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×18th Feb 2020 @ 8 min read

The aufbau principle states: In the ground state of an atom, atomic orbitals are filled by electrons in the order of their increasing energies. In other words, electrons will occupy the lowest-energy orbital first before filling higher-energy orbitals.

According to the aufbau principle, the order of the filling of orbitals is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p…

The IUPAC defines the aufbau principle as:

A rule for building up the electronic configuration of atoms and molecules. It states that a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals are filled before electrons are placed in higher-energy orbitals.

The non-German reader may think aufbau is the name of a scientist, but it is not. The word *aufbau* (*auf-*: up and *bau*: building) is a german word meaning building up or construction.

As the rule says the building up of orbitals takes place with the increasing energy of orbitals. Hence, it is also called as the building-up principle.

As the aufbau principle says lower orbitals are filled first before higher orbitals. The order of the filling is presented in the diagram below.

Each type of orbitals in the above diagram is colored the same and are arranged in the ascending order of the principal quantum number (*n*) from the top to bottom, for example, 2p, 3p, 4p, 5p, 6p… From the left to right, the orbitals are arranged according to the azimuthal quantum number (*l*). *l* = 0 corresponds to the s orbital, *l* = 1 corresponds to the p orbital, *l* = 2 corresponds to the d orbital, and *l* = 3 corresponds to the f orbital.

The order of the filling starts with 1s orbital, which is the smallest and lowest energy orbital. Thus, the first electron enters 1s orbital making the electronic configuration of the hydrogen atom, i.e., 1s^{1}. This is followed by the second electron; it also enters 1s orbital since the s orbital can hold two electrons. This makes the electronic configuration 1s^{2}. It is helium, an inert gas. With every electron that gets added, the atomic number increases by one because a proton also gets added.

After 1s orbital completely filled, the third electron goes into 2s making the electronic configuration 1s^{2} 2s^{1} of the lithium atom, which is directly below hydrogen in the periodic table. Similarly, we have beryllium, 1s^{2} 2s^{2}.

After 2s, the electron fills 2p—follow the arrow in the diagram. 2p can take six electrons. After 2p, we have 3s, 3p, 4s, and so on. The table below lists the electronic configuration of some elements based on the aufbau principle.

Element | Symbol | Atomic number (Z) | Electronic configuration |
---|---|---|---|

Hydrogen | H | 1 | 1s^{1} |

Helium | He | 2 | 1s^{2} |

Lithium | Li | 3 | 1s^{2} 2s^{1} |

Carbon | C | 6 | 1s^{2} 2s^{2} 2p^{2} |

Oxygen | O | 8 | 1s^{2} 2s^{2} 2p^{4} |

Neon | Ne | 10 | 1s^{2} 2s^{2} 2p^{6} |

Sodium | Na | 11 | 1s^{2} 2s^{2} 2p^{6} 3s^{1} |

Magnesium | Mg | 12 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} |

Phosphorus | P | 15 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{3} |

Argon | Ar | 18 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} |

Potassium | K | 19 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{1} |

Calcium | Ca | 20 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} |

Scandium | Sc | 21 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{1} |

Iron | Fe | 26 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{6} |

Zinc | Zn | 30 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} |

Bromine | Br | 35 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{5} |

Krypton | Kr | 36 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{6} |

Zirconium | Zr | 40 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{6} 5s^{2} 4d^{2} |

Tin | Sn | 50 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{6} 5s^{2} 4d^{10} 5p^{2} |

Plutonium | Pu | 94 | 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{6} 5s^{2} 4d^{10} 5p^{6}6s ^{2} 4f^{14} 5d^{10} 6p^{6} 7s^{2} 5f^{6} |

A question that may bother readers is from where did this rule originate. Is there any scientific reason behind the principle? Well, there is no such equation that dictates this rule. But there is a scientific logic behind it, and it is *n* + *l* rule. The energy of an orbital mainly depends on two quantum numbers: the principal quantum number (*n*) and azimuthal quantum number (*l*). The principal quantum number corresponds to the shell while the azimuthal quantum number, subshell. As the value of both quantum numbers increases, the energy of an orbital also increases. One way to express this relation is by *n* + *l* rule, better known as the Madelung rule.

The Madelung rule, named after German physicist Erwin Madelung, states:

- Electrons in an atom fill the orbital with the lowest value of
*n*+*l*. - When two or more orbitals have the same value of
*n*+*l*, the electron will occupy the orbital with the lowest value of*n*.

The rule is straightforward and detailed in the table below.

Orbital | Principal quantum number, n | Azimuthal quantum number, l | n + l |
---|---|---|---|

1s | 1 | 0 | 1 |

2s | 2 | 0 | 2 |

2p | 2 | 1 | 3 |

3s | 3 | 0 | 3 |

3p | 3 | 1 | 4 |

4s | 4 | 0 | 4 |

3d | 3 | 2 | 5 |

4p | 4 | 1 | 5 |

5s | 5 | 0 | 5 |

4d | 4 | 2 | 6 |

5p | 5 | 1 | 6 |

6s | 6 | 0 | 6 |

6s | 6 | 0 | 6 |

4f | 4 | 3 | 7 |

5d | 5 | 2 | 7 |

6p | 6 | 1 | 7 |

7s | 7 | 0 | 7 |

5f | 5 | 3 | 8 |

6d | 6 | 2 | 8 |

7p | 7 | 1 | 8 |

8s | 8 | 0 | 8 |

… | … | … | … |

From the previous table, the order of orbitals concurs with the order of the aufbau principle in the diagram. The energy of the orbital increases with *n* + *l*. Whenever there is a tie in the value, the energy increases with *n*.

In the diagram, each orbital along the diagonal has the same value of *n* + *l* (see below).

The aufbau principle is not a universal rule. Not all atoms obey it, especially transition metals, and lanthanides and actinides. The exceptions are mentioned below.

Around ten transition metals go against the aufbau principle. They are listed in the following table.

Element | Symbol | Atomic number | Aufbau's prediction | Experimental observed |
---|---|---|---|---|

Chromium | Cr | 24 | [Ar] 4s^{2} 3d^{4} | [Ar] 4s^{1} 3d^{5} |

Copper | Cu | 29 | [Ar] 4s^{2} 3d^{9} | [Ar] 4s^{1} 3d^{10} |

Niobium | Nb | 41 | [Kr] 5s^{2} 4d^{3} | [Kr] 5s^{1} 4d^{4} |

Molybdenum | Mo | 42 | [Kr] 5s^{2} 4d^{4} | [Kr] 5s^{1} 4d^{5} |

Ruthenium | Ru | 44 | [Kr] 5s^{2} 4d^{6} | [Kr] 5s^{1} 4d^{7} |

Rhodium | Rh | 45 | [Kr] 5s^{2} 4d^{7} | [Kr] 5s^{1} 4d^{8} |

Palladium | Pd | 46 | [Kr] 5s^{2} 4d^{8} | [Kr] 4d^{10} |

Silver | Ag | 47 | [Kr] 5s^{2} 4d^{9} | [Kr] 5s^{1} 4d^{10} |

Platinum | Pt | 78 | [Xe] 6s^{2} 4f^{14} 5d^{8} | [Xe] 6s^{1} 4f^{14} 5d^{9} |

Gold | Au | 79 | [Xe] 6s^{2} 4f^{14} 5d^{9} | [Xe] 6s^{1} 4f^{14} 5d^{10} |

In each element, the d orbital takes an extra electron from the s orbital, except in palladium where both electrons are consumed by the d orbital.

In lanthanides and actinides, ten elements violate the aufbau principle.

Element | Symbol | Atomic number | Aufbau's prediction | Experimentally observed |
---|---|---|---|---|

Lanthanum | La | 57 | [Xe] 6s^{2} 4f^{1} | [Xe] 6s^{2} 5d^{1} |

Cerium | Ce | 58 | [Xe] 6s^{2} 4f^{2} | [Xe] 6s^{2} 4f^{1} 5d^{1} |

Gadolinium | Gd | 64 | [Xe] 6s^{2} 4f^{8} | [Xe] 6s^{2} 4f^{7} 5d^{1} |

Actinium | Ac | 89 | [Rn] 7s^{2} 5f^{1} | [Rn] 7s^{2} 6d^{1} |

Thorium | Th | 90 | [Rn] 7s^{2} 5f^{2} | [Rn] 7s^{2} 6d^{2} |

Protactinium | Pa | 91 | [Rn] 7s^{2} 5f^{3} | [Rn] 7s^{2} 5f^{2} 6d^{1} |

Uranium | U | 92 | [Rn] 7s^{2} 5f^{4} | [Rn] 7s^{2} 5f^{3} 6d^{1} |

Neptunium | Np | 93 | [Rn] 7s^{2} 5f^{5} | [Rn] 7s^{2} 5f^{4} 6d^{1} |

Curium | Cm | 96 | [Rn] 7s^{2} 5f^{8} | [Rn] 7s^{2} 5f^{7} 6d^{1} |

Lawrencium | Lr | 103 | [Rn] 7s^{2} 5f^{14} 6d^{1} | [Rn] 7s^{2} 5f^{14} 7p^{1} |

In all the above exceptions, the d orbital takes an electron from the f orbital; thorium and lawrencium are special cases. In thorium, 6d consumes both electrons from 5f while in lawrencium 6d is replaced by 7p.

Many elements do not always follow the ordering of orbitals as predicted by the aufbau principle. This is observed in transition metals, lanthanides and actinides. Consider an example of scandium (*Z* = 21). Its electronic configuration is [Ar] 4s^{2} 3d^{1} as per the principle, but this contradicts the spectroscopic observation. When the scandium atom is ionized, we presume the electron will be released from the highest energy orbital—3d orbitals as stated by the principle. However, the electron is released from 4s to form Sc^{+} having the electronic configuration [Ar] 3d^{1} 4s^{1}. This suggests that 3d have more energy than 4s. This behavior is also seen in the remaining transition metals.

A similar trend exists in lanthanides and actinides. For example, the aufbau filling order of neodymium (*Z* = 60) is [Xe] 6s^{2} 4f^{4}, but the order of leaving is [Xe] 4f^{4} 6s^{2}.

For heavier nuclei (*Z* ≥ 120), the aufbau principle becomes invalid. As the nuclear charge increases, the electrons, particularly nearer to the nucleus, experiences a heavy electrostatic force. The electrons of such nuclei have velocities approaching the speed of light. Thus, we need to account for the relativity theory to the quantum mechanic model.

We have already mentioned the exceptions to the rule. But why do we have exceptions? Is there any better rule than this? The answer is no. There is no simple mathematical formula that describes the electronic system. In an atom, we have two electrostatic interactions: the attraction between the positive nucleus and negative electrons, and the repulsion among the negative electrons. Every atom or ion tries to minimize the repulsion and try to reach the lowest energy configuration. And understanding such a system is very complicated, not easy. The explanation of these exceptions is not in the scope of this article.

Several scientists had contributed to the development of the aufbau principle and it was developed over the course of time. The notion of the ordering of orbitals was seen in the old quantum mechanics model, which was proposed by Niels Bohr in early the 1920s. At the end of the decade, Charles Janet, a French engineer, introduced the left-step periodic table based on the *n* + *l* rule. The concept of the *n* + *l* rule was adopted by Erwin Madelung in 1936; he proposed the filling of atomic orbitals with this rule. In 1962, Vsevolod Klechkovsky, a Russian chemist, provided a scientific explanation to the rule using the Thomas–Fermi model.

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