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Aufbau Principle

18th Feb 2020 @ 8 min read

Physical Chemistry

Aufbau principle


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.

the etymology of aufbau
The etymology of aufbau

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.

Aufbau principle diagram
Aufbau principle

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., 1s1. 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 1s2. 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 1s2 2s1 of the lithium atom, which is directly below hydrogen in the periodic table. Similarly, we have beryllium, 1s2 2s2.

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.

The electronic configuration of elements based on the aufbau principle
Element Symbol Atomic number (Z) Electronic configuration
Hydrogen H 1 1s1
Helium He 2 1s2
Lithium Li 3 1s2 2s1
Carbon C 6 1s2 2s2 2p2
Oxygen O 8 1s2 2s2 2p4
Neon Ne 10 1s2 2s2 2p6
Sodium Na 11 1s2 2s2 2p6 3s1
Magnesium Mg 12 1s2 2s2 2p6 3s2
Phosphorus P 15 1s2 2s2 2p6 3s2 3p3
Argon Ar 18 1s2 2s2 2p6 3s2 3p6
Potassium K 19 1s2 2s2 2p6 3s2 3p6 4s1
Calcium Ca 20 1s2 2s2 2p6 3s2 3p6 4s2
Scandium Sc 21 1s2 2s2 2p6 3s2 3p6 4s2 3d1
Iron Fe 26 1s2 2s2 2p6 3s2 3p6 4s2 3d6
Zinc Zn 30 1s2 2s2 2p6 3s2 3p6 4s2 3d10
Bromine Br 35 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p5
Krypton Kr 36 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6
Zirconium Zr 40 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d2
Tin Sn 50 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p2
Plutonium Pu 94 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6
6s2 4f14 5d10 6p6 7s2 5f6

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:

  1. Electrons in an atom fill the orbital with the lowest value of n + l.
  2. 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.

Madelung rule or n + l rule
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 and Madelung rule
The aufbau principle and Madelung rule


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.

Transition metals

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

Exceptions to the aufbau principle in transition metals
Element Symbol Atomic number Aufbau's prediction Experimental observed
Chromium Cr 24 [Ar] 4s2 3d4 [Ar] 4s1 3d5
Copper Cu 29 [Ar] 4s2 3d9 [Ar] 4s1 3d10
Niobium Nb 41 [Kr] 5s2 4d3 [Kr] 5s1 4d4
Molybdenum Mo 42 [Kr] 5s2 4d4 [Kr] 5s1 4d5
Ruthenium Ru 44 [Kr] 5s2 4d6 [Kr] 5s1 4d7
Rhodium Rh 45 [Kr] 5s2 4d7 [Kr] 5s1 4d8
Palladium Pd 46 [Kr] 5s2 4d8 [Kr] 4d10
Silver Ag 47 [Kr] 5s2 4d9 [Kr] 5s1 4d10
Platinum Pt 78 [Xe] 6s2 4f14 5d8 [Xe] 6s1 4f14 5d9
Gold Au 79 [Xe] 6s2 4f14 5d9 [Xe] 6s1 4f14 5d10

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.

Lanthanides and actinides

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

Exceptions to the aufbau principle in lanthanides and actinides
Element Symbol Atomic number Aufbau's prediction Experimentally observed
Lanthanum La 57 [Xe] 6s2 4f1 [Xe] 6s2 5d1
Cerium Ce 58 [Xe] 6s2 4f2 [Xe] 6s2 4f1 5d1
Gadolinium Gd 64 [Xe] 6s2 4f8 [Xe] 6s2 4f7 5d1
Actinium Ac 89 [Rn] 7s2 5f1 [Rn] 7s2 6d1
Thorium Th 90 [Rn] 7s2 5f2 [Rn] 7s2 6d2
Protactinium Pa 91 [Rn] 7s2 5f3 [Rn] 7s2 5f2 6d1
Uranium U 92 [Rn] 7s2 5f4 [Rn] 7s2 5f3 6d1
Neptunium Np 93 [Rn] 7s2 5f5 [Rn] 7s2 5f4 6d1
Curium Cm 96 [Rn] 7s2 5f8 [Rn] 7s2 5f7 6d1
Lawrencium Lr 103 [Rn] 7s2 5f14 6d1 [Rn] 7s2 5f14 7p1

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.

Inconsistency in the ordering of orbitals

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] 4s2 3d1 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] 3d1 4s1. 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] 6s2 4f4, but the order of leaving is [Xe] 4f4 6s2.

Relativistic effects

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.

Why exceptions?

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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10th Jan 2021
...hi..I have studied this aufbau in 1991...after many years your article has put light into the exception and has discriminated the principle from formula with lucid explanation..but when I was studying there was no spectrograph explanation.. thankyou love..
17th Nov 2020
Good Easy to understand
06th Nov 2020
such an amazing explanation, thanks for the great the way their is a small problem in n l chart, there are two 6s there instead of one.
20th Sep 2020
Great explanation mate, atleast I have the guts to appreciate your work here.
15th Aug 2020
It's such a nice information which helps me to improve my knowledge as well as helpful for crack neet
Bold Ninja
22nd Jul 2020
Illustrated diagrams make it super easy to understand the concept of Aufbau Principle.
John Gilbert
21st Jul 2020
This was phenomenal, really easy to understand the ' Aufbau Principle '. Thank you!

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