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Physical Review A

Analysis of causality in time-dependent density-functional theory

Manoj K. Harbola, Arup Banerjee

Published Dec 1, 1999• 15 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.60.5101

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

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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.

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Physical Review A•Oct 6, 2006•1 citation
It is pointed out that the kernel of the inverse response function \ensuremathδv(r,t)/\ensuremathδ\ensuremathρ(r\ensuremath′,t\ensuremath′),\ensuremath{\delta}v(\mathbf{r},t)/\ensuremath{\delta}\ensuremath{\rho}({\mathbf{r}}^{\ensuremath{'}}{,t}^{\ensuremath{'}}),\ensuremathδv(r,t)/\ensuremathδ\ensuremathρ(r\ensuremath′,t\ensuremath′), whose existence is guaranteed by the Runge-Gross theorem, cannot vanish for t<t\ensuremath′.t<{t}^{\ensuremath{'}}.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′{t}^{\ensuremath{'}}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.
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.
We resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory. We do this by defining a new action functional within the Keldysh formalism. By functional differentiation the new functional leads to response functions which are symmetric in the Keldysh time contour parameter, but which become causal when a transition to physical time is made. The new functional is further used to derive the equations of the time-dependent optimized potential method.
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.
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.
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.
A formally exact expression for the interaction density response function $\ensuremath{\chi}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$ exists in terms of (i) its noninteracting counterpart ${\ensuremath{\chi}}_{0}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$ and (ii) an exchange (x)(x)(x)-correlation (c)(c)(c) kernel ${f}_{xc}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$. In the absence of a first-principles theory for the $\ensuremath{\omega}$ dependence of fxc{f}_{xc}fxc​, the adiabatic approximation is most frequently made in this term, to construct a workable time-dependent density-functional theory. In the present study, a proposal is put forward to avoid the adiabatic approximation by working in the exchange-only limit in which fxc{f}_{xc}fxc​ is set equal to ${f}_{x}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$. We then refer to a result for the exchange energy given by Pines and Nozieres to motivate the assumption that ${f}_{x}={f}_{x}[\mathrm{Im}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{0}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})]$. The essential proposal here is therefore that the integral equation to be solved for the interacting density response function $\ensuremath{\chi}$ is in the exchange-only case characterized entirely by the noninteracting response function ${\ensuremath{\chi}}_{0}$.