Adiabatic local-density approximation ALDA

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

The simplest possible approximation of the time-dependent xc potential is the so-called time-dependent or adiabatic local density approximation (ALDA). It employs the functional form of the static LDA with a time-dependent density ... 

Additional steps are required for an efficient implementation in a plane wave code [34]. The adiabatic local density approximation (ALDA) of the TDDFT kernel,... 

The simplest approximation, which also is the one that has been used most often, is the adiabatic local-density approximation (ALDA). In the static case, the local-density approximation amounts to letting e c be a function of the electron density in the point of interest, and in the ALDA this approximation is kept and any time-dependence is ignored. Accordingly,... 

With eqn (7) the time-dependent Kohn-Sham scheme is an exact many-body theory. But, as in the time-independent case, the exchange-correlation action functional is not known and has to be approximated. The most common approximation is the adiabatic local density approximation (ALDA). Here, the non-local (in time) exchange-correlation kernel, i.e., the action functional, is approximated by a time-independent kernel that is local in time. Thus, it is assumed that the variation of the total electron density in time is slow, and as a consequence it is possible to use a time-independent exchange-correlation potential from a ground-state calculation. Therefore, the functional is written as the integral over time of the exchange-... 

The second approach is used by Baerends and co-workers. They use linear response theory, but instead of calculating the full linear response function they use the response function of the noninteracting Kohn-Sham system together with an effective potential. This response function can be calculated from the Kohn-Sham orbitals and energies and the occupation numbers. They use the adiabatic local density approximation (ALDA), and so their exchange correlation kernel, /xc (which is the functional derivative of the exchange correlation potential, Vxc, with respect to the time-dependent density) is local in space and in time. They report frequency dependent polarizabilities for rare gas atoms, and static polarizabilities for molecules. 

We will rely on the so-called adiabatic local density approximation (ALDA) to describe the exchange-correlation time-dependent functional Vxc[n, t), based on the exchange-correlation of a free electron-gas [25-28]. 

Figure 27 Excitation energies for the He atom obtained with different approaches. KS marks the orbital eigenvalues for the exact Kohn-Sham potential, ALDA the results with the adiabatic local-density approximation, and for the TDOEP approaches different orbital-dependent functionals have been used. Both the ALDA and the TDOEP results have been obtained using the time-dependent density-functional theory...

The standard approximations in TDDFT are the local density approximation (LDA) for vXCCT and the related adiabatic LDA (ALDA) for f . Within LDA, the potential is taken from the model of the homogeneous electron gas [43], with the dominant exchange part vxJA of vx A being only the 1/3 power of pa... 

All approximations we study for /xc are adiabatic. The most ubiquitous is ALDA (or more precisley, the adiabatic local spin density approximation) in which... 

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