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arXiv:1509.03071 · Physical Review Letters

Thermal Density Functional Theory: Time-Dependent Linear Response and Approximate Functionals from the Fluctuation-Dissipation Theorem

Aurora Pribram−Jones, Paul Grabowski, Kieron Burke

Published Jun 8, 2016• 80 citations
arXivHTMLPDF

Abstract

The van Leeuwen proof of linear-response time-dependent density functional theory (TDDFT) is generalized to thermal ensembles. This allows generalization to finite temperatures of the Gross-Kohn relation, the exchange-correlation kernel of TDDFT, and fluctuation dissipation theorem for DFT. This produces a natural method for generating new thermal exchange-correlation approximations.

DOI
10.1103/physrevlett.116.233001

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•Jan 1, 2005•0 citations
The van Leeuwen proof of linear-response time-dependent density functional theory (TDDFT) is generalized to thermal ensembles. This allows generalization to finite temperatures of the Gross-Kohn relation, the exchange-correlation kernel of TDDFT, and fluctuation dissipation theorem for DFT. This produces a natural method for generating new thermal exchange-correlation approximations.
This chapter discusses the linear-response framework of time-dependent density-functional theory (TDDFT). It begins with a thorough and self-contained exposition of general linear-response theory for many-body systems. The time-dependent as well as the frequency-dependent response are discussed. Several exact representations and properties of the density-density response function are reviewed, such as the Lehmann representation and fluctuation-dissipation theorem, and the Kramers-Kronig relations. It is shown how several important spectroscopic observables are obtained from linear-response theory. Then, the TDDFT formulation of linear-response theory is derived. The key quantity in this formalism is the so-called exchange-correlation kernel. The remainder of the chapter demonstrates how this linear-response formalism can be used to calculate excitation energies in principle exactly with TDDFT. The Casida equation is derived, as well as several approximate schemes. A comparison is made with time-dependent Hartree-Fock theory.
Time-dependent density functional theory (TDDFT) is a standard approach for calculating optical excitations of molecules and solids, while ensemble DFT (EDFT) is a promising alternative under development. We introduce ensemble TDDFT (ETDDFT), a practical theory that combines the two, generalizing both; we ensemble generalize the Gross-Kohn equation and the exchange-correlation kernel of TDDFT, and generalize EDFT to time-dependent problems. We relate coordinate scaling to the adiabatic connection. The new theory provides multiple avenues for constructing and using approximations. We illustrate these on the two-site Hubbard model. We connect our results to the more general case of nonperturbative time dependence.
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.
This chapter discusses time-dependent current-density-functional theory (TDCDFT). It first asks under what circumstances one should choose the current density as basic variable, instead of the particle density. There are several reasons: a current-based formulation is more suitable for constructing functionals beyond the adiabatic approximation. Furthermore, the current is the natural variable to describe coupling to electromagnetic fields and vector potentials. The basic existence theorems of TDCDFT are proved, and several exact properties and the linear-response regime are discussed. The second half of the chapter deals with the so-called Vignale-Kohn functional, which is a local and frequency-dependent approximation to the exchange-correlation vector potential. The Vignale-Kohn functional is expressed in terms of viscoelastic stresses in the electron liquid. Several applications in the linear-response regime and in real time are presented. The Vignale-Kohn functional works well for describing dissipative effects in collective, plasmon-like electron dynamics.
The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.
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.