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/0406456 · The Journal of Chemical Physics

Time-dependent exchange-correlation current density functionals with memory

Yair Kurzweil, Roi Baer

Published Nov 2, 2004• 53 citations
arXivHTMLPDF

Abstract

Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).

DOI
10.1063/1.1802793

Related papers

Galilean-Invariant Exchange-Correlation Functionals with Quantum Memory

Roi Baer, Yair Kurzweil

A framework for variational time-dependent current-density-functional theory beyond linear response and adiabatic approximations is given. The method allows incorporation of quantum memory into exchange-correlation functionals which respect Vignale’s “Galilean invariance” (GI) principle (Phys. Rev. Lett. 74, 3233 (1995)). The free variables of the theory are the electron velocity-field and density. Gauge-invariant time-dependent (TD) Kohn-Sham equations are derived. The action functionals allow memory effects and are expressed in terms Lagrangian quantities, to ensure GI. General closed-form expressions for the corresponding scalar and vector potentials are derived and applied within a specific functional, based on the homogeneous electron gas (HEG). The scalar potential depends on the future velocity field while the vector potential depends on both future and past velocity fields. These future dependencies naturally arise when non-local temporal effects are allowed and do not violate physical causality.

•Nov 1, 2003•0 citations

The Keldysh formalism in the time-dependent density-functional theory

A. Korostilv

We have considered application the Keldysh formalism in time-dependent \ndensity functional theory. It is shown that effects of electron-electron interaction can \nbe taken systematically into account in the framework of the Kadanoff-Baym \nequations for Green functions determined on the Keldysh contour for the second-order \n47 \nself-energy approximation. With the help of these Green functions we have derived \nthe time-dependent Kohn-Sham potential from an action functional. The Keldysh \nformalism in a similar way leads to response functions that obey the causality \nprinciple that is illustrated on the case of the time-dependent optimized effective \npotential equations.

•Jan 1, 2010•0 citations

Action formalism of time-dependent density-functional theory

Martín A. Mosquera

The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.

arXiv:1303.5679•Physical Review A•Aug 26, 2013•6 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

Introduction to the Keldysh formalism and applications to time-dependent density-functional theory

Robert van Leeuwen, Nils Erik Dahlen, Gianluca Stefanucci +2 more

This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.

arXiv:cond-mat/0506130•Jun 6, 2005•18 citations

A reformulation of time-dependent Kohn–Sham theory in terms of the second time derivative of the density

Walter Tarantino, Carsten A. Ullrich

The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle, exactly by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second-order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.

arXiv:2012.04953•The Journal of Chemical Physics•May 28, 2021•17 citations
Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).
A framework for variational time-dependent current-density-functional theory beyond linear response and adiabatic approximations is given. The method allows incorporation of quantum memory into exchange-correlation functionals which respect Vignale’s “Galilean invariance” (GI) principle (Phys. Rev. Lett. 74, 3233 (1995)). The free variables of the theory are the electron velocity-field and density. Gauge-invariant time-dependent (TD) Kohn-Sham equations are derived. The action functionals allow memory effects and are expressed in terms Lagrangian quantities, to ensure GI. General closed-form expressions for the corresponding scalar and vector potentials are derived and applied within a specific functional, based on the homogeneous electron gas (HEG). The scalar potential depends on the future velocity field while the vector potential depends on both future and past velocity fields. These future dependencies naturally arise when non-local temporal effects are allowed and do not violate physical causality.
We have considered application the Keldysh formalism in time-dependent \ndensity functional theory. It is shown that effects of electron-electron interaction can \nbe taken systematically into account in the framework of the Kadanoff-Baym \nequations for Green functions determined on the Keldysh contour for the second-order \n47 \nself-energy approximation. With the help of these Green functions we have derived \nthe time-dependent Kohn-Sham potential from an action functional. The Keldysh \nformalism in a similar way leads to response functions that obey the causality \nprinciple that is illustrated on the case of the time-dependent optimized effective \npotential equations.
The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.
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.
This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.
The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle, exactly by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second-order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.