Rule of Diagonals

The  rule of diagonals is a construction principle that allows describing the electronic configuration of an atom or ion, according to the energy of each orbital or energy level. In this sense, the electronic distribution of each atom is unique and is given by the quantum numbers.

These numbers define the space where electrons are most likely to be located (called atomic orbitals) and also describe them. Each quantum number is related to a property of atomic orbitals, which helps to understand the characteristics of atomic systems by the arrangement of their electrons within the atom and in their energies.

In the same way, the rule of diagonals (also known as Madelung’s Rule) is based on other principles that obey the nature of electrons, in order to correctly describe their behavior within chemical species.

What is the rule of diagonals for ?

This procedure is based on the Aufbau principle , which states that in the process of integration of the protons to the nucleus (one by one), when the chemical elements are constituted, the electrons are also added to the atomic orbitals.

By occupying the orbitals, the electrons are first placed in the levels that have the lowest energy and are unoccupied, and then they are located in those with the highest energy.

Electronic configurations of chemical species

Similarly, this rule is used to obtain a fairly accurate understanding of the electronic configurations of elemental chemical species; that is, the chemical elements when they are in their fundamental state.

Thus, by gaining an understanding of the configurations of electrons within atoms, the properties of chemical elements can be understood.

Acquiring this knowledge is essential for the deduction or prediction of these properties. Similarly, the information provided by this procedure helps explain why the periodic table agrees so well with investigations of the elements.

Although this rule applies only to atoms that are in their ground state, it works quite well for the elements of the periodic table.

The Pauli exclusion principle is obeyed, which states that two electrons belonging to the same atom are unable to possess the four equal quantum numbers. These four quantum numbers describe each of the electrons found in the atom.

Thus, the principal quantum number (n) defines the energy level (or shell) in which the studied electron is located and the azimuthal quantum number (ℓ) is related to the angular momentum and details the shape of the orbital.

Likewise, the magnetic quantum number (m  ) expresses the orientation that this orbital has in space and the spin quantum number (m s ) describes the direction of rotation that the electron presents around its own axis.

Furthermore, Hund’s rule expresses that the electronic configuration that exhibits the greatest stability in a sublevel is considered the one that has more spins in parallel positions.

By obeying these principles, it was determined that the distribution of electrons complies with the diagram shown below:

In this image the values ​​of n correspond to 1, 2, 3, 4…, according to the energy level; and the values ​​of ℓ are represented by 0, 1, 2, 3…, which are equivalent to as, p, d and f, respectively. So the state of the electrons in the orbitals depends on these quantum numbers.


Taking into account the description of this procedure, some examples for its application are given below.


In the first place, to obtain the electronic distribution of potassium (K), its atomic number must be known, which is 19; that is, the potassium atom has 19 protons in its nucleus and 19 electrons. According to the diagram, its configuration is given as 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 .

The configurations of polyelectronic atoms (which have more than one electron in their structure) are also expressed as the configuration of the noble gas before the atom plus the electrons that follow it.

For example, in the case of potassium it is also expressed as [Ar] 4s 1 , because the noble gas before potassium in the periodic table is argon.


Another example, but in this case it is a transition metal, is that of mercury (Hg) which has 80 electrons and 80 protons in its nucleus (Z = 80). According to the construction scheme, its complete electronic configuration is:

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 .

As with potassium, the configuration of mercury can be expressed as [Xe] 4f 14 5d 10 6s 2 , because the noble gas that precedes it in the periodic table is xenon.


The rule of diagonals is designed to be applied only to atoms that are in a fundamental state and with an electric charge equal to zero; that is, it is very well coupled to the elements of the periodic table.

However, there are some exceptions for which there are significant deviations between the assumed electronic distribution and the experimental results.

This rule is based on the distribution of the electrons when they are located in the sublevels obeying the n + ℓ rule, which implies that the orbitals that have a small magnitude of n + ℓ are filled before those that show a greater magnitude of this parameter.

As exceptions, the elements palladium, chromium and copper are presented, of which electronic configurations are predicted that do not agree with what is observed.

According to this rule, palladium must have an electronic distribution equal to [Kr] 5s 2 4d 8 , but the experiments yielded one equal to [Kr] 4d 10 , which indicates that the most stable configuration of this atom occurs when the subshell 4d is full; that is, it has a lower energy in this case.

Similarly, the chromium atom should have the following electronic distribution: [Ar] 4s 2 3d 4 . However, experimentally it was obtained that this atom acquires the configuration [Ar] 4s 1 3d 5 , which implies that the state of lower energy (more stable) occurs when both subshells are partially filled.

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