Hypernetted Chain Theory of Charged Impurity

Abstract

Applying the inhomogeneous hypernetted chain method we formulate the general impurity problem for a correlated Bose or Fermi fluid as a host. In the present paper we apply the method to an impurity in electron gas. From the Euler-Lagrange equation for the induced density one obtains the screening and cusp conditions. It turns out that these two conditions alone are sufficient to give accurate prediction for the positron annihilation in metals. We construct numerical solution for the Euler-Lagrange equation demanding the two conditions to be satisfied. The resulting energies compare well with existing field theoretical calculations for hydrogen and positronium impurities. The uniform limit approximation is used to derive an expression for the dielectric function. In the present theory the linear response to an external field is determined by one characteristic function which is calculable from the ground state quantities alone. Eliminating this function in favour of the dielectric function which is assumed to be known exactly we obtain an equation for the non linear density response which may have more general validity than the approximations made here suggest.