We now introduce a fixed point charge Q at the origin. The associated charge density is Qδ (r), where δ (r) is the Dirac delta function. After the system has returned to equilibrium, let the change in the electron density and electric potential be Δ ρ (r) and Δ φ (r) respectively. The charge density and electric potential are related by Poisson's equation, which gives − − ∇ ∇ 2 [ Δ Δ ϕ ϕ ( r ) ] = 1 ε ε 0 [ Q δ δ ( r ) − − e Δ Δ ρ ρ ( r ) ] , where ε 0 is the vacuum permittivity.