Pair-correlation function of an electron gas

The pair-correlation function p(0), at r = 0, of an electron gas is derived by a partial-wave, phase-shift analysis. The dependence of p(0) on r(s), the radius of the sphere (in Bohr units) containing on average one electron, is found to be 32/(8 + 3r(s))(2). This analytic result, obtained with the help of a simplifying approximation, is compared with several numerical calculations. An improved formula is also derived. p(O) is an important parameter in theories of charge response or spin response whenever a perturbation, similar to cos qx, has a small wave length, i.e., q much greater than k(F). Electron-electron scattering depends on both response functions and, consequently, also on p(0).