Aufbau Principle - The Madelung Energy Ordering Rule

The Madelung Energy Ordering Rule

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), where orbitals with a lower n + l value are filled before those with higher n + l values. In this context, n represents the principal quantum number and l the azimuthal quantum number; the values l = 0, 1, 2, 3 correspond to the s, p, d, and f labels, respectively.

The rule is based on the total number of nodes in the atomic orbital, n + l, which is related to the energy. In the case of equal n + l values, the orbital with a lower n value is filled first. The fact that most of the ground state configurations of neutral atoms fill orbitals following this n + l, n pattern was obtained experimentally, by reference to the spectroscopic characteristics of the elements.

The Madelung energy ordering rule applies only to neutral atoms in their ground state, and even in that case, there are several elements for which it predicts configurations that differ from those determined experimentally. Copper, chromium, and palladium are common examples of this property. According to the Madelung rule, the 4s orbital (n + l = 4 + 0 = 4) is occupied before the 3d orbital (n + l = 3 + 2 = 5). The rule then predicts the configuration of ₂₉Cu to be 1s22s22p63s2 3p64s23d9, abbreviated 4s23d9 where denotes the configuration of Ar (the preceding noble gas). However the experimental electronic configuration of the copper atom is 4s13d10. By filling the 3d orbital, copper can be in a lower energy state. Similarly, chromium takes the electronic configuration of 4s13d5 instead of 4s23d4. In this case, chromium has a half-full 3d shell. For palladium, the Madelung rule predicts 5s24d8, but the experimental configuration 4d10 differs in the placement of two electrons.