# Orbitals and shells

1. Nov 15, 2010

### mariush

Hey guys!

I didn't know where else to turn to but this forum.
I am trying to figure out what the difference between a orbital and a shell really is, and what an orbital is in reality.
As I understand, E in the Schrödinger equation represents the binding energy for the electron to the nucleus. For an H-atom, it is given by -R_h/n^2, where R_h is the rydberg constant, and n is an positive integer that represents the different possible energy levels for the electron. From what I've read, n represents the "electron shell", and the orbitals are electrons with the same energy within the same shell.

So my question is, what is it that really determines the energy of the electron? Distance from the nucleus or mechanical energy? And how do I tell the difference between a shell and an orbital?

If you know a place i can read up on this, I'll happily do so.
Thanks! Marius

2. Nov 15, 2010

### luriol

The difference has to do with angular momentum. For a hydrogen atom, with only one electron, the different values of n have different energies, but all the possible angular momentum states consistent with that n have the same energy.

For multi-electron atoms, the hydrogenic solutions are still used to name the shells and orbitals, but when there is more than one electron present the energies are not all the same. This is because, while for a hydrogen atom, energy does not depend on angular momentum within a shell, when you include electron-electron interactions in multi-electron atoms this degeneracy is broken.

For a very simple discussion see Halliday Resnick and Walker chapter 40.

3. Nov 16, 2010

### cgk

First, what are orbitals? The term "orbital" has a clear quantitative meaning, namely, an orbital is a one-electron wave function[1]. An atomic shell, however, is more of a qualitative concept to understand how atoms are built (by filling shells at the same formal "n" level).

That being said: For atoms other than hydrogen there is basically a 1:1 relationship between (spatial) orbitals and atomic shells. Each shell (say, 3s,3p,3d,4d...) is represented by one spatial orbital in a Hartree-Fock solution (or Kohn-Sham or other mean-field solution).

The terms should still be distinguished:
o "orbitals" also make sense in molecules, and they come not only in spatial terms, but also as spin-orbitals.

o While in higher elements a shell would typically include the angular momentum specification (a "3s" shell is the third orbital of s symmetry, but it is not really related to the "3p" or "3d" shell at all, so there is no reason to speak of a "3" shell without the s or p), for hydrogen this is not done as discussed by luriol.

o For element other than hydrogen the orbital energies (i) are not static, but depend on the occupancy of *all* the orbitals and the state in question and (ii) the total state energy is *not* given as sum of the orbital energies.

[1]: These are either used as building block for many-electron wave functions (by putting them into determinants or configuration state functions) or for the analysis for many-electron wave functions in terms of one-electron wave functions (natural orbitals and such

4. Nov 18, 2010

### mariush

Alright!
Thanks to both of you for great posts! I picked up a copy of Halliday Resnick and Walker yesterday. It seems like a great book, but it didn't have a 40th chapter, so i tried to look elsewhere. Now, what confuses me is the different definitions I have come across: Chemistry for dummies says: "The principal quantum number n describes the average distance of the orbital form the nucleus (...) Chemists sometimes call the orbital electron shells". My high school chemistry book states: "The areas around the nucleus that are likely to hold electrons, are called shells. (...) Electrons in the same shell with different energies, are called different orbitals".

How can that be correct? Isn't the energy the same for an electron in e.g. 3s and 3d the same since the principal quantum number is the same?

In my mind, I have that the "shell", n, describes the energy of the electron, which in terms is determined by the binding energy to the nucleus and the shape and momentum in it's orbital. I thought that the orbital was described by the number l for the momentum in it's orbital and m_l for the orientation of the density of electrons.

Did I get it all wrong? I feel the more I read, the more confused i get. If you know a book that addresses this topic, that does not require too much background knowledge, please tell me :)

Thanks!

5. Nov 19, 2010

### alxm

It's not really correct. Or rather, it's a bit oversimplified. The reality here is this:
The state of an electron in an atom is an orbital, each orbital represents a unique pattern of motion of the electron (you could say 'orbits', but that's a misleading term from the QM point of view). Each orbital has a corresponding energy, although different orbitals can have the same energy. Each orbital can have two electrons (with opposite 'spin') in them.

An orbital is specified not by one but by three 'quantum numbers': n, l, m. Which are named 'principal', 'angular momentum', 'magnetic' and 'spin', respectively. They follow the rule that n is 1,2,3..., and for each n there's an l which is an integer between 0 and n-1, and for each l there are m values which go from -l to +l.

n=1, l=0, m=0 --> 1s
n=2, l=0, m=0 --> 2s
n=3, l=1, m=-1,0,1 --> 2px, 2py, 2pz

The principal quantum number represents the total linear momentum of the electron, in other words, its kinetic energy. The angular momentum quantum number describes its angular momentum, and the "magnetic" quantum number describes the spatial direction of the angular momentum. (hence the x,y,z) So these three numbers are required to know the energy of the electron.

The 'shell' is determined by the principal quantum number, n=1,2,3.. are the K,L,M shells and so on (note the capital letters - don't confuse the shell designations with the quantum numbers. In practice, the 'shell' concept isn't really used much, though). The orbital 'type' is determined by l. l=0,1,2,3.. are s,p,d,f orbitals. Now, you might not have seen a designation like '2px' before, that includes the magnetic quantum number (the subscripts). The reason is that for a single atom, it's spherically symmetric, all directions are equal, so all orbitals with the same n and l will have the same energy. So there's no need to specify it.

So, the energy is not only determined by 'n' but by 'l' too. Different orbitals in the same shell will have different energies if they have different angular momenta, which you can also see for hydrogen http://upload.wikimedia.org/wikipedia/commons/0/08/Energy_levels_of_hydrogen_atom.png" [Broken])

Now, Rydberg's formula and the Bohr model give the energies as proportional to 1/n^2. What's up with that? The answer is that they're wrong. Or rather, they only describe the energies of the 's'-states, where l=0. As for the description of the geometry, you have to be careful. Unlike the early 'planetary' models of the atom, orbitals (as opposed to 'orbits' or the visual idea of 'shells') http://chemlinks.beloit.edu/Stars/images/orbitals.jpg" [Broken], except for the s-orbitals, and electrons don't stay in a specific location, or specific distance from the nucleus.

It's true that the average distance to the nucleus increases as n increases, but two electrons with the same n and different l values don't share the same average distance, so the previous statement is only true when you're comparing orbitals with the same 'l'. So a shell is 'n', a 'subshell' is the set of orbitals with a given l value, and an orbital has all of n,l,m (although for reasons stated, the orbitals in a particular subshell are not always distinguished, so if you say 'excitation from the 1s to 2p level', you mean any 2p level).

This is how 'orbitals' are usually taught in chemistry, and it's all based on analogy to hydrogen. However, hydrogen is a single-electron atom, which easily leads to some confusions. One is the difference between an orbital and an energy state of the atom. For hydrogen, these are the same. For atoms with more than one electron, they're not. The energetic state depends on the energy of all the electrons. Since every electron interacts with every other electron, the difference between two energy levels in a many-electron atom can't be described as simply as being the difference between the two states the excited electron transitioned between. You can describe it that way as an approximation, but it's not strictly true.

Strictly, the orbital picture is an approximation for many-electron systems (or even more strictly, the picture where each electron occupies one orbital is an approximation). But from the chemistry point of view, it's a good enough approximation to qualitatively describe most chemistry. Much as viewing the solar system as a set of planets independently orbiting the sun is a good enough picture to understand the general concept of planetary motion, even though you can't treat them independently if you want an exact description of what's going on.

Last edited by a moderator: May 5, 2017
6. Nov 20, 2010

### vissh

Heya :D Thanks Alxm [I too was wondering abt these for 2 days hehe]
Here are my conclusion[correct me pls if i am wrong]and some question:-
1) So Angular momentum by Bohr's model
i.e. mvr = nh/2(pie)
applies only for "s" states of hydrogen and Hydrogen like species[He+,Li2+] and By this formulae , Do we Find out the total angular momentum of a single electron in the nth-shell or the total angular momentum of the nth shell ??

2) Orbital angular momentum of an e- is [l(l+1)]1/2[h/2(pie)]
Does it mean e- in 1s , 2s ,3s etc have same orbital angular momentum ?

3) What is total angular momentum of an e- ?

4) The intrinsic spin momentum of an e- is given by
[s(s+1)]1/2[h/2(pie)] where s = 1/2.
What is meant by Spin momentum of a "System"?

Thanks again (^.^)

7. Nov 20, 2010

### Staff: Mentor

The Bohr model formula for orbital angular momentum ($L=n \hbar$) is completely wrong. In general, the magnitude of the orbital angular momentum is

[tex]L = \sqrt{l(l+1)} \hbar[/itex]

where for a given n, l can have values 0,1,...,n-1.

For n = 1, l can only be 0 (s-state), and L can only be 0 also.

For n = 2, l can be 0 (s-state) or 1 (p-state), and L can be 0 or $\sqrt{2} \hbar$.

And similarly for larger values of n.

8. Nov 20, 2010

### vissh

lol Mine teachers never told us that in school haha . I always thought bohr's formula for the L to be right . thx for sharing :D

9. Nov 23, 2010

### mariush

alxm - thanks so much for the clarification! What a great post, you really need to write a book on this, this is the first time I've come accross such a great explanaion..! Thanks again!