Adiabatic local-density approximation

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 quantity ffffr.r. a ) is the frequency-dependent XC kernel for which common approximations are applied frequency-independent (adiabatic) local density approximations (LDA), adiabatic generalized gradient approximations (GGA), hybrid-DFT variants such as the popular functionals B3LYP and PBEO in which Kxc contains an admixture of Hartree-Fock ( exact ) exchange X,... 

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,... 

Note that here and later on r denotes the single-particle coordinate whereas R is still used as abbreviation for all nuclear positions as in Eq. (1). The potential (5) consists, on one hand, of an external potential V(r,R), which in our case is time-dependent owing to the atomic motion R( ). On the other hand, there are electron-electron interaction terms, namely the Hartree and the exchange-correlation term, which depend both via the density p on the functions tpj. The exchange-correlation potential VIC is defined within the so-called adiabatic local density approximation [25] which is the natural extension of the lda from stationary dpt. It is assumed to give reliable results for problems where the time scale of the external potential (in our case typical collision times) is larger than the electronic time scale. 

In Ref. [193] various non-hybrid and hybrid exchange-correlation potentials and suitable adiabatic local density approximations for the exchange-correlation kernel were... 

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,... 

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...

A review of the approximations in any time-depedendent density functional calculation of excitation energies is given. The single-pole approximation for the susceptibility is used to understand errors in popular approximations for the exchange-correlation kernel. A new hybrid of exact exchange and adiabatic local density approximation is proposed and tested on the He and Be atoms. 

Density Functional Theory beyond the Adiabatic Local Density Approximation. 

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]. 

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