The order in which these orbitals are filled is given by the n + ℓ rule (also known as the Madelung rule (after Erwin Madelung), or the Klechkowski rule (after Vsevolod Klechkovsky in some, mostly French and Russian-speaking, countries), or the diagonal rule.
(a) the Bohr–Sommerfeld “solar system” atomic model (with electron spin and the Madelung principle), which provides the magic-number elements that end each row of the table and gives the number of elements in each row,
Klechkovsky also studied theoretical chemistry, and proposed a theoretical justification of the empirical Madelung rule for the ordering of atomic orbital energies.
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Pekka Pyykkö, for example, used computer modeling to calculate the positions of elements up to Z=172, and found that several were displaced from the Madelung energy-ordering rule.