Correlation density-gradient expansions

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 short-range form of the correlation potential has been considered in DFT primarily in terms of density-gradient corrections to the LDA [6,7]. The correlation energy functional given by Eq.(7) here is not expressed explicitly as a functional of the electronic density, but is a formally exact expression which could be used to calibrate proposed functionals in particular cases where an accurate Cl expansion of a correlated ground-state wave function is known. When occupied orbital functions are expressed in a localized representation,... 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

Langreth and Mehl (LM) ° carefully studied the convergence properties of the gradient expansion and suggested a practical scheme which, for spherical atoms, yielded improved densities and correlation energies. The LM correction is... 

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