Aufbau principle

In atomic physics and quantum chemistry, the Aufbau principle, also called the Aufbau rule, states that in the ground state of an atom or ion, electrons first fill subshells of the lowest available energy, then fill subshells of higher energy. For example, the 1s subshell is filled before the 2s subshell is occupied. In this way, the electrons of an atom or ion form the most stable electron configuration possible. An example is the configuration for the zinc atom, meaning that the 1s subshell has 2 electrons, the 2s subshell has 2 electrons, the 2p subshell has 6 electrons, and so on.
The configuration is often abbreviated by writing only the valence electrons explicitly, while the core electrons are replaced by the symbol for the last previous noble gas in the periodic table, placed in square brackets. For zinc, the last previous noble gas is argon, so the configuration is abbreviated to 4s² 3d¹⁰, where signifies the core electrons whose configuration in zinc is identical to that of argon.
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 if multiple orbitals of the same energy are available, electrons will occupy different orbitals singly and with the same spin before any are occupied doubly. If double occupation does occur, the Pauli exclusion principle requires that electrons that occupy the same orbital must have different spins.
Passing from one element to another of the next higher atomic number, one proton and one electron are added each time to the neutral atom.
The maximum number of electrons in any shell is 2n², where n is the principal quantum number.
The maximum number of electrons in a subshell is equal to 2, where the azimuthal quantum number is equal to 0, 1, 2, and 3 for s, p, d, and f subshells, so that the maximum numbers of electrons are 2, 6, 10, and 14 respectively. In the ground state, the electronic configuration can be built up by placing electrons in the lowest available subshell until the total number of electrons added is equal to the atomic number. Thus subshells are filled in the order of increasing energy, using two general rules to help predict electronic configurations:

Electrons are assigned to subshells in order of increasing value of n + .
For subshells with the same value of n + , electrons are assigned first to the subshell with lower n.

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.

Madelung energy ordering rule

In neutral atoms, the approximate order in which subshells are filled is given by the n + rule, also known as the:

Madelung rule
Janet rule
Klechkowsky rule
Wiswesser's rule
Moeller's rubric
Linus Pauling's diagram
aufbau rule or
diagonal rule

Here 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 subshells, respectively. Subshells with a lower n + value are filled before those with higher n + values. In the many cases of equal n + values, the subshell with a lower n value is filled first. The subshell ordering by this rule is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, 8s, 5g,... For example, thallium has the ground-state configuration or in condensed form, 6s² 4f¹⁴ 5d¹⁰ 6p¹.
Other authors write the subshells outside of the noble gas core in order of increasing n, or if equal, increasing n +, such as Tl . They do so to emphasize that if this atom is ionized, electrons leave approximately in the order 6p, 6s, 5d, 4f, etc. On a related note, writing configurations in this way emphasizes the outermost electrons and their involvement in chemical bonding.
In general, subshells with the same n + value have similar energies, but the s-orbitals are exceptional: their energy levels are appreciably far from those of their n + group and are closer to those of the next n + group. This is why the periodic table is usually drawn to begin with the s-block elements.
The Madelung energy ordering rule applies only to neutral atoms in their ground state. There are twenty elements for which the Madelung rule predicts an electron configuration that differs from that determined experimentally, although the Madelung-predicted electron configurations are at least close to the ground state even in those cases.
One inorganic chemistry textbook describes the Madelung rule as essentially an approximate empirical rule although with some theoretical justification, based on the Thomas–Fermi model of the atom as a many-electron quantum-mechanical system.

Exceptions in the d-block

The valence d-subshell "borrows" one electron from the valence s-subshell.

Atom	₂₄Cr	₂₉Cu	₄₁Nb	₄₂Mo	₄₄Ru	₄₅Rh	₄₆Pd	₄₇Ag	₇₈Pt	₇₉Au	₁₀₃Lr
Core electrons									4f¹⁴	4f¹⁴	5f¹⁴
Madelung rule	3d⁴ 4s²	3d⁹ 4s²	4d³ 5s²	4d⁴ 5s²	4d⁶ 5s²	4d⁷ 5s²	4d⁸ 5s²	4d⁹ 5s²	5d⁸ 6s²	5d⁹ 6s²	6d¹ 7s²
Experimental	3d⁵ 4s¹	3d¹⁰ 4s¹	4d⁴ 5s¹	4d⁵ 5s¹	4d⁷ 5s¹	4d⁸ 5s¹	4d¹⁰	4d¹⁰ 5s¹	5d⁹ 6s¹	5d¹⁰ 6s¹	7s² 7p¹

For example, in copper ₂₉Cu, according to the Madelung rule, the 4s subshell is occupied before the 3d subshell. The rule then predicts the electron configuration, abbreviated where denotes the configuration of argon, the preceding noble gas. However, the measured electron configuration of the copper atom is. By filling the 3d subshell, copper can be in a lower energy state.
A special exception is lawrencium ₁₀₃Lr, where the 6d electron predicted by the Madelung rule is replaced by a 7p electron: the rule predicts, but the measured configuration is.

Exceptions in the f-block

The valence d-subshell often "borrows" one electron from the valence f-subshell. For example, in uranium ₉₂U, according to the Madelung rule, the 5f subshell is occupied before the 6d subshell. The rule then predicts the electron configuration where denotes the configuration of radon, the preceding noble gas. However, the measured electron configuration of the uranium atom is.

Atom	₅₇La	₅₈Ce	₆₄Gd	₈₉Ac	₉₀Th	₉₁Pa	₉₂U	₉₃Np	₉₆Cm
Core electrons
Madelung rule	4f¹ 6s²	4f² 6s²	4f⁸ 6s²	5f¹ 7s²	5f² 7s²	5f³ 7s²	5f⁴ 7s²	5f⁵ 7s²	5f⁸ 7s²
Experimental	5d¹ 6s²	4f¹ 5d¹ 6s²	4f⁷ 5d¹ 6s²	6d¹ 7s²	6d² 7s²	5f² 6d¹ 7s²	5f³ 6d¹ 7s²	5f⁴ 6d¹ 7s²	5f⁷6d¹ 7s²

All these exceptions are not very relevant for chemistry, as the energy differences are quite small and the presence of a nearby atom can change the preferred configuration. The periodic table ignores them and follows idealized configurations. They occur as the result of interelectronic repulsion effects; when atoms are positively ionized, most of the anomalies vanish.
The above exceptions are predicted to be the only ones until element 120, where the 8s shell is completed. Element 121, starting the g-block, should be an exception in which the expected 5g electron is transferred to 8p. After this, sources do not agree on the predicted configurations, but due to very strong relativistic effects there are not expected to be many more elements that show the expected configuration from Madelung's rule beyond 120. The general idea that after the two 8s elements, there come regions of chemical activity of 5g, followed by 6f, followed by 7d, and then 8p, does however mostly seem to hold true, except that relativity "splits" the 8p shell into a stabilized part and a destabilized part, and that the 8s shell gets replaced by the 9s shell as the covering s-shell for the 7d elements.

History

The aufbau principle in the new quantum theory

The principle takes its name from German, Aufbauprinzip, "building-up principle", rather than being named for a scientist. It was formulated by Niels Bohr in the early 1920s. 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 subshells 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 have high subshell eccentricity, so that they get closer to the nucleus and feel on average a less strongly screened nuclear charge.
Wolfgang Pauli's model of the atom, including the effects of electron spin, provided a more complete explanation of the empirical aufbau rules.