On the Action Formalism of Time-dependent Density-functional Theory

TL;DR

This paper analyzes the action functional in time-dependent density-functional theory, addressing causality issues by extending the formalism to the Keldysh contour and deriving a causality-consistent variational equation.

Contribution

It introduces a new variational framework using the Keldysh contour to resolve causality paradoxes in TDDFT and connects existing approximations within this formalism.

Findings

The adiabatic local-density approximation is consistent within the new formalism.

A variational equation relating exchange-correlation potential and its kernel is derived.

The approach can lead to new methods for approximating the exchange-correlation potential.

Abstract

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, i.e., 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, as well as 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 approximations methods to the exchange-correlation potential and avoid the causality dilemma.