  It is well known that in the gradient expansion approximation to density
functional theory (DFT) the gradient and Laplacian of the density make
interchangeable contributions to the exchange correlation (XC) energy. This is
an arbitrary "gauge" freedom for building DFT models, normally used to
eliminate the Laplacian from the generalized gradient approximation (GGA) level
of DFT development. We explore the implications of keeping the Laplacian at
this level of DFT, to develop a model that fits the known behavior of the XC
hole, which can only be described as a system average in conventional GGA. We
generate a family of exchange models that obey the same constraints as
conventional GGA's, but which in addition have a finite-valued potential at the
atomic nucleus unlike GGA's. These are tested against exact densities and
exchange potentials for small atoms, and for constraints chosen to reproduce
the SOGGA and the APBE variants of the GGA. The model reliably reproduces
exchange energies of closed shell atoms, once constraints such the local
Lieb-Oxford bound, whose effects depend upon choice of energy-density gauge,
are recast in invariant form.
