Image potential and the exchange-correlation weighted density approximation functional

It is known that the exchange-correlation (XC) potential $v_{\mathrm{XC}}$ obtained by using the weighted density approximation (WDA) does not yield the correct image behavior $-1/4z$ outside a metal surface. Some authors have speculated that this limitation could be overcome by symmetrizing the XC hole, using a two-point scaling factor in its modelization by the WDA. In this work, we show that this symmetrization procedure does not improve the XC potential $v_{\mathrm{XC}},$ and makes it unphysical. Alternatively, we present a WDA scheme based on an anisotropic scaling for the XC hole, constructed to verify both the normalization of the XC hole and the exact value of the second momenta of the XC hole. In this last case, the potencial $v_{\mathrm{XC}}$ has the correct image decay.