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The nonlocal polarizability density of alpha(r,r'; omega) gives the polarization induced at a point r in a quantum mechanical system, due to a perturbing field of frequency that acts at the point r', within linear response; thus it reflects the distribution of polarizability in the system. In order to gain information about the nature and functional form of alpha(r,r'; ), in this work we analyze the nonlocal polarizability density of a well-characterized system, a homogeneous electron gas at zero temperature. We establish a connection between the static, longitudinal component of the nonlocal polarizability density in position space and the dielectric function epsilon(k,o), and then use the connection to obtain results at three levels of approximation to epsilon(k,0): we compare the Thomas-Fermi (TF), random phase approximation (RPA), and Vashishta-Singwi (VS) forms. At TF level, we evaluate the nonlocal polarizability density analytically, while within the RPA we obtain asymptotic analytical results. The RPA and VS results are similar, and qualitatively distinct from the TF results, which diverge as |r-r'| approaches zero. Within the RPA, we find two long-range components in alphaL(r,r';0): The first is a monotonically decreasing component that arises from charge screening in the electron gas, and varies as |r-r'|-3; the second is an oscillatory component with terms of order |r-r'|-n (n 3) associated with Friedel oscillations in the electron density. These results indicate the possibility of long-range, intramolecular terms in the nonlocal polarizability densities of individual molecules.
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