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| Fig. 1: Comparison of the dimensionless part of the transverse current response function (dashed) with the dimensionless part of the density response (solid) for the case of x = 1. The blue and red curves are the real and imaginary parts, respectively. The computer code that generated this graph is available here. I plotted using gnuplot. The plot file is shown here. |
The density-density response function of the electron gas
where
has an expression in terms of simple functions of the form
The rule for the branch cuts is
The current-current response function on the other hand
cannot be expressed in terms of simple functions and so must be evaluated by computer. Actually, only the transverse part of the current response has this problem. Because of spherical symmetry, the response function decomposes into longitudinal and transverse parts
The former is related to the density response by
by virtue of particle conservation (a.k.a. gauge invariance). The transverse part is given by
where
This is a special function that looks worse than it actually is. You can generate it from its simple imaginary part
by numerical hilbert transform. Moreover the effect of this function is rather benign. As can be seen from Fig. 1, it mainly smooths out the kinks of the density response.
The LaTeX source for the above equations is available here.