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My textbook (Chemistry the Central Science, edition 12, Brown et al.) says that effective nuclear charge increases down a column. Indeed, this is consistent with the values provided by Clementi in the 1960s.

Is there intuition or some qualitative explanation for why effective nuclear charge ($Z_\mathrm{eff}$) for the outermost electrons increases down a column (as opposed to being roughly equal)?

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  • $\begingroup$ Do you mean the effective nuclear charge ($\ce{Z_{eff}}$) for the outermost electrons? $\endgroup$ – Shoubhik Raj Maiti Sep 21 at 20:41
  • $\begingroup$ @ShoubhikRajMaiti Yes. $\endgroup$ – Typical Highschooler Sep 21 at 20:41
  • $\begingroup$ One thing might be that for higher principal quantum numbers, the d, f shells are being filled, which are said to be diffuse and not good at shielding. $\endgroup$ – Shoubhik Raj Maiti Sep 21 at 20:43
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Initially, let's take the electrons as probability density fields instead of classical particles.

The increase on the number of protons in the nucleus increases the electric force on all electrons, but the core electrons suffer higher attraction due to higher proximity and lighter electronic blindage, so their probability density field is forced nearer the nucleus.

Reminding of Electric Force: $$\mathbf{F}_\text{E} = \mathbf{k}_\text{C} \frac{q_1q_2}{\mathbf{r}^{2}}$$

Consequently, the outer electrons will suffer lower blindage force from these core electrons, parallelly to the nucleus attraction, allowing them to get closer to the nucleus as well.

As effect, of course, the outer electrons also force the inner electrons towards the nucleus, but their probability densities reach a minimum repulsion energy in such a distance that electrons on the same level are forced closer to the nucleus than on the atomic element above, going down a column in the periodic table.

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A qualitative way to understand this trend is that with each increase in n, going row by row down a column, the electrons of filled inner shells create a screening effect upon the charge of the nucleus. The nuclear charge grows with each added proton, but as electrons are placed into new shells (with larger n) above previous shells, they experience not only the attraction of the nucleus but the repulsion of the inner electrons.

This doesn't explain the increase in Z_eff down a column by itself though. For that, we also need to consider that the radial probability distribution (The likelihood of finding the electron a certain radial distance from the nucleus) of the (n)s, (n)p, (n-1)d, and (n-2)f electrons within a row, changes to reflect the appearance of nodes as n, the energy level, increases.

Consider the s orbitals and the alkali metals/hydrogen. The 1s orbital (Hydrogen) has no radial nodes in the probability distribution, while the 2s (Lithium) and 3s (Sodium) have one and two radial nodes, respectively. The nodes divide up the radial probability into peaks (Technically, I think it's like n - l peaks, but you asked for a qualitative explanation). As n gets larger, the most probable peak moves farther away from the nucleus (Which should hopefully make sense), but the other peaks appear deeper within the atom, closer to the nucleus. These smaller peaks of probability can actually appear deeper (Shorter radial distance) than some inner "Shielding" orbitals and allow the electron described by that probability distribution to feel an effectively less screened nuclear charge. This happens even though this electron is more probable to be found farther away than the shielding layers of electrons underneath it.

I found this image from the cited online textbook to be illustrative of this example I am talking about:

1s,2p,2s electron radial probability distributions

I am referencing this textbook's section on "Electron Penetration": https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/07._Periodic_Properties_of_the_Elements/7.2%3A_Shielding_and_Effective_Nuclear_Charge

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It is due to the less efficient shielding from d and f orbitals.

P orbitals electrons shield less the outermost electrons from the S orbitals electrons increasing the effective nuclear charge.!But p orbitals electrons shield the valence electrons pretty well and we dont take them in account.

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