Density gradient expansions

Sahni V, Gruenebaum J, Perdew JP (1982) Study of the density-gradient expansion for the exchange energy, Phys Rev B, 26 43714377... 

Kinetic Energies Calculated from Density Gradient Expansion... 

Using result (86) to lowest order in a density gradient expansion gives... 

Salvetti formula for the correlation energy in terms of the electron density and the non-interacting kinetic energy density, then replaces the latter by its second-order density-gradient expansion. (The Colle-Salvetti formula itself is derived from a number of theoretical approximations, and is fitted to the correlation energy of the helium atom.) The result is then cast into the GGA form of Eq. (59) via integration by parts[55]. 

Density-gradient expansion (DGE). These are formal analogs of the three-dimensional Taylor expansion of the exchange-correlation energy in derivatives of the density ... 

The second-order density gradient expansion for exchange performs well only in the limit of small reduced density gradients x. The assumption that x is small may be justified for an infinite electron gas but not for finite systems, where x diverges in far-out regions. This can be demonstrated by assuming a spherically symmetric exponential density p(r) =... 

Start with the analytic second-order density gradient expansion for the coupling constant- and angle-averaged exchange hole, u). Convert the diverging DGE hole... 

Simply imposing correct asymptotic limits on the second-order density-gradient expansion (Section 24.7.1) proved to be a very elfective strategy for designing exchange functionals. It is reasonable to assume that the more exact constraints an approximate density functional satisfies, the more accurate and universal it will be. This idea is behind many existing density functional approximations. 

The TF theory and electronic description is considered as the referential for the uniform distribution of electrons in atoms and molecules, respecting which the electronic accumulation in bonding is further described, usually as a perturbation - as in DFT when density gradient expansions are considered, or by general reformulation of the problem in terms of localization - in which case special quantum treatment as provided by stochastic Fokker-Planck modeling is needed these issues will be in next addressed and imfolded. 

Lee, C. Ghosh, S. K. Density-gradient expansion of the kinetic-energy functional for molecules. Phys. Rev. A 1986, 33, 3506-3507. 

R. Peverati, Y. Zhao, andD. G. Truhlar,/. Phys. Chem. Lett., 2,1991-1997 (2011). Generalized Gradient Approximation That Recovers the Second-Order Density-Gradient Expansion with Optimized Across-the-Board Performance. 

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