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Physical Review A

Reply to “Comment on ‘Analysis of causality in time-dependent density-functional theory’ ”

Manoj K. Harbola

Published Apr 11, 2001• 2 citations
Publisher

Abstract

By employing the equation derived by Amusia and Shaginyan in their comment on our work, we show that the potential corresponding to a time-dependent density is a noncausal functional of the density. We further strengthen this conclusion on the basis of the exact equation satisfied by the density.
DOI
10.1103/physreva.63.056502

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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

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

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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.

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Reply to “Comment on ‘Critique of the foundations of time-dependent density-functional theory’ ”

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In the preceding Comment, Maitra, van Leeuwen, and Burke (MLB) [Phys. Rev. A 78, 056501 (2008) attempt to refute our criticism of the foundations of time-dependent density functional theory [see J. Schirmer and A. Dreuw, Phys. Rev. A 75, 022513 (2007)]. However, their arguments miss the essence of our position. This is partly due to an ambiguity concerning the present status of the time-dependent (TD) Kohn-Sham (KS) equations. Based on the first Runge-Gross (mapping) theorem, the TD KS equations have to be established as a fixed-point iteration scheme for potential functionals (PF-FPI). Here, however, the convergence of the FPI is not assured. On the other hand, time propagation of the TD KS equations in their PF-FPI form fails because these equations do not represent physical equations of motion. MLB present a construction of the exact exchange-correlation potential, apparently contradicting one of our findings. We explain that their construction is explicitly based on the solution of the TD Schr\"odinger equation and, for this reason, does not address our objections, let alone refute them.

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Comment on “Analysis of causality in time-dependent density-functional theory”

M. Ya. Amusia, Vasily R. Shaginyan

It has been argued that there exists only one inverse of the linear-response function $\ensuremath{\chi},$ i.e., ${\ensuremath{\chi}}^{\ensuremath{-}1},$ which depends symmetrically on its spatial-times variables [M. K. Harbola and A. Banerjee, Phys. Rev. A 60, 5101 (1999)]. Some brief comments on this consideration are presented here. We show in contrast to Harbola and Banerjee, that it is possible to construct the causal inverse also. At the same time, we confirm the main statement of [Harbola and Banerjee] that in fact there is no contradiction between the symmetry and causality.

arXiv:cond-mat/0006356•Physical Review A•Apr 11, 2001•9 citations

Analysis of causality in time-dependent density-functional theory

Manoj K. Harbola, Arup Banerjee

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 ${f}_{\mathrm{xc}},$ 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.

Physical Review A•Dec 1, 1999•15 citations

Reply to “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’ ”

I. Guedes

In this reply I discuss further the misconception in my paper [Phys. Rev. A 63, 034102 (2001)] related to the solution of the Schr\"odinger equation for a particle in a time-dependent linear potential.

Physical Review A•Jul 8, 2003•18 citations
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.
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 the preceding Comment, Maitra, van Leeuwen, and Burke (MLB) [Phys. Rev. A 78, 056501 (2008) attempt to refute our criticism of the foundations of time-dependent density functional theory [see J. Schirmer and A. Dreuw, Phys. Rev. A 75, 022513 (2007)]. However, their arguments miss the essence of our position. This is partly due to an ambiguity concerning the present status of the time-dependent (TD) Kohn-Sham (KS) equations. Based on the first Runge-Gross (mapping) theorem, the TD KS equations have to be established as a fixed-point iteration scheme for potential functionals (PF-FPI). Here, however, the convergence of the FPI is not assured. On the other hand, time propagation of the TD KS equations in their PF-FPI form fails because these equations do not represent physical equations of motion. MLB present a construction of the exact exchange-correlation potential, apparently contradicting one of our findings. We explain that their construction is explicitly based on the solution of the TD Schr\"odinger equation and, for this reason, does not address our objections, let alone refute them.
It has been argued that there exists only one inverse of the linear-response function $\ensuremath{\chi},$ i.e., ${\ensuremath{\chi}}^{\ensuremath{-}1},$ which depends symmetrically on its spatial-times variables [M. K. Harbola and A. Banerjee, Phys. Rev. A 60, 5101 (1999)]. Some brief comments on this consideration are presented here. We show in contrast to Harbola and Banerjee, that it is possible to construct the causal inverse also. At the same time, we confirm the main statement of [Harbola and Banerjee] that in fact there is no contradiction between the symmetry and causality.
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.