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While I was preparing for my upcoming exams, I stumbled upon this sentence which is bothering me quite a bit:

The contraction of the lanthanoids is due to the imperfect shielding of one electron by another in the same subshell.

Though I am well aware of lanthanoids contraction and the poor shielding effect of the f- and d-orbitals, but still according to what I have studied and understood so far is the shielding effect is the shielding of valence electrons by the electrons between the nucleus and the valence electrons from the nuclear charge.

Then how can an electron shield another electron of the same subshell?

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    $\begingroup$ @Zhe So does this actually effect the atomic radus of the lanthanides $\endgroup$
    – ayush paul
    Commented Feb 9, 2019 at 20:15
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    $\begingroup$ Normally, as you add electrons, after adding several, you'd need to expand to a higher energy -- and larger -- orbital. But with lanthanides, you're basically sticking all of the electrons in the same energy level where shielding is imperfect. So the shielding does not grow commensurate with the increase in nuclear charge. $\endgroup$
    – Zhe
    Commented Feb 9, 2019 at 22:43

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Shielding is simply the cancellation of some of the effective charge of the nucleus.

It's a lot easier to envision an electron to shield another electron that, by time average, is likely to be farther from nucleus, e.g., $1s$ shielding $3p$.

But for 2 electrons in the same subshell, on average, you can expect one to be closer to the nucleus than the other roughly 50% of the time. While it's true that a good deal of the time that other electron may be on the other side of the nucleus, for d- and f- orbitals, we're looking at more than 1 electron, so there's some partial shielding. Partial, because the shielding electrons don't actually shield a good portion of the time due to relative distances from the nucleus.

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Consider that there are two electrons in two different but adjacent circles. Assume that these circles encircle the nucleus.

Now also assume that another circle is encirciling the nucleus and this circle is currently vaccant. Now an electron is passing nearby this whole system This electron gets attracted by the positevely charged nucleus, and gets accomodated initially on the middle circle. This electron starts experiencing repulsive forces from the nearby electrons and get repelled away to the outermost circle. Even though repulsion is still there, it is reduced to some extent currently. At present the electrons in the inner circles are shielding the electron in the outermost circle or they are preventing the electron in the outermost circle from getting a more effective nuclear attraction.

That is an electron is being shielded by another electron. If we assume that the three circles mentioned above form a single subshell then it becomes clear that an electron in a subshell is being shielded by another electron which is present in the same subshell.

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