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arXiv:1103.5702 · Physical Review A

Comment on “Modifying the variational principle in the action-integral-functional derivation of time-dependent density-functional theory”

Giovanni Vignale

Published Apr 27, 2011• 5 citations
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Abstract

In a paper recently published in Phys. Rev. A [arXiv:1010.4223], Schirmer has criticized an earlier work of mine [arXiv:0803.2727], as well as the foundations of time-dependent density functional theory. In Ref.[2], I showed that the so-called "causality paradox" - i.e., the failure of the exchange-correlation potential derived from the Runge-Gross time-dependent variational principle to satisfy causality requirements - can be solved by a careful reformulation of that variational principle. Fortunately, the criticism presented in Ref.[1] is based on elementary misunderstandings of the nature of functionals, gauge transformations, and the time-dependent variational principle. In this Comment I wish to point out and clear these misunderstandings.

DOI
10.1103/physreva.83.046501

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Reply to "Comment on 'Modifying the variational principle in the action-integral-functional derivation of time-dependent density functional theory'"

J. Schirmer

In a recent paper [Phys. Rev. A 77, 062511 (2008)], it was advocated to modify the variational principle for the action-integral functional in the Runge-Gross foundation of time-dependent density-functional theory. This was criticised in a subsequent paper [Phys. Rev. A 62, 052510 (2010)] by the present author. In a Comment [Phys. Rev. A 83, 046501 (2011)], it is argued that the criticism is unfounded. This is a response as given to the four specific points raised in the Comment, clarifying and confirming the essence of the original criticism.

arXiv:1105.2718•May 13, 2011•0 citations

Modifying the variational principle in the action-integral-functional derivation of time-dependent density-functional theory

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According to a recent paper by G. Vignale [Phys. Rev. A 77, 062511 (2008)], the problems arising in the original derivation of time-dependent density-functional theory based on the Runge-Gross (RG) action-integral functional (AIF) are because of an incorrect variational principle (VP). This argument and the proposed modification of the VP are critically analyzed. The more fundamental problem, though, is the indefiniteness of the RG AIF. In contrast to a widely held belief the indefiniteness is not eliminated in the variational procedure, which unwittingly is corroborated by Vignale's initial point.

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Action formalism of time-dependent density-functional theory

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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.

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I show that the so-called causality paradox of time-dependent density-functional theory arises from an incorrect formulation of the variational principle for the time evolution of the density. The correct formulation not only resolves the paradox in real time, but also leads to an expression for the causal exchange-correlation kernel in terms of Berry curvature. Furthermore, I show that all the results that were previously derived from symmetries of the action functional remain valid in the present formulation. Finally, I develop a model functional theory which explicitly demonstrates the workings of the formulation.

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J. Schirmer

The density-based action-integral functional introduced by Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] in their foundation of time-dependent density-functional theory is reexamined. Based on a simple expansion of the original definition, it becomes apparent that the action-integral functional is a trivial construct and, moreover, nonstationary. It cannot be used to establish equations of motion for the time evolution of quantum systems at the density-function level.

arXiv:1203.5052•Physical Review A•Jul 26, 2012•7 citations
In a paper recently published in Phys. Rev. A [arXiv:1010.4223], Schirmer has criticized an earlier work of mine [arXiv:0803.2727], as well as the foundations of time-dependent density functional theory. In Ref.[2], I showed that the so-called "causality paradox" - i.e., the failure of the exchange-correlation potential derived from the Runge-Gross time-dependent variational principle to satisfy causality requirements - can be solved by a careful reformulation of that variational principle. Fortunately, the criticism presented in Ref.[1] is based on elementary misunderstandings of the nature of functionals, gauge transformations, and the time-dependent variational principle. In this Comment I wish to point out and clear these misunderstandings.
In a recent paper [Phys. Rev. A 77, 062511 (2008)], it was advocated to modify the variational principle for the action-integral functional in the Runge-Gross foundation of time-dependent density-functional theory. This was criticised in a subsequent paper [Phys. Rev. A 62, 052510 (2010)] by the present author. In a Comment [Phys. Rev. A 83, 046501 (2011)], it is argued that the criticism is unfounded. This is a response as given to the four specific points raised in the Comment, clarifying and confirming the essence of the original criticism.
According to a recent paper by G. Vignale [Phys. Rev. A 77, 062511 (2008)], the problems arising in the original derivation of time-dependent density-functional theory based on the Runge-Gross (RG) action-integral functional (AIF) are because of an incorrect variational principle (VP). This argument and the proposed modification of the VP are critically analyzed. The more fundamental problem, though, is the indefiniteness of the RG AIF. In contrast to a widely held belief the indefiniteness is not eliminated in the variational procedure, which unwittingly is corroborated by Vignale's initial point.
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.
I show that the so-called causality paradox of time-dependent density-functional theory arises from an incorrect formulation of the variational principle for the time evolution of the density. The correct formulation not only resolves the paradox in real time, but also leads to an expression for the causal exchange-correlation kernel in terms of Berry curvature. Furthermore, I show that all the results that were previously derived from symmetries of the action functional remain valid in the present formulation. Finally, I develop a model functional theory which explicitly demonstrates the workings of the formulation.
It is pointed out that the kernel of the inverse response function $\ensuremath{\delta}v(\mathbf{r},t)/\ensuremath{\delta}\ensuremath{\rho}({\mathbf{r}}^{\ensuremath{'}}{,t}^{\ensuremath{'}}),$ whose existence is guaranteed by the Runge-Gross theorem, cannot vanish for $t<{t}^{\ensuremath{'}}.$ As such the kernel fxc,{f}_{\mathrm{xc}},fxc​, which is the functional derivative of the time-dependent exchange-correlation potential with respect to the density, can be symmetric with respect to an interchange between t and ${t}^{\ensuremath{'}}$ without violating causality. Thus there is no conflict between the symmetry requirement of this kernel and causality in time-dependent density-functional theory.