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

"Chemists familiar with conventional quantum mechanics will applaud and benefit greatly from this particularly instructive, thorough and clearly written exposition of density functional theory: its basis, concepts, terms, implementation, and performance in diverse applications. Users of DFT for structure, energy, and molecular property computations, as well as reaction mechanism studies, are guided to the optimum choices of the most effective methods. Well done!" Paul von RaguE Schleyer "A conspicuous hole in the computational chemist's library is nicely filled by this book, which provides a wide-ranging and pragmatic view of the subject.[...It] should justifiably become the favorite text on the subject for practioneers who aim to use DFT to solve chemical problems." J. F. Stanton, J. Am. Chem. Soc. "The authors' aim is to guide the chemist through basic theoretical and related technical aspects of DFT at an easy-to-understand theoretical level. They succeed admirably." P. C. H. Mitchell, Appl. Organomet. Chem. "The authors have done an excellent service to the chemical community. [...] A Chemist's Guide to Density Functional Theory is exactly what the title suggests. It should be an invaluable source of insight and knowledge for many chemists using DFT approaches to solve chemical problems." M. Kaupp, Angew. Chem.

A Chemist's Guide to Density Functional Theory

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

Time-dependent density functional theory (TDDFT) can be viewed as an exact reformulation of time-dependent quantum mechanics, where the fundamental variable is no longer the many-body wave function but the density. This time-dependent density is determined by solving an auxiliary set of noninteracting Schrodinger equations, the Kohn-Sham equations. The nontrivial part of the many-body interaction is contained in the so-called exchange-correlation potential, for which reasonably good approximations exist. Within TDDFT two regimes can be distinguished: (a) If the external time-dependent potential is "small," the complete numerical solution of the time-dependent Kohn-Sham equations can be avoided by the use of linear response theory. This is the case, e.g., for the calculation of photoabsorption spectra. (b) For a "strong" external potential, a full solution of the time-dependent Kohn-Sham equations is in order. This situation is encountered, for instance, when matter interacts with intense laser fields. In this review we give an overview of TDDFT from its theoretical foundations to several applications both in the linear and in the nonlinear regime.

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Time-Dependent Density Functional Theory

By the Hellmann-Feynman theorem, the density n(r) of many electrons in the presence of external potential v(r) obeys the relationships F${d}^{3}$r n(r)\ensuremath{\nabla}v(r)=0 and F${d}^{3}$r n(r)r\ifmmode\times\else\texttimes\fi{}\ensuremath{\nabla}v(r)=0. By the virial theorem, the interacting kinetic and electron-electron repulsion expectation values obey 2T[n]+${V}_{\mathrm{ee}}$[n]=-F${d}^{3}$r n(r)r\ensuremath{\cdot}\ensuremath{\nabla}[\ensuremath{\delta}T/\ensuremath{\delta}n(r)+\ensuremath{\delta}${V}_{\mathrm{ee}}$/\ensuremath{\delta}n(r)]. The exchange energy functional ${E}_{x}$[n] and potential ${v}_{x}$([n];r)\ensuremath{\equiv}\ensuremath{\delta}${E}_{x}$/\ensuremath{\delta}n(r) must satisfy ${E}_{x}$[n]+F${d}^{3}$r n(r)r\ensuremath{\cdot}\ensuremath{\nabla}${v}_{x}$([n];r)=0, while the correlation energy and potential must satisfy ${E}_{c}$[n]+F${d}^{3}$r n(r)r\ensuremath{\cdot}\ensuremath{\nabla}${v}_{c}$([n];r)0. Somewhat counterintuitively, it is not true that T[${n}_{\ensuremath{\gamma}}$]=${\ensuremath{\gamma}}^{2}$T[n] and ${V}_{\mathrm{ee}}$[${n}_{\ensuremath{\gamma}}$]=\ensuremath{\gamma}${V}_{\mathrm{ee}}$[n], where ${n}_{\ensuremath{\gamma}}$(r)\ensuremath{\equiv}${\ensuremath{\gamma}}^{3}$n(\ensuremath{\gamma}r) is a scaled density with scale factor \ensuremath{\gamma}\ensuremath{\ne}1. In fact, it is impossible to partition the exact Hohenberg-Kohn functional into a piece that scales as ${\ensuremath{\gamma}}^{2}$ and a piece that scales as \ensuremath{\gamma}, even if complete freedom with the partitioning is allowed. Instead there are universal scaling inequalities.For instance, T[${n}_{\ensuremath{\gamma}}$]+${V}_{\mathrm{ee}}$[${n}_{\ensuremath{\gamma}}$]${\ensuremath{\gamma}}^{2}$T[n]+ \ensuremath{\gamma}${V}_{\mathrm{ee}}$[n] and T[${n}_{\ensuremath{\gamma}}$]+\ensuremath{\gamma}${V}_{\mathrm{ee}}$[${n}_{\ensuremath{\gamma}}$]g${\ensuremath{\gamma}}^{2}$(T[n ]+${V}_{\mathrm{ee}}$[n]), and consequent inequalities involving ${E}_{c}$[n]. All the above virial and scaling requisites are universal in that they are independent of external potential and they must hold for arbitrary proper n. In addition, for the ground-state energy (E) and n of any atom or molecule at its equilibrium nuclear configuration, there is the inequality E-${T}_{s}$[n], where ${T}_{s}$ is the noninteracting kinetic energy. In the closed-shell tight-binding limit, the correlation potential obeys F${d}^{3}$r n(r)r\ensuremath{\cdot}\ensuremath{\nabla}${v}_{c}$([n];r)=0, and so cannot be a monotonic function of r for an atom in this limit.Further, (\ensuremath{\partial}/\ensuremath{\partial}\ensuremath{\gamma})${E}_{c}$[${n}_{\ensuremath{\gamma}}$]${\ensuremath{\Vert}}_{\ensuremath{\gamma}=1}$=E $_{c}$[n]+${T}_{c}$[n]=-F${d}^{3}$r n(r)r\ensuremath{\cdot}\ensuremath{\nabla}${v}_{c}$([n];r), which implies that the exact ${E}_{c}$ should be fairly insensitive to scaling. With the help of the ionization-potential theorem, it is argued that the exact ${v}_{c}$([n];r) in an atom often has a positive part. Common approximations to the correlation potential are examined for their effects upon the highest occupied Kohn-Sham orbital energy and the density moment 〈${r}^{2}$〉, and these effects are found to be related. Further improvements needed in the approximate correlation potentials are relatively large, but not nearly so large as those recently suggested for the atoms Ne, Ar, Kr, and Xe: The discrepancy between theoretical values of 〈${r}^{2}$〉 from Hartree-Fock or configuration-interaction calculations, and experimental values from measured diamagnetic susceptibilities, is tentatively resolved in favor of theory.

Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms

It is shown that the density can be used as the basic variable for calculating the properties of excited states. The correspondence is not between an eigenstate and its density, as is the case with the ground state, but between the subspace spanned by the number of lowest-energy eigenstates and the sum of their densities. An extension of the Hohenberg-Kohn-Sham theory (1964-5) for excited states has also been developed. The equations derived are similar in form to those for the ground-state density but the interpretation is different. The lowest-order approximation of the present theory coincides with Slater's 'transition-state' theory (1974).

http://iopscience.iop.org/article/10.1088/0022-3719/12/24/013/pdf

A K Theophilou

Papers

The energy density functional formalism for excited states