# Aufbau principle

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The Aufbau principle states that, hypothetically, electrons orbiting one or more atoms fill the lowest available energy levels before filling higher levels (e.g., 1s before 2s). In this way, the electrons of an atom, molecule, or ion harmonize into the most stable electron configuration possible.

Aufbau is a German noun that means construction or "building-up". The Aufbau principle is sometimes called the building-up principle or the Aufbau rule.

The details of this "building-up" tendency are described mathematically by atomic orbital functions. Electron behavior is elaborated by other principles of atomic physics, such as Hund's rule and the Pauli exclusion principle. Hund's rule asserts that even if multiple orbitals of the same energy are available, electrons fill unoccupied orbitals first, before reusing orbitals occupied by other electrons. But, according to the Pauli exclusion principle, in order for electrons to occupy the same orbital, they must have different spins (−1/2 and 1/2).

A version of the Aufbau principle known as the nuclear shell model is used to predict the configuration of protons and neutrons in an atomic nucleus.[1]

The states crossed by the same arrow have same n + ℓ value. The arrow direction indicates the order of filling states.

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 Janet rule or the Klechkowsky rule (after Charles Janet or Vsevolod Klechkovsky in some, mostly French and Russian-speaking, countries), or the diagonal rule.[2] 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 the azimuthal quantum number; the values = 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.[3] 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.[4]

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.[5] 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 [Ar]4s23d9 where [Ar] denotes the configuration of Ar (the preceding noble gas). However the experimental electronic configuration of the copper atom is [Ar]4s13d10. By filling the 3d orbital, copper can be in a lower energy state. Similarly, chromium takes the electronic configuration of [Ar]4s13d5 instead of [Ar]4s23d4. In this case, chromium has a half-full 3d shell. For palladium, the Madelung rule predicts [Kr]5s24d8, but the experimental configuration [Kr]4d10 differs in the placement of two electrons.

## History

### The Aufbau principle in the new quantum theory

In the old quantum theory, orbits with low angular momentum (s- and p-orbitals) get closer to the nucleus.

The principle takes its name from the German, Aufbauprinzip, "building-up principle", rather than being named for a scientist. In fact, it was formulated by Niels Bohr and Wolfgang Pauli in the early 1920s, and states that:

This was an early application of quantum mechanics to the properties of electrons, and explained chemical properties in physical terms. Each added electron is subject to the electric field created by the positive charge of the atomic nucleus and the negative charge of other electrons that are bound to the nucleus. Although in hydrogen there is no energy difference between orbitals with the same principal quantum number n, this is not true for the outer electrons of other atoms.

In the old quantum theory prior to quantum mechanics, electrons were supposed to occupy classical elliptical orbits. The orbits with the highest angular momentum are 'circular orbits' outside the inner electrons, but orbits with low angular momentum (s- and p-orbitals) have high orbital eccentricity, so that they get closer to the nucleus and feel on average a less strongly screened nuclear charge.

### The n + ℓ energy ordering rule

A periodic table in which each row corresponds to one value of n + ℓ was suggested by Charles Janet in 1927. In 1936, the German physicist Erwin Madelung proposed his empirical rules for the order of filling atomic subshells, based on knowledge of atomic ground states determined by the analysis of atomic spectra, and most English-language sources therefore refer to the Madelung rule. Madelung may have been aware of this pattern as early as 1926.[6] In 1962 the Russian agricultural chemist V.M. Klechkowski proposed the first theoretical explanation for the importance of the sum n + ℓ, based on the statistical Thomas–Fermi model of the atom.[7] Many French- and Russian-language sources therefore refer to the Klechkowski rule. In recent years some authors have challenged the validity of Madelung's rule in predicting the order of filling of atomic orbitals. For example, it has been claimed, not for the first time, that in the case of the scandium atom a 3d orbital is occupied "before" the occupation of the 4s orbital. In addition to there being ample experimental evidence to support this view, it makes the explanation of the order of ionization of electrons in this and other transition metals far more intelligible, given that 4s electrons are invariably preferentially ionized.[8]