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Energy of Atomic Orbital

13th Feb 2020 @ 8 min read

Physical Chemistry

Energy level diagram

Every atomic orbital is associated with a particular amount of energy. The energy of an orbital depends on the shape and size of the orbital. In a multiple-electron system, the shielding effect also influences the orbital's energy.

Energies of orbitals are quantized as per quantum mechanics. Thus, there are only selected energy levels available, which an electron can occupy. An electron residing in a particular orbital has the same energy of that orbital. When an electron descends to a lower energy orbital, it emits electromagnetic radiation. The energy is absorbed when it ascends to a higher energy orbital.

The energy of an electron is always expressed in negative values. The negative sign indicates that we need to supply the energy to release an electron bounded to its nucleus. It is minimum (most negative) at zero radii and reaches zero at infinity. As we move far from the nucleus, the energy of an electron increases. Electrons that are farther away from the nucleus will require a less amount of energy to free it than that are closer to the nucleus.

Energy with the radius
Energy with the radius

The electrons in an atom distribute themselves to reach the lowest possible energy to stabilize the electronic system. This is achieved by filling lower energy orbitals first. The two quantum numbers play a vital role in deciding the energy of orbitals: the principal quantum number n and the azimuthal quantum number l. The principal quantum number decides the size of the orbital (or the nuclear radius) while the azimuthal quantum number is responsible for different shapes of orbitals. For example, l = 0 is the s orbital, l = 1 is the p orbital, l = 2 is the d orbital, l = 3 is the f orbital, and so on.

The general rule is the energy increases with the principal quantum number and azimuthal quantum number; however, there are some exceptions.

Energy of orbitals and shell

Each shell corresponds a value of the principal quantum number, which starts with n = 1. Thus, 1s is the lowest energy orbital and is the nearest to the nucleus. 1s is followed by the second shell, which consists of 2s and 2p. The second has more energy and farther from the nucleus than the first. The second is followed by the third (3s, 3p, and 3d), which is followed by the fourth, and so forth.

The energy of orbital increases with the principal quantum number (or shell).
The energy of an orbital increases with the principal quantum number (or shell).

In the above figure, circular paths represent energy levels, not orbits.

Electrostatic interactions

Before we get into the azimuthal quantum number, it is necessary to revise some basics on the atom.

According to the quantum mechanics model, an atom consists of a central positively charged nucleus and negatively charged electronic cloud. This electronic cloud is held by the electrostatic force of attraction between electrons and its nucleus. In a multiple-electron system besides this attractive force, there is repulsive force among electrons. These electrostatic interactions significantly affect the energy of orbitals in different shells as well as within a shell.

As mentioned earlier, the electrons in an atom distribute themselves such that the energy of the electronic system is the lowest. This is attained by minimizing the repulsive interactions.

Shielding effect

One of the effects due to electrostatic interactions is the shielding effect. It arises because electrons in lower orbitals dampen the attractive force between the nucleus and electrons in higher orbitals. Electrons in higher orbitals experience the less nuclear charge compare to inner electrons. As a consequence, outer orbitals moves farther away from the nucleus. This widens the energy gap between inner and outer orbitals.

Shielding effect
Inner electrons shield the outer electrons from the nucleus.

The nuclear charge experienced by outer electrons is measured in terms of the effective nuclear charge (Zeff).

The shielding effect also depends on the shape of the orbitals. The orbitals that are wider in space provide better shielding. As a result, the s orbital shields more effectively than the p orbital, which shields better than d. The shielding strength (S) is maximum for s and decreases for the rest.

Shielding effect within a shell

Splitting of orbitals in a shell

Within a shell, there are several subshells: s, p, d, f… The electrons in an s subshell, which is the widest in space, experience the strongest electrostatic force of attraction and minimum repulsion than the rest subshells. Furthermore, it also provides the shielding effect to other outer orbitals of a subshell. Consequently, it gets closer to the nucleus than the rest. The same is true for p and other remaining orbitals. This results in the splitting of the orbitals in a shell.

Consider the third shell: 3s, 3p, and 3d. 3s is the closest to the nucleus than 3p, which is closer than 3d.

In addition to the above if we look at the probability density graph of each orbital (see figure below), the 2s orbital has a maximum probability density (Ψ2) at r =0. The probability density is minimum for 3d, and 3p comes in-between.

Probability density of 3s, 3p, and 3d
The probability density of 3s, 3pz, and 3dz2 for the hydrogen atom along the z-axis.

Note: Although the above plot is for the hydrogen atom, the plot for other atoms is similar.

The probability density is the probability of finding an electron per unit volume. Thus at smaller radii, the probability of finding an electron is more in 3s orbital than the other two. Electrons are likely to spend more time closer to the nucleus in comparison to 3p and 3d. Because of this, 3s orbital shields 3p and 3d. In the same manner, 3p shields 3d. Hence, orbitals split and get farther from one another.

Finally, we can conclude that the energy of an orbital increases with the azimuthal quantum number within a shell.

From the above all, we can generalize the energy of orbital as: 1s < 2s < 2s < 3s < 3p < 3d…

Madelung's rule

As we move toward higher orbitals, we will encounter a number of exceptions to what stated above. The reason for such exceptions is the electrons in an atom want to reach the lowest energy level by minimizing the repulsion. One of the solutions to this problem is the Madelung rule, named after Erwin Madelung. It is not a universal rule but very helpful.

The rule states the lower the value of n + l, the lower its energy. In other words, an electron will occupy the orbital with the lowest value of n + l. If two or more orbitals have the same value of n + l, the orbital with the lowest n is considered. Here, n is the principal quantum number and l is the azimuthal quantum number.

See the table below.

Madelung's rule
OrbitalPrincipal quantum number, nAzimuthal quantum number, ln + l
1s101
2s202
2p213
3s303
3p314
4s404
3d325
4p415
5s505
4d426
5p516
6s606

As we see from the above table, the correct order of orbitals with increasing energy is as follows:

The orbital with increasing energy as per the Madelung rule

The energy level diagram below depicts the same.

Energy level diagram as per Madelung's lungs
Energy level diagram as per Madelung's lungs

As mentioned before, this order is not universally true. There are a good number of exceptions, especially in transition metals, lanthanides and actinides. The reason for exceptions is the same: the electronic system tries to reach the lowest energy by minimizing the electron-electron repulsion. There is no simple formula to get a universal rule.

One notable exception is 3d and 4s orbital. If we see the energy level diagram, the energy gap between 3d and 4s is very small. In some cases, the orbitals swap to reach the lowest energy, particularly in transition metals.

Electronic configuration for transition metals.

Effective nuclear charge

The effective nuclear charge is due to the shielding effect. This is discussed in the previous section. In the modern periodic table, the atomic number increases from left to right. The number of protons in the nucleus also increases. As a result, the lower orbitals of heavier atoms experience a strong Coulomb force compare to lighter atoms. Consequently, their orbitals shrinks and the energy also decreases. For example, the energy of 2s orbital in the hydrogen atom is more than the energy of 2s orbital of the lithium atom. The orbital of lithium has more energy than that of potassium.

Energy of 2s orbital of hydrogen, lithium, potassium, and Rubidium

Hydrogen atom

The hydrogen atom is the simplest atom because we have only one electron and one proton. It is a single-electron system. Thus, there is no electron-electron repulsion, and so no shielding effect. The only electrostatic attraction between the electron and proton exists.

In the hydrogen atom, all the orbitals within a shell have the same energy. Such orbitals, which have the same energy are called degenerate orbitals. Thus, s = p = d = f.

The energy of the orbital is independent of the azimuthal quantum number. The principal quantum number solely determines the energy. Therefore, we can write the following:

Electronic configuration in the hydrogen atom

The energy level diagram for the hydrogen atom is shown below.

The energy level diagram for the hydrogen atom
The energy level diagram for the hydrogen atom

If noticed, the energy gap in successive shells decreases with the energy. The orbitals get closer and closer as we move higher.

The electron will mostly spend time in 1s orbital since it is the most stable condition. It can jump to an excited state from the ground state by absorbing energy.

All the above statements are valid for species similar to hydrogen atom. Some examples are He+, Li1+, Be2+, and so on. They all are a single-electron system.

The Bohr model holds good for the hydrogen atom. The energy of a shell according to the Bohr atomic theory is given as:

The energy of the electron for the hydrogen atom as per Bohr model

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