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Physical Review Letters

Time-Dependent Density-Functional Theory in the Linear-Response Regime

Tai Kai Ng, K. S. Singwi

Published Dec 7, 1987• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.59.2627

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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(r,\ensuremathω)=\ensuremath∫d3r\ensuremath′\ensuremathχ(r,r\ensuremath′;\ensuremathω)V(r\ensuremath′,\ensuremathω)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})n(r,\ensuremathω)=\ensuremath∫d3r\ensuremath′\ensuremathχ(r,r\ensuremath′;\ensuremathω)V(r\ensuremath′,\ensuremathω) for all \ensuremathω\ensuremath{\omega}\ensuremathω under a very general thermodynamic condition.
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.
This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schrödinger equation to external potentials. Such a mapping is provided by the Runge-Gross theorem and lies at the very core of time-dependent density functional theory. After introducing the necessary mathematical concepts, the usual mapping "there" - from potentials to wave functions as solutions to the Schrödinger equation - is revisited paying special attention to Sobolev regularity. This is scrutinised further when the question of functional differentiability of the solution with respect to the potential arises, a concept related to linear response theory. Finally, after a brief introduction to general density functional theory, the mapping "back again" - from densities to potentials thereby inverting the Schrödinger equation for a fixed initial state - is defined. Apart from utilising the original Runge-Gross proof this is achieved through a fixed-point procedure. Both approaches give rise to mathematical issues, previously unresolved, which however could be dealt with to some extent within the framework at hand.
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.
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.
Under reasonable assumptions the time-dependent particle density $n(\stackrel{P\vec}{r},t)$ and the current density $\stackrel{P\vec}{j}(\stackrel{P\vec}{r},t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\stackrel{P\vec}{r},t)$ and $\stackrel{P\vec}{A}(\stackrel{P\vec}{r},t)$ and is initially in the quantum state $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$ can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and starting from an initial state $\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{'}}(0)⟩$, which yields the same density and current as $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$. Here we show that given the initial state of this other many-particle system, the potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$, if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.
We propose a memory form of exchange-correlation potential vXC(r,t){v}_{\mathrm{XC}}(\mathbf{r},t)vXC​(r,t) for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our vXC{v}_{\mathrm{XC}}vXC​ is not limited to the linear response regime. The proposed form of vXC{v}_{\mathrm{XC}}vXC​ is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for vXC{v}_{\mathrm{XC}}vXC​ based solely on an existing parametrization of the linear XC response kernel ${f}_{\mathrm{XC}}^{\mathrm{hom}}(n,\ensuremath{\omega})$ of the uniform gas. Application to quantum wells seems promising.