Direct calculation of the electron density via the milne equation

The closed formulas for reduced density matrices, derived in a previous paper, are used to find the electron density distributions in a Coulomb potential and in the Kr atom directly, without finding the eigenvalues and eigenfunctions first. These formulas give accurate results, including the shell structure (missing in the Thomas-Fermi model) through the whole r space, except in a very small region around the nucleus. An ad hoc method is introduced to yield a good density there. The advantages as well as shortcomings of using such a direct method in an SCF scheme of atomic structure calculations have been summarized.