Why do atoms want a full outer shell in quantum mechanics?

[epsilonjon]

Hi.

I have read some QM and am trying to use it to understand why the noble gas configuration is the most desirable for an atom.

It is my understanding that an anti-symmetric spatial wavefunction has a lower energy, since the electrons tend to be further apart. This means that the atom will have lower energy when the spins are parallel, because you have a symmetric spin state and hence an anti-symmetric spatial wavefunction. I think this is the basis for Hund's first rule?

So say we are on the first row of the periodic table and we gradually fill the 2p subshell. With nitrogen we have 3 electrons in the 2p subshell, so it's possible for them to all have parallel spins (so long as they are in different orbitals). By my reasoning above (or equivalently (I think) Hund's first rule), this will be the arrangement that happens in nature since it will help minimize the energy of the atom.

If we now go to oxygen with 4 electrons in the 2p subshell, two of the electrons are forced to be in the same orbital (symmetric spatial wavefunction), so they will have to be in the anti-symmetric spin configuration. The electrons will be closer together on average, so will have more energy. Does this mean that an oxygen atom has more energy than nitrogen atom? Similarly, fluorine has more energy than oxygen, and neon has more energy than fluorine?

I was hoping I would get the exact opposite of this!

Is this analysis correct? If so, why (if not to minimize energy) is a full outer shell more desirable for an atom? If the above is incorrect, where am I going wrong?

Thanks for any help!

 

[epsilonjon]

I think maybe this got no replies because it's been asked/answered before?

Apologies if that is the case. I saw an old thread here from 2003 but it didn't seem to answer my question.

I'd still really appreciate any help as it's bugging me big time.

Thanks, Jon.

 

[Darwin123]

Hi.


If we now go to oxygen with 4 electrons in the 2p subshell, two of the electrons are forced to be in the same orbital (symmetric spatial wavefunction), so they will have to be in the anti-symmetric spin configuration. The electrons will be closer together on average, so will have more energy.

The two electrons may be closer to each other on average. This will increase their mutual energy. However, the two electrons may also be closer to either of the two nuclei on average. This will decrease their mutual energy.
The nuclei are positively charged while the electrons are negatively charged. Therefore, the electrons and the nuclei attract each other. Anything that brings that electrons and nuclei closer together on average will decrease the mutual energy.
The above is a rather obvious generalization. It explains why some configurations have less energy that others. However, that doesn't specifically explain Hunds Rule.
The following is my visualization of the mechanism behind Hund's Rule. I have no reference to support it. However, it is consistent
If you have a half filled orbital, electrons start out symmetrically distributed about the nucleus. The probability antinodes of the atomic wave functions are as far apart as they can get, even before hybridization.
Wave interference causes each pair of separate atomic wave function to hybridize. The wave functions of the individual electrons can constructively interfere in a location close to the nucleus of the atom. The hybrid wave functions peak closer to the nucleus. Thus, their wave interference pulls the electrons closer to the nucleus. The same wave interference can't pull the electrons together as much because the electrons are already as far apart in a symmetrical distribution.
If you add or subtract one electron to the half filled orbital, that electron can't be symmetrically distributed. A hybrid wave function between two electrons always ends up closer to a third electron. It has to be near either one or the other electron. Therefore, wave interference makes the electron (or hole) move closer (or away) from the other electrons.
You asked about atomic wave functions. Each atom has a single nucleus. A slight aside with respect to chemical bonding. The same type of heuristics can be useful for chemical bonding. Except here, we have two nucei instead of one nucleus.
A chemical bond forms by a combination of wave interference, and a decrease in the total energy. Constructive wave interference makes the two electron particles come closer together in a region right between the two nuclei. Electrostatic repulsion raises their energy. Each electron has been dragged farther from its original nucleus. However, it has been dragged closer to the other nucleus. The probability of it being on the opposite side of the atom from the other atom has now decreased. The total result is that the energy in a bonding orbital has decreased.
Other interacts contribute. I haven't included effects such as spin orbit coupling, spin spin coupling, etc. However, these are dipole an quadropole effects. The largest changes in energy are caused by the electrostatic interactions.