Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0008468 · Physical Review Letters

Many-Body Diagrammatic Expansion in a Kohn-Sham Basis: Implications for Time-Dependent Density Functional Theory of Excited States

I. V. Tokatly, Oleg Pankratov

Published Mar 5, 2001• 65 citations
arXivHTMLPDF

Abstract

We formulate diagrammatic rules for many-body perturbation theory which uses Kohn-Sham Green's functions as basic propagators. The diagram technique allows one to study the properties of the dynamic nonlocal exchange-correlation (xc) kernel f(xc). We show that the spatial nonlocality of f(xc) is strongly frequency dependent. In particular, in extended systems the nonlocality range diverges at the excitation energies. This divergency is related to the discontinuity of the xc potential.

DOI
10.1103/physrevlett.86.2078

Related papers

Diagrammatic techniques for time-dependent density-functional theory

Ralf Stubner

The density-functional theory (DFT) is one of the most important ab initio methods for calculating properties of many-body systems. The success of DFT is based on the Kohn-Sham (KS) method and the approximations for the exchange-correlation (xc) effects. These approximations are well understood. However, DFT is first and foremost a theory for the ground state. It is very difficult to obtain properties of excited states. The time-dependent density-functional theory (TDDFT) can overcome this restriction and in the linear-response regime can give access to the properties of excited states. However, the xc effects, which are contained in the xc kernel f_xc , have to be approximated. The approximations for f_xc are not well understood up to now. Green-functions-based conventional many-body theory (MBT) is an alternative approach to the many-body problem. Especially when combined with diagrammatic techniques, MBT allows to derive physically motivated approximations. This work investigates the use of diagrammatic techniques from MBT for TDDFT . The aim of the investigations is to contribute to the development, testing, and understanding of the approximations involved in the time-dependent case. It is shown that it is possible to expand the xc kernel in a diagrammatic series. Explicit diagrammatic rules for constructing f_xc in any order of the perturbation theory are given. The derived diagrammatic rules enable us to study the analytic properties of f_xc. It is shown that f_xc is finite at KS excitation energies in every order of the perturbation theory. If this were not the case, finite-order approximations to f_xc would not give sensible results for the kernel which plays the role of a mass operator connecting the KS density--density response function with the exact density--density response function. However, there is one important difference between f_xc and the self-energy, the latter playing the role of the mass operator connecting free and exact Green function. This becomes apparent when the perturbative expansion for f_xc is used to calculate the correction to the KS excitation energies. This correction is obtained up to the second-order with respect to the interaction and with respect to one- and two-particle irreducible elements. It is shown that if a perturbative approximation for f_xc is used, the correction to the KS excitation energies has to be calculated using a consistent approach based on the same order of perturbation theory for both quantities. One of the most promising applications of TDDFT in the linear response regime is the calculation of electronic excitation spectra of solids, where TDDFT could replace the computationally expensive solution of the BSE , which is normally used for describing the excitonic correlations. It is shown that the xc kernel can be separated into a quasiparticle and a excitonic part in an exact and unambiguous way. The diagrammatic rules derived for f_xc apply to these parts separately. This allows us to express the excitonic part in terms of the three-point function, for which an integral equation is derived. This integral equation is similar to the BSE and gives the exact translation of the BSE into the TDDFT language. There exists the possibility of cancellation effects in the kernel of the integral equation. If these effects are effective, a first-order approximation to the excitonic f_xc is sufficient. The cancellation effects in the equation for Lambda are explicitly studied for a two-band model semiconductor. Using a general short-range interaction between the quasiparticles (QPs) all relevant equations are solved analytically. It is shown that the cancellation is most effective directly at the band-gap energy. Consequently, the first-order and the exact excitonic f_xc are very similar close to the band-gap energy for all interaction strengths. For a weak interaction the first-order approximation to the excitonic f_xc describes the different excitonic correlations in the density©density response function very well. Qualitatively similar results are found when the QPs interact via the long-ranged Coulomb interaction.

•Jan 1, 2005•0 citations

Many-body diagrammatic expansion for the exchange-correlation kernel in time-dependent density functional theory

I. V. Tokatly, Ralf Stubner, Oleg Pankratov

A diagrammatic expansion for the dynamic exchange-correlation kernel ${f}_{\mathrm{xc}}$ of time-dependent density functional theory is formulated. It is shown that ${f}_{\mathrm{xc}}$ has no singularities at Kohn-Sham transition energies in every order of the perturbation theory. However, it may diverge with the system size in extended systems. This signifies that any approximate perturbative substitute for ${f}_{\mathrm{xc}}$ requires a consistent perturbative treatment of the response function to avoid uncontrollable errors in the many-body corrections to excitations energies.

arXiv:cond-mat/0108012•Physical review. B, Condensed matter•Feb 27, 2002•24 citations

TDDFT and many-body theory

Carsten A. Ullrich

This chapter presents various techniques from many-body theory to construct time-dependent exchange-correlation (xc) functionals. The first section introduces a perturbation theory along the so-called adiabatic connection. This leads to a perturbative expansion of the xc potential, with exact exchange representing the lowest order. The following section introduces nonequilibrium Green's functions along the Keldysh contour. A Keldysh action principle is formulated which yields a formally exact representation of the time-dependent xc potential. As an alternative, the xc potential is expressed through the so-called Sham-Schlüter equation. The third section establishes a connection between the xc kernels of time-dependent density-functional theory in linear response and many-body perturbation theory. A diagrammatic expansion of the xc kernel is given. Finally, an expression for the xc kernel is derived starting from the Bethe-Salpeter equation. This xc kernel gives excellent agreement with experimental optical absorption spectra of insulators, including excitonic features.

Oxford University Press eBooks•Dec 22, 2011•0 citations

Generalized Kohn-Sham Density-Functional Theory via Effective Action Formalism

Marat Valiev, Gayanath Fernando

We present a rigorous formulation of generalized Kohn-Sham density-functional theory. This provides a straightforward Kohn-Sham description of many-body systems based not only on particle-density but also on any other observable. We illustrate the formalism for the case of a particle-density based description of a nonrelativistic many-electron system. We obtain a simple diagrammatic expansion of the exchange-correlation functional in terms of Kohn-Sham single-particle orbitals and energies; develop systematic Kohn-Sham formulation for one-electron propagators and many-body excitation energies. This work is ideally suited for practical applications and provides a rigorous basis for a systematic development of the existing body of first-principles calculations in a controllable fashion.

arXiv:cond-mat/9702247•Feb 27, 1997•5 citations

Current-Dependent Exchange-Correlation Potential for Dynamical Linear Response Theory

Giovanni Vignale, Walter Kohn

The frequency-dependent exchange-correlation potential, which appears in the usual Kohn-Sham formulation of a time-dependent linear response problem, is a strongly nonlocal functional of the density, so that a consistent local density approximation generally does not exist. This problem can be avoided by choosing the current density as the basic variable in a generalized Kohn-Sham theory. This theory admits a local approximation which, for fixed frequency, is exact in the limit of slowly varying densities and perturbing potentials.

arXiv:cond-mat/9605052•Physical Review Letters•Sep 2, 1996•492 citations

Communications: On the relation between the scalar and tensor exchange-correlation kernels of the time-dependent density-functional theory

V. U. Nazarov, Giovanni Vignale, Y.-C. Chang

The scalar f(xc) and tensor f(xc) exchange-correlation (xc) kernels are key ingredients of the time-dependent density functional theory and the time-dependent current density functional theory, respectively. We derive a comparatively simple relation between these two kernels under the assumption that the dynamic xc can be considered "weak." A calculation of the frequency-dependent dielectric function of silicon using this relation in conjunction with Vignale-Kohn f(xc) demonstrates a potential of our method to account for the dynamic many-body effects within the rigorous scheme of time-dependent density functional theory. Our formula provides a bridge between the scalar f(xc), which directly enters many applications, and the tensor f(xc) which, due to its locality in space, is much easier to approximate.

arXiv:1001.2795•The Journal of Chemical Physics•Jul 13, 2010•5 citations
We formulate diagrammatic rules for many-body perturbation theory which uses Kohn-Sham Green's functions as basic propagators. The diagram technique allows one to study the properties of the dynamic nonlocal exchange-correlation (xc) kernel f(xc). We show that the spatial nonlocality of f(xc) is strongly frequency dependent. In particular, in extended systems the nonlocality range diverges at the excitation energies. This divergency is related to the discontinuity of the xc potential.
The density-functional theory (DFT) is one of the most important ab initio methods for calculating properties of many-body systems. The success of DFT is based on the Kohn-Sham (KS) method and the approximations for the exchange-correlation (xc) effects. These approximations are well understood. However, DFT is first and foremost a theory for the ground state. It is very difficult to obtain properties of excited states. The time-dependent density-functional theory (TDDFT) can overcome this restriction and in the linear-response regime can give access to the properties of excited states. However, the xc effects, which are contained in the xc kernel f_xc , have to be approximated. The approximations for f_xc are not well understood up to now. Green-functions-based conventional many-body theory (MBT) is an alternative approach to the many-body problem. Especially when combined with diagrammatic techniques, MBT allows to derive physically motivated approximations. This work investigates the use of diagrammatic techniques from MBT for TDDFT . The aim of the investigations is to contribute to the development, testing, and understanding of the approximations involved in the time-dependent case. It is shown that it is possible to expand the xc kernel in a diagrammatic series. Explicit diagrammatic rules for constructing f_xc in any order of the perturbation theory are given. The derived diagrammatic rules enable us to study the analytic properties of f_xc. It is shown that f_xc is finite at KS excitation energies in every order of the perturbation theory. If this were not the case, finite-order approximations to f_xc would not give sensible results for the kernel which plays the role of a mass operator connecting the KS density--density response function with the exact density--density response function. However, there is one important difference between f_xc and the self-energy, the latter playing the role of the mass operator connecting free and exact Green function. This becomes apparent when the perturbative expansion for f_xc is used to calculate the correction to the KS excitation energies. This correction is obtained up to the second-order with respect to the interaction and with respect to one- and two-particle irreducible elements. It is shown that if a perturbative approximation for f_xc is used, the correction to the KS excitation energies has to be calculated using a consistent approach based on the same order of perturbation theory for both quantities. One of the most promising applications of TDDFT in the linear response regime is the calculation of electronic excitation spectra of solids, where TDDFT could replace the computationally expensive solution of the BSE , which is normally used for describing the excitonic correlations. It is shown that the xc kernel can be separated into a quasiparticle and a excitonic part in an exact and unambiguous way. The diagrammatic rules derived for f_xc apply to these parts separately. This allows us to express the excitonic part in terms of the three-point function, for which an integral equation is derived. This integral equation is similar to the BSE and gives the exact translation of the BSE into the TDDFT language. There exists the possibility of cancellation effects in the kernel of the integral equation. If these effects are effective, a first-order approximation to the excitonic f_xc is sufficient. The cancellation effects in the equation for Lambda are explicitly studied for a two-band model semiconductor. Using a general short-range interaction between the quasiparticles (QPs) all relevant equations are solved analytically. It is shown that the cancellation is most effective directly at the band-gap energy. Consequently, the first-order and the exact excitonic f_xc are very similar close to the band-gap energy for all interaction strengths. For a weak interaction the first-order approximation to the excitonic f_xc describes the different excitonic correlations in the density©density response function very well. Qualitatively similar results are found when the QPs interact via the long-ranged Coulomb interaction.
A diagrammatic expansion for the dynamic exchange-correlation kernel fxc{f}_{\mathrm{xc}}fxc​ of time-dependent density functional theory is formulated. It is shown that fxc{f}_{\mathrm{xc}}fxc​ has no singularities at Kohn-Sham transition energies in every order of the perturbation theory. However, it may diverge with the system size in extended systems. This signifies that any approximate perturbative substitute for fxc{f}_{\mathrm{xc}}fxc​ requires a consistent perturbative treatment of the response function to avoid uncontrollable errors in the many-body corrections to excitations energies.
This chapter presents various techniques from many-body theory to construct time-dependent exchange-correlation (xc) functionals. The first section introduces a perturbation theory along the so-called adiabatic connection. This leads to a perturbative expansion of the xc potential, with exact exchange representing the lowest order. The following section introduces nonequilibrium Green's functions along the Keldysh contour. A Keldysh action principle is formulated which yields a formally exact representation of the time-dependent xc potential. As an alternative, the xc potential is expressed through the so-called Sham-Schlüter equation. The third section establishes a connection between the xc kernels of time-dependent density-functional theory in linear response and many-body perturbation theory. A diagrammatic expansion of the xc kernel is given. Finally, an expression for the xc kernel is derived starting from the Bethe-Salpeter equation. This xc kernel gives excellent agreement with experimental optical absorption spectra of insulators, including excitonic features.
We present a rigorous formulation of generalized Kohn-Sham density-functional theory. This provides a straightforward Kohn-Sham description of many-body systems based not only on particle-density but also on any other observable. We illustrate the formalism for the case of a particle-density based description of a nonrelativistic many-electron system. We obtain a simple diagrammatic expansion of the exchange-correlation functional in terms of Kohn-Sham single-particle orbitals and energies; develop systematic Kohn-Sham formulation for one-electron propagators and many-body excitation energies. This work is ideally suited for practical applications and provides a rigorous basis for a systematic development of the existing body of first-principles calculations in a controllable fashion.
The frequency-dependent exchange-correlation potential, which appears in the usual Kohn-Sham formulation of a time-dependent linear response problem, is a strongly nonlocal functional of the density, so that a consistent local density approximation generally does not exist. This problem can be avoided by choosing the current density as the basic variable in a generalized Kohn-Sham theory. This theory admits a local approximation which, for fixed frequency, is exact in the limit of slowly varying densities and perturbing potentials.
The scalar f(xc) and tensor f(xc) exchange-correlation (xc) kernels are key ingredients of the time-dependent density functional theory and the time-dependent current density functional theory, respectively. We derive a comparatively simple relation between these two kernels under the assumption that the dynamic xc can be considered "weak." A calculation of the frequency-dependent dielectric function of silicon using this relation in conjunction with Vignale-Kohn f(xc) demonstrates a potential of our method to account for the dynamic many-body effects within the rigorous scheme of time-dependent density functional theory. Our formula provides a bridge between the scalar f(xc), which directly enters many applications, and the tensor f(xc) which, due to its locality in space, is much easier to approximate.