We find that the multivalued character of the $G$ factor as a function of the reduced gradient ($s$) still exists after accounting for pseudopotential artifacts and the kinetic energy global upper bound. We also find that the VT84F functional indeed exhibits stable convergence and more reasonable results for self-consistent bulk properties compared to other generalized gradient approximation (GGA) kinetic energy density functionals (KEDFs) that we tested earlier. However, VT84F generally yields overestimated equilibrium volumes, which may result from its inability (as with all GGAs) to reproduce the $G-s$ multivalued character. The analogous failure to predict the multivalued character of $G$ as a function of the reduced density $\left(d\right)$ is also likely to be responsible for the inaccuracy of our vWGTF functionals reported earlier. Our multivaluedness analysis therefore does not impugn any particular GGA KEDF. Instead, it merely confirms the importance of pointwise analysis for improving KEDFs by emphasizing the need to resolve the multivaluedness of $G$ with respect to various density variables.

• Received 4 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.117102