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

To use solar irradiation or interior lighting efficiently, we sought a photocatalyst with high reactivity under visible light. Films and powders of TiO 2- x N x have revealed an improvement over titanium dioxide (TiO 2 ) under visible light (wavelength 2 has proven to be indispensable for band-gap narrowing and photocatalytic activity, as assessed by first-principles calculations and x-ray photoemission spectroscopy.

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Visible-Light Photocatalysis in Nitrogen-Doped Titanium Oxides

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

The local density theory is developed by Hohenberg, Kohn and Sham is extended to the spin polarized case. A spin dependent one- electron potential pertinent to ground state properties is obtained from calculations of the total energy per electron made with a 'bubble' (or random phase) type of dielectric function. The potential is found to be well represented by an analytic expression corresponding to a shifted and rescaled spin dependent Slater potential. To test this potential the momentum dependent spin susceptibility of an electron gas is calculated. The results compare favourably with available information from other calculations and from experiment. The potential obtained in this paper should be useful for split band calculations of magnetic materials.

https://iopscience.iop.org/article/10.1088/0022-3719/5/13/012/pdf

A local exchange-correlation potential for the spin polarized case: I

A scheme that reduces the calculations of ground-state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations has obvious advantages. It is not surprising, then, that the density functional formalism, which provides a way of doing this, has received much attention in the past two decades. The quality of the energy surfaces calculated using a simple local-density approximation for exchange and correlation exceeds by far the original expectations. In this work, the authors survey the formalism and some of its applications (in particular to atoms and small molecules) and discuss the reasons for the successes and failures of the local-density approximation and some of its modifications.

The density functional formalism, its applications and prospects

The possibilities of the Hohenberg-Kohn-Sham local density theory are explored in view of recent advances in the theory of the interacting electron gas. The authors discuss and provide numerical data for the effective exchange-correlation potentials mu xc for ground state problems and Vxc for excitation spectra. The potential mu xc(r) is written as a factor beta ( rho (r)), which depends on the local density rho (r), times the Kohn-Sham-Gaspar potential mu x=-e2kF/ pi . The factor beta , which is represented by a simple analytical formula, varies between 1 for very high densities and about 1.5 for very low densities, where mu xc thus approaches the Slater approximation. The potential Vxc is given in the form of a table. The explicit predictions of the approximate theories are mapped in their linearized versions. The theory is also developed for the case when the ion cores are treated separately, and the local density approximation is applied only to the valence electrons.

http://iopscience.iop.org/article/10.1088/0022-3719/4/14/022/pdf

Explicit local exchange-correlation potentials

Publisher Summary: This chapter discusses two major developments that have taken place over the past decade. First is the enormous wealth of energy band calculations that have had tremendous success in explaining the properties of specific solids, but in which the connection with first principles is not always apparent. Second is the spectacular progress of many-body theory applied to the solid state that has given a number of new results, although often of a rather general and formal nature, such as to provide the justification and a formal basis for a one-electron theory. The electron gas problem is treated in some detail here. The problem of the crystal potential is given due attention. It discusses the potential from the ion cores as well as from the valence electrons, and suggests schemes that incorporate essential exchange and correlation effects for the valence electrons. An energy band calculation that properly includes the effects of exchange and correlation describes the elementary excitations called quasi particles. Quasi-particle properties are usually discussed using the remarkable Landau theory of the Fermi liquid. This chapter gives a brief presentation of the theory and reviews the present status of calculations of the Fermi liquid parameters and how they are determined from experiments. (Less)

Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids

By neglecting certain terms in the Hamiltonian (of importance only in the immediate neighbourhood of the threshold energy) we are able to write an exact expression for the photoemission intensity which involves the standard time inverted LEED state and broadened hole-states. If inelastic extrinsic losses are neglected our expression is similar to the one used by e.g. Feibelman and Eastman [1], Pendry [2] and Spanjaard et al. [3]. Our theory also allows inelastic losses (including interference between intrinsic and extrinsic losses) to be evaluated. The expressions are similar to those obtained by Inglesfield [4], but the damping effect no longer has to be put in by hand. Our theory further can handle not only core electron but also valence electron photoemission and it is not restricted to metals but applies to all types of solids. We have also traced why and when Inglesfield's one step theory agrees with a semiclassical calculation.

https://iopscience.iop.org/article/10.1088/0031-8949/32/4/033/pdf

A New Approach to the Theory of Photoemission from Solids

Experimental results for core-electron photoemission ${J}_{\mathbf{k}}(\ensuremath{\omega})$ are often compared with the one-electron spectral function ${A}_{c}({\ensuremath{\epsilon}}_{k}\ensuremath{-}\ensuremath{\omega}),$ where $\ensuremath{\omega}$ is the photon energy, ${\ensuremath{\epsilon}}_{k}$ is the photoelectron energy, and the optical transition matrix elements are taken as constant. Since ${J}_{\mathbf{k}}(\ensuremath{\omega})$ is nonzero only for ${\ensuremath{\epsilon}}_{k}g0,$ we must actually compare it with ${A}_{c}({\ensuremath{\epsilon}}_{k}\ensuremath{-}\ensuremath{\omega})\ensuremath{\theta}({\ensuremath{\epsilon}}_{k}).$ For metals ${A}_{c}(\ensuremath{\omega})$ is known to have a quasiparticle (QP) peak with an asymmetric power-law [theories of Mahan, Nozi\`eres, de Dominicis, Langreth, and others (MND)] singularity due to low-energy particle-hole excitations. The QP peak starts at the core-electron energy ${\ensuremath{\epsilon}}_{c},$ and is followed by an extended satellite (shakeup) structure at smaller $\ensuremath{\omega}.$ For photon energies $\ensuremath{\omega}$ just above threshold, ${\ensuremath{\omega}}_{\mathrm{th}}=\ensuremath{-}{\ensuremath{\epsilon}}_{c}, {A}_{c}({\ensuremath{\epsilon}}_{k}\ensuremath{-}\ensuremath{\omega})\ensuremath{\theta}({\ensuremath{\epsilon}}_{k})$ as a function of ${\ensuremath{\epsilon}}_{k} (\ensuremath{\omega}$ constant) is cut just behind the quasiparticle peak, and neither the tail of the MND line nor the plasmon satellites are present. The sudden (high-energy) limit is given by a convolution of ${A}_{c}(\ensuremath{\omega})$ and a loss function, i.e., by the Berglund-Spicer two-step expression. Thus ${A}_{c}(\ensuremath{\omega})$ alone does not give the correct photoelectron spectrum, neither at low nor at high energies. We present an extension of the quantum-mechanical (QM) models developed earlier by Inglesfield, and by Bardyszewski and Hedin to calculate ${J}_{\mathbf{k}}(\ensuremath{\omega}).$ It includes recoil and damping, as well as shakeup effects and extrinsic losses, is exact in the high-energy limit, and allows calculations of ${J}_{\mathbf{k}}(\ensuremath{\omega})$ including the MND line and multiple plasmon losses. The model, which involves electrons coupled to quasibosons, is motivated by detailed arguments. As an illustration we have made quantitative calculations for a semi-infinite jellium with the density of aluminum metal and an embedded atom. The coupling functions (fluctuation potentials) between the electron and the quasibosons are related to the random-phase-approximation dielectric function, and different levels of approximations are evaluated numerically. The differences in the predictions for the photoemission spectra are found small. We confirm the finding by Langreth that the BS limit is reached only in the keV range. At no photon energy are the plasmon satellites close to being either purely intrinsic or extrinsic. For photoelectron energies larger than a few times the plasmon energy, a semiclassical approximation gives results very close to our QM model. At lower energies the QM model gives a large peak in the ratio between the total intensity in the first plasmon satellite and the main peak, which is not reproduced by the SC expression. This maximum has a simple physical explanation in terms of different dampings of the electrons in the QP peak and in the satellite. For the MND peak ${J}_{\mathbf{k}}(\ensuremath{\omega})$ and ${A}_{c}({\ensuremath{\epsilon}}_{k}\ensuremath{-}\ensuremath{\omega})$ agree well for a range of a few eV, and experimental data can thus be used to extract the MND singularity index. For an embedded atom at a small distance from the surface there are, however, substantial deviations from the large-distance limit. Our model is simple enough to perform quantitative calculations allowing for band-structure and surface details.

Transition from the adiabatic to the sudden limit in core-electron photoemission

The interference between extrinsic and intrinsic losses in x-ray absorption fine structure (XAFS) is treated within a Green's-function formalism, without explicit reference to final states. The approach makes use of a quasiboson representation of excitations and perturbation theory in the interaction potential between electrons and quasibosons. These losses lead to an asymmetric broadening of the main quasiparticle peak plus an energy-dependent satellite in the spectral function. The x-ray absorption spectra (XAS) is then given by a convolution of an effective spectral function over a one-electron cross section. It is shown that extrinsic and intrinsic losses tend to cancel near excitation thresholds, and correspondingly, the strength in the main peak increases. At high energies, the theory crosses over to the sudden approximation. These results thus explain the observed weakness of multielectron excitations in XAS. The approach is applied to estimate the many-body corrections to XAFS, beyond the usual mean-free path, using a phasor summation over the spectral function. The asymmetry of the spectral function gives rise to an additional many-body phase shift in the XAFS formula.

Lars Hedin

Papers

Explicit local exchange-correlation potentials

Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids

A New Approach to the Theory of Photoemission from Solids

Transition from the adiabatic to the sudden limit in core-electron photoemission

Interference between extrinsic and intrinsic losses in x-ray absorption fine structure