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The density of an atom in a state of well-defined angular momentum has aspecific finite spherical harmonic content, without and with interactions.Approximate single-particle schemes, such as the Hartree, Hartree-Fock, andLocal Density Approximations, generally violate this feature. We analyze, bymeans of perturbation theory, the degree of this violation and show that it issmall. The correct symmetry of the density can be assured by aconstrained-search formulation without significantly altering the calculatedenergies. We compare our procedure to the (different) common practice ofspherically averaging the self-consistent potential. Kohn-Sham densityfunctional theory with the exact exchange-correlation potential has the correctfinite spherical harmonic content in its density; but the corresponding exactsingle particle potential and wavefunctions contain an infinite number ofspherical harmonics.Comment: 11 pages, 6 figures. Expanded discussion of spherical harmonic  expansion of Hartree density. Some typos corrected, references adde

Symmetry of the atomic electron density in Hartree, Hartree-Fock, and density-functional theories