| 4:37:04 PM PST - Thu, Mar 6th 2014 |
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Dear Shipenn,
Indeed originally people would calculate the laplacian of the density and evaluate the potential generated by a density functional. The paper you cite by Perdew and Wang (http://dx.doi.org/10.1103/PhysRevB.33.8800) uses this approach. However, the need to evaluate the second derivative of the density led to increased costs in evaluating functionals and was considered cumbersome. In 1992 it was shown that using an integration-by-parts the terms including the second derivatives of the density can be eliminated (provided you want to evaluate matrix elements in a finite basis, if I remember correctly). This is documented in: John A. Pople, Peter M.W. Gill, Benny G. Johnson, "Kohn—Sham density-functional theory within a finite basis set", Chemical Physics Letters, Volume 199, (1992), Pages 557-560, http://dx.doi.org/10.1016/0009-2614(92)85009-Y. So in practice we never evaluate the actual potential we just calculate the required matrix elements. Now, for some functionals we can actually calculate the first and second order partial derivatives of the functional with-respect-to its arguments (i.e. the density and the norm of the gradient of the density). I guess it should be possible to express the potential in those terms but I have never tried.
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