Recent Quantum Monte Carlo data for the exchange-correlation energy density of pseudopotential systems strongly suggest the value of using the Laplacian of the density as a variable for constructing first-order corrections to the local density approximation of density functional theory. We report on an exchange functional built on these observations and extended to the all-electron case. The model keeps the typical properties of constraint-based generalized gradient approximations (GGAs) and also has a finite-valued potential at the nucleus, unlike the GGA. Problems with oscillatory behavior in the potential due to higher order derivatives are controlled by a curvature minimization constraint. The results are tested against exact potentials for the He and Ne atom. A combination of gradient and Laplacian as suggested by a gradient expansion of the exchange hole gives the best overall results. © 2012 Wiley Periodicals, Inc.