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Can one atomic orbital be distinguished from another by its size/volume?

And does this depend on the kind of atom, I mean does it differ from element to element?

Qmechanic
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Yash Sharma
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    To be honest, atomic orbitals are only properly defined for the hydrogen atom. – Superfast Jellyfish Jun 13 '20 at 13:34
  • So is it that it doesn't matter? And that defining the orbital for other atoms would not give us any valuable information or understanding about the atomic structure? – Yash Sharma Jun 13 '20 at 13:44
  • There's a lot of info about orbitals and their shapes at https://en.wikipedia.org/wiki/Atomic_orbital "The diagrams cannot show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space". – PM 2Ring Jun 13 '20 at 13:57

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I'll answer each of your questions one at a time.

1) Atomic orbitals don't really have a 'volume' per se. They are mathematical functions that describe an electron in a hydrogen-like atom (more on that in a second) with a certain energy level and angular momentum. Using this function, you can calculate the probability the electron is found within a specified volume. Let's look at some 2-D representations of some orbitals:

If the electron is more likely to be found in an area, the color is brighter. As you can see, they can be told apart. However, it may be difficult to tell apart certain orbitals, depending on the representation (compare $\psi_{3,2,1}$ and $\psi_{4,3,2}$ in this example). Again, these are not the orbitals; they are representations of where the electron could be found, calculated from the orbitals.

When you see a 3-D representation of an orbital, like you would in a chemistry textbook, you are seeing a surface under which an electron is likely to be found. Look at this, a description of $\psi_{4,3,1}$ (chemistry would call this the 4d orbital) for example:

This surface contains a volume where the electron is 45% likely to be found, but again note that is not the orbital. It is a volume where the electron is decently likely to be found. Good choices of surfaces can also allow one to tell these apart.

2) Orbitals are only properly defined for hydrogen-like atoms, which are atoms with one electron, because those are the only one's we been able to analytically solve so far. However, for all these atoms for which orbitals are defined, you can tell apart good representations of the orbitals, because each is mathematically distinct.

John Dumancic
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  • Minor comment: I'm pretty sure that the (3,2,1) and (4,3,2) orbitals have a different behavior outside the plane where they are projected in your image, and their similarity here is a parallax effect. Which, of course, only reinforces the main thrust of your answer. Very nicely written. – rob Jun 13 '20 at 16:39
  • @rob thanks for the comment! I meant to inform the questioner that, depending on the representation, orbitals could be difficult to tell apart by sight alone. I'll edit for clarity right now. – John Dumancic Jun 13 '20 at 16:41
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    @rob actually, these pictures show the similarity of (3,2,1) and (4,3,2) probability densities faithfully enough: the complete densities are cylindrically-symmetric, so there's nothing much to gain when drawing a 3D plot. The really noticeable difference will be visible if you draw real (or imaginary) parts of the wavefunctions instead: there you'll get the $\cos(im\varphi)$ oscillations, which do differ due to different values of $m$. – Ruslan Jun 13 '20 at 17:34
  • @Ruslan Correct! We are conditioned to think of the amplitude and phase as if they are separate, but they really aren't. Thank you for elaborating on the details of the difference here; I hadn't remembered in as much detail as you. – rob Jun 13 '20 at 17:43
  • @Tesseract I am now clear about what an atomic orbital is. But if the orbitals are only properly defined for Hydrogen atoms, then when we want to calculate probability of where an electron is in heavier elements do we just have to extrapolate it with the orbital for Hydrogen as our reference? Or is it something very different and this question is not valid? – Yash Sharma Jun 14 '20 at 06:40
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    @Yash Sharma Orbitals are defined for atoms with one electron. So, if you have a heavier element with one electron (say $\mathrm{Li}^{2+}$), you can analytically solve that, and you get orbitals for that element. For multi-electron atoms, one can no longer analytically solve for the orbitals, and you enter the realm of quantum chemistry. Check out this question and the answers within. – John Dumancic Jun 14 '20 at 16:39