[mistche20]

Right, so the 3d orbital has no radial nodes, and it's maximum density at a distance r=6 (in atomic units).

3p on the other hand has 1 radial node and displays its maximum at r=9.

My conclusion is that 3d is more contract than 3p.



but then... why is 3p filled first ?

[Jasim]

Why is 3p lower in energy than 3d? And why is 3d higher in energy than both 3p and 4s?

From what I understand it has to do with a combination of the angular momentum and number of nodes in the orbital. The more nodes increases the energy?

[Enthalpy]

Electron density takes a very different aspect, depending on if you define it per volume unit or per radius unit.

Just because the surface of a sphere increases with its radius, it increases the density per radius unit at a bigger radius as compared with the density per volume unit. The radius of maximum density is also larger if you use the density per radius unit.

For instance, the maximum density per volume unit is at the nucleus for s orbitals, while the density per radius unit is zero at the nucleus.

Same story with the Boltzmann distribution of momentum in a gas: the maximum per "volume" of k or p vector is around zero, while it's non-zero per unit of k or p "length".

Then, orbitals define an electrostatic and a kinetic energy, this latter increasing when the electron is more confined. s orbitals confine the electron only over the radius, other orbitals confine it around a plane or a direction, which increases the kinetic energy.

Nice drawings:
http://winter.group.shef.ac.uk/orbitron/

[Jasim]

So you are saying that the volume of the 3d orbital is smaller than both the 3p and 4s orbitals.

[cheese (MSW)]

Let us go back and understand why the 2s AO in Li(0) is lower in energy than the 2p AO and hence Li is [He]1s^2 2s^1.  Go to the wonderful site 
http://winter.group.shef.ac.uk/orbitron/AOs/2p/radial-dist.html
and look at the radial distribution functions (4πr^2 Ψ^2 ) for the 2s and 2p.  The RDFs tell you where the e⁻ in that AO spends its time.  As you can see the maximum probability of finding the 2p e⁻ is closer to the nucleus than that of the 2s AO.  But look closely at the 2s: whoa it can penetrate the 1s AO and actually be at the nucleus! (Bizarre!) The Schrödinger Wave Equation shows this is true for all ns AOs: they have a finite chance of being at the nucleus (explains NMR, Mossbauer, relativity effects).  So what do you think happens to Zeff (Zeff = Z – S where Z is the nuclear charge and S is the screening by the other e⁻s) for the 2s vis-à-vis the Zeff 2p?   If you still can’t understand go to, P. Atkins et al, SHRIVER & ATKINS Inorganic Chemistry 4th ed. p 19, fig 1.20.  Once you realize that AOs with same n penetrate the core e⁻s to different extents (use the orbitron site to give you the RDFs for 3s, 3p, 3d) and hence have different Zeffs leading to 3s<3p<3d the answer should be apparent.  And it all is explained by the SWE.
 I do not like [quantum mechanics], and I am sorry I ever had anything to do with it. Erwin Schrödinger.

[Jasim]

I admit this is the first time I've examined this topic in any detail. If I understand correctly, what you are saying and what the RDFs indicate is that electrons actually permeate the nucleus of an atom? That is somewhat contrary to the simplistic view that I learned in general chemistry. Is that accurate of what is actually occurring at the subatomic level?

[cheese (MSW)]

The probability for a 1s e⁻ being inside a nucleus is as you can expect extremely low (but not 0); but this is how electron capture transmutation is believe to occur (google electron capture).  It is often not appreciated the e⁻ AO with n≥3 spend a significant fraction of their time below the core  e⁻ n= 2s,p ns=1.  The penetration is s>p>d and hence the 3s e⁻ experiences an overall greater nuclear charge the 3p which in turn experiences nuclear charge than the 3d the peneration of which is minor. This despite the fact that the maximum probability (RDF) of the 3s AO
is further from the nucleus.
Have you ever wondered  how a C-13 nucleus in C-13 NMR is so sensitive to its chemical environment (think about the s e⁻ density in an sp^3 C cf an sp^2 C)  There are other effects.

[Enthalpy]

...electrons actually permeate the nucleus of an atom?

Yes for all s orbitals. They even have their maximum density (per volume unit, not per radius unit) right at the nucleus. All others have zero density there.

[Enthalpy]

One energy term not mentioned up to now just results from the orbital moment. It implies a kinetic energy, which increases with the moment.

[cheese (MSW)]

"In January 1926, Schrödinger published in the Annalen der Physik the paper "Quantisierung als Eigenwertproblem" [tr. Quantization as an Eigenvalue Problem] on wave mechanics and what is now known as the Schrödinger equation. In this paper he gave a "derivation" of the wave equation for time independent systems, and showed that it gave the correct energy eigenvalues for the hydrogen-like atom. This paper has been universally celebrated as one of the most important achievements of the twentieth century, and created a revolution in quantum mechanics, and indeed of all physics and chemistry."  http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger
We have been poking the SWE since that time and except for a correction for relativity (Dirac) it has never failed.
Conclusion: The bizarre solns to the SWE are the best model we have for electrons in the vicinity of the nucleus.
The RDFs take into account the probability of finding an e⁻ (α Ψ^2) in the volume of a thin concentric shell a distance r from the nucleus (4πr^2).  As I’ve stated it the best answer a chemist has to the question "But where is the e⁻?"

[Jasim]

Electrons being pulled so close to the nucleus would also help to partially explain the small size of atoms with fewer valance electrons. That is something I always had a little difficulty rationalizing in gen chem. It could also help explain the fact that the valence electrons are primarily the ones involved in reactions. I guess what I'm saying is that I didn't realize the inner shells shrank so much or were pulled in so tightly to the nucleus as atomic number is increased.

Thank you guys for helping to explain! and thanks to the OP for bringing up such a great topic/question.

[juanrga]

I admit this is the first time I've examined this topic in any detail. If I understand correctly, what you are saying and what the RDFs indicate is that electrons actually permeate the nucleus of an atom? That is somewhat contrary to the simplistic view that I learned in general chemistry. Is that accurate of what is actually occurring at the subatomic level?

Yes, electrons have a non-zero probability of being found at nuclei. Said that, Hydrogen-like atomic orbitals such as 3d 3p are not valid for small radii, because those orbitals are derived assuming a point-like nucleus.