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The Keldysh formalism in the time-dependent density-functional theory

A. Korostilv

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

We have considered application the Keldysh formalism in time-dependent \ndensity functional theory. It is shown that effects of electron-electron interaction can \nbe taken systematically into account in the framework of the Kadanoff-Baym \nequations for Green functions determined on the Keldysh contour for the second-order \n47 \nself-energy approximation. With the help of these Green functions we have derived \nthe time-dependent Kohn-Sham potential from an action functional. The Keldysh \nformalism in a similar way leads to response functions that obey the causality \nprinciple that is illustrated on the case of the time-dependent optimized effective \npotential equations.

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We have considered application the Keldysh formalism in time-dependent \ndensity functional theory. It is shown that effects of electron-electron interaction can \nbe taken systematically into account in the framework of the Kadanoff-Baym \nequations for Green functions determined on the Keldysh contour for the second-order \n47 \nself-energy approximation. With the help of these Green functions we have derived \nthe time-dependent Kohn-Sham potential from an action functional. The Keldysh \nformalism in a similar way leads to response functions that obey the causality \nprinciple that is illustrated on the case of the time-dependent optimized effective \npotential equations.
This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.
Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).
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; that is, 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 and 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 approximation methods for the exchange-correlation potential and to avoid the causality dilemma.
We resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory. We do this by defining a new action functional within the Keldysh formalism. By functional differentiation the new functional leads to response functions which are symmetric in the Keldysh time contour parameter, but which become causal when a transition to physical time is made. The new functional is further used to derive the equations of the time-dependent optimized potential method.
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.