The energy density functional formalism for excited states
28 Dec 1979-Journal of Physics C: Solid State Physics (IOP Publishing)-Vol. 12, Iss: 24, pp 5419-5430
TL;DR: In this paper, it was shown that the density of an eigenstate and its density can be used as the basic variable for calculating the properties of excited states and an extension of the Hohenberg-Kohn-Sham theory for excited states has also been developed.
Abstract: 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).
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TL;DR: In this paper, the self-interaction correction (SIC) of any density functional for the ground-state energy is discussed. But the exact density functional is strictly selfinteraction-free (i.e., orbitals demonstrably do not selfinteract), but many approximations to it, including the local spin-density (LSD) approximation for exchange and correlation, are not.
Abstract: 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.
16,027 citations
Book•
01 Sep 2001
TL;DR: A Chemist's Guide to Density Functional Theory should be an invaluable source of insight and knowledge for many chemists using DFT approaches to solve chemical problems.
Abstract: "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.
3,550 citations
TL;DR: In this paper, the authors survey the local density functional formalism and some of its applications and discuss the reasons for the successes and failures of the local-density approximation and some modifications.
Abstract: 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.
3,285 citations
TL;DR: An overview of TDDFT from its theoretical foundations to several applications both in the linear and in the nonlinear regime is given.
Abstract: 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.
1,283 citations
TL;DR: It is impossible to partition the exact Hohenberg-Kohn functional into a piece that scales as ${\ensuremath{\gamma}}^{2}$ and a piece which scales as £1, even if complete freedom with the partitioning is allowed, because there are universal scaling inequalities.
Abstract: 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{
abla}v(r)=0 and F${d}^{3}$r n(r)r\ifmmode\times\else\texttimes\fi{}\ensuremath{
abla}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{
abla}[\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{
abla}${v}_{x}$([n];r)=0, while the correlation energy and potential must satisfy ${E}_{c}$[n]+F${d}^{3}$r n(r)r\ensuremath{\cdot}\ensuremath{
abla}${v}_{c}$([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{
e}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{
abla}${v}_{c}$([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{
abla}${v}_{c}$([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.
745 citations
References
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TL;DR: In this paper, the Hartree and Hartree-Fock equations are applied to a uniform electron gas, where the exchange and correlation portions of the chemical potential of the gas are used as additional effective potentials.
Abstract: From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of $\frac{2}{3}$.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
47,477 citations
TL;DR: In this article, the ground state of an interacting electron gas in an external potential was investigated and it was proved that there exists a universal functional of the density, called F[n(mathrm{r})], independent of the potential of the electron gas.
Abstract: This paper deals with the ground state of an interacting electron gas in an external potential $v(\mathrm{r})$. It is proved that there exists a universal functional of the density, $F[n(\mathrm{r})]$, independent of $v(\mathrm{r})$, such that the expression $E\ensuremath{\equiv}\ensuremath{\int}v(\mathrm{r})n(\mathrm{r})d\mathrm{r}+F[n(\mathrm{r})]$ has as its minimum value the correct ground-state energy associated with $v(\mathrm{r})$. The functional $F[n(\mathrm{r})]$ is then discussed for two situations: (1) $n(\mathrm{r})={n}_{0}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}(\mathrm{r})$, $\frac{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}{{n}_{0}}\ensuremath{\ll}1$, and (2) $n(\mathrm{r})=\ensuremath{\phi}(\frac{\mathrm{r}}{{r}_{0}})$ with $\ensuremath{\phi}$ arbitrary and ${r}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$. In both cases $F$ can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.
38,160 citations
TL;DR: In this article, 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.
Abstract: 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.
3,750 citations
TL;DR: The spin-density-functional (SDF) formalism has been 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 as discussed by the authors.
Abstract: 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.
2,763 citations