N-representability and stationarity in time-dependent density functional theory

TL;DR

This paper develops a new formalism for time-dependent density functional theory that ensures N-representability and stationarity, addressing foundational issues and unifying previous approaches.

Contribution

It introduces a generalized variational principle based on the norm of the functional derivative, ensuring a unique stationary point for the density.

Findings

The formalism guarantees N-representability of densities.

It establishes a unique stationary point corresponding to the Schrödinger equation.

The approach subsumes the original Runge-Gross formulation.

Abstract

To construct an N-representable time-dependent density-functional theory, a generalization to the time domain of the Levy-Lieb (LL) constrained search algorithm is required. That the action is only stationary in the Dirac-Frenkel variational principle eliminates the possibility of basing the search on the action itself. Instead, we use the norm of the partial functional derivative of the action in the Hilbert space of the wave functions in place of the energy of the LL search. The electron densities entering the formalism are $N$-representable, and the resulting universal action functional has a unique stationary point in the density at that corresponding to the solution of the Schr\"{o}dinger equation. The original Runge-Gross (RG) formulation is subsumed within the new formalism. Concerns in the literature about the meaning of the functional derivatives and the internal consistency of the RG formulation are allayed by clarifying the nature of the functional derivatives entering the formalism.