Exact HF exchange

However, studies on the WDA are far from finished yet many important questions can be asked. For example, we still do not know whether it is the WDA XEDF or the WDA OF-KEDF that causes the appearance of the shell structure in heavier atoms (Z > 30). Nor do we know the reason why the shell structure is not evident for lighter atomic species. How does the ad hoc symmetrization scheme - [see Eq. (1 23)] effect the LR behavior How does the individual WDA XEDF compare with the exact HF exchange if the KS and the HF equations are solved Similarly, how good is the WDA OF-KEDF by itself if the TF-HK equation is solved with the LDA XCEDF instead of the WDA XCEDF ... 

In a similar spirit, Poater et al. have reoptimized the B3LYP parameters in order to minimize differences between computed electron densities from this modified DPT level and calculated at the QCISD level for a series of 16 small molecules (Poater, Duran, and Sola 2001). They observed, as already emphasized above, that different molecules require different amounts of exact HF exchange for optimal agreement between the two methods. 

In order to make improvements over the LSDA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not only the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA) and is the second rung on Jacob s Ladder. The third rung, meta-GGA functional, includes a dependence on the Laplacian of the density (V p) or on the orbital kinetic energy density (t). The fourth row, the hyper-GGA or hybrid functionals, includes a dependence on exact (HF) exchange. Finally, the fifth row incorporates the unoccupied Kohn-Sham orbitals. This is most widely accomplished within the so-called double hybrid functionals. 

The use of parentheses around x in the derivative of Exc means that only the explicit dependence on x is differentiated (i.e. the MO coefficients are not differentiated). Note that the above equations are valid for HF as well, if the XC energy is defined to be the exact HF exchange energy. However, taking Exc according to Eq. (15) formally yields... 

Hybrid potentials involve some combination of exact HF exchange mixed into the DFT-GGA potential, the most common being the B3LYP functional. The exchange-correlation energy is ... 

The first implementation of vdW-DFl was done with plane-wave code [56]. In that study, the binding energy of benzene dimers was computed. Since a plane-wave code with pseudopotentials was used, it was checked whether the the core densities affect interaction energies and a negligible difference was found for the systems considered. Nonlocal vdW-DFl is known to severely overestimate intermonomer separations (see, e.g., [56]) and is attributed to the use of " [57]. When exact HF exchange instead of pbe coupled to the vdW-DFl, this... 

When exact HF exchange is used with nonlocal functionals, HF-vdW-DFl and HF-W09 severely overbind the dispersion-bound part of S22 [77]. With LSDA pgj.. subtracted, HF-W09 gives mean deviations close to zero, while HF-vdW-DFl still overbinds. Predictably, the part in vdW-DFl and... 

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