Energy expressions for model exchange potentials: Beyond the Levy--Perdew virial relation

The common way to assign energies to Kohn--Sham exchange potentials is by using the Levy--Perdew virial relation. However, for model potentials that are not functional derivatives, this approach leads to energy expressions that lack translational invariance. We point out that there is a more general procedure for constructing density functionals from model potentials, of which the Levy--Perdew relation is just a special case. Using this generalization we propose a method for converting model potentials into density functionals that ensures translational invariance of the energy. To illustrate our approach we construct a competitively accurate exchange functional from the model potential of van Leeuwen and Baerends.