Gradient-expansion of the inhomogeneous electron-gas revisited

In the present work, we revisit the problem 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 linear response, an approach known as the gradient expansion approximation. To obtain the exchange ($b_x$) and correlation ($b_{c}$) contributions to the coefficient $b_{xc}$, i.e., to the prefactor of the $q^2$ term of the proper-polarization function, we revisited all the previous calculations and expose misconceptions which led to incorrect conclusions. We used various ways to apply a necessary regularization to the singular Coulomb interaction potential. We found that the separate exchange ($b_x$) and correlation ($b_c$) contributions to the coefficient $b_{xc}$ have regularization-scheme dependent values even though the regulator is set to zero at the end of the calculation. This implies that it is impossible to define such a separation meaningfully. On the contrary, we found that when the regulator is set to zero at the end of the calculation, the combination $b_{xc}$ is regularization-scheme independent and, thus, has a unique value. We conclude that it is incorrect to separate those two terms when constructing a generalized-gradient-approximation (GGA) contribution to the density functional. This appears to be a common approach in most popular GGA functionals, where various constraints are applied to each contribution separately.