Adiabatic LDA

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... 

Further, there are asymptotically corrected XC kernels available, and other variants (for instance kernels based on current-density functionals, or for range-separated hybrid functionals) with varying degrees of improvements over adiabatic LDA, GGA, or commonly used hybrid DFT XC kernels [45]. The approximations in the XC response kernel, in the XC potential used to determine the unperturbed MOs, and the size of the one-particle basis set, are the main factors that determine the quality of the solutions obtained from (13), and thus the accuracy of the calculated molecular response properties. Beyond these factors, the quality of the... 

Tables 1-3 show the results of calculations based on Eqs. (362) and (363). The calculation of Table 1 employs the ordinary local density approximation (LDA) for and the adiabatic LDA (188) for /,c (both using the parametriz-ation of Vosko, Wilk and Nusair [90]). In this limit, the kernel G is approximated by [103]...

The jALDAl indicates the first-version adiabatic LDA (ALDA) functioned corrected by incorporating vector potentials (j). The ET-pVQZ basis functions eire used. Excerpt from van Eaassen and de Boeij (2004)... 

In the variety of LDA approximation also called adiabatic LDA (ALDA), the kernel term is local in space as well as in time and maybe written as... 

It should be noted that the above TDLDA picture a priori involves two touchy approxmations. The first one consists in using the LDA which basically relies on the assumption of weakly varying (in space) electron density. This LDX approximation has been widely used in metal clusters arid does not raise problems with respect to the observables we arc interested in. The second approximation is to use in a dynamical context a functional which has been tuned to static problems. The extension of LDA to TDLDA is thus a further approximation which can he considered as adiabatic , in the sense that we are using, at each instant, the energy density as expressed... 

We now turn to our results for nxc and exc. The spherically averaged exchange-correlation hole, nxc(r,s), obtained from our adiabatic calculations is shown in figure 2(a) together with the LDA approximation (28) to this quantity... 

A fourth form of gradient correction has recently been proposed by Becke [107], based the adiabatic connection method (ACM). It uses a linear combination of the HF, LDA and B88 exchange contributions, together with the... 

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 ... 

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. 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

The B3LYP hybrid functional (Becke 1993), the first hybrid functional, is the most frequently used functional (or method) in all functionals (or all theories) in quantum chemistry calculations. This functional uses three parameters as the mixing ratios to form the adiabatic connections between the Hartree-Fock exchange integral and the LDA exchange functional and between the LYP-GGA correlation functional and the LDA correlation functional, and to combine with the attenuated GGA term of the B88 exchange functional. 

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