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Physical Review A

Maitra-Burke example of initial-state dependence in time-dependent density-functional theory

A. Holas, Robert Balawender

Published Mar 1, 2002• 3 citations
Publisher

Abstract

In a recent paper, Maitra and Burke [Phys. Rev. A 63, 042501 (2001); 64, 039901(E) (2001)] have given an interesting and instructive example that illustrates a specific feature of the time-dependent density-functional theory---the dependence of the reconstructed time-dependent potential not only on the electron density, but also on the initial state of the system. However, a concise form of its presentation by these authors is insufficient to reveal all its peculiarities. Our paper represents a very detailed study of this valuable example, intended to facilitate a better understanding and appreciation.

DOI
10.1103/physreva.65.034502

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In a recent paper, Maitra and Burke [Phys. Rev. A 63, 042501 (2001); 64, 039901(E) (2001)] have given an interesting and instructive example that illustrates a specific feature of the time-dependent density-functional theory---the dependence of the reconstructed time-dependent potential not only on the electron density, but also on the initial state of the system. However, a concise form of its presentation by these authors is insufficient to reveal all its peculiarities. Our paper represents a very detailed study of this valuable example, intended to facilitate a better understanding and appreciation.
Time-dependent density functionals depend in principle on the initial state of a system, but this is ignored in functional approximations presently in use. For one electron, it is shown that there is no initial-state dependence: for any density, only one initial state produces a well-behaved potential. For two noninteracting electrons with the same spin in one dimension, an initial potential that makes an alternative initial wave function evolve with the same density and current as a ground state is calculated. This potential is well-behaved, and can be made arbitrarily different from the original potential.
Predictivity of the Kohn-Sham approach to dynamical problems, when regarded as an initial value problem in a time-dependent density functional framework, is analysed for a class of models for which the argument devised in the work of Maitra et al. (Phys. Rev. A 78, 056501 (2008), arXiv:cond-mat/0710.0018) for the standard electronic many-body problem does not apply. The original argument is here extended and revised. As a result, predictivity for this class of problems seems possible only at the price of introducing extra unknown functionals in the corresponding Kohn-Sham equation. Furthermore, the same argument, when applied to original electronic problem, suggests that the Hartree-exchange-correlation potential is not unambiguously identified by the contemporary and past densities and initial states, but also requires knowledge of the divergence of the contemporary Kohn-Sham current.
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.
Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is vvv representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time vvv representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.
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.