrogen molecule, using a minimum basis set, shows that the LSD approximation gives good results for the energy curve for all separations studied, in contrast to the spin-independent local approximation. In particular, the error in the binding energy is only 0.1 eV, and bond breaking is properly described. For solids, the SDF formalism provides a framework for band models of magnetism. An estimate of the splitting between spin-up and spin-down energy bands of a ferromagnetic transition metal shows that the LSD approximation gives a correction of the correct sign and order of magnitude to published $X\ensuremath{\alpha}$ results. To stimulate further use of the SDF formalism in the LSD approximation, the paper is self-contained and describes the necessary formulas and input data for the potentials.

Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism

We find that within local-density schemes for calculating the cohesive energy of simple metals greater sophistication in treating the atom is required. The outermost electron in, e.g., the sodium atom has an unpaired spin. For this and the many similar cases a generalization of the scheme to a spin-density-functional formalism is needed. Application of the local-spin-density approximation gives, e.g., the energy of the hydrogen atom within 1.6% of the exact value, while the local-density approximation is 10% off. The improvement is due to our use of a better model system, i.e., the spin-polarized electron liquid, in the local approximation. We elaborate on the factors leading to the smallness of the error, and we find that there is a systematic partial cancellation between too attractive and too repulsive contributions to the binding for valence electrons in hydrogen and similar atoms. When we extend Tong's calculation for sodium metal along these lines, we find the cohesive energy to lie within 4% of the experimental value. A similar improvement is found for lithium. The spin-density scheme should be a very useful practical method for a large range of applications, including the calculation of chemisorption and charge transfers.

Contribution to the cohesive energy of simple metals: Spin-dependent effect

The momentum distributions and Compton profiles of the conduction electrons in Li and Na are calculated considering effects of orthogonalization of the core electrons, periodic potential, and correlation. Using the orthogonalized-plane-wave method with parameters from Callaway and electron-gas data from an earlier calculation by Lundqvist, we find that the anisotropy of the Compton profile should be measurable for single crystalline Li but not for Na, and that the results compare well with recent experimental data for polycrystalline Li and Na. There is a slight indication of correlation effects beyond those considered here, but more accurate experimental data are needed for a more decisive conclusion.

B. I. Lundqvist

Papers

Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism

Contribution to the cohesive energy of simple metals: Spin-dependent effect

Calculated Momentum Distributions and Compton Profiles of Interacting Conduction Electrons in Lithium and Sodium

Effect of ground-state and excitation potentials on energy levels of Ni metal

Photo-field-emission: A new probe of electron states between the Fermi and vacuum levels