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Diagrammatic techniques for time-dependent density-functional theory

Ralf Stubner

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

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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

Time-dependent density-functional theory for extended systems

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For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.

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I. V. Tokatly, Oleg Pankratov

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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.

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arXiv:cond-mat/0602020•Physical Review A•Feb 22, 2007•84 citations
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.
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.
For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.
We investigate the description of excitonic effects within time-dependent density-functional theory (TDDFT). The exchange-correlation kernel fxc{f}_{xc}fxc​ introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for fxc{f}_{xc}fxc​, we express its excitonic part fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ in terms of the effective vertex function $\ensuremath{\Lambda}$. The latter fulfills an integral equation that thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on the Bethe-Salpeter equation (BSE). The diagrammatic structure of the kernel in the equation for $\ensuremath{\Lambda}$ suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ is sufficient. A potential advantage of TDDFT over the many-body BSE method is thus dependent on the efficiency of the above-quoted cancellation. We explicitly verify this for an analytically solvable two-dimensional two-band model. The calculations confirm that the low-order fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ perfectly describes the bound exciton as well as the excitonic effects in the continuous spectrum in a wide range of the electron-hole coupling strength.
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.
The general expectation that, in principle, the time-dependent density-functional theory (TDDFT) is an exact formulation of the time evolution of an interacting NNN-electron system is critically reexamined. It is demonstrated that the previous TDDFT foundation, resting on four theorems by Runge and Gross (RG) [Phys. Rev. Lett. 52, 997 (1984)], is invalid because undefined phase factors corrupt the RG action integral functionals. Our finding confirms much of a previous analysis by van Leeuwen [Int. J. Mod. Phys. B 15, 1969 (2001)]. To analyze the RG theorems and other aspects of TDDFT, an utmost simplification of the Kohn-Sham (KS) concept has been introduced, in which the ground-state density is obtained from a single KS equation for one spatial (spinless) orbital. The time-dependent (TD) form of this radical Kohn-Sham (rKS) scheme, which has the same validity status as the ordinary KS version, has proved to be a valuable tool for analysis. The rKS concept is used to clarify also the alternative nonvariational formulation of TD KS theory. We argue that it is just a formal theory, allowing one to reproduce but not predict the time development of the exact density of the interacting NNN-electron system. Besides the issue of the formal exactness of TDDFT, it is shown that both the static and time-dependent KS linear response equations neglect the particle-particle (ppp-ppp) and hole-hole (hhh-hhh) matrix elements of the perturbing operator. For a local (multiplicative) operator this does not lead to a loss of information due to a remarkable general property of local operators. Accordingly, no logical inconsistency arises with respect to DFT, because DFT requires any external potential to be local. For a general nonlocal operator the error resulting from the neglected matrix elements is of second order in the electronic repulsion.