Basic Concepts
of Time-Dependent Density-Functional Theory

The starting point for the theoretical description of a nonrelativistic electronic system is the time-dependent Schrödinger equation (TDSE)

which determines the propagation in time of the -particle system evolving from its initial state

Although fully describes the electronic system it is, of course, not observable. In a typical collision problem one usually measures the probabilities of finding some of the electrons in a given state or at a certain place in configuration space. These inclusive probabilities [6] can be expressed in terms of the q-particle density [7]

where denotes coordinates and spin of the -th electron and indicates summation over spin and integration over space coordinates, respectively. With the definition (3) describes the inclusive probability of finding electrons at given positions in configuration space, while the remaining electrons are not detected explicitly. As a special case one obtains for the spin-free one-particle density

which is the key for the understanding of DFT. The total hamiltonian that enters the TDSE (1)

includes the kinetic energy, the mutual Coulomb repulsion between the electrons

and the external potential

which characterizes the geometry as well as the explicit time-dependence of the particular quantum system. This decomposition of the Hamiltonian into a universal, time-independent part (6) and a system-specific time-dependent external potential is a second important ingredient for the conceptual understanding of DFT. Let us, thus, define a map by solving the TDSE for different external potentials but common initial state . As the one-particle density is uniquely determined by the time-dependent many-particle state this obviously defines a second map . The foundation of DFT involves the proof that the map is invertible, i.e., can be obtained as a functional of the one-particle density . As a consequence any observable that can be written as an expectation value of a Hermitian operator would then be uniquely determined by . More precisely the conditions under which is a 1-1 map between the external potential and the one-particle density are formulated by the Runge-Gross theorem [8]:

• For every single-particle potential that can be expanded into a Taylor series with respect to time around there exists a map by solving the TDSE with a fixed initial state . This map can be inverted up to an additive merely time-dependent function in the potential.

At this point a few remarks might be appropriate: (i) As the invertibility of the map between potentials and densities can only be shown with respect to a given initial state , the solution of the TDSE rigorously depends on the density and the initial state. Consequently, any observable of the system is a functional of and . (ii) The potential as a functional of is only determined up to a merely time-dependent function. This corresponds to an ambiguity in up to a time-dependent phase factor, which cancels out for any observable characterized by an operator that is free of time derivatives. (iii) The Runge-Gross theorem can be applied to any system characterized by a given interaction , in particular for . This fact is used in the subsequent section, where a set of effective one-particle Schrödinger equations yielding the exact one-particle density is derived.