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Hund's Rule of Maximum Multiplicity

28th Feb 2020 @ 6 min read

Physical Chemistry

The Hund rule of maximum multiplicity

Statement

The Hund rule of maximum multiplicity states:

  1. When two or more orbitals of equal energy (or very close energy) are available, electrons will fill the orbitals singly before filling doubly.
  2. All the electrons in the orbitals will have the same spin to maximize the multiplicity.

In simple words, the rule says the lowest-energy electronic configuration is attained with the maximum number of parallel electron spins.

Friedrich Hermann Hund
Friedrich Hermann Hund (1896–1997)
[Wikipedia/CC by]

The rule was formulated by Friedrich Hund, a German physicist, in 1925. Hund's contributions to quantum mechanics are significant, particularly in the electronic structure of atoms. He is also known for quantum tunneling.

Explanation

Along with the aufbau principle and Pauli exclusion principle, the Hund is very helpful in understanding the electronic configuration of an atom. The aufbau principle is used to predict the electronic configuration when the energy difference between orbitals is considerably large, and the Pauli exclusion principle restricts the maximum number of electrons in an orbital to two. The Hund rule useful to predict the ground state of atoms or molecules where the equal energy orbitals are available. This is especially seen in the orbitals of the same subshell which are equal in energy. For example, in the p subshell, all its orbitals (px, py, and pz) have the same energy. Such equal energy orbitals are called degenerate orbitals.

As the rule says electrons occupy the same energy orbital singly before pairing. It means the pairing of electrons does not occur until each equal energy orbital has the occupation of one electron. Also, all of these unpaired electrons must have the same spin, i.e., either spin up (↑) or spin down (↓).

Filling of p subshell

Let's consider the filling of 2p orbitals. There are three orbitals in every p subshell, which are equal in energy. The first electron enters either of three orbitals since they are degenerate. Even though the first-occupied orbital is not completely filled and it can take one more electron, the second electron will not occupy it. Instead, it will enter another orbital with the same spin as the first electron. Similarly, the third will occupy the remaining orbital. Thus, the first three electrons occupy all three orbitals with either spin up (shown in the diagram below) or spin down.

The filling of 2p orbitals as per the Hund rule
The filling of 2p orbitals as per the Hund rule

After singly filling, the following three electrons will fill the remaining vacancies, which results in pairing.

Filling of d subshell

The same is true for the d subshell. There are five d orbitals; each has the same energy. So, the first five electrons fill each orbital with the same spin. And the next five will form pairs.

The filling of 3d orbitals as per the Hund rule
The filling of 3d orbitals as per the Hund rule

The Hund rule is not only seen for the orbitals of the same subshell but also for the orbitals of different subshells with a small difference in energy. Chromium is one such example; it is discussed later in the article.

What is multiplicity?

In quantum chemistry, the multiplicity (or spin multiplicity) is defined as the total number of spin orientations and is given by 2S + 1. Here, S is the total spin quantum number, and its value is the sum of all unpaired half spins. According to the Hund rule, the lowest energy configuration is attained when the multiplicity, i.e., 2S + 1, is maximum.

The maximum value of S is obtained only when all the spins are either up or down.

Consider an example of 2p orbitals with three electrons. The possible number of arrangements are shown below.

The combination of spins
The various combination of spins

The maximum value of S is obtained in the first and fourth case, and it is calculated as:

The sum of all spins

From the total spin quantum number, we can find the multiplicity.

Multiplicity of nitrogen

Examples

Carbon

Carbon has six electrons. Its electronic configuration is 1s2 2s2 2p2. The last two electrons are in 2p subshell, and both of them occupy the different orbitals with the same spin.

Energy level diagram of carbon
Energy level diagram of carbon

The total spin quantum number (S) of carbon is Total spin quantum number of carbon. And the multiplicity is 2S + 1 = 3.

Nitrogen

Adding one more electron to carbon, we get the electronic configuration of nitrogen: 1s2 2s2 2p3. In nitrogen, all 2p orbitals are half-filled.

Energy level diagram of nitrogen
Energy level diagram of nitrogen

S is equal to The total spin quantum number of nitrogen. And the multiplicity is 2S + 1 = 4.

Oxygen

Oxygen follows nitrogen and has an extra electron in its electronic configuration: 1s2 2s2 2p4. The fourth electron in 2p subshell forms a pair.

Energy level diagram of oxygen
Energy level diagram of oxygen

S = 1 and the multiplicity is 2S + 1 = 3

Chromium

Chromium, a d block element, has the atomic number of 24. Its electronic configuration is [Ar] 3d54s1. The last two orbitals are 3d and 4s. The difference between the energies of these two orbitals is very small. As we can see from the diagram below, both orbitals are half-filled and all electrons have the same spin as per the rule.

Energy level diagram of chromium
Energy level diagram of chromium

There are six unpaired electrons in chromium. So, The total quantum number of chromiumand the multiplicity is 2S + 1 = 7.

Oxygen molecule

The Hund rule is also used in molecular orbitals. The figure below is the molecular orbital diagram of dioxygen.

Molecular orbital diagram of dioxygen
Molecular orbital diagram of dioxygen

The filling of electrons in the π antibonding orbitals is achieved as per the Hund rule. Both molecular orbitals (π*
x
and π*
y
) are equal in energy and accept one electron each.

Justification

There are several justifications proposed for the explanation of the Hund rule. One of them is the electron-electron repulsion. The stability of the electronic system increases by minimizing the repulsion among electrons. This can be achieved by maximizing the spatial space. In a singly filled orbital, electrons are far apart from one another. This minimizes the repulsion and increases the stability.

The second justification is the shielding effect. In an atom, the columbic attraction between the nucleus and outer electrons is shielded by the inner electrons. Consequently, the outer electrons experience a lesser nuclear charge than the inner electrons. It causes the expansion of outer orbitals and increases the energy of the system. The energy of the system is lowered by decreasing the shielding effect, which is possible by the symmetrical distribution of electrons. Thus, the singly filled orbital has relatively small shielding and the stability is further increases.

The third factor that affects the energy of the system is the exchange energy. Electrons of the same spin residing in degenerate orbitals exchanges their positions. During these exchanges, the energy is released called the exchange energy. The more the exchanges, the more the stability. When degenerate orbitals are filled with unpaired electrons of the same spin, the number of exchanges is the highest, and the system reaches stability.

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