Gradient expansion approximation of the inhomogeneous electron-gas revisited: Higher-order corrections

In our recently published work (our Ref. 1) we revisited the gradient expansion approximation (GEA) of the interacting electron gas, and recalculated the leading-order contribution$-$with respect to the Wigner-Seitz radius $r_s$$-$to the coefficient $B_{xc}[n]$ of the square of the gradient of the electron density in the high-density and slowly varying limits. That work resolved historical controversies regarding these coefficients and demonstrated that serious misconceptions have led to incorrect constraints being imposed on popular functionals within the generalized gradient approximation (GGA). In the present paper, we extend this calculation to obtain the coefficient of the next-to-leading term, which scales as $r_s \ln(r_s)$ relative to the leading order. First, we establish a systematic framework to evaluate the integral expressions for the $b_{xc}$ coefficient of the leading term ($\sim q^2$) of the density-density response function in the long-wavelength limit ($q \to 0$)$-$a prerequisite for computing $B_{xc}[n]$. The significance of the calculation stems from the proof that the coefficient of this $r_s \ln(r_s)$ term receives no corrections from higher-order diagrammatic expressions. Consequently, our derived value serves as an exact, definitive constraint for future GGA functional development; in the high-density slowly-varying limit, any valid functional must reproduce the exact constraints established in both our previous work and the present paper.