Is the Local Density Approximation Exact for Short Wavelength Fluctuations?

A commonly cited reason for the success of the local spin density (LSD) approximation is that it correctly accounts for short wavelength contributions to the exchange-correlation energy. We show that this result, while true in several limits and for several approximations to these fluctuations, is not exact in general, with an analytic demonstration on a specific system (Hooke's atom). Nevertheless, we find that LSD is rather accurate for small separations.