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Physical Review Letters

Density-Functional Theory for Time-Dependent Systems

Erich Runge, Eberhard K. U. Gross

Published Mar 19, 1984• 8,905 citations
Publisher

Abstract

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 $v$-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.

DOI
10.1103/physrevlett.52.997

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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(r\ensuremath→t)v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)v(r\ensuremath→t) leading to a given vvv-representable density n(r\ensuremath→t)n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)n(r\ensuremath→t) is uniquely determined so that the corresponding map v\ensuremath→nv\ensuremath{\rightarrow}nv\ensuremath→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.
We formulate the inverse Kohn–Sham equations based on the density equation theory. The present formulation relies on neither action functionals nor force balance equation, but is constructed by comparing the first-order density equation for the electron interacting real system and that for the electron non-interacting Kohn–Sham system. The derived inverse Kohn–Sham integral equation requires information of an exact ground-state electron density to calculate an exact Kohn–Sham potential for a ground state and it requires an exact time-dependent electron density and a time-derivative of the exact time-dependent first-order density matrix to calculate an exact time-dependent Kohn–Sham potential. A possible algorithm to construct the Kohn–Sham potential for a ground state is suggested as well as a solution algorithm for the time-dependent inverse Kohn–Sham problem without resorting to any approximations. These two solution algorithms, in combination, suggest a way to perform an exact time-dependent density functional calculation as a predictive method.
In the present thesis, a rigorous mathematical formulation of time-dependent density-functional theory for lattice systems is derived, serving a formally exact approach towards solving time-dependent many-particle Schrödinger problems. After introducing the necessary mathematical foundations, a one-to-one mapping from external potentials onto electronic densities obtained by solving the Schrödinger equation is introduced. Its properties and the related Banach spaces of scalar potentials and electronic densities are discussed in detail. It allows to precisely formulate the Kohn-Sham Iteration scheme, mapping a given many-particle to an effective single-particle Schrödinger problem, both generating the exact same density. Employing the Banach fixed point theorem, the Kohn-Sham Iteration scheme is proven to be convergent depending on the chosen initial density.
A time-dependent Kohn-Sham theory is presented for obtaining the time-dependent density which has a periodic dependence on time. A set of coupled single-particle equations $\ensuremath{-}\frac{1}{2}{\ensuremath{\nabla}}^{2}{\ensuremath{\chi}}_{i}+{v}_{\mathrm{eff}}{\ensuremath{\chi}}_{i}={\ensuremath{\epsilon}}_{i}{\ensuremath{\chi}}_{i}$ and $\frac{\ensuremath{\partial}{\ensuremath{\chi}}_{i}^{2}}{\ensuremath{\partial}t}+\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\chi}}_{i}^{2}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}{S}_{i})=0$ are obtained. The ${\ensuremath{\chi}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ and ${S}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ are the phase and amplitude, respectively, of the time-dependent Kohn-Sham orbitals, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ is the time-dependent Kohn-Sham effective potential, and ${\ensuremath{\epsilon}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)=\ensuremath{-}\frac{\ensuremath{\partial}{S}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)}{\ensuremath{\partial}t}$. The density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ is equal to the sum of the squares of the ${\ensuremath{\chi}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$.
A time-dependent density-functional formalism is developed for many-electron systems subjected to external electric and magnetic fields with arbitrary time dependence. The single-particle current density is shown to determine uniquely the time-dependent scalar and vector potentials characterizing the system, and hence also the many-particle wave function. A Levy-type universal functional is defined, and practical schemes for the calculation of electron density and the current density through hydrodynamical as well as a set of single-particle Kohn-Sham-like equations are proposed. The gauge invariance of the present self-consistent formalism is also proved.
Time-dependent [current]-density functional theory for open quantum systems (OQS) has emerged as a formalism that can incorporate dissipative effects in the dynamics of many-body quantum systems. Here, we review and clarify some formal aspects of these theories that have been recently questioned in the literature. In particular, we provide theoretical support for the following conclusions: (1) contrary to what we and others had stated before, within the master equation framework, there is in fact a one-to-one mapping between vector potentials and current densities for fixed initial state, particle-particle interaction, and memory kernel; (2) regardless of the first conclusion, all of our recently suggested Kohn-Sham (KS) schemes to reproduce the current and particle densities of the original OQS, and in particular, the use of a KS closed driven system, remains formally valid; (3) the Lindblad master equation maintains the positivity of the density matrix regardless of the time-dependence of the Hamiltonian or the dissipation operators; (4) within the stochastic Schrödinger equation picture, a one-to-one mapping from stochastic vector potential to stochastic current density for individual trajectories has not been proven so far, except in the case where the vector potential is the same for every member of the ensemble, in which case, it reduces to the Lindblad master equation picture; (5) master equations may violate certain desired properties of the density matrix, such as positivity, but they remain as one of the most useful constructs to study OQS when the environment is not easily incorporated explicitly in the calculation. The conclusions support our previous work as formally rigorous, offer new insights into it, and provide a common ground to discuss related theories.
The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of 4He^{4}\mathrm{He}4He atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.