Trivial Constraints on Orbital-free Kinetic Energy Density Functionals

TL;DR

This paper clarifies misconceptions about constraints on orbital-free kinetic energy density functionals, showing that certain claimed limitations are not valid constraints but only hold for specific minimizing densities.

Contribution

It refutes previous claims that differential virial theorems impose constraints on KEDFs, clarifying the actual scope of these relationships.

Findings

Relationships hold only for minimizing densities, not arbitrary densities.

Claims of constraints from differential virial theorems are invalid.

Validity for all densities is not restored by density-potential bijection.

Abstract

Kinetic energy density functionals (KEDFs) are central to orbital-free density functional theory. Limitations on the spatial derivative dependencies of KEDFs have been claimed from differential virial theorems. We point out a central defect in the argument: the relationships are not true for an arbitrary density but hold only for the minimizing density and corresponding chemical potential. Contrary to the claims therefore, the relationships are not constraints and provide no independent information about the spatial derivative dependencies of approximate KEDFs. A simple argument also shows that validity for arbitrary $v$-representable densities is not restored by appeal to the density-potential bijection.