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Physical Review A

Density-functional theory for time-dependent systems

Asish Kumar Dhara, Swapan Kumar Ghosh

Published Jan 1, 1987• 157 citations
Publisher

Abstract

The time-dependent density-functional theory of Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] is reexamined on the basis of its limitations, and the criticisms raised by Xu and Rajagopal [Phys. Rev. A 31, 2682 (1985)] are addressed, within the imposition of natural boundary conditions of vanishing density and potential at infinity. Also, for a single-particle system characterized by an arbitrary time-dependent potential, the uniqueness of the density-to-potential mapping is established explicitly for both bound and scattering states.

DOI
10.1103/physreva.35.442

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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
The time-dependent density-functional theory of Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] is reexamined on the basis of its limitations, and the criticisms raised by Xu and Rajagopal [Phys. Rev. A 31, 2682 (1985)] are addressed, within the imposition of natural boundary conditions of vanishing density and potential at infinity. Also, for a single-particle system characterized by an arbitrary time-dependent potential, the uniqueness of the density-to-potential mapping is established explicitly for both bound and scattering states.
The possibility of formulating a density-functional theory for time-dependent systems is studied in the linear-response regime. Two results are obtained: (i) The work of Runge and Gross is generalized to a finite time interval with an arbitrary form of the external potential; and (ii) there exists no zero in the mapping $n(\mathrm{r}, \ensuremath{\omega})=\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}\ensuremath{\chi}(\mathrm{r}, {\mathrm{r}}^{\ensuremath{'}};\ensuremath{\omega})V({\mathrm{r}}^{\ensuremath{'}}, \ensuremath{\omega})$ for all $\ensuremath{\omega}$ under a very general thermodynamic condition.
We show that a time-dependent particle density n(rt)n(\mathbf{r}t)n(rt) obtained from a given many-particle system can, under mild restrictions on the initial state, always be reproduced by an external potential ${v}^{\ensuremath{'}}(\mathbf{r}t)$ in a many-particle system with different two-particle interactions. Given the initial state of this other many-particle system, the potential ${v}^{\ensuremath{'}}(\mathbf{r}t)$ is unique up to a purely time-dependent function. As a special case we obtain the well-known Runge-Gross theorem.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.
The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.
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.