LSD approximation exact locally

The first argument was based on the result of a second order (in e2) calculation of Exr(k) for a spin-unpolarized surface. They found that for large k, 2)(k) is 

The second argument came from the density functional version of the random phase approximation[63,46]. Within that scheme, they found that for any spin-unpolarized system, the leading gradient corrections to fD(k) became vanishingly small as k — oo. 

We have recently studied this short wavelength hypothesis in considerable de-tail[18]. We showed that the short wavelength hypothesis is correct for several limiting regimes and for certain approximate treatments of the inhomogeneous gas. However, we also found that the short wavelength hypothesis is not exact in general. 

The key step in our proof is an exact analysis of the large wavevector behavior of an inhomogeneous system. To get a quantity which depends only on k, we define the angle-averaged wavevector decomposition 

A simple Fourier analysis of the asymptotic expansion for large k then yields[65] 

---