where ⃗ r is the position we probe for the charge density, ⃗ r is any point from which the electron propagates to ⃗ r, χ(⃗ r, ⃗ r) is the Lindhard response function describing how the potential at ⃗ r contributes to the charge density at ⃗ r, and finally Vext( ⃗ r) is the external potential (here caused by an external impurity). The Lindhard response function is expressed in the following way