Runge-Gross theorem

The Runge-Gross Theorem (according to Erich Runge and Eberhard KU Gross ) is the formal basis of the time-dependent density functional theory and shows that for a many-particle system at every initial state (wave function at the point in time ) there is a clear mapping between the electron density at any point in time and the external one (time-dependent) potential (except for an additive term that only depends on time) exists. ${\ displaystyle t = t_ {0}}$ ${\ displaystyle n ({\ vec {r}}, t)}$ ${\ displaystyle t}$ ${\ displaystyle v _ {\ text {ex}} ({\ vec {r}}, t)}$

The derivation takes place in two steps:

The external potential is developed as a Taylor series around an initial point in time. With the help of the Ehrenfest theorem, it can be shown that two external potentials that differ by more than one additive constant produce different flow densities.
Using the continuity equation , it is shown that different flow densities also mean different electron densities.

The positive statement about the existence of this mapping makes it possible to calculate the dynamics of quantum mechanical many-body problems with the help of the electron density alone.

The set was published by Runge and Groß in 1984.

Web links

Kenny B. Lipkowitz, Thomas R. Cundari, Peter Elliott, Filipp Furche , Kieron Burke: Excited states from time dependent density functional theory (PDF; 1.1 MB). In: Reviews in Computational Chemistry, Volume 26, doi : 10.1002 / 9780470399545.ch3

Individual evidence

↑ Erich Runge, EKU Gross: Density-Functional Theory for Time-Dependent Systems . In: Phys. Rev. Lett. . 52, No. 12, March 19, 1984, p. 997. doi : 10.1103 / PhysRevLett.52.997 .

[1] Erich Runge, EKU Gross: Density-Functional Theory for Time-Dependent Systems . In: Phys. Rev. Lett. . 52, No. 12, March 19, 1984, p. 997. doi : 10.1103 / PhysRevLett.52.997 .