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We develop a generalized gradient expansion of the inhomogeneous dynamicalmean-field theory method for determining properties of ultracold atoms in atrap. This approach goes beyond the well-known local density approximation andat higher temperatures, in the normal phase, it shows why the local densityapproximation works so well, since the local density and generalized gradientapproximations are essentially indistinguishable from each other (and from theexact solution within full inhomogeneous dynamical mean-field theory). Butbecause the generalized gradient expansion only involves nearest-neighborcorrections, it does not work as well at low temperatures, when the systemsenter into ordered phases. This is primarily due to the problem that orderedphases often satisfy some global constraints which determine the spatialordering pattern, and the local density and generalized gradient approximationsare not able to impose those kinds of constraints; they also overestimate thetendency to order. The theory is applied to phase separation of different massfermionic mixtures represented by the Falicov-Kimball model and to determiningthe entropy per particle of a fermionic system represented by the Hubbardmodel. The generalized gradient approximation is a useful diagnostic for theaccuracy of the local density approximation---when both methods agree, they arelikely accurate, when they disagree, neither is likely to be correct.

Generalized gradient expansion for inhomogeneous dynamical mean-field theory: Application to ultracold atoms in a harmonic trap