The correct gradient expansion approximation for the interacting electron gas

We will present our studies of the inhomogeneous electron gas under the influence of a weak external potential, which allows us to calculate the gradient corrections to the density functional within the linear response, an approach known as the gradient expansion approximation. To obtain the exchange and correlation contributions to the coefficient of the $q^2$ terms of the proper polarization function, we revisited all the previous calculations and expose misconceptions. We used various ways to apply a necessary regularization of the Coulomb interaction potential. We found that when the regulator is set to zero at the end of the calculation, only the combination of the exchange plus correlation contribution is regularization-scheme independent and thus has a unique value. On the contrary, the exchange or the correlation contributions separately do not have regularization-scheme independent values even though the regulator is set to zero at the end of the calculation, and therefore it is impossible to define such a separation meaningfully. We conclude that it is incorrect to separate those two terms when constructing a generalized gradient approximation contribution to the density functional and to apply various constraints on these parts independently.