I recently came across a question comparing the average radius of subshells.

A search on the internet gave the following result for single electron atoms: $$\langle r\rangle_{n,\,l}=\frac{a_0 n^2 \left(\frac{1}{2} \left(1-\frac{l (l+1)}{n^2}\right)+1\right)}{Z}$$

Thus, the average radius should decrease in the order s>p>d for same n

But, I remember the usual saying that goes like s orbitals are more close to the nucleus and d orbitals are more spread out. Or s orbitals have more penetrative power than d orbitals. This same argument is given when computing Zeff, that s shield better than d.

So, are these arguments correct? Also, these arguments come into play in multielectron species. So, does the average radius of subshells follow a different order in multi-electron species? I have tried searching the internet but couldn't find any such data for multi-electron species.

I have already seen this answer and it explains well but it doesn't answer why the above arguments give the wrong result. So, I posted a new question.

  • 2
    $\begingroup$ Comparing avg r may be very misleading, as even for about the same avg r, different radial and angular distribution plays big role. $\endgroup$
    – Poutnik
    Commented Sep 15, 2021 at 12:34
  • $\begingroup$ Ask yourself, for the given avg radius, what happens to the orbital energy, screening and being screened factors, if the electron distribution gets more or less spreaded radially or angularly. The classical electrostatic mental analysis is sufficient for it. Include this analysis to the question. Readers can then guide you by their confirmation or refutation. $\endgroup$
    – Poutnik
    Commented Sep 17, 2021 at 8:00
  • $\begingroup$ Actually, I want to ask that whether the same order follows for multi electron atoms? If yes, then what about the shielding arguments that I gave; if no, then why is it different from single electron atoms. $\endgroup$
    – Govind
    Commented Sep 18, 2021 at 19:07


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