Minimum principal quantum numbers for

In the early days of atomic theory, it was often assumed that only an empirical understanding could be achieved beyond the first few rows of the periodic table, where the simple formulation of the aufbau principle breaks down. The modern view is that shell and subshell filling can be accounted for within the central field approximation, provided the centrifugal barrier effects are included. 

Of particular significance are the long periods which arise when filling the d and / subshells. These rows of the table contain the transition elements for the d subshells and the lanthanides and actinides for the 4/ and 5/ subshells respectively. 

The minimum n value for a given is equal to + 1, and this is a consequence of the existence of a centrifugal barrier, as follows readily from the form of the effective radial potential 

The second term is readily identified ( + l)h2/2mr2 = M2/2mr2, and thus, for circular motion, there is an apparent force  

Veff describes a potential which rises to infinity at r = 0, and crosses the V = 0 axis at r = ( + l)/2. Outside this radius, it has a minimum, 

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