Time-Dependent Kohn-Sham Equations

The essence of DFT is the determination of the exact one-particle density from which any many-particle observable can be derived. Let us assume we know this density and can calculate it from a set of ficticious orbitals

The ansatz (8) suggests that the orbitals fulfil a single-particle Schrödinger equation of the form

where the existence of the single-particle potential is assumed. (This is discussed in the literature under the topic v-representability [9].) If this potential exists, the Runge-Gross theorem guarantees its uniqueness, i.e., there is up to an additive merely time-dependent function exactly one single-particle potential, which together with the time-dependent Kohn-Sham (KS) equations (9) reproduces the exact one-particle density . Essentially, the KS equations represent an exact mapping of the -electron problem onto a set of single-particle problems. The crucial point is, however, that one does not know the KS potential explicitly. Nevertheless, a few general properties of the KS potential can be established: (i) the KS potential is local in space in contrast to the exchange term in Hartree-Fock theory, (ii) by virtue of the Runge-Gross theorem the KS potential must be a unique functional of the exact density for a given initial state and for a given KS determinant . The latter condition can be largely simplified if we assume that the time-dependent electronic system evolves from a non-degenerate ground state of the initially undisturbed system, which, via stationary DFT, is fully determined by its corresponding density . In this case the KS potential is a unique functional of the density alone

Based on the experience with single-particle pictures one usually splits the KS potential into its classical parts - the external (Coulomb) interaction and the Hartree potential , which includes the screening of the external potential due to the electrons - and a genuine quantum part the exchange-correlation potential

However, as nothing is known about the exchange-correlation potential so far, one needs an additional property of the many-particle system to determine the KS potential explicitly. The solution of the TDSE corresponds to a stationary point of the action integral

which essentially should be a functional of as is. The TDSE is then obtained by variation of with respect to . Can one therefore conclude that the exact one-particle density is a stationary point of the action integral as well, thus ? This is obviously not the case, as the Runge-Gross theorem predicts the functional only up to an arbitrary time phase. Because of the time-derivative in the Schrödinger operator the action (12) is, in fact, a functional of and the undetermined phase (for a detailed discussion on appropriate action functionals see [4] and [10]). Consequently, (12) is not useful as an additional source for the derivation of the KS potential. A more pragmatic approach rests on the assumption that the time-dependence of the KS potential is only due to the time-dependence of the density, where the functional dependence on the density is taken from stationary DFT. This is called the adiabatic approximation. The exchange-correlation (xc) potential for a nondegenerate ground state is related to the corresponding energy on the basis of the Rayleigh-Ritz variational principle

where and are the expectation values of the interacting system, denotes the kinetic energy, and the Hartree energy of the KS system. The functional derivative (13) yields the functional dependence of the xc-potential on for any particular approximation of the energy functional . Replacement of the ground-state density by is the essential idea of the adiabatic approximation.


The Local-Density Approximation

The most convenient ansatz for the xc-energy is based on the assumption that the energy functional can be locally approximated by that of the homogeneous electron gas. This local-density approximation (LDA) yields for the exchange part a simple analytical expression

while the correlation part is given in terms of a parametrization [11]. The LDA (or ALDA for adiabatic LDA) should give reasonable results for systems in which the density is slowly varying both in space and time. However, it is important to note that the exchange potential (14) decreases exponentially, giving rise to an asymptotically incorrect compensation of the self-energy contained in the Hartree term. In atomic physics this can be cured by forcing the correct asymptotic decrease of the KS potential [12].


The Optimized-Potential Method

The problems associated with the self-energy effects can be solved more systematically on the basis of orbital-dependent density functionals. As the KS orbitals are also functionals of one can express the xc-potential as a functional of the KS orbitals. Using the chain rule for functional differentiation one obtains

with cc indicating the complex conjugate of the preceding expression. For the x-only term the exact energy functional is known

which together with (15) yields an integral equation for the local exchange part of the KS potential. This scheme, originally introduced by Talman and Shadwick [13] as the optimized (effective) potential method (OEP OPM) has been shown to be very successful in ground-state DFT [14] as well as in time-dependent systems. The local character of the x-potential allows for a compensation of the self-energy not only for the occupied ground-state orbitals - as is the case in Hartree-Fock theory - but also for the virtual orbitals that is of particular interest in time-dependent systems. The first attempts to include correlation [15] are very promising for ground-state problems but are yet too complex for time-dependent systems.