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18.5. A BRIEF HISTORY OF GGA’S                                     155     156                            CHAPTER 18. GENERALIZED GRADIENT APPROXIMATION
        • High-density correlation
          The numerical GGA satisfies this condition. As we show below, many chemical systems
          are close to this, and, by truncating the long-ranged Coulomb correlation hole, numerical
          GGA yields a finite value.
        • Self-interaction error
          Simple density functionals like LDA and GEA naturally have a self-interaction error,
          because they cannot tell when there is only one electron in the system. While GGA
          might improve numerically the value of E X or E C for that one electron, it will still have
          a residual self-interaction error.
        • Symmetry dilemma
          Just as above, nothing in GGAs construction helps here.
        • Potentials
          From the naive sense...
        • Koopman’s theorem No real improvement here.

       18.5  A brief history of GGA’s

       We have seen how the real-space cutoff construction imposed on the gradient expansion
       produces a numerically-defined GGA. This is not unique, in that, e.g. a smoother cutoff will
       lead to slightly different curves for moderate gradients, and perhaps much different curves
       for large gradients.


       18.6  Questions about generalized gradient approximations
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