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Physical Review Letters

Time-Dependent Density-Functional Theory: Conceptual and Practical Aspects

H. Köhl, R. M. Dreizler

Published May 12, 1986• 39 citations
Publisher

Abstract

We sketch the proof of an existence theorem for a time-dependent density-functional theory on the set of $N$-representable densities, extending the constrained-search formulation of Levy into the time-dependent domain, and add a brief comment on practical issues.

DOI
10.1103/physrevlett.56.1993

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We sketch the proof of an existence theorem for a time-dependent density-functional theory on the set of NNN-representable densities, extending the constrained-search formulation of Levy into the time-dependent domain, and add a brief comment on practical issues.
To construct an NNN-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 NNN-representable, and the resulting universal action functional has a unique stationary point in the density at that corresponding to the solution of the Schr\"odinger equation. The original Runge-Gross (RG) formulation is subsumed within the current 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.
Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is vvv representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time vvv representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.
We reformulate the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed-point question for potentials on a given time interval. We show that the unique fixed point, i.e. the unique potential generating a given density, is reached as the limiting point of an iterative procedure. The one-to-one correspondence between densities and potentials is a straightforward result provided that the response function of the divergence of the internal forces is bounded. The existence, i.e. the v -representability of a density, can be proven as well provided that the operator norms of the response functions of the members of the iterative sequence of potentials have an upper bound. The densities under consideration have second time-derivatives that are required to satisfy a condition slightly weaker than being square-integrable. This approach avoids the usual restrictions of Taylor-expandability in time of the uniqueness theorem by Runge and Gross ( Phys. Rev. Lett. , 52 (1984) 997 ) and of the existence theorem by van Leeuwen ( Phys. Rev. Lett. , 82 (1999) 3863 ). Owing to its generality, the proof not only answers basic questions in density-functional theory but also has potential implications in other fields of physics.