A severe test at low density

Levy and Perdew[24] recently derived a very severe low density convexity constraint on Exc. Define the convex functional A[n] by 

From the above analysis, it is expected that virtually all present approximations to Exc, including all the GGA s discussed in this paper, violate the convexity condition, because virtually all approximations contain a fd3r n4/3(r) component, and it is, unfortunately, difficult to overcome the concavity of this component as 7 — 0. Along these lines, it has been shown[24] that the LSD approximation and PW91 do not satisfy Eq. (71) for the one-electron densities i(r) = 7r-1 e-2r and n2(r) = (4tt)-1 (1 + 2r) e 2r. All this means that the satisfaction of the low density convexity requirement represents a tough challenge for the future. 

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