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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