Thomas - Fermi screening I read in Ashcroft & Mermin's Solid State text that for the Thomas-Fermi approximation to be applicable, the external potential needs to be "slowly varying," What does it mean for a function (in this case potential) to be slowly varying? 
And also I would like to ask what is the difference between Hartree-Fock method and the method of Thomas-Fermi?
 A: 1) The Thomas-Fermi method is a useful way to estimate the "screening cloud" surrounding electrons in a metal. A quantification of the screening is the inverse dielectric function of the material:
\begin{equation}
\frac{\phi_{ext}(\textbf{q})}{\epsilon(\textbf{q})} = \phi_{total}(\textbf{q})
\end{equation} 
Slowly-varying in this context means that the wavelength of the perturbation considered is long, i.e. it does not vary much over the lattice spacing or over the Fermi wavelength ($\textbf{q}<<\textbf{k}_F$). 
As the wavelength of the perturbation gets shorter and shorter, the perturbation would start to "see" more of the graininess in the electron gas/liquid. Therefore, with a shorter wavelength, it would be difficult to define a "screening cloud" as the perturbation would be varying inside the "cloud".
2) As for the difference between the Thomas-Fermi method and the Hartree-Fock method, the simplest answer is that they seek to calculate different quantities. The Hartree-Fock method is concerned with calculating the many-body wavefunction and the ground state energies by including the effects of electron-electron interactions. This can lead to predictions of experimental quantities such as the low-temperature electronic specific heat as well as the spin susceptibility.
On the other hand, the Thomas-Fermi method seeks to calculate the dielectric function (the screening parameter) -- a quantity that can be probed by e.g. inelastic electron scattering.
A: The potential has to vary slowly enough so that the electrons have time to respond as if the field were static.  
Risking a semi-educated guess:  I suppose that the potential would have to change less than a few percent in a time on the order of the transit time of an electron across a unit cell.  That is, the Fermi velocity divided by the lattice constant.  
