Nonlinear electron-density distribution around point defects in simple metals. I. Formulation

A modification, which is exact in the limit of long wavelength, of the nonlinear theory of Sj\"olander and Stott of electron distribution around point defects is given. This modification consists in writing a nonlinear integral equation for the Fourier transform γ₁₂ (q) of the induced charge density surrounding the point defect, which includes a term involving the density derivative of γ₁₂ (q). A generalization of the Pauli-Feynman coupling-constant-integration method, together with the Kohn-Sham formalism, is used to exactly determine the coefficient of this derivative term in the long-wavelength limit. The theory is then used to calculate electron-density profiles around a vacancy, an eight-atom void, and a point ion. The results are compared with those of (i) a linear theory, (ii) Sj\"olander-Stott theory, and (iii) a fully self-consistent calculation based on the density-functional formalism of Kohn and Sham. It is found that in the case of a vacancy, the results of the present theory are in very good agreement with those based on Kohn-Sham formalism, whereas in the case of a singular attractive potential of a proton, the results are quite poor in the vicinity of the proton, but much better for larger distances. A critical discussion of the theory vis \`a vis the Kohn-Sham formalism is also given. Some applications of the theory are pointed out.