  We improve on the Thomas-Fermi approximation for the single-particle density
of fermions by introducing inhomogeneity corrections. Rather than invoking a
gradient expansion, we relate the density to the unitary evolution operator for
the given effective potential energy and approximate this operator by a
Suzuki-Trotter factorization. This yields a hierarchy of approximations, one
for each approximate factorization. For the purpose of a first benchmarking, we
examine the approximate densities for a few cases with known exact densities
and observe a very satisfactory, and encouraging, performance. As a bonus, we
also obtain a simple fourth-order leapfrog algorithm for the symplectic
integration of classical equations of motion.
