Coulomb Correlations in Inhomogeneous Many-Particle Systems

The fluctuation-dissipation theorem is applied to the study of the pair correlation functions of the system containing charged particles which is not invariant under spatial translation. The results are expressed in terms of the longitudinal response functions and involve correlation functions which depend only on screened short-range Coulomb interactions $S_0$. In the case of the electron gas, the density-density correlation function is $S(\mathbf{r}{\mathbf{r}}^{\prime };\omega )=\left[\delta \right(\mathbf{r}-\mathbf{R})-K(r{\mathbf{R}}^{\prime \prime };\omega \left)v\right({\mathbf{R}}^{\prime \prime }\mathbf{R}\left)\right]S_0(\mathbf{R}{\mathbf{R}}^{\prime };\omega )\left[\delta \right({\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime })-v({\mathbf{R}}^{\prime }{\mathbf{R}}^{\prime \prime \prime }\left)\tilde{K}\right({\mathbf{R}}^{\prime \prime \prime }{\mathbf{r}}^{\prime };-\omega \left)\right]$