Density Functionals with Nonlocal Correlation Term

2 Density Functionals with Nonlocal Correlation Term 

Density functionals that initially account for van der Waals interaction were being developed in the early 1990s. These early representatives were developed only to account for dispersion interaction, that is, for nonoverlapping densities. Two groups developed a functional for layered structures [54] and this led them to the development of the first functional initially accounting for van der Waals interactions, which is now called the vdW-DFl (Van der Waals density functional) [55]. 

While the choice of a DFT functional to be augmented with the empirical dispersion correction is somewhat arbitrary, the choice of E[ + E in nonlocal exchange-correlation functionals is crucial, because the E[ + E pair strongly affects van der Waals bonding. Nonlocal functionals were often developed on the basis of a GGA functional, and, as mentioned previously, GGA functionals produce spurious bonding when closed shells of some monomers are overlapped. Dispersion Interaction Is a correlation effect [26], therefore, such a spurious attraction should be avoided when a nonlocal functional is developed. 

Dion and coworkers introduced the vdW-DFl functional in 2004 [55]. Since some GGA are known to produce unphysical binding, revPBE , was chosen (as the latter Is found to produce negligible bonding [26]). By construction vdW-DFl seamlessly switches between and , which is represented by an LDA correlation. Thus, the vdW-DFl functional may be represented by the formula  

The functional was defined only for closed-shell systems. In fact, original vdW-DFl computation was just a post-SCF (self-consistent field) procedure, where the revPBE SCF procedure was done first, then the revPBE correlation part except for LDa subtracted from the total energy and J was calculated from the revPBE 

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The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

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. 

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