GGA Nonlocality Its Character, Origins, and Effects

A useful way to visualize and think about gradient-corrected nonlocality, or to compare one GGA with another, is to write [19,87] 

Clearly, the correlation energy of (1.206) can be written in the form of (1.217). To get the exchange energy into this form, apply the spin-scaling relation (1.127) to (1.205), then drop small Vs contributions to find 

The Lieb-Oxford bound of (1.122) will be satisfied for all densities n(r) if and only if 

There is much to be seen and explained [21] in (1.217) and Figs. 1.1 and 1.2. However, the main qualitative features are simply stated When we make a density variation in which Vg decreases, C increases, or s increases everywhere, we find that Ex increases and Ec/Ey, decreases. 

To understand this pattern [21], we note that the second-order gradient expansion for the non-interacting kinetic energy Ts[n -, n, which is arguably its own GGA [81], can be written as 

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