Lindhard function

In Eq. (82), setting fk = 2, replacing the summation by an integration, and doing some algebra, one finally obtains the Lindhard function " ... 

Therefore, it is not relevant to talk about the XC effects on the Lindhard function unless, ofcomse, one starts from some Hamiltonian that includes exchange and/or correlation, like the HF Hamiltonian or the KS Hamiltonian. If the latter step is taken, simple plane waves cannot be used as the zeroth-order orbitals any more. Nonetheless, the Lindhard fimction is ideal for om purpose because it is a pme kinetic model [see Eq. (84)]. 

Table II clearly indicates that none of the previously mentioned OF-KEDF s has the eorreet LR behavior at the FEG limit. Even more interestingly, the TF funetional is supposed to be exact at the FEG limit, but its LR funetion has no momentum dependence. At first glance, one would think that there is some ineonsistency involved. In fact, there is no confliet beeause the TF functional is only the zeroth-order perturbation result, while the Lindhard function is the first-order result. A similar paradox exists for the asymptotic Friedel oscillations in Eq. (87).

Figure 6 Comparing the Lindhard function with the momentum-space LR functions of various model OF-KEDF s at the FEG limit...

This can be demonstrated by replacing the Lindhard function by a rational polynomial, which has the correct low and high q behaviour, and is such that it passes through jXtf at q = 2kF. Such a rational polynomial, correct to third order in both q2 and l/q2 is given by the dashed curve in Fig. 6.3. It has the form (Pettifor and Ward (1984))... 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

With Eq. (84) in hand, we can easily assess the quality of various OF-KEDF s mentioned in previous sections, by comparing their momentum-space LR functions with the Lindhard function. For instance, the momentum-space LR function of the TF functional is just the constant prefactor in Eq. (83),... 

Figure 3. Exchange energy as a function of the wave vector of the magnetization inhomogenity Lindhard function (solid line) and exchange-stiffness or continuum approximation (dashed line). For late 3d elements (Cu), k/2kf = 1 corresponds to modulation wavelength of 0.23 nm...

Figure 3 compares the Lindhard function (solid line) with the exchange-stiffness approximation (dashed line). We see that the exchange-stiffness approximation works well unless k is comparable to kf. 

As far as explicit approximations for the polarisation functions Tltidq) are concerned only very little is known, even in the static limit. The complete frequency dependence is available for the noninteracting limit ( ). i e. the relativistic generalisation of the Lindhard function [95, 114]. In addition to its vacuum part (A.26) one has... 

The response function xJ " of a non-interacting homogeneous system is the well-known Lindhard function. The full response function x "". on the other hand, is not known analytically. However, some exact features of x " are known. From these, the following exact properties of /Jr can be deduced ... 

In Eq. (23), x(r) and Xo O ) are the static response function of the homogeneous liquid and the response function of the noninteracting electrons (namely the Lindhard function [47]). 

For the unperturbed FEG, the calculation of x in the RPA leads to the Lindhard function [23] or some variations of it, such as the Mermin function [24]. The Lindhard function is equivalent to the use of plane waves as one-electron wavefunctions in Eq. (13), before the self-consistent calculation of x... 

Lindhard function [6,47]. Explicit expressions for the real and imaginary parts of the noninteracting quadratic density-response fimction were reported in Refs. [52] and [16,29], respectively, and an extension to imaginary frequencies was later reported in Ref. [53]. 

For free electrons, the summation in the last term can be done analytically, (using the standard replacement — [2fl/(2jr) )/d A, and taking the principal value), which gives the Lindhard function <3o( ) ... 

The exchange and correlation interaction is of short range. Thus, G q) — 0 for 5 — 0. Using the well known limit for the Lindhard function ... 

A few comments need to be made here. First, it turns out that the restriction q / 0 in the integration of Eq. (80) is not a problem at all because the Lindhard funetion is analytic for q = 0. Second, there is a weak logarithmic singularity at T = 1 or q = 2kp where the slope of the Lindhard function is divergent. This singularity ean be attributed to the pole of the denominator of Eq. (82), and is... 

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