Self-consistent Fields for Heavy Positive Atomic Ions

Self-consistent Fields for Heavy Positive Atomic Ions.—Let us immediately turn to the use of equation (3) to establish the self-consistent field in a heavy atomic ion with nuclear charge Ze and total number of electrons N. We merely combine the form (3) with the Poisson equation 

Near the nucleus the self-consistent potential energy condition 

Numerical solutions of equation (10) subject to the boundary conditions (11) and (13) are available (see, for example ref. 4) and hence the self-consistent field V(r) in heavy positive ions is established. 

Gombds, Die Statistiche Theorie des Atoms und Ihre Anwendungen , Springer-Verlag Vienna, 1949, p. 360. 

Type I solution neutral atom, potential and electron density have infinite extent. Type II solution corresponds to positive ions, these have a finite radius. If N is the number of electrons, and Z the atomic number, the construction shown determines NjZ ( 1) for the given solution 

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