# What is the shape of f-orbital???

Jun 22, 2017

Personally, I've never really known which one was which until now... turns out they're right on wikipedia.

4F ORBITAL SHAPES

Curated from Wikipedia, these are the $4 f$ orbitals. Row-wise, these have corresponding magnetic quantum number ${m}_{l}$ values in the set $\left\{- 3 , - 2 , - 1 , 0 , + 1 , + 2 , + 3\right\}$.

n = 4 ORBITAL RADIAL NODES

The radial density distribution of the $4 f$ orbitals could be compared with the $4 s , 4 p ,$ and $4 d$ orbitals:

Regarding their nodes, we can see that:

• The $f$ orbitals in the same quantum level have less radial nodes than other orbitals of lower angular momentum $l$ (where the function dips down to $y = 0$ on the above graph).

• In contrast, these also have more angular nodes than the $d$, $p$, and $s$ orbitals in the same quantum level (not seen in the above graph), as they have the highest $l$ here.

POOR CAPACITY FOR ELECTRON SHIELDING

From the above graph, they are also the least effective at electron shielding, as they are the least penetrating orbitals in their quantum level; the radial electron density tapers off before getting near the nucleus, and so the $4 f$ electrons are usually not near the nucleus.

For example, this is what gives rise to the lanthanide contraction, where the $6 s$ electrons penetrate the core significantly and relativistically contract due to traveling close to the speed of light, but the $4 f$ electrons shield poorly.

This can be observed in the 3rd row transition metals, which have only SLIGHTLY larger atomic radii than the respective 2nd row transition metals: