EFFECTIVE NUCLEAR CHARGE: CONCEPT, HOW TO CALCULATE IT AND EXAMPLES - CHEMISTRY - 2025

The effective nuclear charge (Zef) is the attractive force that the nucleus exerts on any of the electrons after being reduced by the effects of shielding and penetration. If there were no such effects, the electrons would feel the attractive force of the actual nuclear charge Z.

In the lower image we have the Bohr atomic model for a fictitious atom. Its nucleus has a nuclear charge Z = + n, which attracts the electrons that orbit around it (the blue circles). It can be seen that two electrons are in an orbit closer to the nucleus, while the third electron lies further away from it.

The third electron orbits feeling the electrostatic repulsions of the other two electrons, so the nucleus attracts it with less force; that is, the nucleus-electron interaction decreases as a result of the shielding of the first two electrons.

So the first two electrons feel the attractive force of a + n charge, but the third one experiences an effective nuclear charge of + (n-2) instead.

However, this Zef would be valid only if the distances (the radius) to the nucleus of all electrons were always constant and definite, locating their negative charges (-1).

Concept

Protons define the nuclei of chemical elements, and electrons define their identity within a set of characteristics (the groups of the periodic table).

Protons increase the nuclear charge Z at the rate of n + 1, which is compensated by the addition of a new electron to stabilize the atom.

As the number of protons increases, the nucleus becomes “covered” by a dynamic cloud of electrons, in which the regions through which they circulate are defined by the probability distributions of the radial and angular parts of the wave functions (orbitals).

From this approach, the electrons do not orbit in a defined region of space around the nucleus, but rather, like the blades of a rapidly rotating fan, they blur into the shapes of the known s, p, d and f orbitals.

For this reason, the negative charge -1 of an electron is distributed by those regions that the orbitals penetrate; the greater the penetrating effect, the greater the effective nuclear charge that said electron will experience in the orbital.

Penetration and shielding effects

Consistent with the above explanation, the electrons in the inner shells do not contribute a -1 charge to the stabilizing repulsion of the electrons in the outer shells.

However, this kernel (the shells previously filled by electrons) serves as a "wall" that prevents the attractive force of the nucleus from reaching the outer electrons.

This is known as the screen effect or shielding effect. Also, not all the electrons in the outer shells experience the same magnitude of this effect; for example, if you occupy an orbital that has a high penetrating character (that is, that transits very close to the nucleus and other orbitals), then you will feel a higher Zef.

As a result, an order of energy stability arises as a function of these Zef for the orbitals: s

This means that the 2p orbital has higher energy (less stabilized by the charge of the nucleus) than the 2s orbital.

The poorer the penetration effect exerted by the orbital, the less its screen effect on the rest of the external electrons. The d and f orbitals show many holes (nodes) where the nucleus attracts other electrons.

How to calculate it?

Assuming negative charges are localized, the formula for calculating Zef for any electron is:

Zef = Z - σ

In this formula σ is the shielding constant determined by the electrons of the kernel. This is because, theoretically, the outermost electrons do not contribute to the shielding of the inner electrons. In other words, 1s ² shields the 2s ¹ electron, but 2s ¹ does not Z shield the 1s ² electrons.

If Z = 40, neglecting the effects mentioned, then the last electron will experience a Zef equal to 1 (40-39).

Slater's rule

Slater's rule is a good approximation of the Zef values ​​for the electrons in the atom. To apply it, follow the steps below:

1- The electronic configuration of the atom (or ion) should be written as follows:

(1s) (2s 2p) (3s 3p) (3d) (4s 4p) (4d) (4f)…

2- The electrons that are to the right of the one being considered do not contribute to the shielding effect.

3- The electrons that are within the same group (marked by the parentheses) contribute 0.35 the charge of the electron unless it is the 1s group, being 0.30 instead.

4- If the electron occupies a sop orbital, then all the n-1 orbitals contribute 0.85, and all the n-2 orbitals one unit.

5- In the event that the electron occupies a dof orbital, all those to its left contribute one unit.

Examples

Determine Zef for the electrons in the 2s orbital

Following Slater's mode of representation, the electronic configuration of Be (Z = 4) is:

(1s ²) (2s ² 2p ⁰)

Since there are two electrons in the orbital, one of these contributes to the shielding of the other, and the 1s orbital is the n-1 of the 2s orbital. Then, developing the algebraic sum we have the following:

(0.35) (1) + (0.85) (2) = 2.05

The 0.35 came from the 2s electron, and the 0.85 from the two 1s electrons. Now, applying Zef's formula:

Zef = 4 - 2.05 = 1.95

What does this mean? It means that the electrons in the 2s ² orbital experience a charge of +1.95 which draws them towards the nucleus, instead of the actual charge of +4.

Determine Zef for the electrons in the 3p orbital

Again, it continues as in the previous example:

(1s ²) (2s ² 2p ⁶) (3s ² 3p ³)

Now the algebraic sum is developed to determine σ:

(, 35) (4) + (0.85) (8) + (1) (2) = 10.2

So, Zef is the difference between σ and Z:

Zef = 15-10.2 = 4.8

In conclusion, the last 3p ³ electrons experience a charge three times less strong than the real one. It should also be noted that, according to this rule, the 3s ² electrons experience the same Zef, a result that could raise doubts in this regard.

However, there are modifications to Slater's rule that help approximate the calculated values ​​to the actual ones.

References

1. Chemistry Libretexts. (2016, October 22). Effective Nuclear Charge. Taken from: chem.libretexts.org
2. Shiver & Atkins. (2008). Inorganic chemistry. In The elements of group 1. (Fourth edition., Pages 19, 25, 26 and 30). Mc Graw Hill.
3. Slater's rule. Taken from: intro.chem.okstate.edu
4. Lumen. The Shielding Effect and Effective Nuclear Charge. Taken from: courses.lumenlearning.com
5. Hoke, Chris. (April 23, 2018). How to Calculate Effective Nuclear Charge. Sciencing. Taken from: sciencing.com
6. Dr. Arlene Courtney. (2008). Periodic Trends. Western Oregon University. Taken from: wou.edu