# Critique of the foundations of time-dependent density-functional theory

## Abstract

The general expectation that, in principle, the time-dependent density-functional theory (TDDFT) is an exact formulation of the time evolution of an interacting N-electron system is critically reexamined. It is demonstrated that the previous TDDFT foundation, resting on four theorems by Runge and Gross (RG) [Phys. Rev. Lett. 52, 997 (1984)], is invalid because undefined phase factors corrupt the RG action integral functionals. Our finding confirms much of a previous analysis by van Leeuwen [Int. J. Mod. Phys. B 15, 1969 (2001)]. To analyze the RG theorems and other aspects of TDDFT, an utmost simplification of the Kohn-Sham (KS) concept has been introduced, in which the ground-state density is obtained from a single KS equation for one spatial (spinless) orbital. The time-dependent (TD) form of this radical Kohn-Sham (rKS) scheme, which has the same validity status as the ordinary KS version, has proved to be a valuable tool for analysis. The rKS concept is used to clarify also the alternative nonvariational formulation of TD KS theory. We argue that it is just a formal theory, allowing one to reproduce but not predict the time development of the exact density of the interacting N-electron system. Besides the issue of the formal exactnessmore »

- Authors:

- Theoretische Chemie, Physikalisch-Chemisches Institut, Universitaet Heidelberg, D-69120 Heidelberg (Germany)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 20982112

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevA.75.022513; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; ACTION INTEGRAL; BORON 15; DENSITY; DENSITY FUNCTIONAL METHOD; ELECTRONS; FOUNDATIONS; FUNCTIONALS; GROUND STATES; LOSSES; MATRICES; MATRIX ELEMENTS; POTENTIALS; RADICALS; TIME DEPENDENCE

### Citation Formats

```
Schirmer, J., Dreuw, A., and Institut fuer Physikalische und Theoretische Chemie, Universitaet Frankfurt, D-60439 Frankfurt.
```*Critique of the foundations of time-dependent density-functional theory*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.022513.

```
Schirmer, J., Dreuw, A., & Institut fuer Physikalische und Theoretische Chemie, Universitaet Frankfurt, D-60439 Frankfurt.
```*Critique of the foundations of time-dependent density-functional theory*. United States. doi:10.1103/PHYSREVA.75.022513.

```
Schirmer, J., Dreuw, A., and Institut fuer Physikalische und Theoretische Chemie, Universitaet Frankfurt, D-60439 Frankfurt. Thu .
"Critique of the foundations of time-dependent density-functional theory". United States.
doi:10.1103/PHYSREVA.75.022513.
```

```
@article{osti_20982112,
```

title = {Critique of the foundations of time-dependent density-functional theory},

author = {Schirmer, J. and Dreuw, A. and Institut fuer Physikalische und Theoretische Chemie, Universitaet Frankfurt, D-60439 Frankfurt},

abstractNote = {The general expectation that, in principle, the time-dependent density-functional theory (TDDFT) is an exact formulation of the time evolution of an interacting N-electron system is critically reexamined. It is demonstrated that the previous TDDFT foundation, resting on four theorems by Runge and Gross (RG) [Phys. Rev. Lett. 52, 997 (1984)], is invalid because undefined phase factors corrupt the RG action integral functionals. Our finding confirms much of a previous analysis by van Leeuwen [Int. J. Mod. Phys. B 15, 1969 (2001)]. To analyze the RG theorems and other aspects of TDDFT, an utmost simplification of the Kohn-Sham (KS) concept has been introduced, in which the ground-state density is obtained from a single KS equation for one spatial (spinless) orbital. The time-dependent (TD) form of this radical Kohn-Sham (rKS) scheme, which has the same validity status as the ordinary KS version, has proved to be a valuable tool for analysis. The rKS concept is used to clarify also the alternative nonvariational formulation of TD KS theory. We argue that it is just a formal theory, allowing one to reproduce but not predict the time development of the exact density of the interacting N-electron system. Besides the issue of the formal exactness of TDDFT, it is shown that both the static and time-dependent KS linear response equations neglect the particle-particle (p-p) and hole-hole (h-h) matrix elements of the perturbing operator. For a local (multiplicative) operator this does not lead to a loss of information due to a remarkable general property of local operators. Accordingly, no logical inconsistency arises with respect to DFT, because DFT requires any external potential to be local. For a general nonlocal operator the error resulting from the neglected matrix elements is of second order in the electronic repulsion.},

doi = {10.1103/PHYSREVA.75.022513},

journal = {Physical Review. A},

number = 2,

volume = 75,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}