Screening linear response theory

the ionic coulomb potential is damped exponentially within a Thomas-Fermi screening length = 1 /ktf. It follows from eqs (2.41) and (6.10) that 

Screening in metals is very effident even the low-density metal sodium with rs — 4 au has a Thomas-Fermi screening length as small as 1.3 au. 

The screening in metals is also perfect. This follows from the fact that the total number of electrons associated with the screening density, 5p(r), is identically equal to Z, which can be seen by integrating 

Thus the static dielectric constant diverges in the long wavelength limit as q - 0. 

The Thomas-Fermi approximation is, unfortunately, a poor approximation for the sp-valent metals. It is based on the assumption that the potential varies much more slowly than the screening length of the electrons themselves, so that the local approximation for the kinetic energy, eqn (6.6), is valid. In practice, however, the variation in the ionic potential is measured by the core radius, Rc (cf Fig. 5.11), which is not large but of the same size as the screening length, XTF. Thus, we do not satisfy the criterion for the validity 

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The theory of the lattice dynamics of covalent semiconductors has been developed in the late sixties and early seventies [l]. The essence of this theory is the description of the density distribution of the valence electrons between the ions at their arbitrary and instantaneous positions. Originally this part of the electron distribution due to deviations of the ions from their equilibrium positions is derived from linear response theory. Subsequently phonon dispersion curves can be obtained from this dielectric screening method. From 1972 the present authors started their efforts to calculate phonon frequencies using this method. A presentation of linear response and dielectric screening theory is given in these proceedings in the paper by J.T. 

Screening effects are one of the most important manifestations of the existence of electron-electron interactions in solids. To discuss them, we will first consider a spatially homogeneous system, in which the response at a position r to an electric perturbation localized at ro only depends upon r — ro. This is true, for example, in an homogeneous interacting electron gas. The concept of the dielectric constant refers to the response to a weak perturbation. The relationship between the modification of the charge density Sp(q,(a) and the electrostatic potential F(q,co), is linear in this case, which is the range of validity of the linear response theory. It is then possible to define the electronic susceptibility expressed in... 

The frequency dependence of SHG at simple metal surface has been the focus of a recent theoretical study of Liebsch [100]. Time-dependent density functional theory was used in these calculations. The results suggest that the perpendicular surface contribution to the second harmonic current is found to be significantly larger than had been assumed previously. He also concludes that for 2 a> close to the threshold for electron emission, the self-consistently screened nonlinear electronic response becomes resonantly enhanced, analogous to local field enhancement in the linear response near the bulk plasma frequency. 

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